Conservation, Derived Ordering, Equilibrium, and Smooth Hybrid Propagation Geometry in Quantum-Operational Geometry
Conservation, Derived Ordering, Equilibrium, and Smooth Hybrid Propagation Geometry in Quantum-Operational Geometry
Michael TolbertAI Rating: 3.3/5This paper derives a staged propagation geometry from regular finite-dimensional quantum-operational realization structures, combining informational admissibility, canonical minimum-response realization, and provenance-coherent ordering. It constructs a smoothly assembled Lorentz-signature propagation form and associated horizon and boundary structures, while explicitly stopping short of identifying the result with physical spacetime or establishing global causality or gravitational dynamics.
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3.3 out of 5. Audited on TheoryOfEverything.ai. Permanent record: https://theoryofeverything.ai/review-profile/paper/c0fc8f76-a8ee-423e-85fa-c79d2dda0eb7
AI Review Rating
Composite of the review dimensions below, on a 0–5 scale.
This paper is a dense, carefully self-scoped work of finite-dimensional mathematical physics that stages four principles (QII admissibility, LAS minimality, UPCP ordering, and response-threshold propagation) to derive a Lorentz-signature propagation geometry, explicitly stopping short of any claim to physical spacetime. The panel classified this as pure_mathematics, so the falsifiability axis was converted to VERIFIABILITY — independent checkability of theorem-level claims — rather than penalizing the work for lacking laboratory predictions; this conversion is appropriate given the manuscript's own stated scope and its Section 20 recomputable R^4 consistency model. All four math specialists independently verified the core linear-algebraic content as correct: the orthogonal splitting V_o = RT_o ⊕_q H^QH_o, the signature computation for Q_o, the rank-one defect identity D_o = q_o − G_o = (1+c_o^2)Θ̂_o⊗Θ̂_o with its clean Horizon-recovery corollary, and Theorem 5.11's ordered-cone representation. Two concrete, convergent rigor concerns emerged across three of the four math reviewers and should be prioritized by the author. First, Theorem 8.9 asserts domain-level 'smooth intrinsic field' assembly for the propagation cones before smoothness of the pointwise capacity c_o has been established — smoothness is only supplied later, conditionally, by Theorem 17.1 under Regular Local Continuation (Principle 15.1); two specialists flagged this as a scope-of-claim overstatement requiring the theorem to be read fiberwise until the later result is invoked. Second, the manuscript advertises 'Cartan-compatible assembly' (Remark 16.1, Corollary 18.2, Lemma 16.2's CAN1–CAN6 framework) as part of the smoothing architecture, but the actual proof of Theorem 17.1 uses only Regular Local Continuation plus pointwise uniqueness plus a regular-value argument — Lemma 16.2 is never invoked and its hypotheses are never verified for the HRCEL frontier, making this advertised derivational component unsupported as written (though the direct proof survives independently). One specialist (claude-opus-5) raised a more serious concern not fully corroborated by the others: that Theorem 8.7's Assumption 8 (Continuation-Speed Correspondence, equating operational distinguishability speed with the response-metric norm) and Assumption 7's unspecified speed-resource functional ν_o are asserted without derivation yet are load-bearing for the entire finite-threshold conclusion and everything downstream (RCSP, BET, HRCEL). This is a legitimate close reading worth the author's attention, though it was not independently flagged by the other three math reviewers, who rated the same theorem's logic as valid conditional on its stated assumptions — readers should treat this as a flagged-for-verification point of specialist disagreement rather than a settled defect. A minor symbol collision was also noted: Theorem 10.7 redefines A_o(v) as c_o^2 a^2, conflicting with Definition 8.4's A_o(v):=Θ̂_o(v) used throughout Sections 8, 11, 13, and 17; this is self-consistent within Theorem 10.7 but could confuse cross-referencing readers. On completeness, all three sources specialists converge near 4/5: the paper is exceptionally disciplined about labeling principles versus derived results and stating claim boundaries, but several foundational inputs (the additional-structure preorder, the structural restoration-cost functional, RMS, FPRP, and Regular Local Continuation) remain posited rather than constructed, and the Section 20 worked model stipulates rather than derives its capacity function, so it tests downstream algebra but not the paper's heaviest physical premises. The evidence_strength score of 0/5 is expected and appropriate for a paper (not a framework) and should not be read as a quality judgment. Reference verification flagged 12 of 15 citations as unverified in automated databases (e.g., Kobayashi-Nomizu, Maldacena, Van Raamsdonk), but none were marked FABRICATED — these are standard, well-known texts and the appropriate characterization is a citation-metadata/identifier gap, not a scholarly-integrity concern.
This review was generated by AI for research and educational purposes. It is not a substitute for formal peer review. All analyses are advisory; publication decisions are based on numerical score thresholds.
The logical architecture is unusually well policed. Every major theorem is followed by an explicit claim-boundary remark, the dependency chain is stated and acyclic, and the paper repeatedly resists the conflation that would be most tempting (Horizon = propagation boundary); HES7, BET5, HRCEL5 and Corollary 10.3 all prove separation rather than assert it. Sector definitions (C^+, H^QH, K^+, Σ_c, F^+) remain stable across Sections 5–13. Rank-one and rank-≥2 edge cases are handled explicitly (Sec. 6.1, BET5, Theorem 8.9's empty normalized section when dim V_o = 1, Prop 9.4). Two local inconsistencies prevent a 5. (a) Theorem 10.7 redefines A_o(v) := c_o^2 a^2, in direct conflict with A_o(v) := \hatΘ_o(v) used in Def. 8.4, Sec. 11, and Sec. 13; the statement is internally consistent but the symbol collision is a genuine slip in a load-bearing section. (b) In the proof of Theorem 10.1(3), v_ε is asserted to lie outside the closure \bar K^+_o, justified by the sentence 'whose response-threshold description uses Q_o ≤ 0 together with nonnegative advance' — this is correct, but the inequality range 0 < ε < sqrt(q_o(h,h))/c_o gives Q_o(v_ε) > 0 strictly, so the conclusion holds; the wording obscures rather than breaks it. Neither defect propagates to the central conclusions.
The purely linear-algebraic and differential-geometric content is correct and I verified it independently. The orthogonal splitting V_o = RT_o ⊕_q H^QH_o and the identity q_o(v,v)=A_o(v)^2+q_o(v^⊥,v^⊥) follow from \hatΘ_o(T_o)=q_o(T_o,T_o)=1 and q_o(T_o,h)=\hatΘ_o(h)=0. The signature claim for Q_o(aT_o+h)=q_o(h,h)-c_o^2 a^2 is immediate (negative on the advance line, positive on the Horizon), giving (-,+,...,+). The defect identity D_o = q_o - G_o = (1+c_o^2)\hatΘ_o⊗\hatΘ_o checks: q_o(v,w)=\hatΘ_o(v)\hatΘ_o(w)+q_o(v^⊥,w^⊥) minus G_o(v,w)=q_o(v^⊥,w^⊥)-c_o^2\hatΘ_o(v)\hatΘ_o(w) yields (1+c_o^2)\hatΘ_o(v)\hatΘ_o(w). Theorem A's one-dimensionality lemma (a closed convex cone K with K∪(-K)=W, K∩(-K)={0} forces dim W = 1, by connectedness of the unit sphere for dim ≥ 2) is valid. The gluing argument in Theorem 17.1 (local RLC tracks + pointwise uniqueness ⇒ global smooth c) is correct, and the regular-value computation d_vΨ_h[h]=2q_o(h,h)=2c_o^2>0 is right. However, the cap applies: Theorem 8.7, the paper's headline derivation, rests on Assumption 8 (V_o(v)=sqrt(q_o(v,v))) and the unspecified functional ν_o, neither derived nor referenced; and Theorem 9.6's finite capacity is an explicit premise (Assumption 9.5), as Remark 9.7 concedes. Additionally, Theorem 2.1's measurement-sector least-norm solution and the eigenvalue/d scaling argument are sketched rather than fully computed, and Lemma 16.2's CAN2 non-circularity condition is stated as an assumption rather than verified for the HRCEL application.
Verifiability conversion used, appropriate for pure mathematics. Many central consequences are presented in explicit finite-dimensional notation with proof sketches or proofs: the survival-order representation, response-threshold cone, rank-one Lorentz-signature form, and the R^4 consistency model can be independently checked conditional on their premises. The manuscript also states numerous exact scope and failure boundaries, such as fixed-rank restrictions and loss of uniqueness/nondegeneracy. However, independent verification of the full claimed architecture requires substantial reconstruction of inherited Paper 3 geometry and acceptance or separate operational instantiation of several high-level inputs, especially Structural Restoration Cost, RMS, FPRP, Physical Propagation Realizability, and Regular Local Continuation. The included R^4 model selects rather than derives several of these ingredients. Thus the conditional formal claims are checkable, but the full framework has only a partial independent verification path.
The manuscript is carefully organized, provides a dependency map and logical-status table, repeatedly distinguishes assumptions from derived results, and is commendably explicit about scope limitations. These features substantially help audit a long framework paper. However, a graduate-level reader outside the author’s preceding-paper program must absorb a large number of bespoke primitives and acronyms—QII, LAS, UPCP, PC4, RTP, RCSP, BET, HRCEL, RLC, SIA, and SSR—before evaluating the central message. Several later sections restate closely related cone/boundary consequences under new named theorems, which obscures the incremental contribution of each result. The presentation would be clearer with an early plain-language worked example that begins from explicit operational data, a compact glossary, and a table separating primitive modeling assumptions from consequences. Notation appears broadly consistent, so no symbol-redefinition cap is applied.
The staged synthesis is a recognizable original contribution: it separates informational admissibility, minimum-response realization, provenance-based ordering, and a threshold cone construction, then emphasizes the distinction between the UPCP equilibrium hyperplane and the null propagation boundary. The recovery of the Horizon from the rank-one defect and the organization of smooth assembly around a scalar capacity field are also coherent framework-level contributions. Nevertheless, the Lorentzian rank-one deformation, cone geometry, bundle methods, and regular-level-set assembly are acknowledged standard ingredients, while the manuscript offers limited direct comparison with nearby information-geometric, operational-reconstruction, or emergent-causality constructions. It does not yet supply an explicit quantum-operational realization or a new physical consequence that clearly differentiates this synthesis from alternatives. Novelty is therefore credible at the level of architecture and terminology, but not yet demonstrated as a sharply distinguished new mechanism with independently consequential outputs.
The paper is unusually disciplined in stating its own scope and explicitly disclaiming conclusions it does not reach (repeated 'claim boundary' and 'logical status' remarks after nearly every theorem). Each of its three headline derivational claims (derived PC4, derived pointwise RTP, derived smooth HRCEL assembly) is supported by a fully worked proof chain with assumptions clearly labeled as principles/assumptions versus derived theorems. Boundary/edge cases are addressed explicitly in several places (e.g., dim V_o=1 vs ≥2 cases for the Horizon and null boundary, rank-changing loci excluded from claims, the 'model scope' section explicitly noting the tangent-bundle identification is a modeling choice not a general result). Limitations are stated extensively (Section 22, numerous remarks). Some gaps remain: several foundational inputs (the additional-structure preorder ⪯_add, the structural restoration-cost resource functional R_o(Γ), the survival-perturbation set Δ_o, the 'informationally empty-redescription' equivalence) are introduced as abstract assumed data without concrete construction or existence proof, which is acknowledged by the author as an assumption but leaves open whether such data exist in any given model — a secondary but real gap. The one worked example (Section 20) stipulates A^QII_0 = TU and an arbitrary smooth c(o) by fiat rather than deriving them from the QSL-FPRP/restoration-cost chain, so the 'recomputable consistency check' does not independently test the heaviest assumptions (FPRP, RMS, Structural Restoration Cost). These are gaps in secondary/supporting material rather than in the core logical chain connecting the stated premises to the stated conclusions, so they do not rise to a missing-central-derivation issue.','
These are equations, theorem steps, or proof moves where math specialists identified compressed or unverified derivations. They are shown once here as the canonical deduplicated list so the review stays auditable without repeating the same flags in every math specialist report.
- highTheorem 8.7, Assumption 7 (Generalized Speed-Limit Action Bound) - The dynamical speed-resource functional nu_o is described as 'independently specified' but is nowhere specified in the paper; consequently c_{P,o}(v) := inf_G nu_o(G) is an undefined quantity used in a quantitative bound.
If wrong: The quantitative bound c_o <= sqrt(B_{P,o}^2 - 1) is vacuous, and the 'QSL' half of the QSL-FPRP architecture carries no content; the derivation of pointwise RTP reduces to the qualitative closure axioms 1-6 plus a bare finiteness postulate, i.e. essentially Assumption 9.5 restated.
- highTheorem 8.7, Assumption 8 (Continuation-Speed Correspondence) - The identity equating infinitesimal operational distinguishability speed with the response-metric norm is asserted as an assumption with no derivation, no citation, and no connection to the Composite Bures Response Representation of Theorem 2.1, where such a link would plausibly be established.
If wrong: Step 3's properness of J_o fails, so sup J_o may be infinite; then no finite c_o exists, Q_o = q_o(v^perp,v^perp) - c_o^2 \hatTheta_o(v)^2 is not defined, and the Lorentz-signature form G_o, the null boundary (Theorem 10.12, BET), RCSP, HRCEL, and SSR are all unsupported. The paper's central claim that Lorentz signature is derived rather than assumed collapses to the prior Version-4 position in which RTP is adopted.
- mediumLemma 16.2 CAN2 / Remark 16.1 / Theorem 17.1 - CAN2 imposes a non-circularity condition on the normalization map N (it must not be defined using the selected parameter section or assembled congruence), and CAN5 requires dim Lambda = dim W with invertible vertical derivative. Neither condition is verified for the HRCEL application; Remark 16.1 asserts that no separate HRCEL normalization map is required, and the proof of Theorem 17.1 in fact does not invoke Lemma 16.2 at all.
If wrong: If Lemma 16.2's hypotheses fail in the HRCEL setting, nothing in Theorem 17.1 breaks, since the lemma is not used. The issue is a mismatch between the advertised derivational chain ('Cartan-compatible assembly => SIA and SSR') and the proof actually given, which rests on RLC + uniqueness + regular-value theorem alone.
- mediumRemark 16.1 and Corollary 18.2, Cartan-compatible assembly - No Cartan principal bundle, normalization map, or verification of CAN1–CAN6 is supplied for the HRCEL frontier. The asserted Cartan identification is therefore not reproducible as an application of Lemma 16.2.
If wrong: The Cartan-natural congruence characterization remains unsupported; Theorem 17.1's direct smooth-capacity and frontier conclusions do not depend on it.
- mediumSections 16–18; Lemma 16.2 and Theorem 17.1 - The manuscript presents Cartan assembly as a component of the smooth-assembly chain, but the HRCEL Smoothing Law does not apply Lemma 16.2 or verify its CAN1–CAN6 hypotheses for an HRCEL associated bundle, normalization map, or model frontier family.
If wrong: The direct RLC-plus-gluing proof can still support conditional smooth capacity and frontier assembly, but the stronger architectural assertion that Cartan-compatible assembly has been derived or is doing mathematical work is unsupported. Any claimed Cartan naturality beyond the direct bundle construction requires an additional application theorem.
- mediumTheorem 2.1, measurement-sector least-norm solution and the Bures pullback scaling - The least-norm minimization over {\dot W_i} and the subsequent eigenvalue/tangent-entry scaling by 1/d under Omega are asserted ('the least-norm solution gives', 'Hence its Bures pullback is') without the intermediate computation.
If wrong: Theorem 2.1's explicit state-plus-measurement representation of q_o would be incorrect, but the downstream geometry (Sections 3-19) uses only that q is smooth, positive definite, and gauge independent; the main construction would survive. Peripheral to the central claim.
- mediumTheorem 8.9 (Theorem B: Response-Defined Propagation Geometry) - The theorem claims smooth domain-level cone-field assembly before smoothness of the pointwise threshold c_o has been assumed or derived. The proof establishes fiberwise cone properties; its final smoothness sentence itself conditions assembly on smoothness of c, which is only supplied later by Theorem 17.1 under RLC.
If wrong: Without a smooth c field, the total-space propagation cone and Lorentz-signature form need not be smooth over the base. Theorem 8.9 must be read fiberwise; the paper's smooth hybrid-geometry conclusion is supported only later and conditionally on RLC.
- lowLemma 16.2, surjectivity conclusion - The proof invokes nonemptiness of every normalized model fiber, but CAN1–CAN6 do not explicitly require it. A smooth submersion need not be surjective.
If wrong: Lemma 16.2's surjective-submersion conclusion fails for empty selected fibers; its smooth pullback construction can still hold where fibers exist.
- lowTheorem 10.7, definition of A_o(v) - Symbol collision: A_o(v) is redefined as c_o^2 a^2, conflicting with Definition 8.4's A_o(v) := \hatTheta_o(v) used in Sections 8, 11, 13, 17.
If wrong: No mathematical consequence; a reader cross-referencing Theorem 10.7(4) against HRCEL2 could mistakenly read R_o(v) < c_o A_o(v) and F_o(v) < A_o(v) as inconsistent. Purely expositional.
- lowTheorem 17.1 (HRCEL Smoothing Law), Step: 'On an overlap W_i ∩ W_j, both c_i(o) and c_j(o) are HRCEL capacities at the same point. By SL2, that capacity is unique. Hence c_i = c_j on every overlap.' - The proof glues local smooth capacity tracks using pointwise uniqueness. This is a standard argument but relies on the Regular Local Continuation Principle (Principle 15.1) which is a foundational local regularity principle, not a derived theorem.
If wrong: If the Regular Local Continuation Principle fails, the smooth capacity field would not exist. However, the principle is explicitly labeled as a foundational local regularity principle, and the paper's scope limitations acknowledge this.
- lowTheorem 8.7 (QSL-FPRP Derivation of Pointwise RTP), Step 3: 'QSL-FPRP properness' - The proof derives the bound RR_o(v) ≤ sqrt(B_{P,o}^2 - 1) using the Generalized Speed-Limit Action Bound, Continuation-Speed Correspondence, and Finite Process Resource Principle. The derivation is complete but relies on the assumption that the infimal physical-resource requirements have finite supremum (FPRP).
If wrong: If the FPRP assumption fails, the bound would not hold and the threshold c_o might not be finite. However, FPRP is explicitly identified as a new physical finiteness input, not derived from earlier principles.
- lowTheorem 8.9, opening domain-level conclusion - The smooth-field wording precedes the local-continuation premise needed to make the pointwise thresholds vary smoothly. The theorem's qualification directs the reader to the later assembly result.
If wrong: Theorem 8.9 establishes only its fiberwise conclusions at that stage; domain-level smoothness requires Theorem 17.1.
These flags compare the submission's headline wording with what its body establishes. They are not mathematical-error findings.
Claimed: PC4, pointwise RTP, and smooth assembly are derived through the staged architecture.
Established: They are conditional deductions using, respectively, strong order and integration hypotheses, qualitative propagation and finite-resource hypotheses, and a local smooth-continuation principle together with smooth UPCP data.
The headline shorthand can obscure which inputs are independently assumed; the body largely discloses them.
Claimed: The paper presents smooth cone, propagation-form, frontier, and SSR assembly as consequences of the staged informational-to-geometric chain, including Cartan-compatible assembly.
Established: The final smooth statements are conditional on the foundational Regular Local Continuation principle, which directly supplies smooth local capacity tracks; global smoothness then follows by uniqueness and ordinary gluing. The displayed Cartan lemma is not applied to the HRCEL data.
The smoothness conclusion is not derived from the earlier QII/LAS/UPCP/RTP structure alone, and the specific claimed Cartan contribution is not demonstrated in the HRCEL proof.
Claimed: 'pointwise Response–Threshold Propagation is derived from qualitative refinement structure together with a generalized speed-limit action bound, continuation-speed correspondence, physical projective realizability, and the Finite Process Resource Principle' — presented as replacing an adopted axiom by a derivational chain.
Established: Theorem 8.7 proves that twelve assumptions, of which at least four (the unspecified speed-resource functional nu_o, the Continuation-Speed Correspondence V_o(v)=sqrt(q_o(v,v)), FPRP, and relative openness of the propagation-compatible ratio set) carry the entire quantitative content, jointly imply the existence of a unique finite c_o.
The word 'derived' suggests reduction to more primitive or independently established structure. In fact the Continuation-Speed Correspondence is an undefended identification of an operational speed with a specific metric norm, and nu_o is never defined. Remark 8.8 isolates FPRP as 'the explicit physical finiteness input' but does not mention that Assumption 8 is equally load-bearing and equally unsupported. The abstract's framing therefore understates how much substantive content was transferred rather than eliminated.
Claimed: The paper claims to derive a 'smoothly assembled Lorentz-signature propagation form' and associated horizon and boundary structures from regular finite-dimensional quantum-operational realization structures.
Established: The body establishes a smooth Lorentz-signature form on the QII-admissible bundle under the Regular Local Continuation Principle and HRCEL Smoothing Law, but explicitly stops short of identifying the result with physical spacetime or establishing global causality or gravitational dynamics.
The abstract's phrasing 'smoothly assembled Lorentz-signature propagation form' is accurate for the selected regular domain, but the paper's own scope limitations make clear this is a pre-spacetime propagation geometry, not a physical spacetime geometry. The gap is minor because the abstract also includes the explicit disclaimer.
Claimed: Cartan-compatible assembly contributes to the established smooth HRCEL package.
Established: Direct smooth assembly is proved from local continuation and uniqueness, whereas a specific HRCEL application of the Cartan assembly lemma is not verified.
The Cartan characterization is stronger than the demonstrated direct smoothing result.
Claimed: 'the HRCEL Smoothing Law together with Cartan-compatible assembly yields Smooth Informational Assembly'; the dependency map lists 'HRCEL smoothing + Cartan assembly => SIA and SSR'.
Established: The proof of Theorem 17.1 uses RLC, pointwise uniqueness for overlap agreement, gluing, and the regular-value theorem. Lemma 16.2 (Cartan Assembly) is stated and proved but is never invoked in Theorem 17.1 or Corollary 18.1/18.2.
Cartan-compatible assembly is advertised as a component of the derivational chain but does no work in the proof actually given. Remark 16.1 explicitly says 'no separate HRCEL normalization map is required', which is a concession that the Cartan machinery is not used; the abstract and conclusion nonetheless list it as a load-bearing step.
Claimed: Section 9 is titled 'Refinement-Cost Characterization of Pointwise RTP' and is listed in the architecture as part of the derivation of RTP.
Established: Remark 9.7 states plainly: 'Finite Refinement Capacity is an explicit quantitative premise. The theorem decomposes and characterizes pointwise RTP; it does not derive finite propagation capacity from the preceding informational geometry alone.'
The paper is internally honest here, but the abstract and dependency map do not distinguish the Section 9 characterization route (which assumes finite capacity) from the Section 8 QSL-FPRP route (which assumes FPRP plus the speed correspondence). A reader could take the two as independent corroborating derivations when they are alternative assumption packages for the same conclusion.
Claimed: The introduction states that 'the resulting capacity field, Lorentz-signature propagation form, defect tensor, cone field, punctured null boundary, normalized boundary bundle, threshold graph, and canonical bridge representative assemble smoothly on the selected connected fixed-rank QII-regular domain.'
Established: This is established in Section 19 (Strong Smooth Regular Realizability) as a consequence of the HRCEL Smoothing Law, which itself depends on the Regular Local Continuation Principle (a foundational local regularity principle) and pointwise RTP uniqueness.
The claim is conditional on the Regular Local Continuation Principle, which is explicitly labeled as a foundational principle rather than a derived theorem. The introduction's phrasing could be read as unconditional, but the body clearly states the conditionality.
Strengths
- Exceptional, unusually disciplined claim-boundary practice: nearly every theorem is followed by an explicit 'logical status' and 'claim boundary' remark stating precisely what is and is not established, substantially reducing overclaiming risk.
- The rank-one defect identity D_o = q_o − G_o = (1+c_o^2)Θ̂_o⊗Θ̂_o (Theorem 10.9) is independently verified as correct by multiple specialists and yields a genuine rigidity result: ker D_o = H^QH_o with uniqueness up to sign and normalization.
- The paper rigorously maintains the distinction between the Quantum Horizon (UPCP equilibrium separatrix) and the propagation null boundary across HES, BET, and HRCEL — proving the separation (e.g., HES7, BET5, HRCEL5, Corollary 10.3) rather than merely asserting it.
- Theorem 5.11's linear survival-order representation and the orthogonal advance/Horizon splitting underlying the Lorentz-signature construction are correct, standard, finite-dimensional arguments, verified independently by all four math reviewers.
- Section 20 supplies a fully worked, recomputable R^4 consistency model allowing independent readers to hand-check the cone, null boundary, defect tensor, and Horizon/boundary distinction.
Areas for Improvement
- Restrict or explicitly condition Theorem 8.9's domain-level 'smooth intrinsic field' language: as written it precedes the smoothness result for c_o, which is only established later in Theorem 17.1 under Principle 15.1 (Regular Local Continuation); multiple specialists flagged this exact passage.
- Either supply a concrete instantiation of Lemma 16.2's CAN1–CAN6 hypotheses for the HRCEL frontier, or remove/soften the 'Cartan-compatible assembly' framing in Remark 16.1, Corollary 18.2, and the abstract, since Theorem 17.1's actual proof does not invoke the Cartan lemma at all.
- Address the flagged concern (raised specifically by one math reviewer, not corroborated by the other three) that Theorem 8.7's Assumption 8 (Continuation-Speed Correspondence) and the unspecified speed-resource functional ν_o in Assumption 7 are asserted without derivation; given this is load-bearing for the threshold c_o and everything built on it, an explicit justification or citation would strengthen the central derivation chain.
- Correct the symbol collision in Theorem 10.7, where A_o(v) is redefined as c_o^2 a^2, conflicting with the A_o(v):=Θ̂_o(v) convention used in Definition 8.4 and Sections 8, 11, 13, and 17.
- Provide a general existence construction (or explicit discussion of existence conditions) for the abstract inputs posited axiomatically — the additional-structure preorder ⪯_add, the structural restoration-cost functional R_o(Γ), the admissible perturbation set Δ_o — since these currently have stated axiomatic properties but no demonstrated instantiation.
- Consider deriving the Section 20 model's capacity function c(o) and bundle identification from the paper's own restoration-cost/RMS/FPRP chain rather than stipulating them by fiat, so the worked example also tests the heaviest physical premises rather than only the downstream geometric algebra.
- Reduce acronym/named-theorem proliferation (QII, LAS, UPCP, PC4, RTP, RCSP, BET, HRCEL, RLC, SIA, SSR) with a consolidated glossary or earlier plain-language recap, and check for theorem-label reuse (e.g., 'Theorem A'/'B'/'C'/'D' appearing in multiple distinct sections) that could confuse cross-referencing.
Key Equations (3)
Response-threshold quadratic form whose sign classifies propagating, null-critical, and future-oriented non-propagating directions.
Polarized Lorentz-signature propagation form on each QII-admissible continuation fiber.
Response-threshold characterization of propagation on the positively oriented sector.
Other Equations (3)
Defines the intrinsic QII-admissible continuation space as the kernel of the reduced structural defect.
Canonical decomposition of an admissible continuation into UPCP advance and a transverse component in the Quantum Horizon.
Rank-one defect identity, which makes the Quantum Horizon recoverable as the kernel of the defect.
Tags & Keywords
Keywords: quantum-operational geometry, quantum informational identity, Bures response metric, provenance-coherence ordering, response-threshold propagation, Lorentz-signature geometry, null propagation boundary, Cartan assembly
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Conservation, Derived Ordering, Equilibrium, and Smooth Hybrid Propagation Geometry in Quantum-Operational Geometry Paper 4 Version 5 Michael Tolbert October 2026 Contents 1 Introduction4 2 Inherited regular realization geometry8 2.1 Composite Bures response representation . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Structural Quantum Informational Identity11 4 Least Additional Structure15 5 Unified Provenance–Coherence Principle17 5.1 Derived PC4: restoration cost, survival order, and differential representation . . . . . 18 5.2 Standing survival architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.3 Pre-survival structural severity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.4 Restoration-Monotone Survival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.5 Theorem D: survival-set comparability . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.6 Theorem A: linear survival-order representation . . . . . . . . . . . . . . . . . . . . . 20 5.7 Theorem B: smooth local integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.8 Theorem C: assembled PC4 representation . . . . . . . . . . . . . . . . . . . . . . . . 23 5.9 Final PC4 assembly statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.10 Logical boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6 The Local Quantum Horizon25 1
Paper 4 Version 5Canonical Realization and Hybrid Geometry 6.1 Nontriviality qualification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.2 Global scope disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.3 The Quantum Horizon as an Organizing Structure . . . . . . . . . . . . . . . . . . . 27 7 Horizon Equilibrium Structure28 8 Theorem A: Cone Promotion32 8.1 Response-Threshold Propagation Principle . . . . . . . . . . . . . . . . . . . . . . . . 34 8.2 Inherited Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.3 Response-Normalized UPCP Advance and Canonical Splitting . . . . . . . . . . . . . 35 8.4 Response-Threshold Propagation Principle . . . . . . . . . . . . . . . . . . . . . . . . 37 8.5 Theorem B: Response-Defined Propagation Geometry . . . . . . . . . . . . . . . . . 43 8.6 Logical Status and Claim Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 8.7 Adopted Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 9 Refinement-Cost Characterization of Pointwise RTP46 10 Horizon-Organized Hybrid Geometry and Null Boundary Realization47 10.1 Separatrix-to-null-barrier transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 10.2 Horizon uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 10.3 Horizon propagation-failure structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 10.4 Horizon essentiality and unique recovery . . . . . . . . . . . . . . . . . . . . . . . . . 54 10.5 Null Boundary Realization and threshold–radius identification . . . . . . . . . . . . . 57 10.6 Status after pointwise null-boundary realization . . . . . . . . . . . . . . . . . . . . . 58 11 Response-Capacity Saturation Law59 12 Boundary Emergence Theorem61 13 Horizon Response-Capacity Equilibrium Law64 14 Regular Local Continuation, Cartan Assembly, and HRCEL Smoothing67 15 Regular Local Continuation67 16 Cartan assembly68 2
Paper 4 Version 5Canonical Realization and Hybrid Geometry 17 HRCEL smoothing law70 18 Consequences72 19 Strong Smooth Regular Realizability73 20 A Finite-Dimensional Model of the HRCEL Construction74 21 Discussion76 21.1 The Emergent Hybrid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 21.2 Architectural and scope observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 21.3 Lessons Learned from the Refinement Program . . . . . . . . . . . . . . . . . . . . . 77 22 Scope, Limitations, and Open Questions79 23 Conclusion79 3
Paper 4 Version 5Canonical Realization and Hybrid Geometry Abstract We develop a staged propagation geometry on the regular finite-dimensional quantum- operational realization structures inherited from Paper 3. Structural Quantum Informational Identity (QII) supplies an intrinsic admissible continuation space; Formal Surplus Representation and Least Additional Structure (LAS) select the canonical minimum-response realization class; and the Unified Provenance–Coherence Principle (UPCP) supplies the order and orientation structure from which the Quantum Horizon is obtained. Version 5 replaces three previously bundled or adopted components by explicit derivational chains. First, PC4 is derived from Structural Restoration Cost, Restoration-Monotone Survival, survival-set comparability, linear order representation, and smooth local integration. Second, pointwise Response–Threshold Propagation is derived from qualitative refinement structure to- gether with a generalized speed-limit action bound, continuation-speed correspondence, physical projective realizability, and the Finite Process Resource Principle. Third, Smooth Informa- tional Assembly is derived from the Regular Local Continuation Principle, pointwise threshold uniqueness, and Cartan-compatible assembly through the HRCEL Smoothing Law. The resulting capacity field, Lorentz-signature propagation form, defect tensor, cone field, punctured null boundary, normalized boundary bundle, threshold graph, and canonical bridge representative assemble smoothly on the selected connected fixed-rank QII-regular domain. The rank-one passage from positive-definite response data and a normalized covector to a Lorentz- signature form is standard; the contribution claimed here is the information-first derivation and controlled assembly of the inputs. The result remains a pre-spacetime propagation geometry on the QII-admissible bundle. It does not establish tangent-bundle identification, integrability, four-dimensionality, physical spacetime, global causality, or gravitational dynamics. 1 Introduction Quantum theory presents noncommutative operator and measurement data, while familiar causal geometry is formulated in terms of directional, separating, and orientation structures. The purpose of this paper is not to identify the operational manifold with physical spacetime or to combine quantum and gravitational field equations. The narrower objective is to determine which propagation- geometric structures can be obtained intrinsically from regular quantum-operational realization geometry. The work is situated alongside informational reconstructions of quantum theory, structural- realist approaches to scientific representation, and contemporary programs investigating emergent spacetime and causal organization [14, 4, 3, 15, 7, 13, 10, 2]. Paper 3 established a smooth regular operational manifold O reg , a principal realization bundle π :R reg −→O reg , exact compact gauge fibers, a positive framed response metricg fr , and a distinguished minimum- response horizontal distributionH fr . These structures provide canonical horizontal realization transport for a prescribed operational continuation. They do not, by themselves, decide which operational continuations preserve informational identity or how admissible continuations should be ordered. The present paper separates those questions into three roles: 1. QII: conservation and feasibility. Which infinitesimal continuations preserve the repre- sented informational architecture up to admissible intertwining? 4
Paper 4 Version 5Canonical Realization and Hybrid Geometry 2.LAS: canonical variation. For a fixed QII-compatible continuation, which realization representative contains no additional realization structure? 3.UPCP: ordering and orientation. How are QII-compatible continuations ordered by robustness of provenance coherence? The resulting architecture is QII: conservation LAS: canonical variationUPCP: ordering. Their pointwise synthesis yields the Local Quantum Horizon. No preferred ray is required: the UPCP differential structure directly supplies positive, zero, and negative sectors on the QII-admissible continuation space. The principal contributions are separated by logical status. Core pointwise and fiberwise results. 1. Intrinsic admissibility and canonical realization: gauge-descent of the QII kernel, Formal Surplus Representation, and LAS–minimum-response compatibility. 2. Orientation and equilibrium: UPCP, the Quantum Horizon, and the canonical advance– Horizon splitting. 3.Threshold propagation geometry: factorization through attained Refinement Ratios, full ratio attainability on rank-at-least-two fibers, finite response capacity, and the local Lorentz-signature propagation form. Derived ordering, propagation, and regularity extensions. PC4 is no longer retained as a bundled UPCP axiom: its clauses are derived from Structural Restoration Cost, Restoration- Monotone Survival, Theorems D–A–B, and the assembled differential-representation theorem. Pointwise RTP is no longer adopted as an independent propagation principle: it is derived from the QSL–FPRP architecture, with the Finite Process Resource Principle exposed as the physical finiteness input. Smooth Informational Assembly is no longer the load-bearing premise for cross- point regularity: the Regular Local Continuation Principle supplies smooth local capacity tracks, pointwise RTP uniqueness forces overlap agreement, and Cartan-compatible HRCEL assembly yields the global smooth capacity and frontier geometry. Theorem A and RTP provide the domain-level cone promotion and quantitative propagation refine- ment that form the intended endpoint of the present construction. Full tangent-bundle realization, effective dimension, growth laws, global causal reconstruction, and spacetime interpretation remain outside the claims of this manuscript. Novelty and Significance The novelty of the present work does not lie in the isolated algebraic observation that a rank-one deformation of a positive-definite quadratic form can produce Lorentzian signature. That operation 5
Paper 4 Version 5Canonical Realization and Hybrid Geometry is standard finite-dimensional geometry, and no new general rank-one construction is claimed here. The contribution instead lies in the derivational architecture that precedes and constrains that operation. Informational admissibility, canonical minimal realization, provenance-coherent ordering, and response-threshold selection are assembled in stages so that the distinguished data entering the Lorentzian construction are not introduced merely because they recover a desired signature. The resulting Lorentzian structure is therefore the endpoint of the architecture developed here rather than its starting assumption. This distinction makes provenance part of the mathematical content, rather than ancillary bookkeeping. If a preferred direction, deformation scale, threshold, or null structure is selected simply because it produces Lorentzian geometry, the resulting construction provides limited grounds for assigning that geometry further physical significance. The question pursued here is stronger: whether the ingredients required for the Lorentzian transition can be obtained from independently specified informational and realization principles, with the geometric signature arising only after those ingredients have been established. A Lorentz-signature quadratic form can be written down abstractly with little difficulty. The present framework instead asks whether such a form can arise at the end of a controlled passage from informational structure to geometric realization. To the extent that this passage succeeds, the resulting geometry becomes a candidate pre-spacetime realization architecture whose additional structure and possible physical interpretation can be investigated rather than presupposed. No identification with physical spacetime is asserted here. Lorentzian signature alone is insufficient for such an identification, and the present construction does not establish dimensionality, dynamics, empirical adequacy, or other structures ultimately required of a physical spacetime. It establishes an upstream architecture within which those stronger questions can be posed without assuming their answers. In particular, the construction is intended to support later investigation of admissible continuation structure and its independent degrees of freedom, including whether that structure selects a definite dimensionality. The present paper does not claim those downstream results; it establishes the architecture that makes their investigation possible. The broader novelty claim is therefore specific but consequential: the standard Lorentzian alge- bra is not the proposed innovation; the proposed innovation is the staged informational- to-geometric derivation that supplies and constrains its inputs. The goal is not to declare the resulting geometry to be spacetime prematurely, but to determine how far spacetime-like structure can be obtained from antecedent informational and realization principles and subjected to progressively stronger mathematical and physical tests. Validation status. The results established here are mathematical consequences of the inherited realization geometry and the framework’s stated premises. The finite-dimensional model provides a recomputable consistency check, not an empirical or quantum-operational validation. We agree that a nontrivial operator- and measurement-level realization is an important next test of the framework and plan to construct one from explicit quantum-operational data. That analysis is reserved for subsequent work so the present paper’s derivational architecture can be evaluated independently of a particular model. Independent derivation, comparison with other reconstruction programs, and eventual physical testing remain necessary before the resulting 6
Paper 4 Version 5Canonical Realization and Hybrid Geometry geometry could be identified with spacetime. Architecture and Logical Dependency Map The construction is staged. Each stage answers a distinct question and adds only the structure needed for the next stage: Inherited regular realization geometry −→ QII admissibility −→ LAS canonical realization −→ UPCP orientation −→ Quantum Horizon and HES −→ RTP propagation selection −→ RCSP saturation −→ BET boundary emergence −→ HRCEL equilibrium geometry −→ Regular Local Continuation −→ HRCEL Smoothing and Cartan-compatible assembly −→ Smooth Informational Assembly and SSR, with Selector Rigidity retained as a no-autonomous-freedom theorem. The arrows record logical dependence, not temporal evolution, physical causation, or dimensional reduction. The inherited response metricqremains positive definite: it normalizes the UPCP direction, supplies the orthogonal advance–Horizon decomposition, and measures transverse response. Lorentz signature appears only in the distinct propagation form selected by RTP and assembled through HRCEL. LayerQuestionOutput QIIWhich continuations preserve infor- mational identity? Linear admissible continuation bundle. LASWhich realization contains no sur- plus structure? Canonical minimum-response realization. UPCPHow are continuations oriented?Positive, zero, and negative sectors. Horizon/HESWhere does normalized UPCP vari- ation vanish? Equilibrium separatrix and balance manifold. RTP/RCSP/BETWhich responses propagate, satu- rate, and form the boundary? Unique capacity and selected saturation boundary. HRCEL How are advance, response, capacity, and boundary unified? Lorentz-signature propagation geometry. SIA What separate premise controls base-point assembly? Conditional smooth capacity field. SSRDoes the full bridge package assem- ble smoothly under the regularity premise? Conditional global smooth regular representa- tion. Logical Status of the Main Structures The terms principle, assumption, definition, and derived result are used distinctly. The table below records the logical role of the main components. 7
Paper 4 Version 5Canonical Realization and Hybrid Geometry ComponentStatusFunctionDoes not establish Inherited geome- try Imported from Paper 3 Smooth realization, gauge, and response-metric setting QII, orientation, or propa- gation QIIFoundational princi- ple with derived ker- nel theorem Informational admissibilityOrientation or propagation LASFoundational mini- mality principle Canonical no-surplus real- ization Preferred propagation UPCPFoundational orien- tation principle Positive-projective covector and sectors Finite capacity Horizon/HESDerived synthesis ob- ject and theorem Equilibrium separatrix and balance manifold Propagation boundary RTPDerived QSL–FPRP theorem Unique finite pointwise re- sponse threshold Base-point smoothness by itself RCSP/BETDerived pointwise theorems Unique saturation class and boundary Smooth cross-point assem- bly HRCEL Derived synthesis law Unified advance-response equilibrium geometry Smooth selector without SIA Selector RigidityDerived theoremNo autonomous threshold degree of freedom Smoothness by itself Regular Local Continuation Foundational local regularity principle Smooth local HRCEL ca- pacity tracks Global compatibility by it- self SmoothIn- formational Assembly Derived corollarySmooth capacity and HR- CEL assembly Completeness beyond the regular domain GlobalHR- CEL/SSR Derived theoremsSmooth cone, boundary, threshold graph, and bridge Spacetime, gravity, or di- mension selection The central distinction is QSL–FPRP =⇒ unique pointwise RTP threshold, RTP+RCSP+BET =⇒ unique pointwise threshold and boundary, RLC+RTP uniqueness =⇒ global smooth threshold field, HRCEL smoothing+Cartan assembly =⇒ SIA and SSR. 2 Inherited regular realization geometry Let U ⊆O reg be an open regular domain. Let π :R reg | U −→ U be the inherited smooth principal realization bundle with compact gauge groupG. Its vertical subspace at R is V R := kerdπ R . The inherited positive,G-invariant framed response metricg fr determines the smooth gauge- equivariant orthogonal decomposition, using standard smooth, Riemannian, and bundle-geometric 8
Paper 4 Version 5Canonical Realization and Hybrid Geometry machinery [8, 9, 5, 6], T R R reg = H fr R ⊕ V R . For o = π(R), the unique horizontal lift is denoted h fr R : T o U −→ H fr R . Every lift e v ∈ T R R reg of v ∈ T o U decomposes uniquely as e v = h fr R (v) + z, z ∈ V R , and orthogonality gives ∥e v ∥ 2 fr =∥h fr R (v)∥ 2 fr +∥z∥ 2 fr . Thus the horizontal lift is the unique pointwise minimum-response lift of a prescribed operational tangent. This inherited fact is representation geometry; it neither defines QII compatibility nor orders distinct operational continuations. 2.1 Composite Bures response representation The inherited operational response metric admits the following explicit fixed-rank state–POVM representation. This result makes the state and measurement contributions to the LAS-compatible response geometry explicit while preserving the regular-stratum claim boundary. Theorem 2.1 (Composite Bures Response Representation). LetH S be a complex Hilbert space of finite dimension d, and let o = (ρ,E 1 ,...,E n )∈O reg r ρ ,r E :=D r ρ (H S )× POVM reg r E (H S ) lie on a regular fixed-rank state-POVM stratum, where r ρ = rankρ,r E = (r 1 ,...,r n ), r i = rankE i . Let u = ( ̇ρ, ̇ E 1 ,..., ̇ E n )∈ T o O reg r ρ ,r E be an operational tangent. Equivalently, its components satisfy ̇ρ = ̇ρ † ,Tr ̇ρ = 0, N ρ ̇ρN ρ = 0, and ̇ E i
̇ E † i , N i ̇ E i N i = 0, n X i=1 ̇ E i = 0, where N ρ and N i are the kernel projections of ρ and E i , respectively. Letq o denote the operational response quadratic form induced by the unique minimum-norm horizontal lift under the additive framed Hilbert-Schmidt realization metric. Define the normalized 9
Paper 4 Version 5Canonical Realization and Hybrid Geometry labeled measurement-state map Ω(E) := ω E
1 d n X i=1 |i⟩⟨i|⊗ E T i , and let g labB E := Ω ∗ g B denote the pullback of the Bures metric from the corresponding labeled output-state space. Then q o (u) = g B ρ ( ̇ρ, ̇ρ) + dg labB E ( ̇ E, ̇ E). Equivalently, the state and measurement contributions satisfy q state ρ = g B ρ , q meas E = dg labB E . More explicitly, write ρ = d X a=1 λ a |a⟩⟨a|, λ a ≥ 0, and, for each i, E i
d X a=1 μ i,a |a i ⟩⟨a i |, μ i,a ≥ 0. Then q o (u) = 1 2 X λ a +λ b
0 |⟨a| ̇ρ|b⟩| 2 λ a
- λ b
1 2 n X i=1 X μ i,a +μ i,b
0 |⟨a i | ̇ E i |b i ⟩| 2 μ i,a
- μ i,b . Proof.Because the framed realization metric is additive across the purification and labeled Naimark factors, and the realization differential splits accordingly, minimization over the affine lift space separates into independent state and measurement minimizations: q o (u) = q state ρ ( ̇ρ) + q meas E ( ̇ E). For the state sector, letXbe a minimal purification amplitude satisfyingρ=XX † . The state-sector response is q state ρ ( ̇ρ) = min ̇ X n ∥ ̇ X∥ 2 HS : ̇ρ = ̇ XX †
- X ̇ X † o . Orthogonal minimization against the vertical purification-gauge directions yields the Uhlmann- horizontal lift. In an eigenbasis of ρ, its minimum squared norm is q state ρ ( ̇ρ) = 1 2 X λ a +λ b
0 |⟨a| ̇ρ|b⟩| 2 λ a
- λ b . 10
Paper 4 Version 5Canonical Realization and Hybrid Geometry Under the standard normalization of the infinitesimal Bures metric, g B ρ ( ̇ρ, ̇ρ) = 1 2 X λ a +λ b
0 |⟨a| ̇ρ|b⟩| 2 λ a
- λ b , and therefore q state ρ = g B ρ . For the measurement sector, choose minimal labeled amplitudes W i satisfying E i = W † i W i . Then q meas E ( ̇ E) = min { ̇ W i } ( n X i=1 ∥ ̇ W i ∥ 2 HS : ̇ E i = ̇ W † i W i
- W † i ̇ W i ) . The fixed-rank tangent condition eliminates kernel-kernel variation, and the least-norm solution gives q meas E ( ̇ E) = 1 2 n X i=1 X μ i,a +μ i,b
0 |⟨a i | ̇ E i |b i ⟩| 2 μ i,a
- μ i,b . Under the labeled-state representation Ω(E) = 1 d X i |i⟩⟨i|⊗ E T i , the eigenvalues of theith block areμ i,a /d, while the corresponding tangent entries are ( ̇ E T i ) ab /d. Hence its Bures pullback is g labB E ( ̇ E, ̇ E) = 1 2d n X i=1 X μ i,a +μ i,b
0 |⟨a i | ̇ E i |b i ⟩| 2 μ i,a
- μ i,b . Therefore q meas E = dg labB E . Combining the two sector identities proves the claim. Scope. The theorem is stratumwise. It applies to regular fixed-rank state-POVM variations, including support motion tangent to the corresponding fixed-rank strata, but makes no claim of a finite smooth extension through rank-changing directions. The measurement termg labB E is specifically the Bures pullback induced by the normalized maximally entangled, outcome-labeled representation Ω. The theorem does not assert that this construction yields globally optimal one-shot POVM discrimination, nor does it establish uniqueness of the composite response metric among all admissible quantum-information geometries. 3 Structural Quantum Informational Identity The refinement architecture used in this section is restricted to fixed labeled events and fixed coarse events. The state, event values, and correlations carried by that architecture may evolve. Theorem 3.1 (Structural QII Conservation and Gauge Descent). Let π :R reg U −→ U 11
Paper 4 Version 5Canonical Realization and Hybrid Geometry be the regular realization principal bundle, with label-preserving gauge groupGand minimum- response horizontal lift h fr R : T o U −→ T R R reg , R∈ π −1 (o). LetHbe the fixed framed ambient carrier, and let{Q i } i∈I be the fixed labeled event projectors. For each fixed event A⊆ I and each block B α of a fixed coarse partition Γ ={B α }, define Q A := X i∈A Q i , Q Γ α := X i∈B α Q i . The refinement structure used in this theorem consists precisely of these fixed label-event and fixed coarse-event identifications. No additional source-target refinement morphisms are included. For eachR ∈ π −1 (o), letP R and Π R denote the smooth Structural-QII projector assignments appearing in the structural profile:P R represents the generated-carrier structural component and Π R the admitted covariance-algebra structural component, both inside the fixed framed ambient architecture. The theorem requires only their stated smoothness and gauge-equivariance properties; it does not identify them with the fixed ambient carrier projector or fixed ambient covariance projector. All vector spaces and linear maps below are real unless otherwise stated. Define the event-compatible infinitesimal intertwiner algebra by k evt :={K ∈ u(H) : [K,Q i ] = 0 for every i∈ I}. Every K ∈ k evt automatically preserves the fixed event and coarse-event projectors: [K,Q A ] = 0,[K,Q Γ α ] = 0. Identify the relevant fixed-rank Grassmannians with their orthogonal-projector manifolds, and define the structural-defect space F R := T P R Gr(H)⊕ T Π R Gr
u(H) . Define the raw Structural QII variation map B R : T o U → F R by B R (v) := DP R
h fr R v ,DΠ R
h fr R v . For K ∈ k evt , define ad K (X) := [K,X] on u(H), and define the infinitesimal structural-intertwiner map L R : k evt → F R by L R (K) := ([K,P R ], [ad K , Π R ]). Let π R : F R −→ F R / imL R be the canonical quotient map. Define the reduced Structural QII defect by C red QII,R := π R ◦ B R , and define A QII o,0 (R) := kerC red QII,R , A QII o (R) := A QII o,0 (R){0}. Then the following conclusions hold. 12
Paper 4 Version 5Canonical Realization and Hybrid Geometry 1.Well-definedness and linearity. The mapsB R ,L R , andC red QII,R are well-defined finite- dimensional real-linear maps. 2. Equivalent Structural QII conditions. For every v ∈ T o U, the following are equivalent: C red QII,R (v) = 0; B R (v)∈ imL R ; and the existence of an event-compatible infinitesimal intertwiner K ∈ k evt such that DP R
h fr R v = [K,P R ] and DΠ R
h fr R v = [ad K , Π R ]. Thus an operational tangent is Structural-QII admissible precisely when its first-order structural variations are induced by one admitted event-compatible infinitesimal intertwiner. 3. Gauge covariance of the raw and reduced Structural-QII defects. Letg ∈ G, set R ′ :=R· g, and letV g :H→Hbe the unitary implementation of the label-preserving gauge transformation. For the Structural-QII assignments, gauge intertwining gives P R ′ = V g P R V † g and Π R ′ = Ad V g ◦Π R ◦ Ad −1 V g ,Ad V g (X) := V g XV † g . Define Γ g : F R → F R ′ by Γ g (X,Y ) := V g XV † g , Ad V g ◦Y ◦ Ad −1 V g . Then Γ g is a linear isomorphism satisfying B R ′ = Γ g ◦ B R and Γ g (imL R ) = imL R ′ . Consequently, Γ g induces a quotient-space isomorphism Γ g : F R / imL R −→ F R ′ / imL R ′ satisfying C red QII,R ′ =Γ g ◦ C red QII,R . 4. Intrinsic QII descent. For every v ∈ T o U, C red QII,R (v) = 0 ⇐⇒ C red QII,R ′ (v) = 0. Therefore, A QII o,0 (R) = A QII o,0 (R ′ )⊆ T o U. 13
Paper 4 Version 5Canonical Realization and Hybrid Geometry The common kernel is independent of the realization representative and defines the intrinsic Structural QII continuation space A QII o,0 ⊆ T o U. 5.Linear and conic structure. The spaceA QII o,0 is a real linear subspace ofT o U. Consequently, A QII o := A QII o,0 {0} is conic and sign symmetric. 6. Claim boundary. The theorem establishes a representative-independent reduced Structural QII defect, an intrinsic pointwise Structural QII continuation space, its kernel characterization, and its linear, conic, and sign-symmetric structure. It does not by itself establish thatA QII o,0 is nonzero or proper, nor that the pointwise spaces assemble as a smooth fixed-rank vector bundle on an open domain. Those conclusions require separate nondegeneracy and constant-rank conditions. Structural QII is an equality-type conservation and compatibility condition; it supplies no variational objective, preferred sign, propagation orientation, propagation ray, or response threshold. Proof.Smoothness of the Structural-QII assignments and real-linearity ofv 7→ h fr R vimply thatB R is real-linear. The commutator and adjoint-action terms are real-linear inK, soL R is real-linear. Hence imL R is a linear subspace of F R , and C red QII,R is well defined and real-linear. By the defining property of the quotient map, π R (B R v) = 0 ⇐⇒ B R v ∈ imL R . By definition ofL R , this is equivalent to the existence ofK ∈ k evt satisfying the two displayed intertwining equations. BecauseKcommutes with everyQ i , it also commutes with every fixed sum Q A and Q Γ α . Now let R ′ = R· g. Gauge equivariance of the minimum-response horizontal lift gives h fr R ′ (v) = d(R g ) R
h fr R (v) . Differentiating the two Structural-QII equivariance identities at fixed g yields DP R ′
d(R g ) R X = V g DP R (X)V † g and DΠ R ′
d(R g ) R X = Ad V g ◦DΠ R (X)◦ Ad −1 V g . Substituting X = h fr R v gives B R ′ = Γ g ◦ B R . For K ∈ k evt , define K ′ := V g KV † g . Since the gauge action is label preserving, V g Q i V † g = Q i , 14
Paper 4 Version 5Canonical Realization and Hybrid Geometry so K ′ ∈ k evt . Moreover, [K ′ ,P R ′ ] = V g [K,P R ]V † g , ad K ′ = Ad V g ◦ ad K ◦ Ad −1 V g , and [ad K ′ , Π R ′ ] = Ad V g ◦[ad K , Π R ]◦ Ad −1 V g . Therefore, L R ′ (K ′ ) = Γ g
L R (K) . Because K 7→ K ′ is bijective, Γ g (imL R ) = imL R ′ . Hence Γ g induces the quotient isomorphism Γ g , and for every v ∈ T o U, C red QII,R ′ (v) =Γ g
C red QII,R (v) . SinceΓ g is an isomorphism, the reduced defect vanishes forR ′ exactly when it vanishes forR. Thus the two representative-relative kernels agree as subspaces of the sameT o U, establishing intrinsic descent. Linearity, conicity, and sign symmetry follow immediately. Corollary 3.2 (Constraint Rank, Not Preference Rank). Under Theorem 3.1, A QII o,0 = kerC red QII,o = B −1 R (imL R ), and dimA QII o,0 = dimT o U − rankC red QII,o . The rank ofC red QII,o measures the number of independent first-order Structural QII constraints. It defines no preference ranking on the admissible continuation space. Remark 3.3 (QII claim boundary). The theorem proves pointwise intrinsic descent. Properness requires a separate nondegeneracy condition, and smooth bundle assembly requires constant-rank hypotheses on a QII-regular domain. 4 Least Additional Structure Fix o∈ U, v ∈ A QII o , and R∈ π −1 (o). Let L R (v) :={e∈ T R R reg : dπ R (e) = v} and let∼ emp be an independently prior informationally empty-redescription equivalence onL R (v). Define N R (o,v) :=L R (v)/∼ emp , and assume the additional-structure preorder ⪯ add is well defined on these classes. Principle 4.1 (Least Additional Structure). For a fixed QII-compatible continuation, admissible realization uses a minimal class of (N R (o,v),⪯ add ) whenever the ordered necessity class has such an element. 15
Paper 4 Version 5Canonical Realization and Hybrid Geometry Assumption 4.2 (Formal Surplus Representation). Assume a partially ordered surplus space (S R (o,v),≤ S ) with least element 0 S and a map s R :N R (o,v)→S R (o,v) satisfying: FSR1. s R ([h fr R (v)]) = 0 S . FSR2. s R ([e]) = 0 S if and only if [e] = [h fr R (v)]. FSR3. For every [e], baseline order compatibility holds: s R ([h fr R (v)])≤ S s R ([e]) =⇒ [h fr R (v)]⪯ add [e], [e]⪯ add [h fr R (v)] =⇒ s R ([e])≤ S s R ([h fr R (v)]). FSR4. Label-preserving gauge changes induce order isomorphisms on the necessity classes and surplus spaces, intertwine the maps s R , and preserve the least surplus element. Formal Surplus Representation is an order-theoretic interpretation of the inherited horizontal–vertical decomposition; it does not define additional structure by framed norm. Proposition 4.3 (Unique Least-Surplus Class). Under FSR1–FSR3, [h fr R (v)] is the unique least element of (N R (o,v),⪯ add ) and hence the unique minimal class selected by LAS. Under FSR4 this selection is gauge natural. Proof. For arbitrary [e], leastness of 0 S and FSR1 give s R ([h fr R (v)]) = 0 S ≤ S s R ([e]). The forward clause of FSR3 yields [h fr R (v)]⪯ add [e]. Hence the class is least. If [e] is also least, then [e]⪯ add [h fr R (v)]; baseline reflection givess R ([e])≤0 S . Since 0 S is least, antisymmetry gives s R ([e]) = 0 S , and FSR2 gives [e] = [h fr R (v)]. Gauge naturality follows from FSR4 and horizontal equivariance. Theorem 4.4 (LAS Representation and Minimum-Response Compatibility). Assume Structural QII descent, LAS, the framed realization geometry of Paper 3, the well-defined quotient necessity class above, and Formal Surplus Representation. Then every lift has a unique decomposition e = h fr R (v) + z, z ∈ V R , and ∥e∥ 2 fr =∥h fr R (v)∥ 2 fr +∥z∥ 2 fr . Thush fr R (v) is the unique minimum-response lift. By Proposition 4.3, [h fr R (v)] is independently the unique least additional-structure class and therefore the unique class selected by LAS. The LAS-selected class is precisely the necessity-class element containing the unique minimum-response lift, and that lift is its unique zero-vertical-component representative. Both selections are gauge natural. 16
Paper 4 Version 5Canonical Realization and Hybrid Geometry Proof.The inherited direct sumT R R reg =H fr R ⊕ V R gives the unique decomposition. Orthogonality gives the norm identity, and positive definiteness gives equality only forz= 0. This proves metric minimality without using⪯ add . Proposition 4.3 independently proves the unique least class and LAS selects it. FSR4 and horizontal equivariance carry the selected class and representative to their counterparts under gauge change. Remark 4.5 (LAS claim boundary). The theorem does not define⪯ add through norm comparison or derive Formal Surplus Representation from the metric. It identifies the agreement of two distinct selections for one fixed QII-compatible continuation. 5 Unified Provenance–Coherence Principle Let ∆ o be a specified class of admissible perturbations that remains within the regular domain and respects the fixed QII identity and description architecture. Definition 5.1 (Provenance–coherence survival set). Forv ∈ A QII o , letS o (v)⊆∆ o be the perturba- tions under whichvremains a logically coherent continuation of the same informational process. Survival requires preservation of:
- informational-process identity;
- consistency among admitted inputs, intermediate transformations, and outputs;
- compositional consistency of successive processing stages; and
logical integrity, excluding contradictory continuation assignments from equivalent descriptions. Principle 5.2 (Unified Provenance–Coherence Principle). Admissible propagation is continuation that preserves provenance coherence. UPCP constrains structural integrity and is independent of semantic informational content. Definition 5.3 (Operational provenance–coherence relation). For v 1 ,v 2 ∈ A QII o , define v 1 ⪯ pc,o v 2 ⇐⇒ S o (v 1 )⊆ S o (v 2 ). PC1 (Reflexivity). v ⪯ pc,o v for all v ∈ A QII o . PC2 (Transitivity). If v 1 ⪯ pc,o v 2 and v 2 ⪯ pc,o v 3 , then v 1 ⪯ pc,o v 3 . PC3 (Structural stability). The preorder is generated by perturbational robustness of process identity, input–output consistency, compositional consistency, and logical integrity. Equivalently, v 1 ⪯ pc,o v 2 precisely when every admissible perturbation preserving provenance coherence ofv 1 is also tolerated by v 2 . 17
Paper 4 Version 5Canonical Realization and Hybrid Geometry 5.1Derived PC4: restoration cost, survival order, and differential representation 5.2 Standing survival architecture Fix a regular QII point o. Let V o :=A QII o,0 be the finite-dimensional QII-admissible continuation space, let ∆ o be the admitted perturbation set, and let S o : V o −→ 2 ∆ o be the provenance-coherence survival assignment. Define v ⪯ pc,o w ⇐⇒ S o (v)⊆ S o (w), and v ∼ pc,o w ⇐⇒ v ⪯ pc,o w and w ⪯ pc,o v. Thus v ∼ pc,o w if and only if S o (v) = S o (w). 5.3 Pre-survival structural severity Definition 5.4 (Pre-survival structural core). LetP str o denote the pre-survival structural core of the admitted information-bearing process ato. It contains only process structure defined independently of provenance-coherence survival, including the admitted interface and incidence structure, input- output structure, transformation and compositional structure, QII-admissibility structure, and logical-consistency structure. By definition,P str o does not include or depend uponS o (v),⪯ pc,o , Θ o , [Θ o ], UPCP orientation, the Quantum Horizon, propagation, or RTP. Two descriptions belong to the same pre-survival structural equivalence class precisely when they are related by an admissible invertible redescription preserving and reflecting the structure contained in P str o . Definition 5.5 (Structural restoration cost). For each δ ∈ ∆ o , let Corr str P str o (δ) denote the class of admissible correction procedures that return the process description perturbed by δ to the pre-survival structural equivalence class of P str o . Let R o (Γ)∈ [0,∞] be an independently specified nonnegative physical or informational resource cost assigned to an admissible structural correction procedure Γ. Assume: 1. Structural independence:R o is defined solely from the correction procedure and the pre-survival structural architecture, without reference to continuation survival, the provenance-coherence preorder, UPCP orientation, or propagation. 2.Structural-redescription invariance: if Γ and e Γare related by an admissible structure-preserving 18
Paper 4 Version 5Canonical Realization and Hybrid Geometry redescription of P str o , then R o (Γ) =R o ( e Γ). Define κ o (δ) :=inf Γ∈Corr str P str o (δ) R o (Γ). If no admissible structural correction exists, setκ o (δ) := +∞. No attainment of the infimum is assumed. Definition 5.6 (Structural perturbation-severity preorder). For δ 1 ,δ 2 ∈ ∆ o , define δ 1 ⪯ ∆,o δ 2 ⇐⇒ κ o (δ 1 )≤ κ o (δ 2 ). Equal restoration cost establishes severity equivalence, not identity of perturbations. Since [0,∞] is totally ordered,⪯ ∆,o is reflexive, transitive, and total. It is continuation independent becauseκ o is defined before and without reference to S o . Remark 5.7 (Independence boundary). Structural Restoration Cost alone makes no claim about membership in any survival set. In particular, κ o (δ 1 )≤ κ o (δ 2 ) does not by definition imply that survival ofδ 2 entails survival ofδ 1 . That compatibility is the separate RMS principle below. 5.4 Restoration-Monotone Survival Principle 5.8 (Restoration-Monotone Survival (RMS)). For every v ∈ V o and δ 1 ,δ 2 ∈ ∆ o , δ 2 ∈ S o (v) and κ o (δ 1 )≤ κ o (δ 2 )=⇒ δ 1 ∈ S o (v).(RMS) Equivalently, everyS o (v) is a lower set of the independently defined structural perturbation-severity preorder. Remark 5.9 (Sufficient-statistic status). RMS asserts that Structural Restoration Cost is a continuation-independent sufficient severity statistic for provenance-coherence survival. Thus equal- cost perturbations are survival-equivalent across all admissible continuations: κ o (δ 1 ) = κ o (δ 2 ) =⇒ δ 1 ∈ S o (v) ⇐⇒ δ 2 ∈ S o (v) ∀v ∈ V o . Consequently, for every v there is a lower set I o (v)⊆ κ o (∆ o ) such that S o (v) = κ −1 o (I o (v)). RMS is a substantive monotonicity principle; it is not claimed to follow automatically from the definition of κ o . 19
Paper 4 Version 5Canonical Realization and Hybrid Geometry 5.5 Theorem D: survival-set comparability Theorem 5.10 (Theorem D: Survival-Set Comparability). Assume the independently defined continuation-independent total preorder ⪯ ∆,o and RMS. Then, for every v,w ∈ V o , S o (v)⊆ S o (w) or S o (w)⊆ S o (v). Consequently, v ⪯ pc,o w or w ⪯ pc,o v, so the survival preorder is fiberwise complete. Proof. Suppose for contradiction that S o (v) and S o (w) are incomparable. Choose δ v ∈ S o (v)\ S o (w), δ w ∈ S o (w)\ S o (v). Totality gives either δ v ⪯ ∆,o δ w or δ w ⪯ ∆,o δ v . In the first case,δ w ∈ S o (w) and RMS implyδ v ∈ S o (w), contradiction. In the second case, δ v ∈ S o (v) and RMS implyδ w ∈ S o (v), contradiction. Hence the survival sets are nested. The continuation-order conclusion follows from the definition of ⪯ pc,o . 5.6 Theorem A: linear survival-order representation Theorem 5.11 (Theorem A: Linear Survival-Order Representation). Assume: 1.Fiberwise completeness: for everyv,w ∈ V o ,v ⪯ pc,o worw ⪯ pc,o v. This condition is supplied by Theorem D. 2. Translation compatibility: for all u,v,w ∈ V o , v ⪯ pc,o w ⇐⇒ v + u⪯ pc,o w + u. 3. Positive-scale compatibility: for all v,w ∈ V o and λ > 0, v ⪯ pc,o w ⇐⇒ λv ⪯ pc,o λw. 4. Closed positive survival cone: C pc o :={v ∈ V o : 0⪯ pc,o v} is closed in the finite-dimensional topology of V o . 5. Nontriviality: V o /∼ pc,o contains more than one equivalence class. Then there is a nonzero covector Θ o ∈ V ∗ o such that v ⪯ pc,o w ⇐⇒ Θ o (v)≤ Θ o (w). 20
Paper 4 Version 5Canonical Realization and Hybrid Geometry Moreover, v ∼ pc,o w ⇐⇒ Θ o (v) = Θ o (w), ker Θ o ={v ∈ V o : v ∼ pc,o 0}, and the positive-projective class [Θ o ]∈ P + (V ∗ o ) is uniquely determined by the survival preorder. Proof. Write C := C pc o ={v ∈ V o : 0⪯ pc,o v}. Translation compatibility gives v ⪯ pc,o w ⇐⇒ w− v ∈ C.(1) Positive-scale compatibility gives closure under positive scalar multiplication. Ifv,w ∈ C, then 0⪯ wand translation of 0⪯ vbywgivesw ⪯ v+w; transitivity gives 0⪯ v+w. HenceCis a convex cone, and it is closed by hypothesis. Fiberwise completeness gives C∪ (−C) = V o .(2) Let L := C∩ (−C). Then L is a linear subspace and v ∈ L ⇐⇒ v ∼ pc,o 0. More generally, using (1), v ∼ pc,o w ⇐⇒ w− v ∈ L. (3) LetV o :=V o /L, letπ L :V o →V o be the quotient map, and setC:=π L (C). SinceL⊆ C,−L⊆ C, and C is a cone, C + L = C. Therefore π −1 L (C) = π −1 L (π L (C)) = C + L = C. By the quotient topology, C is closed. It is convex, total, and pointed: C∪ (−C) =V o ,C∩ (−C) ={0}. We use a standard elementary lemma. If a nonzero finite-dimensional real vector spaceWcarries a closed convex cone K with K∪ (−K) = W, K∩ (−K) ={0}, thendimW= 1. Indeed, ifdimW ≥2, the unit sphereSis connected. ButA:=K∩ Sand−A are disjoint nonempty closed subsets with S = A∪ (−A), a contradiction. Nontriviality givesV o ̸={0}. Applying the lemma yields dimV o = 1. 21
Paper 4 Version 5Canonical Realization and Hybrid Geometry Choose a linear isomorphism Θ o :V o → R with orientation chosen so that C ={ ̄v :Θ o ( ̄v)≥ 0}. Define Θ o :=Θ o ◦ π L . Then Θ o ̸= 0, and from (1), v ⪯ pc,o w ⇐⇒ Θ o (v)≤ Θ o (w). Equation (3) gives the equivalence and kernel statements. If Θ ′ o is another faithful nonzero linear representative, thenkerΘ ′ o =L=kerΘ o . Both descend to nonzero linear functionals on the one-dimensional quotient, so Θ ′ o =a o Θ o for a uniquea o ̸= 0. The common order orientation forces a o
0. Hence [Θ o ]∈ P
(V ∗ o ) is unique. 5.7 Theorem B: smooth local integration Theorem 5.12 (Theorem B: Smooth Local Integration of the Survival-Order Covector). LetUbe a smooth operational manifold, let o∈ U, and suppose V o ⊆ T o U. Let [Θ o ]∈ P + (V ∗ o ) be supplied by Theorem A; choose Θ o ∈ [Θ o ] and set L o := ker Θ o . Assume: 1.Smooth cooriented extension: on a neighborhoodU o ofothere is a smooth codimension-one distribution x7→L x ⊆ T x U with L o ∩ V o = L o , and a smooth transverse orientation agreeing atowith the positive orientation determined by [Θ o ]. 2. Local integrability: L is integrable on U o . 3.Survival-Leaf Fidelity: let∼ pc be the nearby provenance-coherence survival-equivalence relation defined independently from any scalar representative, through equality of the relevant local survival and process-preservation data. After shrinkingU o if needed, its connected local equivalence classes coincide with the connected integral leaves of L. Then, after shrinking U o if needed, there is a smooth submersion P o : U o → R such that (dP o ) o | V o = Θ o . Its connected local level sets are exactly the local survival-equivalence classes, and increasingP o agrees with the positive survival orientation. 22
Paper 4 Version 5Canonical Realization and Hybrid Geometry If e P o is another admissible smooth local representative with the same connected local leaves and transverse orientation, then locally e P o = φ◦ P o , φ ′
0, and (d e P o ) o | V o = φ ′ (P o (o))Θ o . Thus the first-order survival order atoonV o , its equivalence hyperplane, and its sign sectors are representative independent. Proof. Smooth coorientation and integrability give, on a sufficiently small foliation neighborhood, a smooth submersion Q o : U o → R withker(dQ o ) x =L x and connected local level sets equal to the integral leaves. Survival-Leaf Fidelity identifies those leaves with the independently defined local survival-equivalence classes. At o, ker
(dQ o ) o | V o =L o ∩ V o = L o = ker Θ o . Both restricted covectors are nonzero and have the same codimension-one kernel. Therefore (dQ o ) o | V o = a o Θ o for a unique a o ̸= 0. The chosen transverse orientation gives a o
0. Define P o := a −1 o Q o . Then (dP o ) o | V o = Θ o , and positive rescaling preserves leaves and orientation. For uniqueness, restrictP o and e P o to a common foliation chart with connected plaques. Since e P o is constant on every connectedP o -level plaque, it factors as e P o =φ◦ P o . Both functions are submersions, soφ ′ ̸= 0; agreement of transverse orientation givesφ ′ 0. Differentiating and restricting to V o yields (d e P o ) o | V o = φ ′ (P o (o))Θ o . 5.8 Theorem C: assembled PC4 representation Theorem 5.13 (Theorem C: Derived Provenance-Coherence Differential Representation). Assume Structural Restoration Cost, RMS, Theorem D, and all structural and smooth-integration hypotheses of Theorems A and B. Then:
- the survival preorder is fiberwise complete;
- there is a nonzero covector Θ o ∈ V ∗ o satisfying v ⪯ pc,o w ⇐⇒ Θ o (v)≤ Θ o (w); 23
Paper 4 Version 5Canonical Realization and Hybrid Geometry 3. [Θ o ]∈ P + (V ∗ o ) is uniquely determined; 4. locally there is a smooth submersion P o with Θ o = (dP o ) o | V o ; 5. every other admissible smooth representative satisfies e P o = φ◦ P o , φ ′
0, so the first-order order, kernel, and sign sectors at o on V o are representative independent. Consequently, PC4(a)–(d) follow from the stated upstream structural-restoration, monotonicity, ordered-vector-space, and smooth-integration hypotheses. Proof. Structural Restoration Cost induces a continuation-independent total severity preorder on ∆ o . RMS makes every survival set downward closed in that common order. Theorem D therefore gives fiberwise completeness, establishing PC4(a). Theorem A then derives the nonzero faithful linear representative Θ o , its exact survival-equivalence kernel, and its unique positive-projective class. Theorem B integrates this class to a smooth local scalarP o satisfying Θ o = (dP o ) o | V o and proves positive smooth reparameterization independence. These are PC4(b), PC4(c), and PC4(d), respectively. 5.9 Final PC4 assembly statement Corollary 5.14 (Complete derivation of PC4). Under the hypotheses listed in this note, the four clauses formerly assembled as PC4 are conclusions: PC4(a) V o /∼ pc,o is totally ordered; PC4(b) ∃ P o ∈ C ∞ (U o ,R),Θ o = (dP o ) o | V o , v ⪯ pc,o w ⇐⇒ Θ o (v)≤ Θ o (w); PC4(c) Θ o ̸= 0; PC4(d) e P o = φ◦ P o , φ ′ 0, for admissible representatives. Thus PC4 need not remain a monolithic independent axiom. Its content follows from the explicit chain P str o → κ o →⪯ ∆,o RMS =⇒ Theorem D =⇒ Theorem A =⇒ Theorem B =⇒ Theorem C =⇒ PC4(a)−−(d). 5.10 Logical boundary The derivation does not claim that PC1–PC3 alone imply PC4. Its primitive inputs are now exposed explicitly: the pre-survival structural core, an independent redescription-invariant correction-resource 24
Paper 4 Version 5Canonical Realization and Hybrid Geometry functional, RMS, translation compatibility, positive-scale compatibility, closedness and nontriviality of the survival order, and the smooth cooriented extension, integrability, and independently defined Survival-Leaf Fidelity needed for local integration. The improvement is that PC4 itself is no longer assumed as an opaque bundled axiom; each of its four conclusions is produced by a stated theorem from independently typed premises. Definition 5.15 (Information Separatrices and orientation sectors). Define the ambient and QII-admissible separatrices by Σ o := ker Θ o ⊆ T o U,Σ QII o := A QII o,0 ∩ ker Θ o . Define C + o :={v ∈ A QII o : Θ o (v) > 0}, C − o :={v ∈ A QII o : Θ o (v) < 0}. Theorem 5.16 (UPCP Orientation–Propagation Cone Theorem). Assume the Structural QII Conservation Theorem, the LAS framework for canonical realization representatives, and UPCP in the complete form PC1–PC4. Then, at every regular operational point o:
- Θ o (−v) =−Θ o (v) for every v ∈ T o U.
- Every v ∈ A QII o belongs to exactly one of C
o , Σ QII o {0}, and C − o . 3. C + o and C − o are preserved under positive scaling within A QII o . 4. If v,−v ∈ A QII o and Θ o (v)̸= 0, exactly one lies in C + o and the other in C − o . 5. Both C + o and C − o are nonempty. 6. The separatrices and sign sectors are independent of the admissible scalar representative. Consequently, UPCP determines a nontrivial, representative-independent local orientation– propagation cone structure on the QII-admissible continuation space. Proof.The differential Θ o = (dP o ) o is linear, so it is odd and positively homogeneous. Real-number trichotomy gives the sector decomposition, and positive homogeneity gives scaling invariance. QII sign symmetry exchangesvand−v. PC4(c) suppliesw ∈ A QII o,0 with Θ o (w)̸= 0, and QII sign symmetry supplies−w, proving both strict sectors nonempty. Finally, PC4(d) multiplies Θ o only by a positive scalar, preserving its kernel and strict sign sectors. Remark 5.17 (UPCP claim boundary). The theorem uses the derived PC4 representation es- tablished above. It does not establish global cone persistence, preferred-ray selection, Lorentzian reconstruction, global causal structure, spacetime emergence, or the Quantum Horizon synthesis theorem. 6 The Local Quantum Horizon The Quantum Horizon is not a fourth principle. It is the first synthesis object constructed from structures supplied separately by QII, LAS, and UPCP. 25
Paper 4 Version 5Canonical Realization and Hybrid Geometry Definition 6.1 (Local Quantum Horizon). Assume Theorems 3.1, 4.4, and 5.16. The Local Quantum Horizon at o is H QH o := A QII o,0 ∩ ker Θ o ⊆ T o U. Equivalently,H QH o is the set of QII-admissible infinitesimal continuations whose first-order provenance–coherence variation vanishes. Definition 6.2 (LAS-canonical realized Local Quantum Horizon). For R∈ π −1 (o), define [ H QH R := h fr R (H QH o )⊆ H fr R . This is the realization of the operational Horizon without vertical realization surplus. It records LAS’s contribution without changing the operational definition of H QH o . Definition 6.3 (Local Quantum Horizon structure). The associated pointwise structure is QH o
A QII o,0 ,H QH o ,C + o ,C − o , [h fr R ] , where [h fr R ] denotes the intrinsic LAS-canonical realization assignment under the inherited gauge- equivariant realization geometry. Proposition 6.4 (Pointwise Horizon decomposition). A QII o,0 = C + o ⊔H QH o ⊔ C − o . The two strict sectors are nonempty, closed under positive rescaling withinA QII o , and exchanged by sign reversal. The decomposition is representative independent. Proof. This is the QII-admissible sector decomposition of Theorem 5.16, withH QH o = Σ QII o . Repre- sentative independence follows from PC4(d). 6.1 Nontriviality qualification The zero continuation always belongs to H QH o . Because Θ o | A QII o,0 is a nonzero linear functional, dimH QH o = dimA QII o,0 − 1. Hence a nonzero Horizon subspace requires dimA QII o,0 ≥ 2. If dimA QII o,0 = 1, then H QH o ={0}. 6.2 Global scope disclaimer The adjective Local is substantive. The objectH QH o lies in the single tangent spaceT o Uand is defined from pointwise data. It is not presently a subset ofUand is not an embedded hypersurface of the operational manifold. The definition does not assert: • a hidden shell or boundary wrapping around the operational geometry; 26
Paper 4 Version 5Canonical Realization and Hybrid Geometry • a single global object H⊆ U; • smooth variation of o7→H QH o ; • compatibility or transport between horizons at distinct points; • a global cone field or time orientation; • a causal, event, trapping, or black-hole horizon; • a Lorentzian metric, spacetime geometry, or global causal structure; or • an observer-dependent inaccessible region. A global Quantum Horizon structure would require a separate theorem establishing compatibility of the pointwise data across a regular domain. No such theorem is claimed here. 6.3 The Quantum Horizon as an Organizing Structure At first glance, the Local Quantum Horizon may appear to be merely the zero set of the UPCP orientation functional. Formally, this characterization is correct. The Horizon consists of those QII-admissible continuations whose first-order provenance–coherence variation vanishes. However, the development of the present framework suggests a broader interpretation of its role within the emerging geometry. A significant feature of the construction is that the Horizon is identified before any propagation criterion is introduced. QII first determines the admissible continuation space, and UPCP then decomposes that space into positive, zero, and negative sectors. At this stage, no propagation law has yet been imposed. The resulting structure is therefore not yet a propagation cone or a causal geometry. Rather, the construction first produces a distinguished separating structure: the Horizon itself. This order of construction is conceptually significant. The Horizon is not introduced as the boundary of an already existing propagation geometry. Instead, the Horizon appears as part of the initial orientation architecture generated by UPCP. The first geometric distinction produced by the framework is therefore the separation between positive, negative, and equilibrium directions. From this perspective, the Quantum Horizon functions as a distinguished separating structure. It identifies those admissible continuations that are informationally neutral with respect to first-order provenance–coherence variation and thereby establishes the equilibrium sector of the admissible continuation space. The Horizon is the locus at which the UPCP ordering changes sign and consequently provides a natural reference structure for the orientation geometry. The role of the Horizon becomes even clearer after the introduction of the Response-Threshold Propagation Principle. RTP does not redefine the Horizon, nor does it derive the Horizon from a previously existing cone structure. Instead, RTP is formulated relative to the Horizon decomposition itself. The normalized UPCP covector determines a canonical advance direction, while transverse response is measured entirely within Horizon directions. The resulting propagation cone therefore depends on the Horizon structure rather than determining it. 27
Paper 4 Version 5Canonical Realization and Hybrid Geometry This relationship reveals an important distinction. The Quantum Horizon and the propagation boundary are not the same object. The Horizon remains fixed while RTP refines the positive UPCP sector into propagating and non-propagating directions. Propagation becomes a stronger condition than mere positive orientation, and the propagation boundary emerges only after this additional response-threshold criterion has been imposed. One of the central lessons of the present framework is therefore that equilibrium, orientation, and propagation constitute distinct geometric notions. The resulting picture suggests that the Quantum Horizon occupies a role analogous to a separatrix in dynamical systems. It separates qualitatively different classes of admissible informational evolution and provides the geometric reference structure relative to which propagation is subsequently defined. Whether this organizing role persists at the manifold level, or survives eventual spacetime reconstruction, remains an open question beyond the scope of the present paper. Within the limits of the current construction, however, the Quantum Horizon emerges as the distinguished equilibrium and separating structure produced by the combined action of QII, LAS, and UPCP. RTP subsequently refines the geometry built around that structure, but does not replace it. The Horizon therefore remains the foundational geometric reference object throughout the propagation construction. The structures developed thus far remain local to individual admissible continuation spaces. A natural next question is whether the UPCP orientation sectors assemble coherently across a regular domain. Theorem A addresses this question by promoting the pointwise positive sectors to an intrinsic smooth conic field and thereby providing the first domain-level orientation geometry of the framework. 7 Horizon Equilibrium Structure Letπ:V → Ube the smooth fixed-rank QII-admissible continuation bundle over a QII-regular domain U, with rank n≥ 2. Let q ∈ Γ ∞ (Sym 2 V ∗ ) be the inherited smooth positive-definite response metric. Let [Θ] be the smooth positive-projective UPCP covector field, and let b Θ∈ Γ ∞ (V ∗ ) be its q −1 -unit representative. Thus b Θ is nowhere zero and q −1 ( b Θ, b Θ) = 1. Define T := b Θ ♯ q , H QH := ker b Θ. Then b Θ(T ) = q(T,T ) = 1, V = RT ⊕ q H QH . Define the UPCP sectors C + :={v ∈ V : b Θ(v) > 0}, C − :={v ∈ V : b Θ(v) < 0}. 28
Paper 4 Version 5Canonical Realization and Hybrid Geometry Definition 7.1 (UPCP informational equilibrium). A QII-admissible continuationv ∈ Vis in UPCP informational equilibrium if its first-order provenance-coherence variation vanishes. Equivalently, b Θ(v) = 0. The informational-equilibrium sector is therefore E := ker b Θ. This is a first-order informational notion. It does not assert a dynamical fixed point, attractor, minimizer, or equilibrium of an independently specified flow. Definition 7.2 (UPCP informational equilibrium separatrix). A smooth codimension-one vector subbundle S ⊂ V is a UPCP informational equilibrium separatrix if:
- S is the zero-variation sector of the UPCP differential;
- V decomposes disjointly as V = C
⊔ S⊔ C − ; 3. positive fiberwise scaling preserves S, C + , and C − ; 4. S is representative independent under every admissible positive rescaling of the UPCP covector. Definition 7.3 (Primary invariant balance manifold). Relative to (q, b Θ), a smooth codimension-one vector subbundle M ⊂ V is a primary invariant balance manifold if:
- M is canonically determined by the primitive UPCP orientation structure, without using a propagation predicate, response threshold, propagation boundary, or selector;
- V = RT ⊕ q M, so each v ∈ V has unique longitudinal and transverse components;
- M is invariant under positive fiberwise scaling; 4.the advance/refinement comparison and the balance quantities below are defined relative to this splitting. The term primary denotes logical priority in the decomposition architecture; it does not identifyM with a propagation boundary. Theorem 7.4 (Horizon Equilibrium Structure Theorem). Under the preceding assumptions, the following hold. (HES1) UPCP informational-equilibrium characterization. The Quantum Horizon is exactly the UPCP informational-equilibrium sector: H QH =E = ker b Θ. Thus the Horizon is the locus of vanishing first-order provenance-coherence variation. 29
Paper 4 Version 5Canonical Realization and Hybrid Geometry (HES2) Informational equilibrium separatrix. The Quantum Horizon is the unique UPCP informational equilibrium separatrix. Fiberwise, V = C + ⊔ H QH ⊔ C − , and the sign of the UPCP variation changes from positive to negative across its zero sector. The three sectors are invariant under positive scaling and independent of the admissible positive representative of [Θ]. (HES3) Primary invariant balance manifold. The Quantum Horizon is the unique primary invariant balance manifold determined by (q, [Θ]). It supplies the canonical orthogonal splitting V = RT ⊕ q H QH . (HES4) Canonical advance/refinement balance decomposition. Everyv ∈ C + has the unique representation v = a(v)T + v ⊥ , a(v) := b Θ(v) > 0, v ⊥ := v− b Θ(v)T ∈ H QH . Define r(v) := q q(v ⊥ ,v ⊥ ). Then (a(v),r(v)) is the canonical advance/refinement balance datum ofvrelative to the informational equilibrium separatrix. (HES5) Projective balance law and canonical invariant. The refinement ratio RR(v) := r(v) a(v) , v ∈ C + , is homogeneous of degree zero: RR(λv) = RR(v)(λ > 0). HenceRRdescends to the positive projectivizationP + (V) and is the canonical projective invariant of the advance/refinement proportion induced by the Horizon decomposition. Moreover,RR(v) = RR(w) exactly when v and w have the same transverse refinement magnitude per unit advance. (HES6) Projectively invariant balance classes. For every ρ≥ 0, Σ ρ :={v ∈ C + : RR(v) = ρ} is invariant under positive fiberwise scaling. Equivalently, Σ ρ is the inverse image of a level set of the descended invariant onP + (V). Thus the positive projective sector is partitioned into canonical balance classes indexed by ρ≥ 0. (HES7) Equilibrium-separatrix/propagation-boundary distinction. The Quantum Horizon 30
Paper 4 Version 5Canonical Realization and Hybrid Geometry is not a propagation boundary contained in the positive UPCP sector. Indeed, H QH ∩ C + = ∅. Any propagation boundary contained inC + is therefore additional structure defined within, or relative to, the Horizon-induced balance architecture; it cannot be identified with the Horizon itself. Proof.HES1 is immediate from the definitions: both the Local Quantum Horizon and the informational-equilibrium sector are the kernel of the normalized UPCP differential. For HES2, b Θis a nonzero linear functional on each fiber. Real-number trichotomy gives the disjoint sign decomposition V ={ b Θ > 0}⊔{ b Θ = 0}⊔{ b Θ < 0}. The middle sector isker b Θ=H QH . Linearity gives positive homogeneity, so positive scaling preserves all three sectors. If e Θ=b b Θwithb >0, then e Θhas the same kernel and strict sign sectors. Hence the decomposition is representative independent. Uniqueness follows because the zero-variation sector of the UPCP differential is uniquely its kernel. For HES3, normalization gives T = b Θ ♯ q and b Θ(T ) = 1. If h∈ H QH , then q(T,h) = b Θ(h) = 0. Every v ∈ V can be written v = b Θ(v)T +
v− b Θ(v)T , and the second term lies inker b Θ. ThusV=RT ⊕ q H QH . The kernel is invariant under positive scaling and is canonically determined by the UPCP ray; hence it satisfies the definition of primary invariant balance manifold. Indeed,H QH o =ker b Θ o = (RT o ) ⊥ q o . The orthogonal complement of the fixed lineRT o in the positive-definite inner-product space (V o ,q o ) is unique; hence any balance manifold satisfying Definition 7.3(2) equalsH QH o . Its uniqueness is relative to that primitive UPCP datum. HES4 follows from the direct sum, whose summands have zero intersection. Positive definiteness of q makes r(v) well defined and nonnegative. For λ > 0, linearity gives a(λv) = λa(v),(λv) ⊥ = λv ⊥ , r(λv) = λr(v). Therefore RR(λv) = λr(v) λa(v) = RR(v), which proves HES5 and the descent to positive projective space. If v ∈ Σ ρ , then HES5 gives RR(λv) = ρ for every λ > 0. This proves HES6. Finally,h ∈ H QH implies b Θ(h) = 0, whereasv ∈ C + requires b Θ(v)>0. ThusH QH ∩ C + =∅, proving HES7. Corollary 7.5 (Quantum Horizon Equilibrium-Balance Principle). The Quantum Horizon has two rigorously distinct but compatible structural roles: 31
Paper 4 Version 5Canonical Realization and Hybrid Geometry 1.it is the first-order UPCP informational equilibrium separatrix, because it is the representative- independent zero-variation sector separating positive from negative provenance-coherence variation; 2.it is the primary invariant balance manifold, because it supplies the canonical orthogonal advance/Horizon decomposition from which the advance/refinement balance invariant is defined. Accordingly, Quantum Horizon = informational equilibrium separatrix = primary invariant balance manifold while Quantum Horizon̸= propagation boundary. Any later threshold surface, null-critical locus, admissibility frontier, or propagation boundary must emerge as additional structure inside the positive-sector balance geometry organized by the Quantum Horizon. Remark 7.6 (Claim boundary). The term equilibrium in this theorem means only vanishing first- order UPCP provenance-coherence variation. The theorem does not assert dynamical stationarity, attraction, stability under an independently given flow, energy minimization, boundary emergence, propagation selection, smooth threshold selection, or SSR. It establishes the informational equilibrium sector and the balance architecture relative to which those later questions may be formulated. Clarification on the Successive Trichotomies Several results below exhibit three-way classifications, but these classifications should not be con- flated. UPCP first supplies an informational orientation structure, with the Quantum HorizonH QH o =ker b Θ o separating positive orientation, equilibrium, and negative orientation. RTP subsequently imposes an additional response-capacity condition within the positively oriented sector, refining that orientation geometry to a proper propagation cone; positive orien- tation alone is therefore not sufficient for propagation. The resulting response-capacity relation distinguishes subcritical, saturated, and supercritical regimes according toR o (v)≶ c o A o (v). The later Lorentz-signature construction does not introduce another independent trichotomy, but gives a geometric realization of this threshold relation through the sign of the corresponding quadratic form. Boundary Emergence identifies the saturation locus with the null propagation boundary, while HRCEL synthesizes these relations. Thus the recurring sign structure reflects coherence across informational, threshold, and geometric levels, rather than repeated derivations of one result under different terminology. 8 Theorem A: Cone Promotion The following theorem promotes the pointwise UPCP positive sectors constructed above to a smooth intrinsic conic field on a QII-regular domain. 32
Paper 4 Version 5Canonical Realization and Hybrid Geometry Definition 8.1 (QII-regular domain). An open setU QII ⊆O reg is called QII-regular if the family of QII-admissible continuation spaces A QII := G o∈U QII A QII o is a smooth rank-k vector subbundle of TU QII for some fixed k ≥ 1. For a real vector bundle E → U, write P + (E ∗ ) := (E ∗ \ 0)/R
0 for the bundle of positive projective covectors, where α∼ e α exactly when e α = aα for some a > 0. Theorem 8.2 (Theorem A (Cone Promotion)). LetU QII ⊆ O reg be a QII-regular domain with QII-admissible bundle A QII → U QII . Suppose the UPCP data determine a smooth section [Θ]∈ Γ P
(A QII ) ∗ . Equivalently, there exists an open cover {U i } of U QII and smooth nowhere-zero sections Θ i ∈ Γ (A QII | U i ) ∗ such that on every nonempty overlap U i ∩ U j , Θ j = a ij Θ i for some smooth a ij : U i ∩ U j → R
0 . For each o∈ U QII , define the pointwise UPCP positive sector by C
o :={v ∈ A QII o : Θ i (o)(v) > 0}, o∈ U i . Then this definition is independent of i, and C + := G o∈U QII C + o ={(o,v)∈ A QII : Θ i (o)(v) > 0 for o∈ U i } is a well-defined intrinsic smooth open conic field in A QII . More precisely:
- C
is an open submanifold of the total space A QII ; 2. C + ∩ A QII o = C + o for every o∈ U QII ; 3. every fiber C + o is nonempty, open, convex, and positively homogeneous; 4. C + depends only on the projective UPCP section [Θ], and hence is unchanged under every admissible positive rescaling of a local UPCP representative. Consequently, the pointwise UPCP positive sectors promote canonically to the intrinsic conic field 33
Paper 4 Version 5Canonical Realization and Hybrid Geometry C + : {C + o } o∈U QII =⇒ C + . Proof. For each i, define the smooth evaluation map F i : A QII | U i −→ R, F i (o,v) := Θ i (o)(v). On U i ∩ U j one has F j = a ij F i with a ij
0. Therefore F j (o,v) > 0 ⇐⇒ F i (o,v) > 0, so the local subsetsF −1 i ((0,∞)) agree on overlaps. They consequently glue to a uniquely defined subsetC
⊂ A QII . Each local subset is open becauseF i is continuous and (0,∞) is open. Hence C + is open in the smooth manifold A QII , and thus is an open submanifold. The fiber identityC + ∩A QII o =C + o follows directly from the definition. Since Θ i (o)̸= 0, there exists v ∈ A QII o with Θ i (o)(v)̸= 0; replacingvby−vif necessary gives Θ i (o)(v)>0, soC + o is nonempty. Openness in the fiber follows by continuity of the linear functional Θ i (o). If v,w ∈ C + o and a,b > 0, then linearity gives Θ i (o)(av + bw) = aΘ i (o)(v) + bΘ i (o)(w) > 0. Thusav+bw ∈ C + o , proving convexity and positive homogeneity. Finally, replacing Θ i by any admissible representative e Θ i =a i Θ i witha i
0 preserves the strict sign of every evaluation. Hence C
depends only on [Θ] and is intrinsic to the UPCP sign structure. Remark 8.3 (Scope boundary). Theorem 8.2 is restricted to the QII-regular domainU QII . It proves only that the pointwise UPCP positive sectors assemble into an intrinsic smooth open conic field. It makes no claim concerning extension across QII rank-changing loci, Lorentzian representability, causal structure, metric reconstruction, growth dimension, or spacetime interpretation. Theorem A establishes that the UPCP positive sectors assemble into a smooth intrinsic conic field. This result, however, concerns orientation rather than propagation. Positive orientation alone does not distinguish directions that genuinely support propagation from those that do not. The existence of an orientation geometry therefore does not by itself determine a propagation geometry. A further principle is required to identify which positively oriented continuations should be regarded as propagationally admissible. The Response-Threshold Propagation Principle is introduced to supply this additional selection criterion and thereby refine the UPCP orientation structure into a genuine propagation cone. 8.1 Response-Threshold Propagation Principle 8.2 Inherited Structures LetU QII be a QII-regular domain on which the QII-admissible spaces form a smooth finite-rank vector bundle A QII 0 −→ U QII . 34
Paper 4 Version 5Canonical Realization and Hybrid Geometry For o∈ U QII , write V o := A QII o,0 . Paper 3 supplies the smooth, positive-definite, gauge-independent operational response metric q o (v,w) := g fr,R h fr R (v),h fr R (w) , R∈ π −1 (o), whose value is independent of the selected realization representative R. Paper 4 supplies a smooth positive-projective UPCP covector field [Θ o ]∈ P + (V ∗ o ),Θ o ̸= 0, together with the informational future half-space and Local Quantum Horizon C + o :={v ∈ V o : Θ o (v) > 0}, H QH o := ker Θ o . The roles remain distinct: QII supplies admissibility, LAS supplies the minimum-response realization geometry, and UPCP supplies ordering and orientation. 8.3 Response-Normalized UPCP Advance and Canonical Splitting Definition 8.4 (Response-Normalized UPCP Advance). Letobe a regular QII point, letV o be its QII-admissible continuation fiber, letq o be the inherited positive-definite response metric onV o , and let [Θ o ]⊂ V ∗ o be the positive projective UPCP covector class representing the local provenance–coherence ordering. For any representative Θ o ∈ [Θ o ], define its q o -normalized representative by b Θ o := Θ o q q −1 o (Θ o , Θ o ) . The UPCP advance of an admissible continuation v ∈ V o is A o (v) := b Θ o (v). The definition is independent of the chosen positive representative of [Θ o ]. The signs A o (v) > 0, A o (v) = 0, A o (v) < 0 determine, respectively, positive UPCP advance, UPCP equilibrium, and negative UPCP advance. In particular, kerA o = ker b Θ o = H QH o , where H QH o is the Quantum Horizon at o. The quantityA o (v) is a response-normalized informational orientation quantity. It is not, by definition, physical time, energy, a dynamical evolution parameter, or propagation. Positive UPCP advance determines positive UPCP orientation; propagation remains a separate selection supplied 35
Paper 4 Version 5Canonical Realization and Hybrid Geometry by RTP. Proposition 8.5 (Canonical UPCP Advance Decomposition). Under the hypotheses of the preced- ing definition, let T o := b Θ ♯ q o o be the q o -metric dual of b Θ o . Then:
- T o is uniquely characterized by q o (T o ,w) = b Θ o (w) for every w ∈ V o .
- T o has unit response norm and unit UPCP advance: q o (T o ,T o ) = 1, A o (T o ) = 1.
- The Quantum Horizon is the q o -orthogonal complement of the canonical advance direction: H QH o = ker b Θ o = T ⊥ q o o .
- Every v ∈ V o admits the unique decomposition v = A o (v)T o
- v ⊥ , v ⊥ ∈ H QH o . Equivalently, V o = RT o ⊕ q o H QH o .
- The response norm satisfies the orthogonal identity q o (v,v) = A o (v) 2
- q o (v ⊥ ,v ⊥ ). Proof. Because q o is positive definite, the musical map ♯ q o : V ∗ o −→ V o is an isomorphism. Hence T o = b Θ ♯ q o o exists uniquely and satisfies q o (T o ,w) = b Θ o (w) for every w ∈ V o . Since b Θ o is q −1 o -normalized, q −1 o ( b Θ o , b Θ o ) = 1. By metric duality, q o (T o ,T o ) = b Θ o (T o ) = q −1 o ( b Θ o , b Θ o ) = 1, and therefore A o (T o ) = b Θ o (T o ) = 1. 36
Paper 4 Version 5Canonical Realization and Hybrid Geometry For any w ∈ V o , w ∈ H QH o ⇐⇒ b Θ o (w) = 0 ⇐⇒ q o (T o ,w) = 0. Thus H QH o = T ⊥ q o o . Given v ∈ V o , define v ⊥ := v− A o (v)T o . Then b Θ o (v ⊥ ) = b Θ o (v)− A o (v) b Θ o (T o ) = A o (v)− A o (v) = 0. Hence v ⊥ ∈ H QH o , proving the stated decomposition. Its uniqueness follows from RT o ∩ H QH o ={0}, which is immediate fromA o (T o ) = 1. Finally, the two summands areq o -orthogonal andq o (T o ,T o ) = 1, so q o (v,v) = A o (v) 2 q o (T o ,T o ) + q o (v ⊥ ,v ⊥ ) = A o (v) 2
- q o (v ⊥ ,v ⊥ ). Corollary 8.6 (Refinement Ratio in Advance Form). For every positively oriented continuation v ∈ V o with A o (v) > 0, define its transverse response by R o (v) := q q o (v ⊥ ,v ⊥ ). Then the Refinement Ratio is RR o (v) = R o (v) A o (v) = q q o (v ⊥ ,v ⊥ ) A o (v) . ThusRR o (v) is theq o -transverse response per unit of response-normalized UPCP advance. This identity supplies no propagation criterion by itself; RTP separately selects the positively oriented continuations satisfying the finite response threshold. 8.4 Response-Threshold Propagation Principle Theorem 8.7 (QSL–FPRP Derivation of Pointwise Response–Threshold Propagation). Letobe a regular QII point whose QII-admissible continuation fiberV o has finite rankdimV o ≥2. Letq o be the inherited positive-definite response metric, let [Θ o ]∈ P
(V ∗ o ) be the positive-projective UPCP covector class, and let b Θ o := Θ o q q −1 o (Θ o , Θ o ) 37
Paper 4 Version 5Canonical Realization and Hybrid Geometry be its q −1 o -normalized representative. Define A o (v) := b Θ o (v), T o := b Θ ♯ q o o . Then q o (T o ,T o ) = 1, A o (T o ) = 1, and every v ∈ V o admits the unique q o -orthogonal decomposition v = A o (v)T o
- v ⊥ , v ⊥ ∈ H QH o := ker b Θ o . On the positive UPCP sector C
o :={v ∈ V o : A o (v) > 0}, define R o (v) := q q o (v ⊥ ,v ⊥ ), RR o (v) := R o (v) A o (v) . Let Prop o : V o {0}−→{0, 1} be the qualitative propagation predicate. Assume the following. I. Qualitative refinement structure.
- Positive orientation of propagation. Prop o (v) = 1 =⇒ A o (v) > 0.
- Positive-projective invariance. For every v ∈ C
o and every λ > 0, Prop o (λv) = Prop o (v). 3. Two-data quantitative closure. There exists G o : (0,∞)× [0,∞)−→{0, 1} such that Prop o (v) = G o
A o (v),R o (v) (v ∈ C + o ). 4. Zero-refinement compatibility. The attained ratio 0 is propagation-compatible. 5. Downward refinement closure. If an attained ratior ≥0 is propagation-compatible and 0≤ s < r, then the attained ratio s is propagation-compatible. 6. Relative openness. The set of propagation-compatible attained ratios is relatively open in [0,∞). II. Physical realization and resource structure. 38
Paper 4 Version 5Canonical Realization and Hybrid Geometry 7.Generalized Speed-Limit Action Bound. Fix an admitted information-bearing physical processP. For each physically realizable continuationv, letG P,o (v) be the class of admissible physical generators realizingvwhile preservingP, letν o be the independently specified dynamical speed-resource functional of the applicable generalized speed-limit relation, and define c P,o (v) :=inf G∈G P,o (v) ν o (G). The infinitesimal operational distinguishability speed V o (v) satisfies V o (v)≤ c P,o (v). 8. Continuation-Speed Correspondence. V o (v) = q q o (v,v). 9. Nonempty unit-advance realization class. Define K (1) P,o :={v ∈ V o : v admits a physical implementation preserving P, A o (v) = 1}. Assume K (1) P,o ̸= ∅. 10. Finite Process Resource Principle. The infimal physical-resource requirements of same- process unit-advance realizations have finite supremum: B P,o := sup v∈K (1) P,o c P,o (v) <∞. This is the new physical finiteness input; it is not claimed to follow from QII, LAS, UPCP, the Composite Bures Response Representation, or QSL–A alone. 11. Physical Propagation Realizability. Prop o (v) = 1 =⇒ v admits a physical implementation preserving P. 12.Physical Projective Realizability. Ifvadmits a physical implementation preservingP, then every positive reparameterizationλv,λ >0, of the same continuation ray admits a physical implementation preserving P. Then B P,o
1, and there exists a unique constant c o ∈ (0,∞) such that, on the positive UPCP sector, Prop o (v) = 1 ⇐⇒ RR o (v) < c o (v ∈ C
o ). 39
Paper 4 Version 5Canonical Realization and Hybrid Geometry For every nonzero v ∈ V o , the globally typed equivalent statement is Prop o (v) = 1 ⇐⇒ A o (v) > 0 and q o (v ⊥ ,v ⊥ ) < c 2 o A o (v) 2 . Moreover, c o ≤ q B 2 P,o − 1. Hence pointwise Response–Threshold Propagation holds at o. Proof.Step 1: Full Refinement-Ratio attainability. BecausedimV o ≥2, the Horizon subspace H QH o = ker b Θ o has positive dimension. Let r ≥ 0. Choose h r ∈ H QH o with q o (h r ,h r ) = r 2 , and set v r := T o
- h r . Then A o (v r ) = 1, v ⊥ r = h r , R o (v r ) = r, RR o (v r ) = r. Thus RR o (C
o ) = [0,∞). Step 2: Factorization through the Refinement Ratio. Letv,w ∈ C + o satisfyRR o (v) = RR o (w). Define v (A) := v A o (v) , w (A) := w A o (w) . Then A o (v (A) ) = A o (w (A) ) = 1 and R o (v (A) ) = RR o (v) = RR o (w) = R o (w (A) ). Positive-projective invariance and two-data quantitative closure give Prop o (v) = G o
1,RR o (v) = G o
1,RR o (w) = Prop o (w). Full ratio attainability therefore yields a unique classifier F o : [0,∞)−→{0, 1} such that Prop o (v) = F o (RR o (v))(v ∈ C + o ). Define J o := F −1 o (1). Assumptions 4–6 imply 0∈ J o , that J o is downward closed, and that J o is relatively open in [0,∞). 40
Paper 4 Version 5Canonical Realization and Hybrid Geometry Step 3: QSL–FPRP properness. LetvsatisfyProp o (v) = 1. Positive orientation givesA o (v)>0. Set v (A) := v A o (v) . Physical Propagation Realizability and Physical Projective Realizability imply thatv (A) admits a physical implementation preserving P. Since A o (v (A) ) = 1, v (A) ∈K (1) P,o . Positive homogeneity gives RR o (v (A) ) = RR o (v). The unit-advance decomposition and orthogonality yield q o (v (A) ,v (A) ) = 1 + RR o (v) 2 . By the Continuation-Speed Correspondence and the Generalized Speed-Limit Action Bound, q 1 + RR o (v) 2 =V o (v (A) )≤ c P,o (v (A) ). Because v (A) ∈K (1) P,o , FPRP gives c P,o (v (A) )≤ B P,o . Hence RR o (v)≤ q B 2 P,o − 1. Since v was arbitrary among propagating continuations, J o ⊆ h 0, q B 2 P,o − 1 i . Thus J o is bounded and proper: J o ̸= [0,∞). Step 4: Strict positivity of the process-resource envelope. Because 0∈ J o andJ o is relatively open in [0,∞), there exists ε > 0 such that [0,ε)⊆ J o . Chooser ∈(0,ε). By full ratio attainability there isv r ∈ C + o withRR o (v r ) =r, and becauser ∈ J o one has Prop o (v r ) = 1. Applying Step 3 to its unit-advance representative gives p 1 + r 2 ≤ c P,o (v (A) r )≤ B P,o . Since r > 0, it follows that B P,o
1. Step 5: Existence and uniqueness of the finite threshold. Set c o := supJ o . 41
Paper 4 Version 5Canonical Realization and Hybrid Geometry The inclusion [0,ε)⊆ J o gives c o
0, while boundedness gives c o <∞. Let 0≤ r < c o . By the defining property of the supremum, there existss ∈ J o withr < s ≤ c o . Downward closure gives r ∈ J o . Hence [0,c o )⊆ J o . The definition of c o gives J o ⊆ [0,c o ]. If c o ∈ J o , relative openness supplies δ > 0 such that (c o − δ,c o
- δ)∩ [0,∞)⊆ J o , which contains values greater thanc o and contradicts the definition ofc o as an upper bound. Therefore c o /∈ J o , and J o = [0,c o ). The endpoint is unique because c o = supJ o . Step 6: Pointwise Response–Threshold Propagation. For v ∈ C
o , Prop o (v) = 1 ⇐⇒ RR o (v)∈ J o ⇐⇒ RR o (v) < c o . If A o (v)≤ 0, positive orientation implies Prop o (v) = 0. Thus, for every nonzero v ∈ V o , Prop o (v) = 1 ⇐⇒ A o (v) > 0 and q o (v ⊥ ,v ⊥ ) < c 2 o A o (v) 2 . Finally, J o ⊆ h 0, q B 2 P,o − 1 i implies c o = supJ o ≤ q B 2 P,o − 1. This completes the proof. Remark 8.8 (RTP status after derivation). Pointwise RTP is a conclusion of Theorem 8.7. The Finite Process Resource Principle is the explicit physical finiteness input; it is not claimed to follow from QII, LAS, UPCP, the response metric, or the generalized speed-limit action bound alone. The theorem is pointwise and makes no base-point regularity claim about o7→ c o . Define the transverse response energy E o (v) := q o (v ⊥ ,v ⊥ ) and the response-threshold quadratic functional Q o (v) := q o (v ⊥ ,v ⊥ )− c 2 o b Θ o (v) 2 . The response-defined future propagation cone is K + o := n v ∈ V o : b Θ o (v) > 0, Q o (v) < 0 o . 42
Paper 4 Version 5Canonical Realization and Hybrid Geometry Equivalently, K + o
v ∈ V o : b Θ o (v) > 0, q q o (v ⊥ ,v ⊥ ) < c o b Θ o (v) . The scale-invariant response ratio ρ o (v) := p q o (v ⊥ ,v ⊥ ) b Θ o (v) , v ∈ C + o , satisfies ρ o (λv) = ρ o (v) for every λ > 0, and RTP is equivalently ρ o (v) < c o . 8.5 Theorem B: Response-Defined Propagation Geometry Theorem 8.9 (Theorem B (Response-Defined Propagation Geometry)). Assume the inherited QII-admissible bundle, the operational response metricq, the smooth positive-projective UPCP orientation [Θ], and the derived pointwise RTP theorem. Fiberwise conclusions hold pointwise; smooth domain-level assembly follows later from the HRCEL Smoothing Law. Then the setsK + o form a smooth intrinsic field of nonempty, open, convex, positively homogeneous, full-dimensional proper cones, in the standard convex-geometric sense [12]. Their closures are salient, and whenever H QH o ̸={0}, K + o ⊊ C + o . Moreover, Q o is a nondegenerate quadratic form of signature (−, +,..., +), and ∂K + o {0} = n v ∈ V o : b Θ o (v) > 0, Q o (v) = 0 o . The nonzero boundary is a smooth conic hypersurface, and every normalized boundary section n v ∈ ∂K + o : b Θ o (v) = 1 o is a strongly convex q o -ellipsoid in T o
- H QH o . Proof. Nonemptiness follows becauseλT o ∈ K
o for everyλ >0. Openness follows from continuity of b Θ o and Q o . Positive homogeneity follows from Q o (λv) = λ 2 Q o (v), b Θ o (λv) = λ b Θ o (v). For convexity, let v,w ∈ K + o and 0 < t < 1. The transverse projection is linear, so ((1− t)v + tw) ⊥ = (1− t)v ⊥
- tw ⊥ . The triangle inequality gives q q o (((1− t)v + tw) ⊥ , ((1− t)v + tw) ⊥ )≤ (1− t) q q o (v ⊥ ,v ⊥ ) + t q q o (w ⊥ ,w ⊥ ) < c o
(1− t) b Θ o (v) + t b Θ o (w) , so (1− t)v+tw ∈ K + o . A nonempty open subset of the finite-dimensional vector spaceV o is full-dimensional. 43
Paper 4 Version 5Canonical Realization and Hybrid Geometry We first characterize the closure: K + o
n v ∈ V o : b Θ o (v)≥ 0, Q o (v)≤ 0 o . The forward inclusion follows by continuity of b Θ o andQ o . Conversely, letvsatisfy the two weak inequalities. If b Θ o (v)>0 andQ o (v)<0, thenv ∈ K + o . If b Θ o (v)>0 andQ o (v) = 0, write v = A o (v)T o
- v ⊥ and set v ε = (A o (v) + ε)T o
- v ⊥ . Thenv ε → v, b Θ o (v ε )>0, andQ o (v ε )<0 for everyε >0, sov ∈K
o . If b Θ o (v) = 0, then Q o (v) = q o (v,v)≤ 0; positive definiteness forces v = 0, and 0 is approached by εT o ∈ K + o . For saliency, suppose v ∈K + o ∩ (−K + o ). The closure characterization gives b Θ o (v)≥0 and− b Θ o (v)≥0, hence b Θ o (v) = 0. It also gives Q o (v)≤ 0. On the Horizon, Q o (v) = q o (v,v), so positive definiteness implies v = 0. Therefore K + o ∩ (−K + o ) ={0}. By definition,K + o ⊂ C + o . IfH QH o ̸ ={0}, choose nonzeroh ∈ H QH o and setv ε =εT o +h. Then b Θ o (v ε ) = ε > 0, while Q o (v ε ) = q o (h,h)− c 2 o ε 2
0 for sufficiently small ε > 0. Hence v ε ∈ C
o \K + o , proving strict inclusion. On the advance line, Q o (aT o ) =−c 2 o a 2 , while for nonzero h∈ H QH o , Q o (h) = q o (h,h) > 0. The orthogonal splitting therefore shows that Q o has signature (−, +,..., +). Its differential is dQ o | v (w) = 2q o (v ⊥ ,w ⊥ )− 2c 2 o b Θ o (v) b Θ o (w). Ifv ̸= 0,Q o (v) = 0, anddQ o | v = 0, choosingw=T o gives b Θ o (v) = 0. The null equation then gives v ⊥ = 0, so v = 0, a contradiction. Thus the nonzero null set is a smooth hypersurface. Finally, on the normalized section b Θ o (v) = 1, writev=T o +hwithh ∈ H QH o . The boundary equation becomes q o (h,h) = c 2 o . IfdimV o ≥2, thenH QH o has positive dimension and this is the nonemptyq o -sphere of radiusc o in the Horizon hyperplane, hence a smooth strongly convex ellipsoid in arbitrary linear coordinates. IfdimV o = 1, thenH QH o ={0}and the equation has no solution becausec o
0; the normalized boundary section is empty. Smooth field assembly on the rank-at-least-two domain follows from smoothness of q, b Θ, T, the orthogonal projection, and c. Corollary 8.10 (Response-induced causal trichotomy). For every nonzerov ∈ C
o , exactly one of 44
Paper 4 Version 5Canonical Realization and Hybrid Geometry the following occurs: Q o (v) < 0 ⇐⇒ propagating, Q o (v) = 0 ⇐⇒ critical propagation, Q o (v) > 0 ⇐⇒ future-oriented but non-propagating. Corollary 8.11 (Fiberwise Lorentzian representation). The polarization G o (v,w) := q o (v ⊥ ,w ⊥ )− c 2 o b Θ o (v) b Θ o (w) is a symmetric nondegenerate bilinear form of Lorentzian signature onV o ; under the later HRCEL Smoothing Law, the familyo7→ G o is smooth, in the standard semi-Riemannian sense [11,1], and K + o ={v ∈ V o : G o (v,v) < 0, b Θ o (v) > 0}. 8.6 Logical Status and Claim Boundary Remark 8.12 (Axiomatic status). Pointwise RTP is derived by Theorem efthm:qsl-fprp-rtp. Section 9 subsequently gives its pointwise refinement-cost characterization under positive-projective invariance, two-data quantitative closure, and finite refinement capacity. The response metric exists independently in Paper 3; UPCP supplies orientation independently in Paper 4; RTP links them by identifying propagation with bounded transverse response relative to advance. Remark 8.13 (Consequences of RTP). Under RTP pt , properness, saliency, strict inclusion in the half-space, fiberwise smoothness of the nonzero null boundary, strong convexity, and Lorentzian quadratic representability follow pointwise from the response-threshold construction. Smooth variation of these objects over the base additionally requires RTP sm . Remark 8.14 (What RTP adds). RTP pt adds the substantive assertion that each propagation cone is a response-threshold cone with ellipsoidal normalized sections. The quantitative law is no longer retained as an adopted axiom. Its threshold form follows from the qualitative refinement hypotheses and the QSL–FPRP physical-resource architecture. Remark 8.15 (Manifold-level limitation). The theorem constructs a Lorentz-signature bilinear form on the QII-admissible bundleA QII 0 . It does not by itself prove thatA QII 0 equals the full tangent bundle of the operational domain, that it is integrable as the tangent bundle of effective leaves, that the effective dimension is four, or that the resulting geometry is physical spacetime. 8.7 Adopted Formulation The adopted formulation of RTP is: Propagation admissibility is exactly bounded transverse minimum response relative to normalized UPCP advance. Equivalently, v ∈ K + o ⇐⇒ b Θ o (v) > 0 and q o (v ⊥ ,v ⊥ ) < c 2 o b Θ o (v) 2 . 45
Paper 4 Version 5Canonical Realization and Hybrid Geometry The QSL–FPRP theorem provides the pointwise quantitative formulation of propagation refinement. Its unique pointwise cutoff and the identity between qualitative and response-defined propagation are established above. Domain-level smooth assembly is derived later from Regular Local Continuation and the HRCEL Smoothing Law. The response-threshold construction accomplishes more than the introduction of a quantitative propagation rule. By combining the UPCP orientation structure with the inherited response geometry, RTP reveals that the framework contains two distinct but related geometric layers. The first layer is the orientation geometry supplied by UPCP. The second is the propagation geometry supplied by RTP through response-threshold refinement. Understanding the relationship between these structures is essential for interpreting the geometric significance of the framework. The following discussion examines the geometric consequences of this layered architecture and the role it plays in the emergence of the resulting propagation geometry. 9 Refinement-Cost Characterization of Pointwise RTP This section decomposes the quantitative content of pointwise RTP into explicit constituents. Positive-projective invariance and two-data quantitative closure imply factorization through the Refinement Ratio on attained values; the QII–UPCP splitting proves full ratio attainability on fibers of rank at least two; Finite Refinement Capacity supplies the remaining substantive threshold premise. The results characterize pointwise RTP and do not derive finite propagation capacity from QII, LAS, or UPCP alone. Assumption 9.1 (Positive-projective invariance). For v ∈ C + o and λ > 0, Prop o (λv) = Prop o (v). Assumption 9.2 (Two-data quantitative closure). There existsG o : (0,∞)×[0,∞)→{0,1}such that Prop o (v) = G o (A o (v),R o (v)). Lemma 9.3 (Projective Factorization Through the Attained Refinement Ratio). LetR o := RR o (C + o ). There exists a unique F o :R o →{0, 1} such that Prop o (v) = F o (RR o (v)) for every v ∈ C + o . Proof. IfRR o (v) =RR o (w), normalize byA o (v),A o (w)>0. Positive homogeneity gives the same normalized pair (1, RR o (v)). Two-data closure and projective invariance then giveProp o (v) = Prop o (w). Hence the classifier is well defined on each attained ratio. Uniqueness follows by choosing a realizing vector for each r ∈R o . Proposition 9.4 (Refinement-Ratio Attainability). Fix a regular QII–UPCP point, withV o =A QII o,0 , positive-definite q o , nonzero normalized covector b Θ o , T o
b Θ ♯ q o o , H QH o = ker b Θ o , and C + o ={v ∈ V o : b Θ o (v) > 0}, the full UPCP-positive sector prior to RTP selection. Forv=A o (v)T o +v ⊥ , defineA o (v) = b Θ o (v), 46
Paper 4 Version 5Canonical Realization and Hybrid Geometry R o (v) = p q o (v ⊥ ,v ⊥ ), and RR o (v) = R o (v)/A o (v). Then RR o (C + o ) = {0},dimV o = 1, [0,∞), dimV o ≥ 2. Proof. IfdimV o = 1, thenH QH o ={0}and every transverse response vanishes. IfdimV o ≥2, choose aq o -unitu ∈ H QH o . For arbitraryr ≥0, setv r =T o +ru. Then b Θ o (v r ) = 1, sov r ∈ C + o , while A o (v r ) = 1 and R o (v r ) = r. Hence RR o (v r ) = r. Assumption 9.5 (Finite Refinement Capacity). On rank-at-least-two fibers letJ o :=F −1 o (1)⊆ [0,∞). Assume 0∈ J o , J o is downward closed, proper, and relatively open in [0,∞). Theorem 9.6 (Pointwise Refinement-Capacity Characterization). IfdimV o ≥2, there is a unique c o ∈ (0,∞) such that J o = [0,c o ). Hence pointwise propagation is equivalent to RR o (v) < c o . Proof.By Proposition 9.4, every nonnegative ratio is attained, and Lemma 9.3 gives the unique classifier on [0,∞). The stated order-topological hypotheses makeJ o a proper relatively open initial interval containing zero; therefore J o = [0,c o ) for the unique endpoint c o = supJ o ∈ (0,∞). Remark 9.7 (Characterization boundary). Finite Refinement Capacity is an explicit quantitative premise. The theorem decomposes and characterizes pointwise RTP; it does not derive finite propagation capacity from the preceding informational geometry alone. 10Horizon-Organized Hybrid Geometry and Null Boundary Real- ization The pointwise refinement-cost characterization identifies the qualitative propagation set with the response-threshold cone in every admissible fiber. The results in this section sharpen the structural role of the Local Quantum Horizon within that propagation geometry. They remain fiberwise unless a domain-level assertion is stated explicitly, and they do not presume the unresolved smooth assembly of the pointwise capacity relation. 10.1 Separatrix-to-null-barrier transition Theorem 10.1 (Separatrix-to-Null-Barrier Transition). Under the standing structure above, the UPCP orientation geometry and the response-threshold propagation geometry satisfy the following. 1.Informational separatrix. The UPCP functional gives the representative-independent decomposition V o = C + o ⊔ H QH o ⊔ C − o , C − o :={v ∈ V o : b Θ o (v) < 0}. Thus H QH o is the equilibrium separatrix at which first-order UPCP orientation changes sign. 47
Paper 4 Version 5Canonical Realization and Hybrid Geometry 2.Proper threshold refinement. The propagation cone satisfiesK + o ⊆ C + o . IfH QH o ̸={0}, then K + o ⊊ C + o . Hence positive UPCP orientation is necessary but not sufficient for propagation. 3. Quantitative half-space exclusion. For every 0̸= h∈ H QH o and every 0 < ε < p q o (h,h) c o , the vector v ε := εT o
- h satisfies v ε ∈ C
o
K
+
o
.
Thus every nonzero Horizon direction generates positively oriented continuations that are
excluded even from the closed propagation cone.
4. Lorentz-signature propagation form. The polarization
G
o
(v,w) := q
o
(v
⊥
,w
⊥
)− c
2
o
b
Θ
o
(v)
b
Θ
o
(w)(1)
is a symmetric nondegenerate bilinear form of signature (−,+,...,+), withQ
o
(v) =G
o
(v,v),
and
K
+
o
={v ∈ V
o
: G
o
(v,v) < 0,
b
Θ
o
(v) > 0}.
5.Null-barrier formation. The nonzero propagation boundary is precisely the future null
cone of G
o
:
∂K
+
o
{0} ={v ∈ V
o
:
b
Θ
o
(v) > 0, G
o
(v,v) = 0}.
It is a smooth conic hypersurface, and every normalized section
{v ∈ ∂K
+
o
:
b
Θ
o
(v) = 1}
is a strongly convex q
o
-ellipsoid in the affine space T
o
- H QH o .
- Genuine propagation-type transition. For every v ∈ ∂K
o {0}, dQ o | v ̸= 0. Consequently,Q o is a local defining function for the boundary, and every sufficiently small neighborhood ofvinC + o contains directions withQ o <0 and directions withQ o
0. The boundary therefore separates propagating directions from future-oriented non-propagating directions by a genuine first-order sign change. 7.Barrier, not wall. The propagation boundary is not an external terminal edge ofV o and does not terminate QII admissibility or positive UPCP orientation. Rather, withinC
o it is the intrinsic null separatrix in the trichotomy C + o ={ b Θ o
0,Q o < 0}⊔{ b Θ o 0,Q o = 0}⊔{ b Θ o 0,Q o 0}. Across it, propagation status changes while positive orientation persists. 48
Paper 4 Version 5Canonical Realization and Hybrid Geometry Accordingly, the Local Quantum Horizon and the propagation boundary are not identified. The Horizon is the informational equilibrium separatrix of the half-space-like orientation geometry; response-threshold refinement uses the Horizon decomposition to produce a stricter proper cone whose boundary is a genuine null barrier in the induced Lorentz-signature propagation geometry. Proof. Linearity and nondegeneracy of b Θ o give the disjoint UPCP sign decomposition V o = C + o ⊔ ker b Θ o ⊔ C − o . This proves (1). The response-threshold definition immediately gives K + o ⊆ C + o . Assume H QH o ̸={0} and choose 0̸= h∈ H QH o . For v ε := εT o
- h, the canonical splitting gives b Θ o (v ε ) = ε,(v ε ) ⊥ = h. Hence v ε ∈ C
o for every ε > 0, while Q o (v ε ) = q o (h,h)− c 2 o ε 2 .(2) Becauseq o is positive definite andh̸= 0, one hasq o (h,h)>0. Therefore the displayed threshold equation is positive whenever 0 < ε < p q o (h,h) c o . For suchε,v ε lies in the positive UPCP half-space but outsideK + o , whose response-threshold description usesQ o ≤0 together with nonnegative advance. This proves (2), (3), and the strict inclusion. Relative to the q o -orthogonal decomposition V o = RT o ⊕ q o H QH o , the quadratic form satisfies Q o (aT o
- h) = q o (h,h)− c 2 o a 2 . It is negative on the nonzero advance line and positive on every nonzero Horizon direction. There- fore its displayed polarization is nondegenerate of signature (−,+,...,+), andQ o (v) =G o (v,v). Combining this identity with the response-threshold definition proves (4). The nonzero boundary of the strict negative cone in the chosen future sector is the displayed future null set. Its differential is dQ o | v (w) = 2q o (v ⊥ ,w ⊥ )− 2c 2 o b Θ o (v) b Θ o (w). Suppose v ̸= 0, Q o (v) = 0, and dQ o | v = 0. Taking w = T o yields 0 = dQ o | v (T o ) =−2c 2 o b Θ o (v), so b Θ o (v) = 0. The null equation then givesq o (v ⊥ ,v ⊥ ) = 0, and positive definiteness givesv ⊥ = 0. The canonical splitting forcesv= 0, a contradiction. ThusdQ o | v ̸= 0 at every nonzero null vector. The regular-value theorem makes the nonzero null set a smooth conic hypersurface. 49
Paper 4 Version 5Canonical Realization and Hybrid Geometry On the normalized section b Θ o (v) = 1, writev=T o +hwithh ∈ H QH o . The boundary equation becomes q o (h,h) = c 2 o , which is aq o -sphere and hence a strongly convex ellipsoid in arbitrary linear coordinates. This proves (5). BecausedQ o | v ̸= 0, the submersion theorem gives local coordinates centered at each nonzero boundary vector in whichQ o itself is one coordinate. Both signs therefore occur on opposite local sides of the zero set. WithinC + o , the negative side is propagating and the positive side is future-oriented but non-propagating. This establishes (6). Finally, all three response-threshold sectors in (7) lie within the same positive UPCP sector. Crossing Q o = 0 therefore changes propagation type without crossing the informational equilibrium separatrix b Θ o = 0. The null boundary is consequently an intrinsic propagation barrier, not an external wall and not the Quantum Horizon itself. This proves (7) and the concluding distinction. Remark 10.2 (Logical status and claim boundary). The theorem consolidates consequences already established in Paper 4 under RTP: properness, saliency, strict inclusionK + o ⊊ C + o for nontrivial Horizon, Lorentz signature, smooth null boundary, and the response-induced trichotomy. It does not derive RTP, smooth regular representability of the bridge class, tangent-bundle identification, four-dimensionality, spacetime reconstruction, or physical dynamics. The phrase “separatrix-to-null- barrier transition” names a staged refinement of geometric roles, not an equality betweenH QH o and ∂K + o . Corollary 10.3 (No Horizon-tangent null generators). If 0̸= h∈ H QH o , then G o (h,h) = q o (h,h) > 0. Hence no nonzero Horizon-tangent vector is a null generator of the propagation boundary. Every future null generator has strictly positive UPCP advance and nonzero transverse response satisfying q o (v ⊥ ,v ⊥ ) = c 2 o b Θ o (v) 2 . Remark 10.4 (Interpretive formulation). The informational separatrix supplies the advance– equilibrium–retreat organization. The response threshold then restricts the advance sector by requiring transverse response to remain bounded relative to normalized advance. Saturation of that bound produces a null ray. The resulting boundary is therefore a barrier in propagation type: timelike-propagating directions lie inside, null-critical directions lie on it, and positively oriented non-propagating directions lie outside. 10.2 Horizon uniqueness Lemma 10.5 (Horizon Uniqueness). LetV o be the QII-admissible continuation space ato, equipped with the normalized UPCP covector b Θ o , the inherited positive-definite response metricq o , and the pointwise response capacity c o
0. Define H QH o := ker b Θ o , T o := b Θ ♯ q o o , 50
Paper 4 Version 5Canonical Realization and Hybrid Geometry where ♯ q o : V ∗ o → V o is the musical isomorphism induced by q o . Assume the normalization q −1 o ( b Θ o , b Θ o ) = 1, so that b Θ o (T o ) = q o (T o ,T o ) = 1. Then the following hold. 1.Unique UPCP decomposition. The normalized UPCP orientation structure uniquely determines the equilibrium subspace H QH o = ker b Θ o and the canonical q o -orthogonal decomposition V o = RT o ⊕ q o H QH o . Consequently, every v ∈ V o admits the unique representation v = b Θ o (v)T o
- v ⊥ , v ⊥ ∈ H QH o .
- Unique transverse response reference space. Define F o (v) := q o (v ⊥ ,v ⊥ ). ThenF o depends only on the canonicalq o -orthogonal projection ofvontoH QH o . Hence every transverse response contribution entering the response-threshold propagation criterion is measured relative to the Horizon component determined by the normalized UPCP structure and q o . 3.Horizon determination of the propagation law. The response-threshold quadratic form is Q o (v) = F o (v)− c 2 o b Θ o (v) 2 . Accordingly, the alternativesQ o (v)<0,Q o (v) = 0, andQ o (v)>0 are determined entirely by
b Θ o (v),F o (v) . Thus the response-threshold propagation law uses normalized UPCP advance together with transverse response measured through the canonical Horizon projection; no additional decom- position of V o enters this law. Therefore the Local Quantum Horizon is the unique equilibrium subspace induced by the normalized UPCP orientation and the unique transverse reference space appearing in the response-threshold propagation geometry. Proof. The UPCP orientation structure determines a positive-projective covector ray [Θ o ]. Normal- 51
Paper 4 Version 5Canonical Realization and Hybrid Geometry ization by the positive-definite dual metric q −1 o gives b Θ o
Θ o q q −1 o (Θ o , Θ o ) , which is unchanged under every admissible positive rescaling of Θ o . Hence b Θ o andH QH o =ker b Θ o are uniquely determined by [Θ o ] and q o . The metric dual T o
b Θ ♯ q o o is likewise unique. For h∈ H QH o , q o (T o ,h) = b Θ o (h) = 0. Since b Θ o (T o ) = 1, T o /∈ H QH o ; since H QH o has codimension one, V o = RT o ⊕ q o H QH o . Forv ∈ V o , setv ⊥ =v− b Θ o (v)T o . Then b Θ o (v ⊥ ) = 0, sov ⊥ ∈ H QH o , and directness gives uniqueness. Thus F o (v) = q o (v ⊥ ,v ⊥ ) is uniquely determined by the canonical Horizon component. Finally, Q o (v) = q o (v ⊥ ,v ⊥ )− c 2 o b Θ o (v) 2 = F o (v)− c 2 o b Θ o (v) 2 . Its sign is therefore determined by normalized UPCP advance and the transverse response of the canonical Horizon projection, with no further decomposition entering the law. Remark 10.6 (Claim boundary). The lemma is internal to the adopted response-threshold archi- tecture. It proves uniqueness of the Horizon decomposition and of the transverse reference space appearing in that law. It does not assert that no alternative propagation theory could be formulated, nor does it derive the response-threshold law from Horizon uniqueness. The lemma is down- stream of UPCP normalization, the inherited response metric, and the pointwise response-threshold characterization, and must not be used circularly to justify those inputs. 10.3 Horizon propagation-failure structure Theorem 10.7 (Horizon Propagation-Failure Structure). LetV o be a QII-admissible continuation space equipped with the normalized UPCP covector b Θ o , the inherited positive-definite response metric q o , the pointwise response capacity c o
0, and the canonical decomposition V o = RT o ⊕ q o H QH o , H QH o = ker b Θ o . For v = aT o
- h, a = b Θ o (v), h = v ⊥ ∈ H QH o , define F o (v) := q o (h,h), A o (v) := c 2 o a 2 , and Q o (v) := F o (v)− A o (v). Assume c o
0 and, when stated below, H QH o ̸={0}. 52
Paper 4 Version 5Canonical Realization and Hybrid Geometry Then:
- The transverse response functional F o (v) = q o (v ⊥ ,v ⊥ ) depends only on the canonical Horizon projection ofv. Every transverse response contribution entering the propagation law is therefore measured relative to H QH o .
- For every nonzero Horizon vector h∈ H QH o , Q o (h) = F o (h) = q o (h,h) > 0. Hence no nonzero Horizon vector is propagating or null. 3.No Horizon vector lies in the strict propagation cone, and the Horizon meets the closed propagation cone only at the origin: K
o ∩ H QH o = ∅,K + o ∩ H QH o ={0}. Consequently, every nonzero Horizon vector lies outsideK + o . 4. Inside the future-oriented sector C + o ={v : b Θ o (v) > 0} one has the trichotomy K + o ={v : F o (v) < A o (v)}, N + o ={v : F o (v) = A o (v)}, F + o ={v : F o (v) > A o (v)}, with C + o = K + o ⊔ N + o ⊔ F + o . 5. The future null barrier consists exactly of the critical balances q o (h,h) = c 2 o a 2 , a > 0. Every nonzero null generator therefore possesses both a nonzero advance component and a nonzero Horizon component. 6. The Local Quantum Horizon performs two distinct structural roles: (a) it is the equilibrium separatrix of the UPCP orientation geometry; (b) it is the canonical transverse response reference space appearing in the response-threshold propagation law. Consequently, propagation admissibility, criticality, and non-propagation are classified through the balance between normalized advance and Horizon-measured transverse response. 53
Paper 4 Version 5Canonical Realization and Hybrid Geometry Proof.The first statement follows from the Horizon Uniqueness Lemma:v ⊥ is the canonical Horizon projection, soF o (v) =q o (v ⊥ ,v ⊥ ) depends only on that projection. For nonzeroh∈ H QH o one has a = 0, hence Q o (h) = q o (h,h) > 0 by positive definiteness of q o , proving (2). Ifh∈ H QH o , then b Θ o (h) = 0, whereas everyv ∈ K + o satisfies b Θ o (v)>0; henceK + o ∩ H QH o =∅. If h∈K + o ∩ H QH o , the closure characterization from Theorem 8.9 givesQ o (h)≤0. On the Horizon, Q o (h) =q o (h,h), so positive definiteness givesh= 0. Conversely,εT o ∈ K + o for everyε >0 and εT o →0, so 0∈K + o . ThereforeK + o ∩ H QH o ={0}. Statement (4) is a direct rewriting of the sign alternatives forQ o =F o −A o . Statement (5) follows from the equality caseQ o = 0. Finally, the first role of the Horizon follows fromH QH o =ker b Θ o , while the second follows because the propagation law depends on transverse response only through the Horizon projection. Therefore all propagation classifications are expressed through the balance between normalized advance and Horizon-measured transverse response. Remark 10.8 (Claim boundary). The theorem is internal to the adopted response-threshold architecture. It does not assert that no alternative propagation theory could be formulated, nor that the Horizon exhausts the entire non-propagating sector. The theorem only establishes that the current propagation law classifies admissibility, criticality, and non-propagation through the balance of normalized advance and Horizon-measured transverse response. 10.4 Horizon essentiality and unique recovery Theorem 10.9 (Horizon Essentiality and Unique Recovery). LetV o be a finite-dimensional real QII-admissible continuation space withdimV o ≥1. Letq o be the inherited positive-definite response metric, let [Θ o ] + be the positive-projective UPCP orientation ray, and letc o ∈(0,∞) be the pointwise response capacity. Let b Θ o ∈ V ∗ o {0} be the q −1 o -unit representative of [Θ o ] + and define T o := b Θ ♯ q o o , H QH o := ker b Θ o . Write V o = RT o ⊕ q o H QH o . For v = b Θ o (v)T o
- v ⊥ , define G o (v,w) := q o (v ⊥ ,w ⊥ )− c 2 o b Θ o (v) b Θ o (w). Let the exact hybrid metric data be H o = (V o ,q o ,G o , [Θ o ]
). Then:
- The defect form D o := q o − G o 54
Paper 4 Version 5Canonical Realization and Hybrid Geometry satisfies D o = (1 + c 2 o ) b Θ o ⊗ b Θ o . Hence D o is nonzero, symmetric, positive-semidefinite, and rank one. 2. The Quantum Horizon is intrinsically recoverable from the completed hybrid metric data: H QH o = kerD o = ker(q o − G o ). Equivalently, H QH o ={h∈ V o : q o (h,w) = G o (h,w) for all w ∈ V o }. Thus H QH o is the unique maximal linear subspace on which q o and G o agree. 3. Define the defect map D ♭ o : V o → V ∗ o , D ♭ o (v) = D o (v,·). Then imD ♭ o = R b Θ o . Thus (q o ,G o ) recovers the unoriented covector line. Together with the retained positive UPCP orientation, the positive-projective ray [Θ o ] + is recovered. 4.(Uniqueness of the normalized response-threshold representation.) Suppose the same exact pair (q o ,G o ) admits a second response-threshold representation determined by data e Θ o ∈ V ∗ o {0}, e c o
0, with the same metric normalization q −1 o ( e Θ o , e Θ o ) = 1, and associated decomposition V o = R e T o ⊕ q o e H o , e H o := ker e Θ o , for which G o (v,w) = q o ( e v ⊥ , e w ⊥ )− e c 2 o e Θ o (v) e Θ o (w). Then e H o = H QH o , e Θ o =± b Θ o , e c o = c o . If the same positive UPCP orientation ray is preserved, then e Θ o = b Θ o . Accordingly, the Local Quantum Horizon is structurally essential to the exact hybrid geometry: it is generated by the orientation structure, used in the propagation construction, intrinsically recoverable from the completed metric data, and unique among all equally normalized response- threshold representations. 55
Paper 4 Version 5Canonical Realization and Hybrid Geometry Proof. Using the canonical decomposition, q o (v,w) = b Θ o (v) b Θ o (w) + q o (v ⊥ ,w ⊥ ). Subtracting the propagation form gives (q o − G o )(v,w) = (1 + c 2 o ) b Θ o (v) b Θ o (w), proving the defect identity. Since 1 + c 2 o
0, a vector lies in kerD o exactly when b Θ o (v) = 0. Hence kerD o = ker b Θ o = H QH o . The agreement characterization is immediate from D o = q o − G o . The defect map satisfies D ♭ o (v) = (1 + c 2 o ) b Θ o (v) b Θ o , so its image is exactly R b Θ o . For the uniqueness claim, the alternative normalized representation yields D o = (1 + e c 2 o ) e Θ o ⊗ e Θ o . Since D o ̸= 0 is rank one, the generating covectors span the same line: e Θ o = λ b Θ o . The common q o -normalization implies 1 = q −1 o ( e Θ o , e Θ o ) = λ 2 q −1 o ( b Θ o , b Θ o ) = λ 2 , so λ =±1. Hence e Θ o =± b Θ o , giving e H o = ker e Θ o = ker b Θ o = H QH o . Comparing coefficients in the defect identities yields 1 + e c 2 o = 1 + c 2 o , whence e c o =c o . If the positive UPCP orientation ray is fixed, the negative sign is excluded. This proves the theorem. Remark 10.10 (Claim boundary). The theorem establishes recoverability and uniqueness only for the exact normalized hybrid metric data (V o ,q o ,G o ,[Θ o ]
). It does not claim recovery from a null cone, conformal class, propagation cone alone, or an unrelated propagation theory. Nor does it derive RTP from Horizon essentiality. Remark 10.11 (Defect regularity and logical priority). The defectD o =q o − G o is the primary recovery invariant of the completed pointwise hybrid geometry, but it is not yet an independently 56
Paper 4 Version 5Canonical Realization and Hybrid Geometry primary input of the derivation. Since q and b Θ are already smooth and D o = (1 + c 2 o ) b Θ o ⊗ b Θ o , the following regularity statements are equivalent on a fixed-rank QII-regular domain: D assembles smoothly ⇐⇒ λ(o) := tr(♯ q o ◦ D ♭ o ) is smooth ⇐⇒ c is smooth, whereλ(o) = 1 +c 2 o . Thus all unresolved regularity is concentrated in the scalar defect strengthλ. Horizon recoverability does not by itself prove smooth defect assembly or make the Response-Capacity Regularity argument superfluous. 10.5 Null Boundary Realization and threshold–radius identification The preceding results distinguish the Local Quantum Horizon from the propagation boundary while showing that the latter is constructed from the canonical advance–Horizon splitting. This subsection identifies the normalized qualitative propagation boundary with the normalized future null barrier and proves that its response radius is exactly the independently derived refinement cutoff. The construction is pointwise and does not presume base-point smoothness of the capacity field. LetU QII be a fixed-rank QII-regular domain with QII-admissible bundleV=A QII 0 → U QII of rank k ≥ 2. Fix o∈ U QII and use the canonical data V o = RT o ⊕ q o H QH o , H QH o = ker b Θ o . For v ∈ C + o define ρ o (v) := p q o (v ⊥ ,v ⊥ ) b Θ o (v) . Let c o
0 be the unique pointwise cutoff with J o = [0,c o ), and define S
o :={v : b Θ o (v) = 1} = T o
- H QH o , D prop o := K
o ∩ S + o , B o := ∂ S + o D prop o . Set r o (T o
- h) := p q o (h,h). Theorem 10.12 (Null Boundary Realization and Threshold–Radius Identification). For every o∈ U QII , D prop o ={T o
- h : q o (h,h) < c 2 o } and B o ={T o
- h : q o (h,h) = c 2 o } = ∂K
o ∩ S + o . Consequently:
- B o is a nonempty compact smoothq o -sphere of radiusc o inT o +H QH o and bounds the strongly convex ball D prop o ;
- B o ∼ = S dimH QH o −1 , so it is a two-point 0-sphere whendimH QH o = 1 and is connected when dimH QH o ≥ 2;
- r o | B o ≡ c o ; 57
Paper 4 Version 5Canonical Realization and Hybrid Geometry 4. the common geometric radius of B o is exactly the unique informational refinement cutoff; 5.the sectorsr o < c o ,r o =c o , andr o
c o are respectively propagating, null-critical, and future-oriented non-propagating. Thus the normalized propagation boundary is a nonempty compact smoothq o -sphere whose radius is exactly the unique pointwise refinement cutoffc o , and the sign ofQ o realizes the propagating/null- critical/non-propagating trichotomy on the normalized slice. Proof.Everyv ∈ S
o has the unique formv=T o +hwithh∈ H QH o , andρ o (T o +h) = p q o (h,h). The pointwise Refinement-Cost Characterization therefore gives D prop o ={T o
- h : q o (h,h) < c 2 o }. Taking the relative boundary inS
o yields the displayed sphere formula forB o . On the normalized slice the response-threshold quadratic form is Q o (T o
- h) = q o (h,h)− c 2 o , so the sphere is exactly the normalized future null barrier. The sphere description proves the dimension-sensitive topology, and for v = T o
- h∈B o one has r o (v) = q q o (h,h) = c o . Sincec o is independently the unique endpoint ofJ o = [0,c o ), the geometric radius and informational cutoff coincide. The sign of Q o = r 2 o − c 2 o gives the final trichotomy. Remark 10.13 (Domain-level assembly status). The theorem is pointwise and does not by itself establish that B = G o∈U QII B o assembles smoothly overU QII . Smooth boundary assembly is a separate domain-level question controlled in the present architecture by Smooth Informational Assembly and the subsequent regularity results. 10.6 Status after pointwise null-boundary realization The preceding results establish, on every fixed-rank QII-regular domain of rank at least two, the sphere structure of the normalized propagation boundary, radial response isotropy, identification of its radius with the unique informational cutoff, and the pointwise sign trichotomy. The domain-level assembly obligation is discharged below by Regular Local Continuation and the HRCEL Smoothing Law. Smooth capacity, boundary, propagation-form, and defect assembly then follow together on the selected regular domain. 58
Paper 4 Version 5Canonical Realization and Hybrid Geometry 11 Response-Capacity Saturation Law Fix a regular operational pointo. LetV o be the finite-dimensional QII-admissible continuation space, letq o be the inherited positive-definite response metric, and let b Θ o ∈ V ∗ o be the normalized UPCP covector. Define T o := b Θ ♯ q o o , H QH o := ker b Θ o , C + o :={v ∈ V o : b Θ o (v) > 0}. Assume the Horizon Equilibrium Structure Theorem, so thatH QH o is the first-order UPCP informational-equilibrium separatrix and primary invariant balance manifold, with canonical orthog- onal splitting V o = RT o ⊕ q o H QH o . Every v ∈ V o therefore has the unique decomposition v = A o (v)T o
- v ⊥ , A o (v) := b Θ o (v), v ⊥ ∈ H QH o . For v ∈ C
o , define the Horizon-measured transverse response burden and the refinement ratio by R o (v) := q q o (v ⊥ ,v ⊥ ),RR o (v) := R o (v) A o (v) . Assume the pointwise Response-Threshold Propagation Principle and its refinement-cost characteri- zation. Thus there exists a unique response capacity c o
0 such that v ∈ K
o ⇐⇒ v ∈ C + o and RR o (v) < c o . Definition 11.1 (Response-capacity saturation). A continuationv ∈ C + o is response-capacity saturated when its Horizon-measured transverse response burden exactly equals the response capacity associated with its normalized advance: R o (v) = c o A o (v). The response-capacity saturation class is Σ c o :={v ∈ C + o : R o (v) = c o A o (v)}. Theorem 11.2 (Response-Capacity Saturation Principle). Under the preceding assumptions, the following hold. (RCSP1) Unique saturation class. The Horizon-induced balance relation possesses the unique response-capacity saturation class Σ c o ={v ∈ C + o : RR o (v) = c o } ={v ∈ C + o : R o (v) = c o A o (v)}. (RCSP2) Equivalent quadratic characterization. Forv ∈ C + o , the saturation condition is equivalent to q o (v ⊥ ,v ⊥ ) = c 2 o b Θ o (v) 2 . 59
Paper 4 Version 5Canonical Realization and Hybrid Geometry Equivalently, if Q o (v) := q o (v ⊥ ,v ⊥ )− c 2 o b Θ o (v) 2 , then v ∈ Σ c o ⇐⇒ Q o (v) = 0. (RCSP3) Capacity trichotomy. Everyv ∈ C + o belongs to exactly one of the three disjoint classes R o (v) < c o A o (v), R o (v) = c o A o (v), R o (v) > c o A o (v). Equivalently, RR o (v) < c o ,RR o (v) = c o ,RR o (v) > c o . These are respectively the sub-capacity, saturation, and super-capacity regimes. (RCSP4) Projective invariance. For every λ > 0, A o (λv) = λA o (v), R o (λv) = λR o (v),RR o (λv) = RR o (v). Consequently Σ c o is invariant under positive rescaling and defines a projective balance class. (RCSP5) Critical selection within the Horizon balance geometry. The unique cutoffc o selects precisely one member of the Horizon-induced family of balance classes Σ ρ :={v ∈ C + o : RR o (v) = ρ}, ρ≥ 0. The selected member is Σ c o , the unique class at which the response burden saturates the available capacity associated with normalized advance. Proof. For v ∈ C + o , one has A o (v) > 0. Hence R o (v) = c o A o (v) ⇐⇒ R o (v) A o (v) = c o ⇐⇒RR o (v) = c o . This proves the equality of the descriptions in RCSP1. Uniqueness follows from the uniqueness of the pointwise RTP cutoff c o . Because R o (v)≥ 0, c o
0, and A o (v) > 0, squaring preserves equality and gives R o (v) 2 = c 2 o A o (v) 2 . Using R o (v) 2 = q o (v ⊥ ,v ⊥ ), A o (v) = b Θ o (v), yields the quadratic equality in RCSP2, which is exactly Q o (v) = 0. RCSP3 follows from real-number trichotomy applied toRR o (v) andc o . SinceA o (v)>0, multiplica- tion byA o (v) preserves the three inequalities and gives the equivalent burden-capacity formulation. For λ > 0, linearity gives A o (λv) = λA o (v),(λv) ⊥ = λv ⊥ . 60
Paper 4 Version 5Canonical Realization and Hybrid Geometry Positive homogeneity of the norm induced by q o gives R o (λv) = λR o (v), and therefore RR o (λv) = RR o (v). This proves RCSP4. Finally, the Horizon decomposition defines the family of ratio level classes Σ ρ . The unique RTP capacity c o determines exactly one level, namely ρ = c o . This proves RCSP5. Corollary 11.3 (Compatibility with RTP and BET). The pointwise RTP classification may be written K + o ={v ∈ C + o : R o (v) < c o A o (v)}, while RCSP identifies the exact saturation class Σ c o ={v ∈ C + o : R o (v) = c o A o (v)}. Under the Boundary Emergence Theorem, ∂K + o {0} = Σ c o . Thus RCSP supplies the unique saturation condition used by BET, while BET supplies its geometric realization as the nonzero null propagation boundary. Remark 11.4 (Logical status and claim boundary). RCSP is a pointwise saturation theorem. It does not identify the saturation class with the UPCP informational-equilibrium sector, and it does not replace HES or BET. HES supplies the informational-equilibrium separatrix and primary balance manifold. RCSP supplies the unique response-capacity saturation class. BET identifies that class with the null propagation boundary. RCSP does not prove smooth base-point variation ofc o , global boundary assembly, a dynamical equilibrium flow, or SSR. 12 Boundary Emergence Theorem Fix a regular operational pointo. LetV o be the finite-dimensional QII-admissible continuation space, letq o be the inherited positive-definite response metric, and let b Θ o ∈ V ∗ o be the normalized UPCP covector. Define T o := b Θ ♯ q o o , H QH o := ker b Θ o , C + o :={v ∈ V o : b Θ o (v) > 0}. Assume the Horizon Equilibrium Structure Theorem (HES):H QH o is the first-order UPCP informational-equilibrium separatrix and primary invariant balance manifold. Thus V o = RT o ⊕ q o H QH o , and every v ∈ V o has the unique decomposition v = a o (v)T o
- v ⊥ , a o (v) := b Θ o (v), v ⊥ ∈ H QH o . For v ∈ C
o , define the Horizon-measured transverse response and refinement ratio by r o (v) := q q o (v ⊥ ,v ⊥ ),RR o (v) := r o (v) a o (v) . 61
Paper 4 Version 5Canonical Realization and Hybrid Geometry Assume the pointwise Response-Threshold Propagation Principle (RTP) and its refinement-cost characterization: there is a unique capacity c o
0 such that v ∈ K
o ⇐⇒ v ∈ C + o and RR o (v) < c o . Define the response-threshold quadratic functional Q o (v) := q o (v ⊥ ,v ⊥ )− c 2 o b Θ o (v) 2 = a o (v) 2
RR o (v) 2 − c 2 o for v ∈ C + o , and define the saturation class Σ c o :={v ∈ C + o : RR o (v) = c o } ={v ∈ C + o : Q o (v) = 0}. Theorem 12.1 (Boundary Emergence Theorem). Under HES, pointwise RTP, the refinement-cost characterization, and the conclusions of the Separatrix-to-Null-Barrier Transition Theorem, the following hold. (BET1) Two distinct sign structures. HES supplies the UPCP orientation trichotomy V o ={ b Θ o
0}⊔ H QH o ⊔{ b Θ o < 0}, whereH QH o ={ b Θ o = 0}is the informational-equilibrium separatrix. RTP supplies, insideC
o , the independent propagation trichotomy C + o ={Q o < 0}⊔{Q o = 0}⊔{Q o
0}. (BET2) Capacity refinement of the advance sector. The propagation cone and its comple- mentary positive-sector region are K
o ={v ∈ C + o : RR o (v) < c o } ={v ∈ C + o : Q o (v) < 0}, F + o ={v ∈ C + o : RR o (v) > c o } ={v ∈ C + o : Q o (v) > 0}. Thus RTP refines, rather than replaces, the HES orientation geometry. (BET3) Null-boundary emergence. The nonzero boundary of the propagation cone is exactly the response-capacity saturation class: ∂K + o {0} = Σ c o ={v ∈ C + o : Q o (v) = 0}. Equivalently, the null boundary emerges as the unique interface inC + o between propagating directions Q o < 0 and future-oriented non-propagating directions Q o
0. (BET4) Genuine local separation. For every v ∈ Σ c o , dQ o | v ̸= 0. Consequently Σ c o is a smooth conic hypersurface, and every sufficiently small neighborhood ofvin C
o contains points of both K + o and F + o . 62
Paper 4 Version 5Canonical Realization and Hybrid Geometry (BET5) Horizon–boundary separation. The Quantum Horizon contains no nonzero propagation- boundary vector: H QH o ∩ (∂K + o {0}) = ∅. Every nonzerov ∈ ∂K + o satisfies b Θ o (v)>0, whereas everyh∈ H QH o satisfies b Θ o (h) = 0. Thus no nonzero propagation-boundary vector lies on the Quantum Horizon, for any admissible rank. If dimV o ≥ 2, then ∂K + o {0}̸= ∅, and consequently H QH o ̸= ∂K + o . For dimV o = 1, H QH o = ∂K + o ={0}, so full-set nonidentity is not asserted. (BET6) Normalized boundary realization. On the normalized slice S + o :={v ∈ V o : b Θ o (v) = 1} = T o
- H QH o , the boundary is B o := (∂K
o )∩ S + o ={T o
- h : h∈ H QH o , q o (h,h) = c 2 o }. ThusB o is theq o -sphere of radiusc o in the affine Horizon slice and is the normalized geometric realization of the unique refinement cutoff. Proof.HES gives the representative-independent UPCP decomposition into the positive, zero, and negative sectors, with H QH o = ker b Θ o . This proves the first trichotomy in BET1. For v ∈ C
o , one has a o (v) > 0 and Q o (v) = q o (v ⊥ ,v ⊥ )− c 2 o a o (v) 2 = a o (v) 2
RR o (v) 2 − c 2 o . Becausea o (v) 2
0 andc o 0, the signs ofQ o (v) andRR o (v)−c o agree. The pointwise refinement- cost characterization therefore yields Q o (v) < 0⇐⇒ RR o (v) < c o ⇐⇒ v ∈ K
o , and similarly identifiesQ o (v) = 0 withRR o (v) =c o andQ o (v)>0 withRR o (v)> c o . This proves BET1 and BET2. The Separatrix-to-Null-Barrier Transition Theorem identifies the nonzero boundary of the strict future propagation cone with the future null set of Q o . Hence ∂K + o {0} ={v ∈ C + o : Q o (v) = 0} = Σ c o , which proves BET3. The same theorem provesdQ o | v ̸= 0 at every nonzero null vector. The regular-value theorem therefore makes Σ c o a smooth conic hypersurface. The submersion theorem gives local coordinates in whichQ o is a coordinate transverse to the zero set, so both signs occur locally on opposite sides. This proves BET4. Ifh ∈ H QH o , then b Θ o (h) = 0, whereas every nonzero boundary vector in BET3 lies inC + o and 63
Paper 4 Version 5Canonical Realization and Hybrid Geometry therefore has strictly positive UPCP advance. HenceH QH o ∩(∂K + o {0}) =∅. WhendimV o ≥2, the normalized null-boundary equation has nonzero solutions, so the full sets cannot agree. In rank one both full sets equal {0}. This proves BET5. Finally, every v ∈ S + o has the unique form v = T o
- h with h∈ H QH o . On this slice, Q o (T o
- h) = q o (h,h)− c 2 o . Therefore Q o = 0 is equivalent to q o (h,h) = c 2 o , which proves BET6. Corollary 12.2 (Null Boundary Emergence). Within the HES-organized orientation geometry, RTP introduces a second sign-changing classification inside the positive UPCP sector. The saturation class of that classification emerges as the null propagation boundary: HES + RTP + Theorem 10.1 =⇒ Null Boundary Emergence. The Quantum Horizon is the informational-equilibrium separatrix; the null boundary is the response- capacity saturation interface. The latter is generated within the balance geometry organized by the former, but the two are not the same object. Remark 12.3 (Logical status and claim boundary). BET is a synthesis theorem. It does not derive HES, RTP, the unique pointwise capacityc o , or the Separatrix-to-Null-Barrier Transition Theorem. It consolidates their consequences into the precise null-boundary-emergence statement. The result is pointwise. It does not prove smooth base-point assembly ofo7→ c o , of the boundary fibersB o , or of the global propagation geometry, and it does not establish SSR. 13 Horizon Response-Capacity Equilibrium Law Fix a regular operational pointo. LetV o be the finite-dimensional QII-admissible continuation space, q o the inherited positive-definite response metric, and b Θ o ∈ V ∗ o the normalized UPCP covector. Define T o := b Θ ♯ q o o , H QH o := ker b Θ o , C
o :={v ∈ V o : b Θ o (v) > 0}. Assume the following locked results. 1. Horizon Equilibrium Structure Theorem (HES). The Quantum HorizonH QH o is the first-order UPCP informational-equilibrium separatrix and the primary invariant balance manifold. In particular, V o = RT o ⊕ q o H QH o . 2. Response-Capacity Saturation Principle (RCSP). For every v ∈ C + o , writing v = A o (v)T o
- v ⊥ , A o (v) := b Θ o (v), v ⊥ ∈ H QH o , and defining R o (v) := q q o (v ⊥ ,v ⊥ ),RR o (v) := R o (v) A o (v) , 64
Paper 4 Version 5Canonical Realization and Hybrid Geometry the unique RTP response capacity c o
0 selects the unique saturation class Σ c o ={v ∈ C
o : R o (v) = c o A o (v)} ={v ∈ C + o : RR o (v) = c o }. 3.Boundary Emergence Theorem (BET). The saturation class is exactly the nonzero null propagation boundary: ∂K + o {0} = Σ c o . Define the response-threshold quadratic functional Q o (v) := q o (v ⊥ ,v ⊥ )− c 2 o b Θ o (v) 2 . For v ∈ C + o , Q o (v) = A o (v) 2
RR o (v) 2 − c 2 o . Theorem 13.1 (Horizon Response-Capacity Equilibrium Law). Under HES, RCSP, and BET, the following statements hold. (HRCEL1) Informational-equilibrium organization. The Quantum Horizon H QH o = ker b Θ o is the representative-independent first-order informational-equilibrium separatrix. It separates the positive, zero, and negative UPCP sectors and supplies the canonical transverse reference space for the advance/response decomposition. (HRCEL2) Canonical response-capacity balance. Everyv ∈ C + o is governed by the comparison between normalized informational advanceA o (v) and Horizon-measured transverse responseR o (v). The unique response capacity c o yields the trichotomy R o (v) < c o A o (v), R o (v) = c o A o (v), R o (v) > c o A o (v). Equivalently, RR o (v) < c o ,RR o (v) = c o ,RR o (v) > c o . These are respectively the sub-capacity, saturation, and super-capacity regimes. (HRCEL3) Propagation classification. The three response-capacity regimes coincide with the propagation trichotomy: R o (v) < c o A o (v) ⇐⇒ Q o (v) < 0 ⇐⇒ v ∈ K + o , R o (v) = c o A o (v) ⇐⇒ Q o (v) = 0 ⇐⇒ v ∈ ∂K + o {0}, R o (v) > c o A o (v) ⇐⇒ Q o (v) > 0 ⇐⇒ v ∈ C + o \K + o . (HRCEL4) Null-boundary realization of saturation. The unique saturation class is geometri- cally realized as the null propagation boundary: Σ c o ={v ∈ C + o : R o (v) = c o A o (v)} ={v ∈ C + o : Q o (v) = 0} = ∂K + o {0}. 65
Paper 4 Version 5Canonical Realization and Hybrid Geometry Thus exact response-capacity saturation is the interface between propagating and future-oriented non-propagating directions. (HRCEL5) Equilibrium-separatrix/null-boundary distinction. H QH o ∩ (∂K + o {0}) = ∅. HereH QH o =ker b Θ o is the UPCP informational-equilibrium separatrix, whereas∂K + o {0}is the nonzero null propagation boundary selected by response-capacity saturation. For dimV o ≥ 2, the nonzero propagation boundary is nonempty, and therefore H QH o ̸= ∂K + o . FordimV o = 1, both full sets reduce to{0}; accordingly, HRCEL asserts nonzero-boundary separation universally and full-set nonidentity only in rank at least two. (HRCEL6) Normalized geometric form. On the normalized slice S + o :={v ∈ V o : b Θ o (v) = 1} = T o
- H QH o , the null boundary is B o ={T o
- h : h∈ H QH o , q o (h,h) = c 2 o }. Therefore the normalized boundary is the q o -sphere of radius c o in the affine Horizon slice. Proof.HRCEL1 is the HES conclusion. The Horizon is the kernel of the normalized UPCP differential, hence the first-order informational-equilibrium sector, and theq o -orthogonal splitting makes it the canonical transverse reference space. HRCEL2 is RCSP. Forv ∈ C
o , one hasA o (v)>0, so comparison ofR o (v) withc o A o (v) is equivalent to comparison of RR o (v) with c o . Real-number trichotomy yields the three disjoint regimes. For HRCEL3, use Q o (v) = R o (v) 2 − c 2 o A o (v) 2 = A o (v) 2
RR o (v) 2 − c 2 o . SinceA o (v)>0 andc o
0, the sign ofQ o (v) agrees with the sign ofRR o (v)− c o , and pointwise RTP identifies the negative regime with propagation. BET identifies the zero regime with the nonzero boundary; the positive regime is the future-oriented non-propagating region beyond the closed propagation cone. HRCEL4 is the joint RCSP–BET identification of the saturation class with the null boundary. HRCEL5 is exactly BET5. Every Horizon vector has zero UPCP advance, while every nonzero propagation-boundary vector has strictly positive advance. The rank qualification follows from nonemptiness of the nonzero boundary when dimV o ≥ 2. For HRCEL6, everyv ∈ S
o has the unique formv=T o +hwithh∈ H QH o . The saturation equation becomes R o (T o
- h) = c o , 66
Paper 4 Version 5Canonical Realization and Hybrid Geometry or equivalently q o (h,h) = c 2 o . This is the stated q o -sphere. Corollary 13.2 (Compressed Horizon Response-Capacity Equilibrium Law). For every positively oriented QII-admissible continuation, R o (v) < c o A o (v)⇐⇒ propagating, R o (v) = c o A o (v)⇐⇒ null-critical boundary, R o (v) > c o A o (v)⇐⇒ future-oriented but non-propagating. The Quantum Horizon supplies the informational-equilibrium separatrix and balance architecture; response-capacity saturation supplies the unique critical class; the Boundary Emergence Theorem realizes that class as the null propagation boundary. Remark 13.3 (Logical status and claim boundary). HRCEL is an audit-clean synthesis law assembled from the HES, RCSP, and BET. It does not identify response-capacity saturation with UPCP informational equilibrium; these remain distinct structures. It does not derive the pointwise capacityc o , smooth base-point variation ofc o , smooth assembly of the boundary fibers, a dynamical equilibrium flow, global causal structure, or SSR. Its conclusion is pointwise and concerns the organization, saturation, and null-boundary realization of the response-capacity balance law. 14Regular Local Continuation, Cartan Assembly, and HRCEL Smoothing 15 Regular Local Continuation Principle 15.1 (Regular Local Continuation). LetU QII be a connected fixed-rank QII-regular smooth domain. Suppose that, at everyo ∈ U QII , the QII–LAS–UPCP–HRCEL architecture determines a unique intrinsic response capacity c o ∈ (0,∞). Assume that throughout U QII :
- the determining QII–LAS–UPCP geometry remains smooth;
- the HRCEL balance remains unique;
- propagation coherence is preserved;
- the selected balance remains nondegenerate; and
- the geometric type of the selected propagation structure remains unchanged. 67
Paper 4 Version 5Canonical Realization and Hybrid Geometry Then the pointwise HRCEL response capacity admits smooth local continuation. Explicitly, for every o 0 ∈ U QII , there exist an open neighborhood o 0 ∈ W ⊆ U QII and a unique smooth function c W : W −→ (0,∞) such that c W (o) = c o for every o∈ W. No assertion is made across loci at which the QII rank changes, the HRCEL balance loses uniqueness or nondegeneracy, propagation coherence fails, competing branches emerge, a bifurcation occurs, or the geometric type changes. Remark 15.2 (Scope and logical status). Regular Local Continuation is a foundational local regularity principle. It does not assume a globally smooth capacity field, a globally assembled propagation frontier, a smooth propagation form, or Smooth Informational Assembly. Its sole conclusion is the existence of smooth local HRCEL capacity tracks on the selected regular stratum. Global agreement and geometric assembly are derived below. 16 Cartan assembly Remark 16.1 (Relation of RLC to Cartan assembly). In the HRCEL application, Principle 15.1 supplies smooth local capacity tracks. Pointwise RTP uniqueness forces those tracks to agree on overlaps. The Cartan geometry then identifies the local frontier models as representations of one intrinsic congruence. Thus no separate HRCEL normalization map is required in the smoothing theorem below. Lemma 16.2 (Cartan Assembly of an Intrinsically Normalized Congruence). LetUbe a connected regular smooth manifold. Let P G −→ U be the principal bundle of a regular Cartan geometric-differential structureGwith residual structure group G, and let π E : E := P G × G E typ −→ U be a smooth associated bundle. Let Λ be a finite-dimensional smoothG-manifold andWa finite-dimensional representation ofG. Define L := P G × G Λ, W := P G × G W. Suppose that each parameter λ∈L o determines an embedded model submanifold M o (λ)⊆ E o . Assume the following. 68
Paper 4 Version 5Canonical Realization and Hybrid Geometry CAN1. Smooth natural model family. Define the incidence space M := n (λ,x)∈L× U E : x∈ M π L (λ) (λ) o . Assume that M is a smooth embedded submanifold of L× U E and that π M : M−→L,(λ,x)7−→ λ, is a smooth submersion. Assume also that the model family is natural under Cartan equivalence. CAN2. Smooth intrinsic differential normalization. The complete Cartan structural and differential invariants determine a smooth natural fiber-preserving map N :L−→W, π W ◦N = π L . In local associated-bundle trivializations overU i ⊆ U, the normalization is represented by a smooth map N i : U i × Λ−→ W. Its dependence on the Cartan structural and differential invariants and onλis smooth and of finite order. NeitherNnor its local representatives are defined using the selected parameter section, the assembled normalized congruence, or an equivalent encoding of either object. CAN3. Cartan equivariance. Ifg ij :U i ∩ U j → Gis a transition function between local Cartan representations, then N j (o,g ij (o)· λ) = g ij (o)·N i (o,λ). The G-action on W fixes 0. CAN4. Unique intrinsic normalization. For everyo∈ U, there exists a uniqueλ o ∈L o such that N (λ o ) = 0 o . Define X o := M o (λ o ). CAN5. Nondegenerate determined normalization. AssumedimΛ =dimW. For every o∈ U, the vertical derivative D λ N o (λ o ) : T λ o L o −→W o is a linear isomorphism. CAN6. Regular-stratum condition. AssumeUis a connected regular stratum on which the Cartan structure type, residual structure group, associated-bundle types ofE,L,W, and the dimensions and codimensions of the model submanifolds remain constant. Then the uniquely normalized parameters form a smooth section λ∈ Γ ∞ (L), λ(o) = λ o . Moreover, X := G o∈U X o 69
Paper 4 Version 5Canonical Realization and Hybrid Geometry is a smooth Cartan-natural embedded submanifold of E, and π X := π E | X : X −→ U is a smooth surjective submersion. Proof. In a local Cartan trivialization, the normalization is a smooth map N i : U i × Λ−→ W. At (o 0 ,λ o 0 ), CAN4 givesN i (o 0 ,λ o 0 ) = 0, and CAN5 makes the derivative in theλ-variable invertible. The implicit-function theorem therefore gives a unique smooth local solutionλ i (o). Pointwise uniqueness identifies this solution with λ o . On overlaps, CAN3 sends a local zero to the corresponding local zero. CAN4 therefore forces the local branches to satisfy the transition law ofL, so they define a unique global smooth section λ∈ Γ ∞ (L). Define j λ : E −→L× U E, j λ (x) =
λ(π E (x)),x . Since π M : M→L is a submersion, j λ is transverse to M. Hence X := j −1 λ (M) is a smooth embedded submanifold of E. Fiberwise, X ∩ E o = M o (λ o ) = X o . Equivalently,Xis the pullbackU × L Malongλ. Thereforeπ X :X → Uis the pullback of a submersion and is itself a smooth submersion. Surjectivity follows from nonemptiness of every normalized fiber. Naturality makes the construction independent of local Cartan representation. 17 HRCEL smoothing law Theorem 17.1 (HRCEL Smoothing Law). LetU QII be a connected fixed-rank QII-regular smooth domain, and let π V : V −→ U QII be the smooth QII-admissible continuation bundle of rank n≥ 2. Let q ∈ Γ ∞ (Sym 2 V ∗ ) be the smooth positive-definite response metric, and let b Θ∈ Γ ∞ (V ∗ ) 70
Paper 4 Version 5Canonical Realization and Hybrid Geometry be the smooth q −1 -unit UPCP covector field. Define T := b Θ ♯ q , H QH := ker b Θ. Then V = RT ⊕ q H QH . For o∈ U QII and v ∈ C + o :={v ∈ V o : b Θ o (v) > 0}, write v = A o (v)T o
- v ⊥ , v ⊥ ∈ H QH o , and define A o (v) := b Θ o (v), R o (v) := q q o (v ⊥ ,v ⊥ ),RR o (v) := R o (v) A o (v) . Assume the following. SL1. Smooth Equilibrium Architecture. The Horizon bundleH QH =ker b Θis the distinguished smooth HRCEL equilibrium structure, and V = RT ⊕ q H QH is the intrinsic smooth advance-response decomposition. SL2. Unique Pointwise Balance. At everyo∈ U QII , derived pointwise RTP supplies a unique c o ∈ (0,∞) such that Prop o (v) = 1 ⇐⇒ A o (v) > 0 and RR o (v) < c o . Set σ o := c 2 o . The unique normalized saturation frontier is F o :={h∈ H QH o : q o (h,h) = σ o }. Equivalently, the positive saturation class is Σ c o :={v ∈ C
o : R o (v) = c o A o (v)} ={v ∈ C + o : RR o (v) = c o }. No competing capacity, saturation branch, or alternative critical balance class exists in the fiber. SL3. Unique Interface Realization. The saturation class is the unique HRCEL interface between the sub-capacity and super-capacity propagation regimes: Σ c o = ∂K + o {0}. Assume that Principle 15.1 applies to the unique HRCEL balance on U QII . Then the pointwise capacities assemble into a unique smooth field c : U QII −→ (0,∞), c(o) = c o . 71
Paper 4 Version 5Canonical Realization and Hybrid Geometry Moreover, with σ := c 2 , the frontiers assemble into a smooth embedded hypersurface F = G o∈U QII F o ⊂ H QH , and π F :F −→ U QII is a smooth surjective submersion. Proof.By Principle 15.1, every point ofU QII has a neighborhoodW i carrying a smooth local continuation c i : W i −→ (0,∞) with c i (o) = c o . On an overlapW i ∩ W j , bothc i (o) andc j (o) are HRCEL capacities at the same point. By SL2, that capacity is unique. Hencec i =c j on every overlap. The local branches therefore glue uniquely to a global smooth field c∈ C ∞ (U QII , (0,∞)), c(o) = c o . Set σ := c 2 and define Ψ : H QH −→ R,Ψ(h) := q π H (h) (h,h)− σ(π H (h)). Then Ψ is smooth and F = Ψ −1 (0). At h∈F o , d v Ψ h [h] = 2q o (h,h) = 2c 2 o
0. Thus 0 is a regular value along F, so F is a smooth embedded hypersurface of H QH . To show thatπ F is a submersion, work in a local bundle trivialization. Givenu ∈ T o U QII , the tangent condition for (u,v) at h∈F o is D o Ψ h [u] + D h Ψ h [v] = 0. BecauseD h Ψ h [h] = 2c 2 o 0, the vertical derivative is surjective ontoR. Hencevcan be chosen for any prescribedu. Thusdπ F is surjective. Surjectivity ofπ F follows because everyF o is nonempty. 18 Consequences Corollary 18.1 (HRCEL Regularity). Every HRCEL object whose only unresolved regularity is carried by c inherits smooth assembly through its dependence on q, b Θ, T, H QH , c. 72
Paper 4 Version 5Canonical Realization and Hybrid Geometry In particular, G = q− (1 + c 2 ) b Θ⊗ b Θ defines a smooth symmetric bilinear-form field onV. Its positive propagation sector and normalized frontier assemble smoothly. The punctured null interface ∂K + \ 0 V is a smooth hypersurface on the selected regular domain, where 0 V denotes the zero section. Corollary 18.2 (Smooth Informational Assembly). OnU QII , the unique pointwise response thresh- olds and corresponding HRCEL propagation structures assemble smoothly. Hence the smooth- threshold and domain-level regularity conclusion of Smooth Informational Assembly follows from Regular Local Continuation + derived pointwise RTP/HRCEL +Cartan-compatible assembly =⇒ Smooth Informational Assembly Remark 18.3 (Final dependency and claim boundary). Regular Local Continuation is the sole local-continuation principle. It supplies smooth local HRCEL capacity tracks but assumes neither their global compatibility nor any globally assembled HRCEL object. Pointwise RTP uniqueness forces overlap agreement. The HRCEL Smoothing Law then derives the global smooth capacity field, and regular level-set geometry yields the assembled HRCEL frontier and downstream smooth propagation structures. No claim is made across rank-changing, branch-colliding, bifurcating, propagation-incoherent, degenerate, or geometric-type-changing loci. 19 Strong Smooth Regular Realizability The HRCEL Smoothing Law suppliesc∈ C ∞ (U QII ,(0,∞)). Consequently the smooth orthogonal projection, propagation form, defect tensor, cone field, punctured null hypersurface, normalized boundary sphere bundle, threshold graph, and canonical bridge representative are obtained by the same constructions already established pointwise in the preceding sections. In particular, G = q− (1 + c 2 ) b Θ⊗ b Θ, D = q− G = (1 + c 2 ) b Θ⊗ b Θ, and K + ={(o,v) : b Θ o (v) > 0, G o (v,v) < 0} assemble smoothly on the selected regular domain. The punctured null set is a smooth hypersurface, the normalized boundary is the smooth sphere congruence supplied by the HRCEL Smoothing Law, and Σ J ={(o,c(o)) : o∈ U QII },Φ c (o,r) = r− c(o) are smooth with∂ r Φ c = 1. Hence Strong Smooth Regular Realizability holds on the selected connected fixed-rank QII-regular domain. Remark 19.1 (Scope). Here, global means global on the selected connected fixed-rank QII-regular domain. No extension across rank-changing, branch-colliding, bifurcating, propagation-incoherent, degenerate, or geometric-type-changing loci is asserted. 73
Paper 4 Version 5Canonical Realization and Hybrid Geometry 20 A Finite-Dimensional Model of the HRCEL Construction The following model displays the completed architecture in concrete coordinates. It is a directly recomputable consistency model, not an empirical or quantum-operational realization. It does not identify the operational base or continuation bundle with physical spacetime, and none of the paper’s proofs depends on it. Let U ⊆ R 4 be a connected open regular domain. Using the canonical tangent trivialization, TU ∼
U × R 4 , take the model QII-admissible continuation bundle to be V := A QII 0 = TU. Thus, for every o∈ U, V o = A QII o,0 = T o U ∼
R 4 , and the required tangent-bundle relationship A QII 0 ⊆ TU is satisfied as equality in this model. Write v = (a,x,y,z)∈ T o U and w = (a ′ ,x ′ ,y ′ ,z ′ )∈ T o U. Define q o (v,w) = aa ′
- xx ′
- yy ′
- zz ′ , b Θ o (a,x,y,z) = a. These fields are independent ofoin the chosen tangent trivialization and therefore vary smoothly over U. The q o -dual is T o = (1, 0, 0, 0), and H QH o = ker b Θ o ={(0,x,y,z)}, V o = RT o ⊕ q o H QH o . For v = (a,x,y,z), A o (v) = a, v ⊥ = (0,x,y,z), R o (v) = q x 2
- y 2
- z 2 , and for a > 0, RR o (v) = p x 2
- y 2
- z 2 a . This ratio is invariant under positive rescaling. Letσ:U →(0,∞) be smooth and setc(o) = p σ(o). The accepted squared-cost interval is J (2) o = [0,σ(o)), and propagation is x 2
- y 2
- z 2 < c(o) 2 a 2 , a > 0. The propagation form is G o (v,w) = xx ′
- yy ′
- zz ′ − c(o) 2 aa ′ , with matrix [G o ] = diag(−c(o) 2 , 1, 1, 1). Thus G o has signature (−, +, +, +) while q o remains positive definite. 74
Paper 4 Version 5Canonical Realization and Hybrid Geometry The future propagation cone and nonzero null boundary are K + o ={(a,x,y,z) : a > 0, x 2
- y 2
- z 2 < c(o) 2 a 2 }, N
o ={(a,x,y,z) : a > 0, x 2
- y 2
- z 2 = c(o) 2 a 2 }. The Quantum Horizon and nonzero null propagation boundary are disjoint because Horizon vectors have a = 0 while vectors in N
o have a > 0. On the normalized slice a = 1, D prop o ={(1,x,y,z) : x 2
- y 2
- z 2 < c(o) 2 }, B o ={(1,x,y,z) : x 2
- y 2
- z 2 = c(o) 2 } ∼ = S 2 . At each boundary point h = (0,x,y,z), d h [q o (h,h)][h] = 2q o (h,h) = 2c(o) 2
0, so the selected boundary is a regular level set. Set λ(o) = 1 + c(o) 2 and define D o := q o − G o = λ(o) b Θ o ⊗ b Θ o . Then D o is positive semidefinite and rank one, with kerD o = ker b Θ o = H QH o . The threshold graph and canonical bridge are Σ J ={(o,c(o)) : o∈ U},Φ c (o,r) = r− c(o), with Φ −1 c (0) = Σ J and ∂ r Φ c = 1. Model Scope For this consistency model, the QII-admissible continuation bundle is chosen to equal the tangent bundle: A QII 0 = TU. This choice satisfies the required tangent-bundle typing but is not asserted to hold generally. It does not assert that Structural QII universally admits every operational tangent, that the continuation bundle generally equals the full tangent bundle, or that the base U is physical spacetime. The smooth positive functioncis selected as model data. The example does not independently realize the Structural Restoration Cost/RMS chain, FPRP, or Regular Local Continuation from explicit operator and measurement data. That separate quantum-operational realization remains a subsequent test of the framework. The choice of rank four is made only for visualization. In rankn,dimH QH o =n−1 and, for 75
Paper 4 Version 5Canonical Realization and Hybrid Geometry n≥2,B o ∼
S n−2 . No dimensional preference, four-dimensionality theorem, general tangent-bundle identification, or physical-spacetime interpretation follows from this example. 21 Discussion 21.1 The Emergent Hybrid Geometry One of the most striking outcomes of the present framework is that it produces two distinct geometric structures rather than a single unified geometry. The first orientation structure arises from the action of UPCP on the QII-admissible continuation space. Through the positive-projective orientation covector, the admissible continuation space is separated into positive, zero, and negative sectors. This produces an orientation geometry whose fundamental organization is order-theoretic. The resulting orientation geometry is organized by the sign structure of the UPCP functional and therefore possesses a half-space-like character. At this stage, the theory distinguishes advance, equilibrium, and retreat, but it does not yet distinguish propagation from non-propagation. The second structure arises only after the introduction of the Response-Threshold Propagation Principle. RTP imposes an additional criterion inside the positive UPCP sector by comparing transverse minimum response with normalized advance. The result is a smaller propagation region contained within the larger orientation geometry. Positive orientation alone is no longer sufficient. Propagation becomes a stronger condition requiring bounded transverse response. This distinction reveals that orientation and propagation are not the same geometric notion. A continuation may be positively oriented while failing to satisfy the propagation criterion. Conversely, every propagating continuation is necessarily positively oriented. Orientation therefore provides a necessary condition for propagation, while RTP supplies the additional structure required to identify which oriented directions genuinely support propagation. The distinction between the Quantum Horizon and the propagation boundary is a direct manifestation of this hybrid architecture. The Horizon belongs to the orientation geometry and marks the equilibrium sector of the UPCP decomposition. The propagation boundary belongs to the RTP geometry and separates propagating from non-propagating positively oriented continuations. Because these structures arise at different stages of refinement, they need not coincide. Indeed, one of the principal conclusions of the present framework is that the Horizon is not the propagation boundary. The Horizon remains fixed while RTP refines the positive sector into distinct classes of positively oriented continuations. The resulting propagation geometry possesses a significantly different character from the underlying orientation geometry. RTP defines a quadratic functional whose associated bilinear form has Lorentzian signature on each QII-admissible fiber. The inherited response metric remains positive definite and continues to provide the normalization, orthogonal decomposition, and transverse geometry. The Lorentz-signature structure is carried by the response-defined quadratic form rather than by the underlying response metric itself. Consequently, two geometries coexist within the framework. The orientation geometry is generated by UPCP and organizes the admissible continuation space into positive, zero, and negative sectors. 76
Paper 4 Version 5Canonical Realization and Hybrid Geometry The propagation geometry is defined through RTP from the inherited response metric and the UPCP orientation structure. The former is fundamentally order-theoretic. The latter admits a Lorentzian quadratic representation. This coexistence motivates the description hybrid geometry. The term does not indicate the simulta- neous presence of two unrelated mathematical structures. Rather, it reflects a staged refinement process in which a propagation geometry emerges inside a previously established orientation geometry. The propagation geometry does not replace the orientation geometry. It refines it. In this sense, the framework suggests a layered geometric architecture. Orientation appears first through UPCP. The Quantum Horizon emerges as the equilibrium separating structure associated with that orientation geometry. RTP then introduces a quantitative propagation criterion relative to the Horizon decomposition. The resulting propagation cones inherit the orientation structure while exhibiting additional Lorentzian-type organization not present at the earlier stage. The present paper does not establish that this hybrid architecture extends to a Lorentzian manifold, a spacetime manifold, or a physical spacetime geometry. Nor does it determine the effective dimension of the resulting structure. The significance of the result is narrower. The framework demonstrates that an order-theoretic orientation geometry and a Lorentz-signature propagation geometry can arise as distinct but related stages within a common informational construction. Within the present framework, the path to Lorentzian-type propagation structure does not begin with a Lorentzian metric. It begins with informational admissibility, canonical realization, ordering, and refinement. The Lorentz-signature propagation form appears, downstream of derived pointwise RTP, only after those earlier structures have been established. 21.2 Architectural and scope observations The architecture assigns distinct roles to its principal components. QII supplies a gauge-invariant admissible continuation space but no preference. LAS supplies a unique no-surplus realization of a fixed admissible continuation but no choice among continuations. UPCP supplies a local order and orientation but neither conservation nor variational realization. The Local Quantum Horizon is the equilibrium sector that becomes meaningful only when these three structures are imposed together. The positive and negative sectors do not require a preferred ray. The resulting response-threshold form has Lorentzian signature on each QII-admissible fiber, but the present work stops short of constructing a Lorentzian manifold. It provides no tangent-bundle identification, integrability theorem, geodesic or curvature dynamics, global causal order, four-dimensional reduction, or identification with physical spacetime. This stopping point is intentional. It isolates a rigorous local operational geometry without presupposing the stronger global structures that later work may investigate. 21.3 Lessons Learned from the Refinement Program The original objective of this program was not to construct a specific spacetime geometry. It was to determine how much geometric structure could be uncovered before stronger organizing principles became necessary. The development has produced the following lessons. 77
Paper 4 Version 5Canonical Realization and Hybrid Geometry Lesson 1: The Quantum Horizon is not the propagation boundary. The Quantum Horizon arises from the UPCP orientation structure as the equilibrium sector of the admissible continuation space. RTP introduces a distinct propagation criterion within the positive UPCP sector. The Horizon forms the equilibrium sector separating the positive and negative UPCP sectors, whereas the propagation boundary separates propagating and non-propagating positively oriented continuations. Lesson 2: Orientation is weaker than propagation. UPCP identifies positive, zero, and negative sectors, but positive orientation alone does not guarantee propagation. A continuation may be positively oriented while exceeding the response threshold. Positive orientation is necessary, but not sufficient, for propagation. Lesson 3: Propagation structure emerges as selection. QII identifies admissible contin- uations, UPCP orders them, and RTP selects the subclass satisfying the response criterion. The propagation cone is therefore obtained through refinement within a larger space of admissible orientations rather than introduced as a primitive object. Whether a complete causal structure ultimately emerges remains a question for later work. Lesson 4: RTP provides a quantitative realization of refinement. The qualitative frame- work distinguishes admissibility, orientation, and propagation but does not specify a quantitative selection rule. RTP supplies a candidate quantitative criterion implementing that refinement and determines which positively oriented continuations lie inside the propagation cone. Lesson 5: The resulting geometry is hybrid. UPCP supplies a half-space-like orientation geometry. RTP defines a propagation geometry whose associated quadratic form has Lorentzian signature on each QII-admissible fiber. These structures are distinct but related: the propagation geometry refines and depends upon the orientation geometry. Lesson 6: Stronger principles become more intelligible after the relevant structures emerge. The staged construction makes the contribution of each principle visible. QII supplies admissibility, LAS supplies canonical realization, UPCP supplies orientation, and RTP supplies propagation selection. The geometry is not assumption-free; it is built through progressively stronger refinements whose roles remain distinguishable. Lesson 7: The response metric is not the Lorentzian geometry. The response metric inherited from Paper 3 remains positive definite. It normalizes the UPCP advance direction, supplies the orthogonal advance–Horizon decomposition, and measures transverse response. Lorentz signature appears only through the distinct response-threshold quadratic form. Thus the framework does not assume a Lorentzian response metric; it derives a Lorentz-signature propagation form downstream of derived pointwise RTP. Taken together, these lessons provide a consistent picture of geometric emergence while preserving the claim boundary of the present paper. Whether this architecture contributes to a reconstruction of spacetime remains open. 78
Paper 4 Version 5Canonical Realization and Hybrid Geometry 22 Scope, Limitations, and Open Questions The completed derivational chain establishes PC4, pointwise RTP, and Smooth Informational Assembly from their now-exposed structural and physical inputs. PC4 follows from Structural Restoration Cost, RMS, ordered-vector-space representation, and smooth local integration. Pointwise RTP follows from qualitative refinement structure and the QSL–FPRP resource architecture. Smooth HRCEL assembly follows from Regular Local Continuation, pointwise RTP uniqueness, and Cartan- compatible assembly. These conclusions remain restricted to the selected connected fixed-rank QII-regular domain. The construction does not establish that the QII-admissible bundle equals the full tangent bundle of the operational domain, that it is integrable as the tangent bundle of effective leaves, or that its effective dimension is four. It does not establish physical spacetime, geodesic or curvature dynamics, gravitational field equations, reachability and growth laws, or global causal reconstruction. No extension through rank-changing, branch-colliding, bifurcating, propagation-incoherent, degenerate, or geometric-type-changing loci is asserted. 23 Conclusion Paper 4 Version 5 strengthens the informational-to-geometric architecture by deriving three com- ponents that Version 4.6 treated as bundled, adopted, or conditionally regularized. First, PC4 is derived through Structural Restoration Cost, Restoration-Monotone Survival, survival-set com- parability, linear survival-order representation, and smooth local integration. Second, pointwise Response–Threshold Propagation is derived through the QSL–FPRP architecture, with FPRP isolated as the substantive physical finiteness input. Third, Regular Local Continuation supplies smooth local HRCEL capacity tracks; pointwise uniqueness glues those tracks globally; and the HRCEL Smoothing Law together with Cartan-compatible assembly yields Smooth Informational Assembly. The full chain is therefore QII/LAS =⇒ derived PC4 and UPCP orientation =⇒ Quantum Horizon/HES =⇒ derived pointwise RTP =⇒ RCSP/BET/HRCEL =⇒ RLC and HRCEL smoothing =⇒ SIA and SSR. On the selected regular domain, the capacity field, Lorentz-signature propagation form, defect tensor, propagation cone, punctured null boundary, normalized boundary congruence, threshold graph, and canonical bridge representative all assemble smoothly. The resulting geometry remains a smooth Lorentz-signature propagation geometry on the QII- admissible bundle, not a completed reconstruction of physical spacetime. Tangent-bundle identifica- tion, effective four-dimensionality, global causal structure, spacetime interpretation, and gravitational dynamics remain beyond the present claims. 79
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