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: 3.5/5Develops a staged derivation of propagation geometry from regular quantum-operational realization structures, using informational identity, canonical minimum-response realization, and provenance-coherent ordering to obtain a quantum horizon, response-bounded propagation thresholds, and a Lorentz-signature propagation form. The construction assembles these structures smoothly while remaining a pre-spacetime geometry and making no claim of physical spacetime identification or empirical validation.
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3.5 out of 5. Audited on TheoryOfEverything.ai. Permanent record: https://theoryofeverything.ai/review-profile/paper/b6fbb6d7-86cf-4377-b8b2-03ec8d3c527b
AI Review Rating
Composite of the review dimensions below, on a 0–5 scale.
This paper presents a staged, conditional construction in finite-dimensional bundle geometry that derives a Lorentz-signature propagation form from quantum-operational admissibility, minimal-realization, and provenance-coherence principles, while explicitly and repeatedly disclaiming any identification with physical spacetime or empirical validation. The panel was split on whether to classify this as a first_principles_derivation (evaluating whether each transition is forced from a minimal premise) or pure_mathematics (evaluating independent checkability of theorem-level claims). This split is not a defect of the paper — it reflects a genuine ambiguity in the work's own framing: it is presented as a 'staged derivation' suggesting necessity, but its own Logical Status table candidly labels QII, LAS, UPCP, the generalized speed-limit datum, and FPRP as independent primitive assumptions rather than forced consequences. Because the structural-warrant trace did not pass its independent fidelity audit, the panel ultimately scored the work as pure_mathematics, converting falsifiability to verifiability (independent checkability of theorem-level claims) rather than necessity. Readers should understand the mid-range falsifiability/verifiability score (3/5) reflects this conversion, not a judgment that the paper lacks testable physical predictions — such predictions are explicitly out of scope by the authors' own design.
Four independent math specialists converged on high confidence that the core algebra is correct: the orthogonal splitting v = A_o(v)T_o + v⊥, the Lorentzian signature of Q_o and G_o, the convexity/closure/saliency of the propagation cone, the regularity of the null boundary, and the rank-one defect identity D_o = (1+c_o²)Θ̂⊗Θ̂ with its recovery/uniqueness consequences (Theorem 10.9) were all independently verified. However, specialists also converged on a cluster of compressed or under-specified steps that readers should be aware of: Theorem 5.8's proof of the one-dimensionality of the survival-order covector ('connectedness of the unit sphere forces the nontrivial quotient to be one-dimensional') is reconstructible but stated in four compressed sentences despite being load-bearing for the entire downstream Horizon/UPCP architecture; Theorem 5.9 rests on three undischarged hypotheses (smooth cooriented extension, local integrability, Survival-Leaf Fidelity) never jointly verified anywhere in the paper; Theorem 8.7's construction of the classifier F_o from the qualitative propagation predicate is asserted rather than explicitly constructed; and Theorem 17.1 (HRCEL Smoothing Law) omits the rank(V)≥2 hypothesis needed for its nonempty-fiber and surjectivity claims — one specialist showed concretely that for rank 1, Ψ(0) = −c(o)² < 0, so F_o is empty and π_F cannot be surjective as unconditionally stated. A related statement/proof mismatch appears in Theorem 8.10, which asserts K_o^+ is unconditionally full-dimensional with ellipsoidal boundary sections even though its own proof concedes the normalized boundary is empty when dim V_o = 1. Definition 5.12 also introduces an ambient kernel Σ_o := ker Θ_o ⊆ T_oU for a covector Θ_o defined only on V_o, without specifying the required extension — a typing gap that does not infect the paper's core fiberwise (V_o) results but should be repaired. None of these issues were judged to invalidate the central rank-≥2 fiberwise construction; they are flagged as repairable precision gaps rather than refutations.
On completeness, three sources specialists independently and consistently noted that the paper is exceptionally disciplined about its own logical status (claim-boundary remarks after nearly every theorem, an explicit dependency map and status table), but that several 'primitive' inputs — the structural restoration-cost functional, RMS, the generalized speed-limit resource datum, Continuation-Speed Correspondence, and FPRP — are postulated rather than instantiated in any concrete quantum-operational example; the paper's own worked model in Section 20 explicitly disclaims realizing this chain. This is a self-disclosed scope limitation rather than a hidden gap, and is treated accordingly. The evidence_strength score of 0/5 is a direct, mechanical consequence of this being a paper submission (not a framework) rather than any failure to supply data; papers of this conditional-mathematics type are not evaluated for empirical evidence roadmaps.
On novelty and clarity, the staged separation between the UPCP-derived Quantum Horizon (informational equilibrium) and the RTP-derived null propagation boundary (response-capacity saturation) was repeatedly identified as a genuinely distinctive conceptual move, argued via multiple independent uniqueness theorems. Clarity was capped at 3/5 by the science specialists partly due to a real notation hazard (G_o reused for both the Section 9 classifier and the Lorentz-signature propagation bilinear form) and a very dense, acronym-heavy presentation (QII, LAS, UPCP, PC4, RMS, HES, RTP, RCSP, BET, HRCEL, RLC, SIA, SSR) with several near-identical restated trichotomies across Sections 7, 10, 11, 12, and 13.
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 stage carries an explicit claim-boundary remark (Remarks 3.3, 4.5, 5.11, 5.14, 8.3, 8.8, 8.13–8.16, 9.7, 10.2, 10.6, 10.8, 10.10, 10.13, 11.4, 12.3, 13.3, 19.1), and the dependency map in the Introduction matches the actual order of use. The two potentially confusable trichotomies (UPCP orientation sign vs. RTP capacity sign) are explicitly separated in the 'Clarification on the Successive Trichotomies' box, in HES7, in BET1/BET5, and in HRCEL5, and I found no place where they are conflated. The positive-definiteness of q and the Lorentzian signature of G are kept disjoint consistently. Rank-one and rank-≥2 edge cases are handled explicitly in BET5, HRCEL5, Prop. 9.4, and Theorem 10.12(2). Two minor local issues prevent a 5. (i) Theorem 8.10's statement asserts unconditionally that 'every normalized boundary section ... is a strongly convex q_o-ellipsoid' and that K_o^+ is a full-dimensional proper cone, while its own proof concedes that for dim V_o = 1 the normalized boundary section is empty; the dim ≥ 2 hypothesis is present in Theorem 8.7 but is not carried into the statement of Theorem 8.10. (ii) Section 8.1/8.4 duplicate the heading 'Response-Threshold Propagation Principle' for what is both an assumption label and a derived theorem, and Remark 8.13 contains a broken cross-reference ('Theorem efthm:qsl-fprp-rtp'). Neither affects any downstream conclusion.
The algebra I could check is correct. Verified independently: q(v,v) = A_o(v)² + q(v⊥,v⊥) under the normalization q^{-1}(Θ̂,Θ̂)=1 with T_o = Θ̂^♯ (Prop. 8.5); Q_o(aT_o+h) = q(h,h) − c²a², giving signature (−,+,…,+) on the orthogonal splitting; convexity of K_o^+ via linearity of the transverse projection plus the triangle inequality; the closure characterization {Θ̂ ≥ 0, Q ≤ 0} and the saliency argument; dQ_o|v(T_o) = −2c²Θ̂(v) forcing Θ̂(v)=0 and then v=0, so the nonzero null set is a regular hypersurface; D_o = q_o − G_o = (1+c²)Θ̂⊗Θ̂ (since q = Θ̂⊗Θ̂ + q|{H⊥} and G = q|_{H⊥} − c²Θ̂⊗Θ̂), with ker D_o = ker Θ̂ and im D_o^♭ = RΘ̂; the uniqueness computation 1 = λ²q^{-1}(Θ̂,Θ̂) ⇒ λ = ±1 and 1+c̃² = 1+c² ⇒ c̃ = c; G = q − (1+c²)Θ̂⊗Θ̂ in Section 19; and the Theorem 2.1 measurement-sector computation, including the ½-with-full-double-sum bookkeeping (diagonal term |X_aa|² = |D_aa|²/(4μ_a) correctly matches the a=b term of ½Σ|D_ab|²/(μ_a+μ_b)) and the 1/d factor from eigenvalues μ/d and tangent entries Ḋ/d. The QSL–FPRP bound sqrt(1+RR²) ≤ B ⇒ RR ≤ sqrt(B²−1) is correct given Assumptions 7–10. Deductions: several supporting proofs are compressed. Theorem 5.8's proof ('connectedness of the unit sphere forces the nontrivial quotient to be one-dimensional') is a one-line sketch; I was able to reconstruct it (pointedness gives C∩S and (−C)∩S disjoint closed and covering S, so S is disconnected unless dim = 1), but as written it is not reproducible without that reconstruction, and the covector Θ_o it produces is load-bearing for everything downstream. Theorem 5.9 leans on three undischarged hypotheses (smooth cooriented extension, local integrability, Survival-Leaf Fidelity) that are declared but never shown to be jointly satisfiable. Theorem 17.1's submersion claim ('the nonzero vertical derivative permits solving the tangent equation for any prescribed base tangent') is asserted rather than computed. Finally, Section 19 states SSR as a conclusion in running prose rather than as a proved theorem, relying on 'the same constructions already established pointwise'.
Necessity rubric applied (first_principles_derivation). The paper is unusually explicit about which structures are 'foundational principles/assumptions' versus 'derived results' (see the Logical Status table in the Introduction), and several of the derived steps include genuine uniqueness/forcing arguments that rule out alternatives — e.g., Theorem 5.8's derivation of a unique covector ray from total-order axioms, the Horizon Uniqueness Lemma (10.5), and Theorem 10.9's proof that (q,G) determine the Horizon and capacity uniquely up to sign. These are load-bearing transitions that do exclude alternative representations. However, the overall chain is not a chain of forced necessity from a single minimal premise: QII, LAS, UPCP, and the RTP/QSL-FPRP resource architecture are each introduced as independent primitive principles or physical data (Formal Surplus Representation, the admissible perturbation class, the structural restoration-cost functional, the generalized speed-limit datum, FPRP) whose necessity relative to the prior stage is not argued — only their consequences once assumed. The paper's own 'Does not establish' columns candidly acknowledge this at nearly every stage. The honesty about assumption-vs-derivation is a strength, but it means the headline framing ('staged derivation... each stage adds only the structure needed for the next') overstates the forcing character of several major transitions (e.g., Section 5.2's explicit disclaimer that 'the theory does not assert that every operational model canonically admits data satisfying these hypotheses' directly undercuts a necessity reading of PC4). This mix of strong local uniqueness proofs and multiple unargued foundational injections supports a middle score.
The paper is internally well organized, with a logical dependency map, a glossary, and a 'Logical Status of the Main Structures' table that substantially aid navigation, and symbols are used consistently (no redefinition flags were found). However, clarity is significantly taxed by an extremely dense acronym and symbol apparatus (QII, LAS, UPCP, PC4, RMS, HES, RTP, RCSP, BET, HRCEL, RLC, SIA, SSR, FSR1-4, CAN1-6, etc.), long chains of nested assumptions stated as numbered hypotheses (e.g., Theorem 8.7 lists seven numbered physical/structural premises), and repeated near-identical restatements of the same trichotomy (UPCP sign sectors vs. RTP propagation sectors vs. response-capacity regimes) across multiple sections (7, 10, 11, 12, 13) that could have been consolidated. A graduate-trained reader could follow the argument with considerable effort, but the sheer volume of interlocking named objects impedes rather than aids comprehension in several places.
The staged synthesis is distinctive: it connects an admissible-continuation kernel, a positive-projective ordering covector, a response-normalized splitting, and a finite threshold into a Lorentz-signature cone construction, while carefully separating the informational horizon from the null propagation boundary. The explicit rank-one defect/recovery viewpoint and smooth-assembly packaging are potentially useful framework contributions. Nevertheless, the core geometric endpoint—a Lorentzian-signature form obtained from a positive metric plus a distinguished covector and scalar threshold—is acknowledged to be standard, and the manuscript does not compare its bespoke QII/UPCP/RTP structures closely with the nearest existing work in ordered vector spaces, cone geometry, information geometry, or operational quantum frameworks. The extent to which the named principles yield a mechanism beyond a tailored axiomatic synthesis is therefore not yet sharply established.
The paper is highly structured and unusually explicit about its logical dependencies, claim boundaries, and scope. All major variables and structures are defined or explicitly inherited from Paper 3. The central derivational chains (PC4, pointwise RTP, HRCEL smoothing) are each given with stated premises and proofs. Boundary conditions and edge cases are addressed: rank-one vs. rank-at-least-two fibers, the trivial Horizon when dim=1, the distinction between pointwise and domain-level smoothness, and the non-identification of the Quantum Horizon with the propagation boundary. Limitations are extensively documented in Section 22 and throughout via 'claim boundary' remarks. The worked downstream model is clearly labeled as a consistency model that does not instantiate the full upstream architecture. Minor gaps: the paper relies on several primitive inputs (restoration-cost functional, generalized speed-limit resource datum, Continuation–Speed Correspondence, FPRP) that are declared as independently supplied rather than constructed, and the paper does not provide a concrete quantum-operational realization of these data. This is an explicit scope choice rather than an omission, but it means the full chain from quantum operations to geometry is not closed within this paper. The reference verification report shows 13 of 16 citations unverified, but these are standard textbooks and well-known papers, and the report does not label any as fabricated. The unverified status is noted but does not indicate a completeness failure. Overall, the core argument is fully developed within its stated axiom set, with only secondary gaps in the realization of primitive inputs.
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.
- mediumTheorem 17.1 (HRCEL Smoothing Law) - The asserted nonempty frontier fibers and surjectivity of π_F require a nonzero Horizon fiber, hence rank(V)≥2. That condition is not among this theorem’s stated hypotheses.
If wrong: Without the omitted rank condition, the claimed smooth nonempty normalized boundary/frontier assembly and surjective-submersion conclusion fail on allowed rank-one data. Downstream SSR claims relying on a sphere congruence need the same restriction.
- mediumTheorem 5.8 (Linear Survival-Order Representation), proof - The existence and uniqueness of the order-representing covector Θ_o is proved in four compressed sentences. The decisive step — that the quotient of the closed convex total cone by its lineality space is one-dimensional — is asserted via 'connectedness of the unit sphere' without the explicit argument. Reconstructible but not shown.
If wrong: If the quotient were higher-dimensional, no single faithful linear representative Θ_o would exist, PC4(b)–(d) would fail, the Quantum Horizon H^QH = ker Θ̂ would not be a well-defined codimension-one object, and HES, RTP, RCSP, BET, and HRCEL would all lose their defining covector. Load-bearing for the entire construction.
- mediumTheorem 5.9 (Smooth Local Integration of the Survival-Order Covector) - The theorem is conditional on three hypotheses — smooth cooriented extension of ker Θ_o, local integrability, and 'independently defined Survival-Leaf Fidelity' — none of which is constructed or shown to be jointly satisfiable anywhere in the paper. The worked model of Section 20 explicitly declines to instantiate this chain.
If wrong: PC4(c) (local smooth scalar representation) and PC4(d) (positive reparameterization freedom) would be unsupported. Theorem 5.13(6) and Prop. 6.4's representative-independence cite PC4(d) directly; the smooth positive-projective field [Θ] assumed in Section 7 and Theorem 8.2 would lack its claimed derivation.
- mediumTheorem 8.7 (QSL–FPRP Derivation of Pointwise RTP), step 'Projective invariance and two-data closure therefore produce a unique classifier F_o: [0,∞)→{0,1} and a compatible-ratio set J_o = F^{-1}_o(1) that contains zero, is downward closed, and is relatively open.' - The derivation of the classifier F_o from the qualitative propagation predicate and the stated assumptions (positive orientation, positive-projective invariance, two-data quantitative closure, zero-refinement compatibility, downward refinement closure, relative openness) is compressed. The paper does not show the explicit construction of F_o from these assumptions.
If wrong: If the classifier F_o cannot be constructed from the stated qualitative assumptions, the derivation of the unique threshold c_o would fail, and the central RTP result would be unsupported. However, the step is plausible and the assumptions are explicitly stated.
- lowDefinition 5.12 - The ambient separatrix requires an extension of Θ_o from V_o to T_oU; none is specified in this definition.
If wrong: The claimed intrinsic ambient separatrix is not defined. The admissible Horizon ker(Θ_o|V_o) and the subsequent fiberwise propagation calculations remain available.
- lowLemma 16.2 - The optional assembly proof depends on CAN1–CAN6, but their exact statements are not supplied, preventing verification of the implicit-function and pullback steps against the required hypotheses.
If wrong: The optional Cartan-compatible assembly conclusion would be unsupported; the paper's direct HRCEL Smoothing Law would be unaffected.
- lowSection 19 (Strong Smooth Regular Realizability) - SSR is stated as a conclusion in running prose ('Hence Strong Smooth Regular Realizability holds...') rather than as a formally stated and proved theorem; it relies on the assertion that all objects follow 'by the same constructions already established pointwise'.
If wrong: The paper's final assembly claim would require restatement; the pointwise results (Theorems 8.10, 10.9, 12.1, 13.1) and the smoothness of c (Theorem 17.1) would be unaffected.
- lowTheorem 10.1 proof, following equation (2) - The proof uses the strict-cone symbol K_o^+ for the weak-inequality closure of that cone. This is a local definition/notation mismatch, not a failure of the stated exclusion result.
If wrong: The local inference remains valid, but retaining the incorrect equality could confuse later uses of strict versus closed propagation cones and boundary membership.
- lowTheorem 17.1 (HRCEL Smoothing Law), proof - The claim that π_F : F → U_QII is a submersion is asserted in one sentence ('The nonzero vertical derivative permits solving the tangent equation for any prescribed base tangent') rather than computed via the implicit function theorem on the bundle.
If wrong: If π_F were not a submersion, the smooth frontier/boundary congruence claimed in Corollary 18.1 and Section 19 (normalized boundary sphere bundle) would not assemble, though the smoothness of c itself — the main content of the law — would be unaffected.
- lowTheorem 2.1 (Composite Bures Response Representation), constraint equation - The displayed constraint for the measurement sector appears to omit complex conjugation on the transposed entries of X_i. This is consistent with PDF extraction loss of overlines; the subsequent least-norm solution and the final summation formula are internally consistent, so this is flagged as source-fidelity inconclusive rather than as an author error.
If wrong: If the omission were substantive rather than typographical, the real-case formula would still hold but the complex-case Bures coefficient would need restating. Theorem 2.1 is an explicit side result (Remark 2.2 limits its scope) and is not used in any later derivation, so no downstream theorem depends on it.
- lowTheorem 5.8 (Linear Survival-Order Representation) - The proof asserts that the induced cone is closed, pointed, and total, and that connectedness of the unit sphere forces the nontrivial quotient to be one-dimensional. The step from 'closed, pointed, total cone' to 'one-dimensional quotient' is compressed; the paper does not spell out the topological argument that a closed pointed total cone in a finite-dimensional real vector space with connected unit sphere must be one-dimensional.
If wrong: If the one-dimensionality claim failed, the UPCP covector would not be unique up to positive scale, and the entire orientation structure (Horizon, sectors) would be underdetermined. However, the claim is standard and the gap is minor.
- lowTheorem 8.10 (Theorem B), statement vs. proof - The theorem statement asserts without rank qualification that K_o^+ is a nonempty, full-dimensional proper cone with strongly convex ellipsoidal normalized boundary sections, but the proof concedes that for dim V_o = 1 the normalized boundary section is empty. The dim ≥ 2 hypothesis appears in Theorem 8.7 but is not propagated into Theorem 8.10's hypotheses.
If wrong: Only a scope/statement correction is needed; adding 'dim V_o ≥ 2' to the hypotheses repairs it. No downstream result is affected, since all downstream domains assume rank k ≥ 2.
These flags compare the submission's headline wording with what its body establishes. They are not mathematical-error findings.
Claimed: The construction 'obtains a quantum horizon, response-bounded propagation thresholds, and a Lorentz-signature propagation form' as derived results.
Established: The body derives these only under explicitly stated foundational principles (QII, LAS, UPCP, QSL–FPRP, RLC) and independently supplied structural data; the abstract's phrasing can be read as unconditional derivation.
The abstract does not foreground the conditional nature of the derivation, while the body repeatedly emphasizes that the results are conditional on supplied structural data and physical premises.
Claimed: The construction is staged... Each stage answers a distinct question and adds only the structure needed for the next stage, and the paper is presented throughout as deriving the propagation geometry from inherited structure.
Established: Several stages (LAS's minimality preorder, UPCP's perturbation class and restoration-cost functional, RTP's generalized speed-limit datum and Finite Process Resource Principle) are introduced as independent primitive assumptions/principles whose necessity given the prior stage is not argued — only their logical consequences once granted are derived.
The framing implies a forced, minimal-premise derivation chain, but multiple load-bearing primitives are freely stipulated rather than shown to be the unique or necessary addition at that stage; the paper itself acknowledges this via repeated 'does not establish' and 'independently supplied' disclaimers.
Claimed: 'Three principal derivational chains organize the construction. First, PC4 follows from Structural Restoration Cost, Restoration-Monotone Survival, survival-set comparability, linear order representation, and smooth local integration.'
Established: PC4 follows conditionally on primitive structural data (an admissible perturbation class, a redescription-invariant restoration-cost functional, a cooriented integrable extension, and Survival-Leaf Fidelity) whose existence is never established for any model, including the paper's own worked example.
The word 'follows' in the abstract reads as unconditional derivation, whereas Remark 5.11 states the theorem 'is conditional on the independently supplied structural, order, and integration data' and Section 5.2 states 'The theory does not assert that every operational model canonically admits data satisfying these hypotheses.' The body is scrupulous; the abstract phrasing is looser.
Claimed: Section 19 and the Conclusion state that 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'.
Established: Smooth assembly is proved only under the Regular Local Continuation Principle (Principle 15.1), which is itself a declared foundational regularity assumption, not a derived result, and is not verified in the worked model of Section 20.
The Conclusion's chain does list 'RLC and HRCEL smoothing' as a step, so the dependency is disclosed; but the flat assertion that everything 'assembles smoothly' in Section 19 does not repeat the conditionality on RLC at the point of statement. This is a presentation gap, not a mathematical overreach.
Claimed: The full chain 'QII/LAS =⇒ derived PC4 and UPCP orientation =⇒ ... =⇒ SIA and SSR' is presented as established.
Established: The body establishes each link only under the stated premises; the conclusion's arrow notation can be read as logical implication rather than conditional derivation.
The conclusion compresses the conditional nature of the derivation into an arrow chain without restating the premises.
Strengths
- Exceptionally disciplined logical bookkeeping: nearly every theorem carries an explicit 'claim boundary' remark, and a summary table classifies each component as principle, assumption, or derived result with an explicit 'does not establish' column.
- The core Lorentz-signature construction is algebraically sound and independently verified by multiple specialists: the orthogonal splitting, the signature of Q_o/G_o, convexity/closure/saliency of the propagation cone, regularity of the null boundary, and the rank-one defect recovery theorem (Theorem 10.9) all check out under stated premises.
- A conceptually sharp and consistently maintained distinction between the UPCP informational-equilibrium separatrix (Quantum Horizon) and the RTP-derived null propagation boundary, reinforced by independent uniqueness arguments (Horizon Uniqueness Lemma 10.5, Theorem 10.9).
- Theorem 8.7 (QSL-FPRP) cleanly isolates the substantive physical finiteness premise (FPRP) rather than claiming finite propagation capacity follows from informational geometry alone.
- A worked finite-dimensional downstream model (Section 20) makes the abstract construction concrete and recomputable, while explicitly and honestly disclaiming that it instantiates the upstream restoration-cost/FPRP chain.
Areas for Improvement
- Theorem 5.8's proof of the covector's one-dimensionality ('connectedness of the unit sphere forces the nontrivial quotient to be one-dimensional') should be expanded; it is load-bearing for the entire Horizon/UPCP architecture but is currently a four-sentence sketch.
- Theorem 5.9 depends on three hypotheses (smooth cooriented extension of ker Θ_o, local integrability, Survival-Leaf Fidelity) that are never shown to be jointly satisfiable in any example; the paper's own worked model explicitly declines to instantiate this chain.
- Theorem 17.1 (HRCEL Smoothing Law) should add the rank(V)≥2 / positive-dimensional-Horizon hypothesis explicitly; as stated, its claimed nonempty frontier fibers and surjective π_F fail for rank-1 data (Ψ(0) = −c(o)² < 0).
- Theorem 8.10's statement should carry the dim V_o ≥ 2 qualification from Theorem 8.7 into its own hypotheses, since its proof concedes the normalized boundary section is empty in rank 1.
- Definition 5.12's ambient kernel Σ_o := ker Θ_o ⊆ T_oU requires a specified extension of Θ_o from V_o to T_oU, or should be restated entirely within V_o.
- Resolve the notation collision between the Section 9 classifier G_o and the Lorentz-signature propagation bilinear form G_o; more broadly, the dense acronym/symbol apparatus (QII, LAS, UPCP, PC4, RMS, HES, RTP, RCSP, BET, HRCEL, RLC, SIA, SSR) and repeated near-identical trichotomy restatements (Sections 7, 10-13) would benefit from consolidation.
- The Optional Cartan-Compatible Assembly Lemma (16.2) invokes CAN1-CAN6 without stating their precise hypotheses, preventing independent verification of its implicit-function-theorem step; since it is explicitly unused in Theorem 17.1, consider removing it or fully specifying the conditions.
- A concrete worked instantiation of the primitive physical/structural inputs (restoration-cost functional, speed-limit datum, Continuation-Speed Correspondence, FPRP) in an actual quantum-operational model would substantially strengthen confidence that the full chain is non-vacuous.
- Consider a direct comparative discussion against neighboring ordered-vector-space, cone-geometry, or information-geometric constructions to sharpen the novelty claim beyond broad citation.
Key Equations (3)
Canonical orthogonal decomposition into normalized UPCP advance and Horizon-transverse response.
Response-threshold quadratic form whose negative sector defines propagation and whose zero set gives the null boundary.
Future propagation cone defined by positive UPCP orientation and sub-threshold transverse response.
Other Equations (3)
Defines admissible infinitesimal continuations as the kernel of the reduced Structural QII defect.
Polarized propagation form, shown to have Lorentzian signature on each admissible fiber.
Rank-one defect tensor whose kernel recovers the Quantum Horizon.
Tags & Keywords
Keywords: quantum-operational geometry, quantum informational identity, Bures response metric, provenance-coherence ordering, response-threshold propagation, Lorentz-signature geometry, emergent propagation cones, smooth bundle geometry
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Conservation, Derived Ordering, Equilibrium, and Smooth Hybrid Propagation Geometry in Quantum-Operational Geometry Paper 4 Version 5.1 Michael Tolbert October 2026 Contents 1 Introduction4 2 Inherited regular realization geometry9 2.1 Composite Bures response representation . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Structural Quantum Informational Identity11 4 Least Additional Structure15 5 Unified Provenance–Coherence Principle16 5.1 Derived PC4: restoration cost, survival order, and differential representation . . . . . 17 5.2 Admissible Structural Data and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 Standing survival architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.4 Pre-survival structural severity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6 The Local Quantum Horizon20 6.1 Nontriviality qualification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 Global scope disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 The Quantum Horizon as an Organizing Structure . . . . . . . . . . . . . . . . . . . 21 7 Horizon Equilibrium Structure23 8 Theorem A: Cone Promotion27 1
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 8.1 Response-Threshold Propagation Principle . . . . . . . . . . . . . . . . . . . . . . . . 29 8.2 Inherited Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.3 Response-Normalized UPCP Advance and Canonical Splitting . . . . . . . . . . . . . 30 8.4 Response-Threshold Propagation Principle . . . . . . . . . . . . . . . . . . . . . . . . 30 8.5 Theorem B: Response-Defined Propagation Geometry . . . . . . . . . . . . . . . . . 32 8.6 Logical Status and Claim Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.7 Adopted Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9 Refinement-Cost Characterization of Pointwise RTP36 10 Horizon-Organized Hybrid Geometry and Null Boundary Realization37 10.1 Separatrix-to-null-barrier transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 10.2 Horizon uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 10.3 Horizon propagation-failure structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 10.4 Horizon essentiality and unique recovery . . . . . . . . . . . . . . . . . . . . . . . . . 44 10.5 Null Boundary Realization and threshold–radius identification . . . . . . . . . . . . . 47 10.6 Status after pointwise null-boundary realization . . . . . . . . . . . . . . . . . . . . . 48 11 Response-Capacity Saturation Law48 12 Boundary Emergence Theorem51 13 Horizon Response-Capacity Equilibrium Law54 14 Regular Local Continuation, HRCEL Smoothing, and Optional Cartan Assembly57 15 Regular Local Continuation57 16 Optional Cartan-Compatible Assembly57 17 HRCEL Smoothing Law58 18 Consequences58 19 Strong Smooth Regular Realizability59 20 Worked Downstream Geometric Model59 2
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 21 Discussion61 21.1 The Emergent Hybrid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 21.2 Relation to Neighboring Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 21.3 Architectural and scope observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 21.4 Lessons Learned from the Refinement Program . . . . . . . . . . . . . . . . . . . . . 63 22 Scope, Limitations, and Open Questions64 23 Conclusion65 3
Paper 4 Version 5.1Canonical 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. Three principal derivational chains organize the construction. First, PC4 follows from Structural Restoration Cost, Restoration-Monotone Survival, survival-set comparability, linear order representation, and smooth local integration. Second, pointwise Response–Threshold Propagation follows from qualitative refinement structure together with an independently specified generalized speed-limit resource datum, Continuation–Speed Correspondence, physical projective realizability, and the Finite Process Resource Principle. Third, Smooth Informational Assembly follows from Regular Local Continuation, pointwise HRCEL uniqueness, gluing, and regular- value/submersion geometry 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 [15, 4, 3, 16, 8, 14, 11, 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 5.1Canonical 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 follows from Structural Restoration Cost, Restoration-Monotone Survival, Theorems D–A–B, and the assembled differential- representation theorem. Pointwise RTP follows from the QSL–FPRP architecture, with the Finite Process Resource Principle exposed as the physical finiteness input. 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 While rank-one deformation of a positive-definite quadratic form can produce Lorentzian signature, the novelty of the present work is the derivation of the structures that make that deformation non-arbitrary within the framework. Informational admissibility, canonical minimal realization, 5
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry provenance-coherent ordering, and response-resource constraints successively determine the admis- sible continuation structure, orientation data, response capacity, and propagation threshold from which the Lorentz-signature form is constructed. The Lorentzian geometry is therefore a derived endpoint of the architecture rather than an independently selected starting structure. More specifically, the construction separates and connects three stages that are ordinarily available only after additional geometric structure has been supplied: an informationally derived orien- tation sector, a resource-bounded propagation threshold within that sector, and the resulting Lorentz-signature propagation geometry. Regular Local Continuation then promotes the pointwise construction to smooth capacity and frontier fields. The contribution is this dependency-controlled informational-to-geometric derivation and the explicit identification of the structural and physical premises sufficient for each step. Validation status. The results established here are mathematical consequences of the inherited realization geometry and the stated premises. The finite-dimensional model is a recomputable downstream consistency model, not an empirical or full quantum-operational realization. Concrete realization of the primitive structural and physical data, and any eventual physical testing, remain separate application problems. 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 Law −→ Smooth Informational Assembly −→ 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. 6
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry LayerQuestionOutput QIIWhich continuations preserve in- formational identity? Linear admissible continuation bundle. LASWhich realization contains no surplus structure? Canonical minimum-response realization. UPCPHow are continuations oriented?Positive, zero, and negative sectors. Horizon/HESWhere does normalized UPCP variation vanish? Equilibrium separatrix and balance mani- fold. RTP/RCSP/BETWhich responses propagate, sat- urate, and form the boundary? Unique capacity and selected saturation boundary. HRCELHow are advance, response, ca- pacity, and boundary unified? Lorentz-signature propagation geometry. SIAWhat separate premise controls base-point assembly? Conditional smooth capacity field. SSRDoes the full bridge package as- semble smoothly under the regu- larity premise? Conditional global smooth regular represen- tation. 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 5.1Canonical 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 =⇒ SIA and SSR. Compact Glossary QII Structural Quantum Informational Identity. LAS Least Additional Structure. UPCP Unified Provenance–Coherence Principle. PC4 Differential provenance-coherence representation package. RMS Restoration-Monotone Survival. HES Horizon Equilibrium Structure. RTP Response–Threshold Propagation. 8
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry RCSP Response-Capacity Saturation Principle. BET Boundary Emergence Theorem. HRCEL Horizon Response-Capacity Equilibrium Law. RLC Regular Local Continuation. SIA Smooth Informational Assembly. SSR Strong Smooth Regular Realizability. How to read the construction. QII determines which infinitesimal continuations preserve informational identity; LAS selects canonical realizations; the provenance-coherence architecture supplies orientation; and the QSL–FPRP architecture derives a finite pointwise response threshold that selects propagation inside the positive orientation sector. The threshold produces the fiberwise Lorentz-signature propagation geometry. RLC is a separate regularity input used later to promote the pointwise capacities and frontiers to smooth fields. Optional Cartan-compatible assembly is not part of that direct derivation. 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 machinery [9, 10, 6, 7], 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. 9
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 have dimensiond, leto= (ρ,E 1 ,...,E n ) lie on a regular fixed-rank state–POVM stratum, and letu= ( ̇ρ, ̇ E 1 ,..., ̇ E n ) be tangent to that stratum. Under the additive framed Hilbert–Schmidt realization metric, the minimum-response quadratic form is q o (u) = g B ρ ( ̇ρ, ̇ρ) + dg labB E ( ̇ E, ̇ E). If ρ = P a λ a |a⟩⟨a| and E i
P a μ i,a |a i ⟩⟨a i |, 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. The additive realization metric separates the state and measurement minimizations. The state sector is the standard Uhlmann-horizontal Hilbert–Schmidt minimization and yields the displayed Bures term. For the measurement sector choose positive amplitudesW i =E 1/2 i . In an eigenbasis ofE i , write X i = ̇ W i and D i = ̇ E i . The constraint D i = X † i W i
- W † i X i has entries (D i ) ab = √ μ i,a (X i ) ab
√ μ i,b (X i ) ba . Fora=b, least norm gives (X i ) aa = (D i ) aa /(2 √ μ i,a ). Fora̸=b, pairwise constrained minimization gives (X i ) ab
√ μ i,a μ i,a
- μ i,b (D i ) ab , (X i ) ba = √ μ i,b μ i,a
- μ i,b (D i ) ab . Consequently, q meas E ( ̇ E) = 1 2 n X i=1 X μ i,a +μ i,b
0 |(D i ) ab | 2 μ i,a
- μ i,b . The fixed-rank tangent condition removes kernel–kernel variation. Now define Ω(E) = 1 d n X i=1 |i⟩⟨i|⊗ E T i . POVM normalization makes Ω(E) a density operator. Its labeled-block eigenvalues areμ i,a /dand its tangent entries are ( ̇ E T i ) ab /d. Each spectral Bures summand therefore acquires the factor 1/d 2 1/d = 1 d . 10
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry Fixed labels exclude cross-label tangent terms, and transposition leaves the full double sum un- changed. Hence g labB E ( ̇ E, ̇ E) = 1 2d X i,a,b |⟨a i | ̇ E i |b i ⟩| 2 μ i,a
- μ i,b , with the sum restricted to positive denominators, and therefore q meas E = dg labB E . Remark 2.2 (Scope). The calculation is stratumwise, assumes fixed outcome labels and POVM normalization, and uses the displayed Bures convention. It does not claim globally optimal one-shot POVM discrimination or uniqueness among all admissible quantum-information metrics. 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 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 11
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 . 12
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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. 13
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 , 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 ), 14
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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. 15
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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: 16
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry
- informational-process identity;
- consistency among admitted inputs, intermediate transformations, and outputs;
- compositional consistency of successive processing stages; and 4.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 . 5.1Derived PC4: restoration cost, survival order, and differential representation 5.2 Admissible Structural Data and Scope The provenance-coherence construction is conditional on independently supplied structural data. At each regular pointo, these include an admissible perturbation class ∆ o , a structural restoration-cost functional on the specified correction-procedure domain, and the additional-structure preorder ⪯ add on the necessity classes to which LAS applies. Their domains, invariance properties, order requirements, and compatibility conditions are stated in the definitions below. These objects are primitive structural inputs. They are not defined by the survival sets, the resulting survival order, the UPCP covector, the response threshold, or the Lorentz-signature propagation geometry. RMS constrains the relation between independently defined structural severity and survival; it does not construct the restoration-cost functional. The severity/order data used below must likewise be defined independently of the survival relation they represent. The theory does not assert that every operational model canonically admits data satisfying these hypotheses. The corresponding conclusions apply to models in which such data are supplied and verified. Existence, uniqueness, or concrete physical realization of the data in a particular operational theory is a separate application problem. 17
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 5.3 Standing survival architecture Fix a regular QII pointo. LetV o =A QII o,0 , let ∆ o be the independently supplied perturbation class, and let S o : V o → 2 ∆ o be the survival assignment. Define v ⪯ pc,o w iff S o (v)⊆ S o (w). 5.4 Pre-survival structural severity Definition 5.4 (Structural restoration cost). LetP str o be the process structure defined independently of survival, orientation, propagation, and RTP. For eachδ ∈∆ o , letCorr str P str o (δ) be the admissible procedures restoring the perturbed description to the structural equivalence class of P str o . Let R o : Corr str P str o (∆ o )→ [0,∞] be an independently defined redescription-invariant resource functional, and set κ o (δ) :=inf Γ∈Corr str P str o (δ) R o (Γ), with value +∞ when no correction exists. Define δ 1 ⪯ ∆,o δ 2 iff κ o (δ 1 )≤ κ o (δ 2 ). Remark 5.5 (Independence and existence boundary). The domains ∆ o andCorr str P str o (∆ o ), the functionalR o , and the resulting severity preorder are primitive structural data. They are defined without the survival sets or the relation they are used to represent. No universal or canonical existence theorem for these data is asserted. Principle 5.6 (Restoration-Monotone Survival (RMS)). For every v ∈ V o and δ 1 ,δ 2 ∈ ∆ o , δ 2 ∈ S o (v), κ o (δ 1 )≤ κ o (δ 2 )=⇒ δ 1 ∈ S o (v). RMS does not constructR o ; it makes the independently defined restoration cost a continuation- independent sufficient severity statistic for survival. Theorem 5.7 (Survival-Set Comparability). Assume the continuation-independent total severity preorder and RMS. Then, for everyv,w ∈ V o , eitherS o (v)⊆ S o (w) orS o (w)⊆ S o (v). Hence the survival preorder is fiberwise complete. Proof.If the survival sets were incomparable, chooseδ v ∈ S o (v)\ S o (w) andδ w ∈ S o (w)\ S o (v). Totality orders the pair, and RMS places the lower-severity perturbation in both survival sets, a contradiction. Theorem 5.8 (Linear Survival-Order Representation). Assume fiberwise completeness, translation compatibility, positive-scale compatibility, closedness ofC pc o :={v: 0⪯ pc,o v}, and nontriviality. Then there is a nonzero covector Θ o ∈ V ∗ o such that v ⪯ pc,o w ⇐⇒ Θ o (v)≤ Θ o (w), and its positive-projective class [Θ o ] is unique. Proof.Translation and scale compatibility makeC pc o a closed convex total cone. Quotient by its 18
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry lineality space. The induced cone is closed, pointed, and total; connectedness of the unit sphere forces the nontrivial quotient to be one-dimensional. Pulling back an oriented linear coordinate gives Θ o , unique up to positive scale. Theorem 5.9 (Smooth Local Integration of the Survival-Order Covector). Assume a smooth cooriented extension ofkerΘ o , local integrability, and independently defined Survival-Leaf Fidelity identifying connected survival-equivalence classes with the integral leaves. Then locally there is a smooth submersion P o such that Θ o = (dP o ) o | V o . Any other admissible representative has the form e P o = φ◦ P o with φ ′
0. Proof. A foliation chart supplies a local defining submersion. Kernel agreement and coorientation fix its positive scalar ato. Survival-Leaf Fidelity identifies its connected level sets with the independently defined survival classes. Any second such submersion factors through the first by a strictly increasing local reparameterization. Theorem 5.10 (Derived Provenance-Coherence Differential Representation). Under Structural Restoration Cost, RMS, Theorem 5.7, and the hypotheses of Theorems 5.8 and 5.9, PC4(a)–(d) hold: the quotient survival order is total; it has a faithful nonzero linear representative; the representative integrates locally to a smooth scalar; and admissible scalar representatives differ by positive smooth reparameterization. Remark 5.11 (Logical boundary). The theorem is conditional on the independently supplied structural, order, and integration data. It does not claim that PC1–PC3 alone imply PC4 or that every operational model supplies such data. Definition 5.12 (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 5.13 (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. 19
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 5.14 (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. Definition 6.1 (Local Quantum Horizon). Assume Theorems 3.1, 4.4, and 5.13. 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. 20
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry Proof.This is the QII-admissible sector decomposition of Theorem 5.13, 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; • 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 21
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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. 22
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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}. 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: 23
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry
- Mis 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. (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. 24
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry (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 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 25
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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: 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. 26
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 . 27
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 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. 28
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 . 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. 29
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 8.3 Response-Normalized UPCP Advance and Canonical Splitting Definition 8.4 (Response-Normalized UPCP Advance). For Θ o ∈ [Θ o ], define b Θ o := Θ o q q −1 o (Θ o , Θ o ) , A o (v) := b Θ o (v). This is independent of the positive representative of [Θ o ]. Proposition 8.5 (Canonical UPCP Advance Decomposition). Let T o := b Θ ♯ q o o . Then q o (T o ,T o ) = A o (T o ) = 1, H QH o = ker b Θ o = (RT o ) ⊥ q o , and every v ∈ V o has the unique decomposition v = A o (v)T o
- v ⊥ , v ⊥ ∈ H QH o . Moreover, q o (v,v) = A o (v) 2
- q o (v ⊥ ,v ⊥ ). Proof. Metric duality and the normalization of b Θ o give the unit identities. The vectorv ⊥ := v− A o (v)T o lies in the kernel, and orthogonality gives the norm decomposition. Corollary 8.6 (Refinement Ratio in Advance Form). ForA o (v)>0, defineR o (v) := p q o (v ⊥ ,v ⊥ ). Then RR o (v) = R o (v) A o (v) . This identity supplies no propagation criterion by itself. 8.4 Response-Threshold Propagation Principle Theorem 8.7 (QSL–FPRP Derivation of Pointwise Response–Threshold Propagation). Let dimV o ≥2, with the response metric, normalized UPCP advance, and canonical splitting above. LetProp o be the qualitative propagation predicate. Assume positive orientation of propaga- tion, positive-projective invariance, two-data quantitative closure, zero-refinement compatibility, downward refinement closure, and relative openness. Fix an admitted information-bearing physical processP. Assume the following physical realization and resource data. 7.Generalized Speed-Limit Resource Datum. There are a primitive classG P,o of admissible infinitesimal generators preservingP, a realization mapR P,o :G P,o → V o , a resource functional ν P,o :G P,o →[0,∞], and an independently defined operational speedV o onV phys P,o :=imR P,o . For v ∈ V phys P,o set G P,o (v) :={G∈ G P,o :R P,o (G) = v},r P,o (v) :=inf G∈G P,o (v) ν P,o (G), 30
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry and assume the generalized speed-limit bound V o (v)≤ r P,o (v). The generator class, realization map, and resource functional are primitive physical inputs defined without using Prop o ,V o , the q o -norm, r P,o , B P,o , RR o , J o , c o , K + o , Q o , or G o . 8. Continuation–Speed Correspondence. On V phys P,o , V o (v) = q q o (v,v). The two sides arise independently; this equality is a compatibility premise and is not claimed to follow from Theorem 2.1 alone. 9. Nonempty Unit-Advance Realization Class. With K (1) P,o :={v ∈ V o : v ∈ V phys P,o , A o (v) = 1}, assume K (1) P,o ̸= ∅. 10. Finite Process Resource Principle. B P,o := sup v∈K (1) P,o r P,o (v) <∞. 11. Propagating directions admit physical implementations preserving P. 12. Positive reparameterizations of a physically implemented continuation ray are physically implementable. Then there exists a unique c o ∈ (0,∞) such that Prop o (v) = 1 ⇐⇒ A o (v) > 0 and RR o (v) < c o , and c o ≤ q B 2 P,o − 1. Proof. Rank at least two gives full attainment ofRR o (C + o ) = [0,∞). Projective invariance and two-data closure therefore produce a unique classifierF o : [0,∞)→{0,1}and a compatible-ratio set J o = F −1 o (1) that contains zero, is downward closed, and is relatively open. For a propagatingv, physical realization and projective realizability put its unit-advance normaliza- tion v (A) = v/A o (v) in K (1) P,o . Orthogonality gives q o (v (A) ,v (A) ) = 1 + RR o (v) 2 . Assumptions 7–10 imply q 1 + RR o (v) 2 =V o (v (A) )≤ r P,o (v (A) )≤ B P,o , 31
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry soRR o (v)≤ q B 2 P,o − 1. HenceJ o is proper and bounded. Its order-topological properties imply J o = [0,c o ) for the unique endpoint c o = supJ o ∈ (0,∞). Remark 8.8 (QSL–FPRP claim boundary). The theorem derives the finite pointwise threshold from the stated qualitative, realization, speed-resource, compatibility, and finiteness premises. It does not construct the generalized speed-limit datum, derive Continuation–Speed Correspondence, or derive FPRP. Remark 8.9 (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 . 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 8.10 (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, at every regular pointo,K + o ⊂ V o is a nonempty, open, convex, positively homogeneous, full-dimensional proper cone, in the standard convex-geometric sense [13]. Their closures are salient, and whenever H QH o ̸={0}, K + o ⊊ C + o . 32
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 33
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. All preceding conclusions are fiberwise. If, in addition,o7→ c o is smooth on a fixed-rank QII-regular domain, then smoothness ofq, b Θ,T, the orthogonal projection, and cimplies smooth assembly of the propagation cones and their nonzero null boundaries. Under Regular Local Continuation, the required smoothness ofcis supplied later by the HRCEL Smoothing Law. Corollary 8.11 (Response-induced causal trichotomy). For every nonzerov ∈ C
o , exactly one of the following occurs: Q o (v) < 0 ⇐⇒ propagating, Q o (v) = 0 ⇐⇒ critical propagation, Q o (v) > 0 ⇐⇒ future-oriented but non-propagating. Corollary 8.12 (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 [12,1], and K + o ={v ∈ V o : G o (v,v) < 0, b Θ o (v) > 0}. 34
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 8.6 Logical Status and Claim Boundary Remark 8.13 (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 8.14 (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 8.15 (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 follows from the qualitative refinement hypotheses and the QSL–FPRP physical-resource architecture. Remark 8.16 (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 . 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. 35
Paper 4 Version 5.1Canonical Realization and Hybrid 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), 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 36
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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) 37
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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) 38
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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,Q o (v ε )>0. SinceK + o ={v: b Θ o (v)≥0, Q o (v)≤0}, it follows directly thatv ε /∈K + o . 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. 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 39
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 , 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 . 40
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 .
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- 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. 41
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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), and Q o (v) := F o (v)− c 2 o a 2 . Assume c o
0 and, when stated below, H QH o ̸={0}. 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.
No Horizon vector lies in the strict propagation cone, and the Horizon meets the closed 42
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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) < c 2 o a 2 }, N + o ={v : F o (v) = c 2 o a 2 }, F + o ={v : F o (v) > c 2 o a 2 }, 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. 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 8.10 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 − c 2 o a 2 . 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 43
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 satisfies D o = (1 + c 2 o ) b Θ o ⊗ b Θ o . Hence D o is nonzero, symmetric, positive-semidefinite, and rank one.
- 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. 44
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 . 45
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 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. 46
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 ;
- the common geometric radius of B o is exactly the unique informational refinement cutoff;
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). 47
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 . 48
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 . 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 . 49
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 ⊥ . 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)}. 50
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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) . 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}. 51
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 . (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. 52
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry (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 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 53
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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) , 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 . 54
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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}. 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. 55
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 , 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, 56
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 14 Regular Local Continuation, HRCEL Smoothing, and Optional Cartan Assembly 15 Regular Local Continuation Principle 15.1 (Regular Local Continuation (RLC)). On the selected connected fixed-rank QII- regular domain, every point has a neighborhood on which the admissible HRCEL capacity is represented by a smooth positive local track. Pointwise HRCEL uniqueness applies on overlaps. 16 Optional Cartan-Compatible Assembly Remark 16.1 (Relation to HRCEL smoothing). The HRCEL Smoothing Law below does not require the Cartan assembly lemma. RLC supplies smooth local capacity tracks, pointwise HRCEL uniqueness forces overlap agreement, and direct regular-level-set geometry assembles the frontier. The Cartan result is an independent sufficient criterion available only when an application supplies the associated-bundle data and verifies CAN1–CAN6. Lemma 16.2 (Optional Cartan-Compatible Normalized Congruence Assembly). LetP G → Ube a principalG-bundle and letE,L,Wbe associated bundles carrying aG-equivariant model family M o (λ) and normalization mapN:L → W. Assume CAN1–CAN3 and CAN5–CAN6 as stated for smooth incidence, equivariance, intrinsic normalization, vertical nondegeneracy, and constant geometric type. Replace CAN4 by: CAN4. Unique Intrinsic Normalization and Realized Normalized Model. For every o∈ U, there exists a unique λ o ∈ L o such that N (λ o ) = 0 o , and M o (λ o )̸= ∅. Define X o := M o (λ o ). Then the normalized parameters assemble smoothly, the normalized models form a smoothG-natural incidence submanifold X ⊂ E, and π X : X → U is a smooth surjective submersion. 57
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry Proof.CAN5 and the implicit-function theorem give the smooth normalized parameter section. Pullback of the model-incidence submersion gives the submersionπ X . The nonemptiness clause in CAN4 ensures that every base point has a normalized model, so π X is surjective. Remark 16.3 (Application status). Lemma 16.2 is not used in the proof of the HRCEL Smoothing Law. No Cartan-natural identification of the HRCEL frontier is claimed unless a concrete application verifies CAN1–CAN6. 17 HRCEL Smoothing Law Theorem 17.1 (HRCEL Smoothing Law). Assume RLC and pointwise HRCEL uniqueness. Then the local capacity tracks agree on overlaps and define a unique c∈ C ∞ (U QII , (0,∞)). Let H QH → U QII be the smooth Horizon bundle and define Ψ(h) := q π H (h) (h,h)− c(π H (h)) 2 . ThenF:= Ψ −1 (0) is a smooth embedded hypersurface andπ F :F → U QII is a smooth surjective submersion. Proof.On every overlap, two local tracks are capacities for the same pointwise HRCEL problem; uniqueness makes them equal, so they glue smoothly. If h∈ F o , then d v Ψ h [h] = 2q o (h,h) = 2c 2 o
0. Thus zero is a regular value andFis smooth. The nonzero vertical derivative permits solving the tangent equation for any prescribed base tangent, soπ F is a submersion. Every fiber is the nonempty q o -sphere of radius c o , hence π F is surjective. 18 Consequences Corollary 18.1 (Smooth Informational Assembly). Regular Local Continuation, pointwise HRCEL uniqueness, gluing, and Theorem 17.1 yield the unique smooth capacity field and frontier assembly. Every downstream object whose unresolved regularity is carried byctherefore assembles smoothly on the selected regular domain. Remark 18.2 (Dependency boundary). The direct SIA derivation does not depend on Lemma 16.2. Cartan-compatible assembly remains an optional stronger route when CAN1–CAN6 and application- specific bundle data are independently verified. 58
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 20 Worked Downstream Geometric Model The following model begins at the downstream geometric stage of the construction. The smooth positive capacity field and the upstream structures required to define the propagation geometry are supplied data. The example displays the response-threshold quadratic form, propagation cone, null boundary, normalized frontier, and their smooth assembly. It does not instantiate the structural restoration-cost architecture, RMS, the generalized speed-limit resource datum, Continuation–Speed Correspondence, or FPRP, 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. 59
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 . 60
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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 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. 61
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 62
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry RTP, only after those earlier structures have been established. 21.2 Relation to Neighboring Approaches The present construction differs in target from operational reconstructions of quantum theory, which seek the quantum formalism from operational principles, and from thermodynamic, entanglement- based, or holographic programs whose targets include Einstein dynamics or spacetime connectivity [4, 3,5,14,11]. Here the target is narrower: under the stated informational, realization, provenance- coherence, response-resource, compatibility, and regularity premises, the construction yields an oriented response architecture, a finite pointwise propagation threshold, the associated fiberwise Lorentz-signature propagation geometry, and, under Regular Local Continuation, its smooth assembly. The claimed contribution is the dependency-controlled placement of these ingredients within that derivation. 21.3 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. 21.4 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. 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 63
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry 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. 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 64
Paper 4 Version 5.1Canonical Realization and Hybrid Geometry HRCEL assembly follows from Regular Local Continuation, pointwise HRCEL uniqueness, gluing, and regular-value/submersion geometry. 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 The informational-to-geometric architecture establishes three principal derivational chains. First, PC4 is derived through Structural Restoration Cost, Restoration-Monotone Survival, survival- set comparability, 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 yields Smooth Informational Assembly by direct gluing and regular-value/submersion geometry. 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. References [1] John K. Beem, Paul E. Ehrlich, and Kevin L. Easley. Global Lorentzian Geometry, volume 202 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York, 2 edition, 1996. 65
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