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Conservation, Canonical Realization, Equilibrium, and Hybrid Propagation Geometry in Quantum-Operational Geometry Paper 4 Version 4.3 Michael Tolbert October 2026 Contents 1 Introduction4 2 Inherited regular realization geometry8 3 Structural Quantum Informational Identity9 4 Least Additional Structure13 5 Unified Provenance–Coherence Principle14 5.1 PC4: local completeness and smooth differential representability . . . . . . . . . . . 15 6 The Local Quantum Horizon17 6.1 Nontriviality qualification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2 Global scope disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.3 The Quantum Horizon as an Organizing Structure . . . . . . . . . . . . . . . . . . . 18 7 Horizon Equilibrium Structure19 8 Theorem A: Cone Promotion24 8.1 Response-Threshold Propagation Principle . . . . . . . . . . . . . . . . . . . . . . . . 26 8.2 Inherited Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.3 Normalized Orientation and Canonical Splitting . . . . . . . . . . . . . . . . . . . . . 26 8.4 Response-Threshold Propagation Principle . . . . . . . . . . . . . . . . . . . . . . . . 27 1

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 8.5 Theorem B: Response-Defined Propagation Geometry . . . . . . . . . . . . . . . . . 28 8.6 Logical Status and Claim Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.7 Adopted Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 9 Refinement-Cost Characterization of Pointwise RTP31 10 Horizon-Organized Hybrid Geometry and Null Boundary Realization32 10.1 Separatrix-to-null-barrier transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 10.2 Horizon uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 10.3 Horizon propagation-failure structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 10.4 Horizon essentiality and unique recovery . . . . . . . . . . . . . . . . . . . . . . . . . 39 10.5 Null Boundary Realization and threshold–radius identification . . . . . . . . . . . . . 41 10.6 Status after pointwise null-boundary realization . . . . . . . . . . . . . . . . . . . . . 43 11 Response-Capacity Saturation Law43 12 Boundary Emergence Theorem46 13 Horizon Response-Capacity Equilibrium Law49 14 Response-Capacity Regularity51 14.1 Theorem 1: Informational Refinement-Capacity Assembly . . . . . . . . . . . . . . . 52 14.2 Information–Geometry Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 14.3 Theorem 2: Response-Capacity Regularity . . . . . . . . . . . . . . . . . . . . . . . . 53 15 Smooth Informational Assembly and Selector Rigidity54 15.1 Why Smooth Informational Assembly Is the Appropriate Regularity Principle . . . . 56 16 Conditional HRCEL Global Smoothness and Strong Smooth Regular Realizabil- ity57 17 A Finite-Dimensional Model of the HRCEL Construction62 18 Discussion63 18.1 The Emergent Hybrid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 18.2 Architectural and scope observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 18.3 Lessons Learned from the Refinement Program . . . . . . . . . . . . . . . . . . . . . 65 2

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 19 Scope, Limitations, and Open Questions66 20 Conclusion67 3

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Abstract We develop a staged local 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 a unique least-surplus realization class; the Unified Provenance–Coherence Principle (UPCP) supplies projective orientation and the Quantum Horizon; and the Response-Threshold Propagation architecture characterizes propagation through an attained Refinement Ratio and a finite response capacity. These pointwise ingredients yield a proper cone with a Lorentz-signature quadratic representation. The rank-one passage from positive-definite data and a normalized covector to a Lorentz- signature form is standard and is not claimed as a new general construction. The contribution claimed here is the information-first provenance of the inputs. The paper proves pointwise and fiberwise consequences of this staged architecture and, only under the explicitly stated Smooth Informational Assembly and Selector Rigidity hypotheses, derives its smooth fixed-rank bundle consequences and Strong Smooth Regular Realizability. This is a structural stage in an ongoing reconstruction program, not a completed spacetime the- ory. The result does not establish tangent-bundle identification, integrability, four-dimensionality, physical spacetime, global causality, or gravitational dynamics. 1 Introduction Quantum theory presents noncommutative operator and measurement data, while familiar causal geometry is formulated in terms of directional, separating, and orientation structures. The purpose of this paper is not to identify the operational manifold with physical spacetime or to combine quantum and gravitational field equations. The narrower objective is to determine which propagation- geometric structures can be obtained intrinsically from regular quantum-operational realization geometry. The work is situated alongside informational reconstructions of quantum theory, structural- realist approaches to scientific representation, and contemporary programs investigating emergent spacetime and causal organization [14, 4, 3, 15, 7, 13, 10, 2]. Paper 3 established a smooth regular operational manifold O reg , a principal realization bundle π :R reg −→O reg , exact compact gauge fibers, a positive framed response metricg fr , and a distinguished minimum- response horizontal distributionH fr . These structures provide canonical horizontal realization transport for a prescribed operational continuation. They do not, by themselves, decide which operational continuations preserve informational identity or how admissible continuations should be ordered. The present paper separates those questions into three roles: 1.QII: conservation and feasibility. Which infinitesimal continuations preserve the repre- sented informational architecture up to admissible intertwining? 2.LAS: canonical variation. For a fixed QII-compatible continuation, which realization representative contains no additional realization structure? 4

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 3.UPCP: ordering and orientation. How are QII-compatible continuations ordered by robustness of provenance coherence? The resulting architecture is QII: conservationLAS: 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. Conditional regularity extension. Under the additional Smooth Informational Assembly premise, complete propagation-state determination, absence of autonomous selector freedom, and the Selector Rigidity hypotheses, the pointwise capacities and associated geometric fields assemble smoothly. This fourth layer is deliberately separated because it rests on a foundational regularity premise not derived from the pointwise architecture. Theorem A and RTP provide the domain-level cone promotion and quantitative propagation refine- ment that form the intended endpoint of the present construction. Full tangent-bundle realization, effective dimension, growth laws, global causal reconstruction, and spacetime interpretation remain outside the claims of this manuscript. Novelty and Significance The novelty of the present work does not lie in the isolated algebraic observation that a rank-one deformation of a positive-definite quadratic form can produce Lorentzian signature. That operation is standard finite-dimensional geometry, and no new general rank-one construction is claimed here. The contribution instead lies in the derivational architecture that precedes and constrains that operation. Informational admissibility, canonical minimal realization, provenance-coherent ordering, and response-threshold selection are assembled in stages so that the distinguished data entering the Lorentzian construction are not introduced merely because they recover a desired signature. The resulting Lorentzian structure is therefore the endpoint of the architecture developed here rather than its starting assumption. 5

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry This distinction makes provenance part of the mathematical content, rather than ancillary bookkeeping. If a preferred direction, deformation scale, threshold, or null structure is selected simply because it produces Lorentzian geometry, the resulting construction provides limited grounds for assigning that geometry further physical significance. The question pursued here is stronger: whether the ingredients required for the Lorentzian transition can be obtained from independently specified informational and realization principles, with the geometric signature arising only after those ingredients have been established. The aim is thus not to replace a standard algebraic operation with a new one, but to explain why the data entering that operation are present and distinguished in the first place. This changes the significance of the construction. A Lorentz-signature quadratic form can be written down abstractly with little difficulty. The present framework instead asks whether such a form can arise at the end of a controlled passage from informational structure to geometric realization. To the extent that this passage succeeds, the resulting geometry is not merely an arbitrary pseudo-Riemannian example. It becomes a candidate pre-spacetime realization architecture whose additional structure and possible physical interpretation can be investigated rather than presupposed. No identification with physical spacetime is asserted here. Lorentzian signature alone is plainly insufficient for such an identification, and the present construction does not establish the dimension- ality, dynamical content, empirical adequacy, or other structures that a physical spacetime would ultimately require. Those limitations define the next questions rather than diminishing the present one. The purpose of this paper is to establish an upstream geometric architecture within which progressively stronger questions can be posed without assuming their answers. In particular, the construction developed here is intended as a foundation for subsequent investigation of the admissible continuation structure and its independent degrees of freedom, including whether that structure selects a definite dimensionality. Such results, if established, would move the framework beyond the existence of Lorentzian realization toward a more constrained candidate geometry on which genuinely physical questions could eventually be formulated. The present paper does not claim those downstream results. It establishes the architecture that makes their investigation possible. The broader novelty claim is therefore deliberately specific but consequential: the standard Lorentzian algebra is not the proposed innovation; the proposed innovation is the staged informational-to-geometric derivation that supplies and constrains its inputs. The goal is not to assume spacetime and reconstruct familiar geometry within it, nor to declare the resulting geometry to be spacetime prematurely. It is to determine how far spacetime-like structure can be obtained from antecedent informational and realization principles, and to make the remaining steps sufficiently explicit that they can be subjected to further mathematical and, ultimately, physical tests. Validation status. The results established here are mathematical consequences of the inherited realization geometry and the explicitly stated principles and assumptions of the present framework. The finite-dimensional model provides a directly recomputable consistency check of the resulting construction, but it is not an empirical validation of the framework. Likewise, citations to standard differential, convex, semi-Riemannian, and bundle geometry support the mathematical tools used in the derivations but do not independently establish the program-specific informational princi- ples from which the construction begins. The present work should therefore be understood as 6

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry establishing the internal mathematical architecture and its conditional consequences. Independent derivation, comparison with other reconstruction frameworks, and eventual empirical or physical tests remain separate validation requirements for any future identification of the resulting geometry with spacetime. Architecture and Logical Dependency Map The construction is staged. Each stage answers a distinct question and adds only the structure needed for the next stage: Inherited regular realization geometry −→ QII admissibility −→ LAS canonical realization −→ UPCP orientation −→ Quantum Horizon and HES −→ RTP propagation selection −→ RCSP saturation −→ BET boundary emergence −→ HRCEL equilibrium geometry −→ Selector Rigidity −→ Smooth Informational Assembly −→ Global smooth HRCEL realization and SSR. The arrows record logical dependence, not temporal evolution, physical causation, or dimensional reduction. The inherited response metricqremains positive definite: it normalizes the UPCP direction, supplies the orthogonal advance–Horizon decomposition, and measures transverse response. Lorentz signature appears only in the distinct propagation form selected by RTP and assembled through HRCEL. LayerQuestionOutput QIIWhich continuations preserve infor- mational identity? Linear admissible continuation bundle. LAS Which realization contains no sur- plus structure? Canonical minimum-response realization. UPCPHow are continuations oriented?Positive, zero, and negative sectors. Horizon/HESWhere does normalized UPCP vari- ation vanish? Equilibrium separatrix and balance manifold. RTP/RCSP/BETWhich responses propagate, satu- rate, and form the boundary? Unique capacity and selected saturation boundary. HRCEL How are advance, response, capacity, and boundary unified? Lorentz-signature propagation geometry. SR/SIAWhat separate premise controls base-point assembly? Conditional smooth capacity field. SSRDoes the full bridge package assem- ble smoothly under the regularity premise? Conditional global smooth regular representa- tion. Logical Status of the Main Structures The terms principle, assumption, definition, and derived result are used distinctly. The table below records the logical role of the main components. 7

Paper 4 Version 4.3Canonical 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 RTPAdopted propaga- tion principle Bounded refinement-cost se- lection Physical causality or light speed 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 SmoothIn- formational Assembly Foundational regu- larity principle Smooth assembly of eligible intrinsic observables 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 RTP+RCSP+BET =⇒ unique pointwise threshold and boundary, Selector Rigidity+SIA =⇒ smooth threshold field, smooth threshold field =⇒ global smooth HRCEL realization and SSR. 2 Inherited regular realization geometry Let U ⊆O reg be an open regular domain. Let π :R reg | U −→ U be the inherited smooth principal realization bundle with compact gauge groupG. Its vertical subspace at R is V R := kerdπ R . The inherited positive,G-invariant framed response metricg fr determines the smooth gauge- equivariant orthogonal decomposition, using standard smooth, Riemannian, and bundle-geometric machinery [8, 9, 5, 6], T R R reg = H fr R ⊕ V R . 8

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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. 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}. 9

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Every K ∈ k evt automatically preserves the fixed event and coarse-event projectors: [K,Q A ] = 0,[K,Q Γ α ] = 0. Identify the relevant fixed-rank Grassmannians with their orthogonal-projector manifolds, and define the structural-defect space F R := T P R Gr(H)⊕ T Π R Gr

u(H)
. Define the raw Structural QII variation map B R : T o U → F R by B R (v) :=  DP R

h fr R v
,DΠ R

h fr R v
 . For K ∈ k evt , define ad K (X) := [K,X] on u(H), and define the infinitesimal structural-intertwiner map L R : k evt → F R by L R (K) := ([K,P R ], [ad K , Π R ]). Let π R : F R −→ F R / imL R be the canonical quotient map. Define the reduced Structural QII defect by C red QII,R := π R ◦ B R , and define A QII o,0 (R) := kerC red QII,R , A QII o (R) := A QII o,0 (R){0}. Then the following conclusions hold. 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 10

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry and Π R ′ = Ad V g ◦Π R ◦ Ad −1 V g ,Ad V g (X) := V g XV † g . Define Γ g : F R → F R ′ by Γ g (X,Y ) :=  V g XV † g , Ad V g ◦Y ◦ Ad −1 V g  . Then Γ g is a linear isomorphism satisfying B R ′ = Γ g ◦ B R and Γ g (imL R ) = imL R ′ . Consequently, Γ g induces a quotient-space isomorphism Γ g : F R / imL R −→ F R ′ / imL R ′ satisfying C red QII,R ′

Γ g ◦ C red QII,R . 4. Intrinsic QII descent. For every v ∈ T o U, C red QII,R (v) = 0 ⇐⇒ C red QII,R ′ (v) = 0. Therefore, A QII o,0 (R) = A QII o,0 (R ′ )⊆ T o U. 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 v imply 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. 11

Paper 4 Version 4.3Canonical 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 ), 12

Paper 4 Version 4.3Canonical 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. 13

Paper 4 Version 4.3Canonical 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: 14

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry

1. informational-process identity;
2. consistency among admitted inputs, intermediate transformations, and outputs;
3. 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 . 5.1 PC4: local completeness and smooth differential representability Define v 1 ∼ pc,o v 2 ⇐⇒ v 1 ⪯ pc,o v 2 and v 2 ⪯ pc,o v 1 , equivalently S o (v 1 ) = S o (v 2 ). PC4(a) (Local completeness). The induced order on A QII o /∼ pc,o is total. PC4(b) (Faithful smooth differential representability). There are an open neighborhood U o ⊆ U and a smooth function P o : U o → R such that, with Θ o := (dP o ) o ∈ T ∗ o U, one has for all v 1 ,v 2 ∈ A QII o , v 1 ⪯ pc,o v 2 ⇐⇒Θ o (v 1 )≤ Θ o (v 2 ). Thus provenance equivalence is represented by equality of Θ o values. UPCP is formulated at the local first-order level; higher-order provenance distinctions are outside the scope of the present theorem. 15

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry PC4(c) (Nondegeneracy). Θ o | A QII o,0 ̸= 0. PC4(d) (Representative independence). If e P o is another admissible representative, then locally e P o = φ◦ P o , where φ is a smooth orientation-preserving local diffeomorphism with φ ′

0. Consequently, (d e P o ) o = φ ′ (P o (o))Θ o , with positive factor, so the order, zero set, and sign sectors are representative independent. Remark 5.4 (Axiomatic status). PC4 is an explicit UPCP axiom. It is not claimed to follow from PC1–PC3, QII, or LAS. It completes UPCP’s ordering role without transferring conservation or variational work into UPCP. Definition 5.5 (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 5.6 (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:

1. Θ o (−v) =−Θ o (v) for every v ∈ T o U.
2. Every v ∈ A QII o belongs to exactly one of C

o , Σ QII o {0}, and C − o . 3. C + o and C − o are preserved under positive scaling within A QII o . 4. If v,−v ∈ A QII o and Θ o (v)̸= 0, exactly one lies in C + o and the other in C − o . 5. Both C + o and C − o are nonempty. 6. The separatrices and sign sectors are independent of the admissible scalar representative. Consequently, UPCP determines a nontrivial, representative-independent local orientation– propagation cone structure on the QII-admissible continuation space. Proof.The differential Θ o = (dP o ) o is linear, so it is odd and positively homogeneous. Real-number trichotomy gives the sector decomposition, and positive homogeneity gives scaling invariance. QII sign symmetry exchangesvand−v. PC4(c) suppliesw ∈ A QII o,0 with Θ o (w)̸= 0, and QII sign symmetry supplies−w, proving both strict sectors nonempty. Finally, PC4(d) multiplies Θ o only by a positive scalar, preserving its kernel and strict sign sectors. 16

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Remark 5.7 (UPCP claim boundary). The theorem does not establish derivability of PC4, 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 5.6. The Local Quantum Horizon at o is H QH o := A QII o,0 ∩ ker Θ o ⊆ T o U. Equivalently,H QH o is the set of QII-admissible infinitesimal continuations whose first-order provenance–coherence variation vanishes. Definition 6.2 (LAS-canonical realized Local Quantum Horizon). For R∈ π −1 (o), define [ H QH R := h fr R (H QH o )⊆ H fr R . This is the realization of the operational Horizon without vertical realization surplus. It records LAS’s contribution without changing the operational definition of H QH o . Definition 6.3 (Local Quantum Horizon structure). The associated pointwise structure is QH o

A QII o,0 ,H QH o ,C + o ,C − o , [h fr R ]
, where [h fr R ] denotes the intrinsic LAS-canonical realization assignment under the inherited gauge- equivariant realization geometry. Proposition 6.4 (Pointwise Horizon decomposition). A QII o,0 = C + o ⊔H QH o ⊔ C − o . The two strict sectors are nonempty, closed under positive rescaling withinA QII o , and exchanged by sign reversal. The decomposition is representative independent. Proof.This is the QII-admissible sector decomposition of Theorem 5.6, 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}. 17

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 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 18

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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. 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. 19

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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:

1. S is the zero-variation sector of the UPCP differential;
2. 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:

1. Mis canonically determined by the primitive UPCP orientation structure, without using a propagation predicate, response threshold, propagation boundary, or selector;
2. V = RT ⊕ q M, so each v ∈ V has unique longitudinal and transverse components;
3. 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. 20

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry (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. (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 21

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 Θ has the same kernel and strict sign sectors. Hence the decomposition is representative independent. Uniqueness follows because the zero-variation sector of the UPCP differential is uniquely its kernel. For HES3, normalization gives T = b Θ ♯ q and b Θ(T ) = 1. If h∈ H QH , then q(T,h) = b Θ(h) = 0. Every v ∈ V can be written v = b Θ(v)T +

v− b Θ(v)T
, and the second term lies inker b Θ. ThusV=RT ⊕ q H QH . The kernel is invariant under positive scaling and is canonically determined by the UPCP ray; hence it satisfies the definition of primary invariant balance manifold. Indeed,H QH o =ker b Θ o = (RT o ) ⊥ q o . The orthogonal complement of the fixed lineRT o in the positive-definite inner-product space (V o ,q o ) is unique; hence any balance manifold satisfying Definition 7.3(2) equalsH QH o . Its uniqueness is relative to that primitive UPCP datum. HES4 follows from the direct sum, whose summands have zero intersection. Positive definiteness of q makes r(v) well defined and nonnegative. For λ > 0, linearity gives a(λv) = λa(v),(λv) ⊥ = λv ⊥ , r(λv) = λr(v). Therefore RR(λv) = λr(v) λa(v) = RR(v), which proves HES5 and the descent to positive projective space. If v ∈ Σ ρ , then HES5 gives RR(λv) = ρ for every λ > 0. This proves HES6. Finally,h ∈ H QH implies b Θ(h) = 0, whereasv ∈ C + requires b Θ(v)>0. ThusH QH ∩ C + =∅, 22

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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. 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 yet another independent trichotomy, but pro- vides a geometric realization of this threshold relation through the sign of the corresponding quadratic form. Boundary Emergence then identifies the saturation locus with the null propagation boundary, while the later equilibrium law synthesizes these previously established relations. Thus, the recurrence of positive/critical/negative sign structure reflects structural coherence across successive informational, threshold, and geometric levels, rather than repeated derivations of a single result under different terminology. 23

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 . 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:

1. 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; 24

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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. 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. 25

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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. 8.3 Normalized Orientation and Canonical Splitting Let q −1 o denote the dual inner product on V ∗ o . Define the normalized UPCP covector b Θ o := Θ o q q −1 o (Θ o , Θ o ) . This is independent of the positive representative of [Θ o ]. If e Θ o =a o Θ o witha o

0, then b e Θ o = b Θ o . Define the canonical advance vector T o := b Θ ♯ q o , where ♯ q : V ∗ o → V o is the musical isomorphism induced by q o . Then q o (T o ,v) = b Θ o (v), q o (T o ,T o ) = b Θ o (T o ) = 1. Every v ∈ V o has the unique decomposition v = b Θ o (v)T o

• v ⊥ , v ⊥ := v− b Θ o (v)T o . 26

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Since b Θ o (v ⊥ ) = 0 and q o (T o ,v ⊥ ) = 0, one obtains the intrinsic orthogonal splitting V o = RT o ⊕ q o H QH o . Thus UPCP supplies the advance orientation, while the response metric converts the orientation covector into a canonical advance vector and equips the Horizon directions with transverse geometry. 8.4 Response-Threshold Propagation Principle Principle 8.4 (Pointwise Response-Threshold Propagation Principle (RTP pt )). For everyo∈ U QII there exists a positive scalarc o ∈(0,∞) such that a nonzero QII-admissible continuationv ∈ V o is propagationally admissible if and only if it has positive normalized UPCP advance and its transverse minimum response is strictly below the local response capacity: b Θ o (v) > 0, q q o (v ⊥ ,v ⊥ ) < c o b Θ o (v). (RTP) The scalarc o is called the local response capacity or cone-aperture. Pointwise RTP makes no base-point regularity claim about o7→ c o . RTP is the substantive bridge premise. It does not transfer the distinct roles of QII, LAS, and UPCP into one another. QII determines admissibility; Paper 3 and LAS supply the intrinsic response geometry; UPCP supplies orientation; RTP states how response and orientation jointly determine propagation. 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 . 27

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 8.5 Theorem B: Response-Defined Propagation Geometry Assumption 8.5 (Smooth-RTP regularity (RTP sm )). The pointwise capacities supplied by RTP pt assemble into a smooth positive function c : U QII −→ (0,∞). Equivalently, relative to the already smooth fieldsqand b Θ, the pointwise quadratic formsQ o , propagation formsG o , and defect formsD o :=q o − G o assemble smoothly. This is an additional domain-level regularity condition and is not part of RTP pt . Theorem 8.6 (Theorem B (Response-Defined Propagation Geometry)). Assume the inherited QII-admissible bundle, the operational response metricq, the smooth positive-projective UPCP orientation [Θ], RTP pt , and RTP sm . Then the setsK + o form a smooth intrinsic field of nonempty, open, convex, positively homogeneous, full-dimensional proper cones, in the standard convex- geometric sense [12]. Their closures are salient, and whenever H QH o ̸={0}, K + o ⊊ C + o . Moreover, Q o is a nondegenerate quadratic form of signature (−, +,..., +), and ∂K + o {0} = n v ∈ V o : b Θ o (v) > 0, Q o (v) = 0 o . The nonzero boundary is a smooth conic hypersurface, and every normalized boundary section n v ∈ ∂K + o : b Θ o (v) = 1 o is a strongly convex q o -ellipsoid in T o

• H QH o . Proof.Nonemptiness follows becauseλT o ∈ K

o for everyλ >0. Openness follows from continuity of b Θ o and Q o . Positive homogeneity follows from Q o (λv) = λ 2 Q o (v), b Θ o (λv) = λ b Θ o (v). For convexity, let v,w ∈ K + o and 0 < t < 1. The transverse projection is linear, so ((1− t)v + tw) ⊥ = (1− t)v ⊥

• tw ⊥ . The triangle inequality gives q q o (((1− t)v + tw) ⊥ , ((1− t)v + tw) ⊥ )≤ (1− t) q q o (v ⊥ ,v ⊥ ) + t q q o (w ⊥ ,w ⊥ ) < c o

(1− t) b Θ o (v) + t b Θ o (w)
, so (1− t)v+tw ∈ K + o . A nonempty open subset of the finite-dimensional vector spaceV o is full-dimensional. For saliency, supposev ∈K + o ∩(−K + o ). Then b Θ o (v)≥0 and− b Θ o (v)≥0, hence b Θ o (v) = 0. The closed threshold inequality givesq o (v ⊥ ,v ⊥ )≤0. Positivity ofq o impliesv ⊥ = 0, and the splitting 28

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry then gives v = 0. By definition,K + o ⊂ C + o . IfH QH o ̸={0}, choose nonzeroh ∈ H QH o and setv ε =εT o +h. Then b Θ o (v ε ) = ε > 0, while Q o (v ε ) = q o (h,h)− c 2 o ε 2

0 for sufficiently small ε > 0. Hence v ε ∈ C

o \K + o , proving strict inclusion. On the advance line, Q o (aT o ) =−c 2 o a 2 , while for nonzero h∈ H QH o , Q o (h) = q o (h,h) > 0. The orthogonal splitting therefore shows that Q o has signature (−, +,..., +). Its differential is dQ o | v (w) = 2q o (v ⊥ ,w ⊥ )− 2c 2 o b Θ o (v) b Θ o (w). Ifv ̸= 0,Q o (v) = 0, anddQ o | v = 0, choosingw=T o gives b Θ o (v) = 0. The null equation then gives v ⊥ = 0, so v = 0, a contradiction. Thus the nonzero null set is a smooth hypersurface. Finally, on the normalized section b Θ o (v) = 1, writev=T o +hwithh ∈ H QH o . The boundary equation becomes q o (h,h) = c 2 o , which is theq o -sphere of radiusc o in the Horizon hyperplane, hence a smooth strongly convex ellipsoid in arbitrary linear coordinates. Smooth field assembly follows from smoothness ofq, b Θ,T, the orthogonal projection, and c. Corollary 8.7 (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 8.8 (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 RTP sm , the family o7→ G o is smooth, in the standard semi-Riemannian sense [11, 1], and K + o ={v ∈ V o : G o (v,v) < 0, b Θ o (v) > 0}. 8.6 Logical Status and Claim Boundary Remark 8.9 (Axiomatic status). RTP pt is introduced first as an explicit quantitative bridge principle. Section 9 subsequently gives its pointwise refinement-cost characterization under positive- projective invariance, two-data quantitative closure, and finite refinement capacity. The response 29

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 8.10 (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 8.11 (What RTP adds). RTP pt adds the substantive assertion that each propagation cone is a response-threshold cone with ellipsoidal normalized sections. If this principle is adopted axiomatically, Paper 4 openly posits the quantitative law that makes the local propagation cone quadratic. If RTP is later derived from a quantitative provenance-coherence survival theory, the same theorem becomes a derivation rather than an axiomatic reconstruction. Remark 8.12 (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 . RTP pt provides an explicit pointwise quantitative formulation of propagation refinement. Its unique pointwise cutoff and the identity between qualitative and response-defined propagation are established in Section 9. RTP sm is the separate domain-level assertion that these pointwise capacities, propagation forms, or equivalently rank-one defect forms assemble smoothly. Smooth regularity of the capacity field remains conditional in Section 14. 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. 30

Paper 4 Version 4.3Canonical 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 31

Paper 4 Version 4.3Canonical 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) 32

Paper 4 Version 4.3Canonical 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 .

1. 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) 33

Paper 4 Version 4.3Canonical 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,v ε lies in the positive UPCP half-space but outsideK + o , whose response-threshold description usesQ o ≤0 together with nonnegative advance. This proves (2), (3), and the strict inclusion. Relative to the q o -orthogonal decomposition V o = RT o ⊕ q o H QH o , the quadratic form satisfies Q o (aT o

• h) = q o (h,h)− c 2 o a 2 . It is negative on the nonzero advance line and positive on every nonzero Horizon direction. There- fore its displayed polarization is nondegenerate of signature (−,+,...,+), andQ o (v) =G o (v,v). Combining this identity with the response-threshold definition proves (4). The nonzero boundary of the strict negative cone in the chosen future sector is the displayed future null set. Its differential is dQ o | v (w) = 2q o (v ⊥ ,w ⊥ )− 2c 2 o b Θ o (v) b Θ o (w). Suppose v ̸= 0, Q o (v) = 0, and dQ o | v = 0. Taking w = T o yields 0 = dQ o | v (T o ) =−2c 2 o b Θ o (v), so b Θ o (v) = 0. The null equation then givesq o (v ⊥ ,v ⊥ ) = 0, and positive definiteness givesv ⊥ = 0. The canonical splitting forcesv= 0, a contradiction. ThusdQ o | v ̸= 0 at every nonzero null vector. The regular-value theorem makes the nonzero null set a smooth conic hypersurface. 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 34

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Horizon, Lorentz signature, smooth null boundary, and the response-induced trichotomy. It does not derive RTP, smooth regular representability of the bridge class, tangent-bundle identification, four-dimensionality, spacetime reconstruction, or physical dynamics. The phrase “separatrix-to-null- barrier transition” names a staged refinement of geometric roles, not an equality betweenH QH o and ∂K + o . Corollary 10.3 (No Horizon-tangent null generators). If 0̸= h∈ H QH o , then G o (h,h) = q o (h,h) > 0. Hence no nonzero Horizon-tangent vector is a null generator of the propagation boundary. Every future null generator has strictly positive UPCP advance and nonzero transverse response satisfying q o (v ⊥ ,v ⊥ ) = c 2 o b Θ o (v) 2 . Remark 10.4 (Interpretive formulation). The informational separatrix supplies the advance– equilibrium–retreat organization. The response threshold then restricts the advance sector by requiring transverse response to remain bounded relative to normalized advance. Saturation of that bound produces a null ray. The resulting boundary is therefore a barrier in propagation type: timelike-propagating directions lie inside, null-critical directions lie on it, and positively oriented non-propagating directions lie outside. 10.2 Horizon uniqueness Lemma 10.5 (Horizon Uniqueness). LetV o be the QII-admissible continuation space ato, equipped with the normalized UPCP covector b Θ o , the inherited positive-definite response metricq o , and the pointwise response capacity c o

0. Define H QH o := ker b Θ o , T o := b Θ ♯ q o o , 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.

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 . 35

Paper 4 Version 4.3Canonical 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 .

1. 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. 36

Paper 4 Version 4.3Canonical 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), A o (v) := c 2 o a 2 , and Q o (v) := F o (v)− A o (v). Assume c o

0 and, when stated below, H QH o ̸={0}. Then:

1. 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 .
2. For every nonzero Horizon vector h∈ H QH o , Q o (h) = F o (h) = q o (h,h) > 0. Hence no nonzero Horizon vector is propagating or null.
3. If H QH o ̸={0}, then K

o ∩ H QH o ={0}. 37

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Thus the Horizon is disjoint from all nontrivial propagating and critical directions. 4. Inside the future-oriented sector C + o ={v : b Θ o (v) > 0} one has the trichotomy K + o ={v : F o (v) < A o (v)}, N + o ={v : F o (v) = A o (v)}, F + o ={v : F o (v) > A o (v)}, with C + o = K + o ⊔ N + o ⊔ F + o . 5. The future null barrier consists exactly of the critical balances q o (h,h) = c 2 o a 2 , a > 0. Every nonzero null generator therefore possesses both a nonzero advance component and a nonzero Horizon component. 6. The Local Quantum Horizon performs two distinct structural roles: (a) it is the equilibrium separatrix of the UPCP orientation geometry; (b) it is the canonical transverse response reference space appearing in the response-threshold propagation law. Consequently, propagation admissibility, criticality, and non-propagation are classified through the balance between normalized advance and Horizon-measured transverse response. 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 QH o ̸={0}andh ̸= 0, thenQ o (h)>0, so no nonzero Horizon vector satisfies the closed propagation inequalityQ o ≤0; thusK + o ∩ H QH o ={0}. Statement (4) is a direct rewriting of the sign alternatives forQ o =F o − A o . Statement (5) follows from the equality caseQ o = 0. Finally, the first role of the Horizon follows fromH QH o =ker b Θ o , while the second follows because the propagation law depends on transverse response only through the Horizon projection. Therefore all propagation classifications are expressed through the balance between normalized advance and Horizon-measured transverse response. Remark 10.8 (Claim boundary). The theorem is internal to the adopted response-threshold architecture. It does not assert that no alternative propagation theory could be formulated, nor that the Horizon exhausts the entire non-propagating sector. The theorem only establishes that the current propagation law classifies admissibility, criticality, and non-propagation through the balance of normalized advance and Horizon-measured transverse response. 38

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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:

1. 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.
2. The Quantum Horizon is intrinsically recoverable from the completed hybrid metric data: H QH o = kerD o = ker(q o − G o ). Equivalently, H QH o ={h∈ V o : q o (h,w) = G o (h,w) for all w ∈ V o }. Thus H QH o is the unique maximal linear subspace on which q o and G o agree.
3. Define the defect map D ♭ o : V o → V ∗ o , D ♭ o (v) = D o (v,·). Then imD ♭ o = R b Θ o . Thus (q o ,G o ) recovers the unoriented covector line. Together with the retained positive UPCP orientation, the positive-projective ray [Θ o ]

is recovered. 4.(Uniqueness of the normalized response-threshold representation.) Suppose the same exact 39

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 . 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 . 40

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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. 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 41

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 o := K

o ∩ S + o , B o := ∂ S + o D 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 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:

1. B o is a nonempty compact smoothq o -sphere of radiusc o inT o +H QH o and bounds the strongly convex ball D o ;
2. 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;
3. r o | B o ≡ c o ;
4. the common geometric radius of B o is exactly the unique informational refinement cutoff; 5.the sectorsr o < c o ,r o =c o , andr o

c o are respectively propagating, null-critical, and future-oriented non-propagating. Thus NBR2 ′′ , NBR3, NBR4, and pointwise NBR5 hold. Proof.Everyv ∈ S

o has the unique formv=T o +hwithh∈ H QH o , andρ o (T o +h) = p q o (h,h). The pointwise Refinement-Cost Characterization therefore gives D 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 . 42

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 (Remaining assembly obligation). The theorem is pointwise and does not prove thatB= F o B o assembles smoothly overU QII . Establishing this without assuming smoothness of o7→ c o is the remaining NBR1 obligation. 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 remaining regularity obligation is domain-level: one must prove that the fiberwise normalized boundaries assemble smoothly without presupposing smoothness of the capacity assignment. Equivalently, one must establish smooth variation of the scalar defect strengthλ(o) = 1 +c 2 o . Boundary assembly, smooth capacity assembly, smooth propagation-form assembly, and smooth defect assembly are therefore equivalent regularity formulations within the locked pointwise architecture. A proof must derive one of them independently, rather than assume another. This is the next target required to remove the temporary smooth regular representability assumption in Section 14. 11 Response-Capacity Saturation Law Fix a regular operational pointo. LetV o be the finite-dimensional QII-admissible continuation space, letq o be the inherited positive-definite response metric, and let b Θ o ∈ V ∗ o be the normalized UPCP covector. Define T o := b Θ ♯ q o o , H QH o := ker b Θ o , C + o :={v ∈ V o : b Θ o (v) > 0}. Assume the Horizon Equilibrium Structure Theorem, so thatH QH o is the first-order UPCP informational-equilibrium separatrix and primary invariant balance manifold, with canonical orthog- onal splitting V o = RT o ⊕ q o H QH o . Every v ∈ V o therefore has the unique decomposition v = A o (v)T o

• v ⊥ , A o (v) := b Θ o (v), v ⊥ ∈ H QH o . For v ∈ C

o , define the Horizon-measured transverse response burden and the refinement ratio by R o (v) := q q o (v ⊥ ,v ⊥ ),RR o (v) := R o (v) A o (v) . Assume the pointwise Response-Threshold Propagation Principle and its refinement-cost characteri- zation. Thus there exists a unique response capacity c o

0 such that v ∈ K

o ⇐⇒ v ∈ C + o and RR o (v) < c o . 43

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 . 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 44

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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)}. 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. 45

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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}. 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}. 46

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry (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 nonidentity. The informational-equilibrium separatrix and the emergent propagation boundary are distinct: H QH o ∩ C + o = ∅, ∂K + o {0}⊂ C + o . Hence H QH o ̸= ∂K + o . The Horizon organizes the advance/refinement decomposition; RTP selects the saturation class within the positive sector; Theorem 10.1 identifies that class as the null propagation boundary. (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
. 47

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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. Thus the Horizon and the nonzero propagation boundary are disjoint and cannot be identified. This proves BET5. Finally, every v ∈ S + o has the unique form v = T o

• h with h∈ H QH o . On this slice, Q o (T o
• h) = q o (h,h)− c 2 o . Therefore Q o = 0 is equivalent to q o (h,h) = c 2 o , which proves BET6. Corollary 12.2 (Null Boundary Emergence). Within the HES-organized orientation geometry, RTP introduces a second sign-changing classification inside the positive UPCP sector. The saturation class of that classification emerges as the null propagation boundary: HES + RTP + Theorem 10.1 =⇒ Null Boundary Emergence. The Quantum Horizon is the informational-equilibrium separatrix; the null boundary is the response- capacity saturation interface. The latter is generated within the balance geometry organized by the former, but the two are not the same object. Remark 12.3 (Logical status and claim boundary). BET is a synthesis theorem. It does not derive HES, RTP, the unique pointwise capacityc o , or the Separatrix-to-Null-Barrier Transition Theorem. It consolidates their consequences into the precise null-boundary-emergence statement. The result is pointwise. It does not prove smooth base-point assembly ofo7→ c o , of the boundary fibersB o , or of the global propagation geometry, and it does not establish SSR. 48

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 . 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). 49

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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) Horizon-boundary distinction. The informational-equilibrium separatrix and the null propagation boundary are structurally distinct: H QH o ∩ C + o = ∅, ∂K + o {0}⊂ C + o . Hence H QH o ̸= ∂K + o . The Horizon organizes the balance architecture; RCSP selects the saturation class; BET realizes that class as the boundary. (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. 50

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 follows because every Horizon vector has zero UPCP advance, whereas every nonzero boundary vector belongs to the positive UPCP sector. Thus the two objects are disjoint and cannot be identified. 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, 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 Response-Capacity Regularity This section supplies the final regularity upgrade. It separates the informational assembly of pointwise capacities from their geometric realization. 51

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 14.1 Theorem 1: Informational Refinement-Capacity Assembly Theorem 14.1 (Informational Refinement-Capacity Assembly). LetU QII be a fixed-rank QII- regular domain withV=A QII 0 → U QII of rankk ≥2. Assume the canonical QII–LAS–UPCP quantitiesA o ,R o , RR o are defined throughout the domain, with the LAS response quantity inherited from Theorem 4.4; assume Positive-Projective Invariance, Two-Data Quantitative Closure, and Finite Refinement Capacity. Then for every o:

1. RR o (C

o ) = [0,∞); 2. there is a unique F o : [0,∞)→{0, 1} with Prop o (v) = F o (RR o (v)); 3. J o = F −1 o (1) is uniquely determined; 4. there is a unique c o ∈ (0,∞) with J o = [0,c o ) and c o = supJ o ; 5. A o ,R o , RR o vary continuously on the total positive sector; no denominator singularity occurs because A o (v) > 0 there; 6. the capacities assemble set-theoretically as Σ J ={(o,c o ) : o∈ U QII }. No continuity or smoothness of o7→ c o is asserted. Proof.Proposition 9.4 yields full ratio attainability becausek ≥2. Lemma 9.3 therefore gives the unique classifier on [0,∞). Finite Refinement Capacity makesJ o a proper relatively open initial interval containing zero, henceJ o = [0,c o ) for the unique finite positive endpoint. Continuity of the pre-capacity quantities follows from the smooth fixed-rank bundle data and positivity of the denominator. Uniqueness of eachc o gives the set-theoretic graph; smoothness is a later conditional question. Remark 14.2 (Claim boundary). This theorem characterizes the pointwise cutoff from explicit constituent premises. Finite Refinement Capacity remains substantive, and smooth capacity assembly is not concluded. 14.2 Information–Geometry Bridge Definition 14.3 (Information–Geometry Bridge; locked Version 1.2). For the relation Σ J above, define the canonical informational sign representative Φ J (o,r) :=        −1, 0 < r < c o , 0, r = c o , 1, r > c o . A scalar map Φ : U QII × (0,∞)→ R is a bridge realization if sgn Φ(o,r) = Φ J (o,r). Two realizations are bridge-equivalent, written Φ∼ J Ψ, when they induce the same sign partition. The intrinsic bridge object is the equivalence class [Φ] J . 52

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Proposition 14.4 (Existence and zero-set projection). The class [Φ] J is nonempty, and every representative satisfies Φ −1 (0) = Σ J . The zero-set projection Z([Φ] J ) := Σ J is therefore well defined. Proof.The trichotomyr < c o ,r=c o , orr > c o makes Φ J a well-defined representative. Sign- equivalent representatives have identical negative, zero, and positive sets, so they share the zero set Σ J . 14.3 Theorem 2: Response-Capacity Regularity Assumption 14.5 (Temporary smooth regular representability). The bridge class [Φ] J admits a globally smooth representative Φ∈ C ∞ (U QII × (0,∞)) such that ∂ r Φ(o,c o )̸= 0 for every (o,c o )∈ Σ J . Theorem 14.6 (Response-Capacity Regularity; locked Version 1.1). Under the temporary smooth regular representability assumption:

1. Σ J = Φ −1 (0) is a smooth embedded codimension-one hypersurface;
2. Σ J is transverse to every vertical refinement-cost fiber;
3. the threshold assignment c(o) := c o is a unique smooth positive function, c∈ C ∞ (U QII );
4. the canonical smooth boundary normal form is Φ c (o,r) := r− c(o),Φ −1 c (0) = Σ J , ∂ r Φ c = 1. Proof.Because∂ r Φ(o,c o )̸= 0, one hasdΦ (o,c o ) ̸= 0 along the zero set. Thus 0 is a regular value, and the Regular Value Theorem makes Σ J = Φ −1 (0) a smooth embedded hypersurface of codimension one. The same cost-direction derivative gives transversality to the vertical fibers. At each (o,c o ), the Implicit Function Theorem supplies a neighborhoodW o and a unique smooth local functionc (o) :W o →(0,∞) satisfying Φ(x,c (o) (x)) = 0. The informational assembly theorem gives exactly one threshold above every base point, so the local functions agree on overlaps and patch into a global smooth functionc. The stated canonical normal form follows immediately. 53

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Corollary 14.7 (Smooth response-capacity realization of propagation). Define Q c o (v) := q o (v ⊥ ,v ⊥ )− c(o) 2 b Θ o (v) 2 . Then Q c is a smooth intrinsic field of fiberwise quadratic forms, and K + o = K + o,c ={v ∈ V o : b Θ o (v) > 0, Q c o (v) < 0} assembles into the smooth intrinsic response-capacity realization of the propagation geometry. Remark 14.8 (Remaining open regularity question). The RCR conclusion is conditional only on smooth regular representability of [Φ] J . A future theorem deriving that property from the informational architecture may be inserted before the Response-Capacity Regularity Theorem without changing the informational assembly theorem, bridge definition, or geometric proof. Remark 14.9 (Conditional bridge status). The response-capacity regularity theorem is conditional on Temporary Smooth Regular Representability. It proves that such representability is sufficient for smoothness ofc; it does not derive that premise. The next section gives a separate conditional discharge under Smooth Informational Assembly and Selector Rigidity. 15 Smooth Informational Assembly and Selector Rigidity The pointwise refinement-cost results determine a unique finite positive threshold in every admissible fiber. The remaining domain-level issue is whether those thresholds assemble smoothly. The following principle states the load-bearing regularity commitment explicitly and independently of the HRCEL threshold itself. Principle 15.1 (Smooth Informational Assembly). Let p : U QII −→P be the complete propagation-state assignment, constructed from the primitive QII–LAS–UPCP– refinement and qualitative-propagation data without using an independently prescribed threshold field, capacity field, propagation form, boundary bundle, bridge representative, or SSR structure. Assume:

1. p varies smoothly on the selected fixed-rank QII-regular domain;

every admitted datum capable of changing propagation classification or threshold selection is represented in p; 3. every intrinsic observable under consideration is invariant under QII, LAS, gauge, and threshold- redundant equivalences; 4. no autonomous residual mode can change such an observable while leavingpunchanged; and 5.the observable controls a regular distinction in the architecture, so an irregularity would create an omitted distinction, residual mode, branch, or structural singularity. 54

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Then every single-valued intrinsic observable uniquely determined bypinherits the smooth regularity of the complete state assembly. For each o∈ U QII , let J (2) o = [0,σ o ), σ o := c 2 o

0 be the accepted squared refinement-cost set and its unique endpoint. On the normalized slice S

o = T o

• H QH o , the squared refinement cost is C o (T o
• h) = q o (h,h),Hess(C o | S

o ) = 2q o | H QH o . Definition 15.2 (Autonomous selector freedom). An autonomous selector freedom is a nontrivial admissible parameter or residual mode that changesσ o while leaving the complete propagation state p(o) unchanged and survives the declared QII, LAS, gauge, threshold-equivalence, and no-surplus identifications. Remark 15.3 (Conditional status). Smooth Informational Assembly is an explicit foundational regularity premise, not a consequence of QII, LAS, UPCP, RTP, or pointwise capacity uniqueness. The following theorem proves only the implication from SIA and the stated Selector Rigidity hypotheses to smooth capacity assembly. Theorem 15.4 (Conditional Smooth Informational Assembly and Selector Rigidity). Assume Principle 15.1. Suppose further that:

1. the endpoint σ o is uniquely determined by the complete propagation state p(o);
2. no autonomous selector freedom exists; and
3. the selected frontier is fiberwise regular: whenever q o (h,h) = σ o , d h [q o (h,h)][h] = 2σ o

Then, conditional on Principle 15.1, there exists a unique intrinsic endpoint functional S : p(U QII )−→ (0,∞) such that σ = S◦ p. Moreover, σ ∈ C ∞ (U QII , (0,∞)), c := √ σ ∈ C ∞ (U QII , (0,∞)). Consequently the pointwise capacity relation assembles into a smooth positive response-capacity field. Proof. Endpoint uniqueness and complete determination imply that equal propagation states have equal accepted squared-cost sets and hence equal endpoints. Thusσis constant on every fiber ofp, so there is a unique functionalSwithσ=S◦ p. No autonomous selector freedom excludes any additional admissible variable capable of changing σ while p remains fixed. 55

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry The selector is single valued, intrinsic, equivalence-invariant, free of autonomous residual modes, and controls a regular frontier. It therefore satisfies the Smooth Informational Assembly Principle and inherits the smooth regularity of the complete propagation-state architecture. Henceσis smooth. Because σ > 0, smooth functional calculus gives c = √ σ ∈ C ∞ (U QII , (0,∞)). Corollary 15.5 (Discharge of smooth regular representability). The smooth capacity field supplied by Theorem 15.4 gives the canonical global bridge representative Φ c (o,r) := r− c(o),Φ −1 c (0) = Σ J , ∂ r Φ c = 1. Thus the temporary smooth regular representability assumption in Section 14 is discharged on every domain satisfying Principle 15.1 and the hypotheses of Theorem 15.4. Remark 15.6 (Noncircularity and claim boundary). Selector rigidity alone proves factorization and absence of independent selector freedom; it does not by itself prove continuity or smoothness. Smoothness is supplied only by the Smooth Informational Assembly Principle, which is an explicit foundational regularity premise rather than a consequence of the pointwise architecture. The principle is stated independently ofc,σ, the HRCEL boundary package, and SSR, and applies only to intrinsic observables satisfying its completeness, invariance, no-surplus, and regularity conditions. It is the sole new load-bearing regularity assumption for the global smooth and SSR conclusions in this version. 15.1Why Smooth Informational Assembly Is the Appropriate Regularity Prin- ciple Smooth Informational Assembly is an explicit foundational principle. It is not derived from pointwise threshold existence, endpoint uniqueness, or the regularity of individual cost spheres. Those facts determine one threshold in each fiber but do not alone determine how the thresholds depend on the base point. The principle addresses a different question: when a complete informational state varies smoothly, under what conditions should an intrinsic observable uniquely determined by that state also vary smoothly? The adopted answer requires four structural features: complete determination, invariance under informationally empty changes, absence of autonomous residual freedom, and control of a regular rather than bifurcating or degenerate distinction. Selector Rigidity provides the factorization σ = S◦ p. This proves that the threshold is not independent of the complete propagation state, but it does not by itself make the endpoint functionalSsmooth. Smooth Informational Assembly supplies the architecture-wide regularity rule under which eligible intrinsic derived observables inherit the smooth structure of the complete state. Possible failures are thereby classified. A threshold-changing distinction omitted frompviolates completeness. A threshold change generated by realization surplus conflicts with LAS or the declared no-surplus quotient. A change at fixedpis an autonomous selector mode and violates Selector Rigidity. Switching between competing local thresholds violates endpoint uniqueness. Fiberwise 56

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry frontier degeneration is excluded because on the normalized slice C o (T o

• h) = q o (h,h),Hess(C o | S

o ) = 2q o | H QH o , and at the selected level d h [q o (h,h)][h] = 2c 2 o

0. If QII rank changes, the UPCP ray ceases to be smooth, the response metric degenerates, or the complete state ceases to vary smoothly, then the assumptions fail; such points lie outside the selected fixed-rank QII-regular domain. Thus the principle does not claim that every set-theoretic function of smooth data is smooth. Its scope is restricted to intrinsic observables of a complete smooth informational state after omitted information, realization surplus, autonomous freedom, branch switching, and frontier degeneration have been excluded. Remark 15.7 (Claim boundary). Smooth Informational Assembly does not prove completeness of the selected state description and does not extend smoothness across rank-changing, degenerate, or bifurcating loci. Completeness is a foundational commitment of the architecture, and the theorem applies only on the connected fixed-rank QII-regular domain. Remark 15.8 (Regularity boundary and conditional assembly). The preceding informational, pointwise, and fiberwise results—including QII conservation, LAS realization, UPCP orientation, Horizon structure, pointwise RTP, fiberwise Lorentz-signature representation, response-capacity saturation, boundary emergence, and HRCEL—do not depend on an unconditional globally smooth capacity assignment. The domain-level smooth cone and Horizon results remain valid under their explicitly stated smooth-bundle hypotheses. Smooth assembly of the capacity field, propagation form, defect tensor, null-boundary bundle, threshold graph, bridge representative, and the SSR endpoint requires a separate regularity premise or theorem controlling base-point dependence. In the present version, Principle 15.1 is that explicit premise, and Theorem 15.4 is conditional upon it. Thus no pointwise uniqueness or selector-rigidity assertion is used by itself to infer smooth base-point variation. 16 Conditional HRCEL Global Smoothness and Strong Smooth Regular Realizability Theorem 16.1 (Conditional HRCEL Global Smoothness and SSR). Let π : V −→ U QII be a smooth fixed-rank QII-admissible vector bundle of rank n≥ 2. Assume:

1. the inherited response metric is smooth and positive definite, q ∈ Γ ∞ (Sym 2 V ∗ ); 57

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 2. the normalized UPCP covector is a smooth nowhere-vanishing section, b Θ∈ Γ ∞ (V ∗ ), q −1 ( b Θ, b Θ) = 1; 3. T := b Θ ♯ q , H QH := ker b Θ; 4.the HES, RCSP, BET, and HRCEL hold onU QII , together with Principle 15.1 and Theo- rem 15.4. Then the complete HRCEL propagation architecture admits a global smooth regular realization over U QII . In particular, c∈ C ∞ (U QII , (0,∞)); the propagation formG, defect tensorD, propagation-cone fieldK + , nonzero null-boundary hypersurfaceN + , normalized boundary bundleB, threshold graph Σ J , and canonical bridge representative Φ c all assemble smoothly; and Strong Smooth Regular Realizability holds. Proof. 1. Smooth Horizon decomposition. Since q and b Θ are smooth, T = b Θ ♯ q is smooth. Since b Θ is nowhere zero and has constant rank one, H QH = ker b Θ is a smooth codimension-one subbundle. Define the transverse projection P ⊥ (v) := v− b Θ(v)T. Because T and b Θ are smooth, P ⊥ is a smooth bundle endomorphism. Moreover, b Θ(P ⊥ v) = 0, q(T,P ⊥ v) = 0. Therefore V = RT ⊕ q H QH is a smooth orthogonal splitting. 2. Smooth advance and transverse-response functions. On the positive sector C + :={(o,v)∈ V : b Θ o (v) > 0}, define A(o,v) := b Θ o (v) and the squared transverse response R 2 (o,v) := q o (P ⊥ o v,P ⊥ o v). 58

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry Both functions are smooth on the total space of V . Hence RR 2 (o,v) := R 2 (o,v) A(o,v) 2 is smooth on C + . The unsquared ratio RR(o,v) := p R 2 (o,v) A(o,v) is smooth whereverR(o,v)>0. The quadratic formulation below remains smooth along the advance ray R = 0. 3. Smooth global saturation functional. By Theorem 15.4, c∈ C ∞ (U QII , (0,∞)). Define Q(o,v) := R 2 (o,v)− c(o) 2 A(o,v) 2 . Every constituent of Q is smooth; therefore Q∈ C ∞ (V ). Its fiberwise zero set in C + is precisely the RCSP saturation relation Σ c :={(o,v)∈ C + : Q(o,v) = 0}. Thus the pointwise saturation classes assemble into a global zero-set relation defined by a smooth function. 4. Smooth propagation form. Polarizing Q fiberwise gives G o (v,w) := q o (P ⊥ o v,P ⊥ o w)− c(o) 2 b Θ o (v) b Θ o (w). Since q, P ⊥ , c, and b Θ are smooth, G∈ Γ ∞ (Sym 2 V ∗ ). Relative to V o = RT o ⊕ q o H QH o , write v = aT o

• h with h∈ H QH o . Then G o (aT o
• h,aT o
• h) = q o (h,h)− c(o) 2 a 2 . Therefore every G o is nondegenerate with Lorentzian signature (−, +,..., +). Since the rank of V is fixed, the signature is constant across U QII . 59

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 5. Smooth defect tensor. Using the orthogonal splitting, q o (v,w) = b Θ o (v) b Θ o (w) + q o (P ⊥ o v,P ⊥ o w). Define D := q− G. Then D = (1 + c 2 ) b Θ⊗ b Θ. Because c and b Θ are smooth, D ∈ Γ ∞ (Sym 2 V ∗ ). Moreover, kerD o = ker b Θ o = H QH o . Thus the exact hybrid defect assembles smoothly and globally recovers the smooth Quantum Horizon over U QII . 6. Smooth propagation-cone field. Define K + :={(o,v)∈ V : b Θ o (v) > 0, Q(o,v) < 0}. Because b Θ and Q are smooth, K + is open in the total space of V . Fiberwise, K + o ={v ∈ V o : b Θ o (v) > 0, G o (v,v) < 0}. Hence the pointwise propagation cones assemble into a smooth intrinsic open conic field. 7. Smooth null boundary. By BET, ∂K + o {0} ={v ∈ C + o : Q o (v) = 0}. For a nonzero future-null vector v, the fiber derivative is d v Q o (w) = 2q o (v ⊥ ,w ⊥ )− 2c(o) 2 b Θ o (v) b Θ o (w). If d v Q o = 0, choosing w = T o gives b Θ o (v) = 0. The null equation then gives q o (v ⊥ ,v ⊥ ) = 0. Positive definiteness of q o gives v ⊥ = 0, hence v = 0, a contradiction. Therefore d v Q o ̸= 0 on the nonzero future-null set. Thus 0 is a regular value there, and N + :={(o,v)∈ C + : Q(o,v) = 0} is a smooth conic hypersurface in V \ 0. 60

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 8. Smooth normalized boundary assembly. Define the normalized affine subbundle S + :={(o,v)∈ V : b Θ o (v) = 1}. Every v ∈ S + o has the unique form v = T o

• h, h∈ H QH o . On S

, Q o (T o

• h) = q o (h,h)− c(o) 2 . Therefore the normalized boundary is B :={(o,T o
• h) : h∈ H QH o , q o (h,h) = c(o) 2 }. Sinceq,T,H QH , andcare smooth, and the fiber derivative in every nonzero radial direction is nonzero, B is a smooth sphere subbundle of S

. Its fiber is exactly B o ={T o

• h : q o (h,h) = c 2 o }. Thus the normalized null boundaries assemble smoothly.

1. Smooth threshold graph and bridge normal form. The capacity graph Σ J :={(o,c(o)) : o∈ U QII } is the graph of a smooth function and therefore a smooth embedded codimension-one submanifold of U QII × (0,∞). Define Φ c (o,r) := r− c(o). Then Φ c ∈ C ∞ (U QII × (0,∞)),Φ −1 c (0) = Σ J , ∂ r Φ c = 1. Hence the information–geometry bridge has a global smooth representative with vertically nonde- generate zero crossing.
2. Strong Smooth Regular Realizability. The smooth HRCEL realization consists of:
3. the smooth equilibrium-separatrix bundle H QH ;
4. the smooth canonical advance field T;
5. the smooth response metric q;
6. the smooth capacity field c;
7. the smooth Lorentz-signature propagation form G;
8. the smooth rank-one defect tensor D;
9. the smooth proper cone field K

; 61

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 8. the smooth nonzero null hypersurface N + ; 9. the smooth normalized boundary sphere bundle B; 10. the smooth threshold graph Σ J ; and 11. the smooth vertically nondegenerate bridge representative Φ c . Therefore the pointwise HRCEL architecture admits a global smooth regular realization overU QII . In the locked terminology of the program, this is precisely Strong Smooth Regular Realizability. Hence Smooth Informational Assembly + Selector Rigidity =⇒ SSR. Moreover, c, G, D, K + , N + , B, Σ J , Φ c all assemble smoothly over U QII . Remark 16.2 (Scope). Here, global means global over the entire selected fixed-rank QII-regular domain U QII . No extension across QII rank-changing loci is asserted. 17 A Finite-Dimensional Model of the HRCEL Construction The following model displays the completed architecture in concrete coordinates. It is not an empirical model, does not identify the continuation bundle with physical spacetime, and is not used in the proofs. Let U ⊆ R be a connected open interval and let V = U × R 4 . Write v = (a,x,y,z) and define q o (v,w) = aa ′

• xx ′
• yy ′
• zz ′ , b Θ o (a,x,y,z) = a. Then T o = (1, 0, 0, 0), H QH o ={(0,x,y,z)}, V o = RT o ⊕ q o H QH o . For v = (a,x,y,z), v ⊥ = (0,x,y,z), A o (v) = a, 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 v 7→ λv for λ > 0. Let the complete propagation state select a smooth positive fieldσ:U →(0,∞), and setc(o) = p σ(o). The accepted squared-cost interval is J (2) o = [0,σ(o)), so 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 ′ , 62

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry with matrix [G o ] = diag(−c(o) 2 , 1, 1, 1). Thus G o has signature (−, +, +, +) even though the response metric q o remains Euclidean. 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 }. On the normalized slice a = 1, D norm 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 D o = λ(o) b Θ o ⊗ b Θ o . Then G = q− D. 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. This example illustrates the advance–Horizon splitting, the scale-free refinement ratio, the distinction betweenqandG, the saturation boundary, the normalized sphere bundle, smooth defect assembly, and the bridge normal form. The choicen= 4 is made only for visualization. In rankn, one has dimH QH o = n− 1 and B o ∼ = S n−2 ; no dimensional preference follows from the example. 18 Discussion 18.1 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. 63

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry The second structure arises only after the introduction of the Response-Threshold Propagation Principle. RTP imposes an additional criterion inside the positive UPCP sector by comparing transverse minimum response with normalized advance. The result is a smaller propagation region contained within the larger orientation geometry. Positive orientation alone is no longer sufficient. Propagation becomes a stronger condition requiring bounded transverse response. This distinction reveals that orientation and propagation are not the same geometric notion. A continuation may be positively oriented while failing to satisfy the propagation criterion. Conversely, every propagating continuation is necessarily positively oriented. Orientation therefore provides a necessary condition for propagation, while RTP supplies the additional structure required to identify which oriented directions genuinely support propagation. The distinction between the Quantum Horizon and the propagation boundary is a direct manifestation of this hybrid architecture. The Horizon belongs to the orientation geometry and marks the equilibrium sector of the UPCP decomposition. The propagation boundary belongs to the RTP geometry and separates propagating from non-propagating positively oriented continuations. Because these structures arise at different stages of refinement, they need not coincide. Indeed, one of the principal conclusions of the present framework is that the Horizon is not the propagation boundary. The Horizon remains fixed while RTP refines the positive sector into distinct classes of positively oriented continuations. The resulting propagation geometry possesses a significantly different character from the underlying orientation geometry. RTP defines a quadratic functional whose associated bilinear form has Lorentzian signature on each QII-admissible fiber. The inherited response metric remains positive definite and continues to provide the normalization, orthogonal decomposition, and transverse geometry. The Lorentz-signature structure is carried by the response-defined quadratic form rather than by the underlying response metric itself. Consequently, two geometries coexist within the framework. The orientation geometry is generated by UPCP and organizes the admissible continuation space into positive, zero, and negative sectors. 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. 64

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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, conditional on RTP, only after those earlier structures have been established. 18.2 Architectural and scope observations The architecture assigns distinct roles to its principal components. QII supplies a gauge-invariant admissible continuation space but no preference. LAS supplies a unique no-surplus realization of a fixed admissible continuation but no choice among continuations. UPCP supplies a local order and orientation but neither conservation nor variational realization. The Local Quantum Horizon is the equilibrium sector that becomes meaningful only when these three structures are imposed together. The positive and negative sectors do not require a preferred ray. The resulting response-threshold form has Lorentzian signature on each QII-admissible fiber, but the present work stops short of constructing a Lorentzian manifold. It provides no tangent-bundle identification, integrability theorem, geodesic or curvature dynamics, global causal order, four-dimensional reduction, or identification with physical spacetime. This stopping point is intentional. It isolates a rigorous local operational geometry without presupposing the stronger global structures that later work may investigate. 18.3 Lessons Learned from the Refinement Program The original objective of this program was not to construct a specific spacetime geometry. It was to determine how much geometric structure could be uncovered before stronger organizing principles became necessary. The development has produced the following lessons. Lesson 1: The Quantum Horizon is not the propagation boundary. The Quantum Horizon arises from the UPCP orientation structure as the equilibrium sector of the admissible continuation space. RTP introduces a distinct propagation criterion within the positive UPCP sector. The Horizon forms the equilibrium sector separating the positive and negative UPCP sectors, whereas the propagation boundary separates propagating and non-propagating positively oriented continuations. Lesson 2: Orientation is weaker than propagation. UPCP identifies positive, zero, and negative sectors, but positive orientation alone does not guarantee propagation. A continuation may be positively oriented while exceeding the response threshold. Positive orientation is necessary, but not sufficient, for propagation. Lesson 3: Propagation structure emerges as selection. QII identifies admissible contin- uations, UPCP orders them, and RTP selects the subclass satisfying the response criterion. The propagation cone is therefore obtained through refinement within a larger space of admissible orientations rather than introduced as a primitive object. Whether a complete causal structure ultimately emerges remains a question for later work. 65

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 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 conditional on 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. 19 Scope, Limitations, and Open Questions The informational, pointwise, and fiberwise conclusions are unconditional relative to their stated local hypotheses. A global smooth HRCEL realization over the selected fixed-rank QII-regular domain is established only conditionally on Smooth Informational Assembly and the hypotheses of the Selector Rigidity theorem. Under that separate regularity premise, the HES supplies the smooth informational-equilibrium separatrix and balance architecture; RCSP identifies the unique saturation class; BET identifies its null-boundary realization; HRCEL organizes these structures; and the Global Smooth HRCEL Realization theorem establishes SSR and smooth assembly ofc,G, D, K + , N + , B, Σ J , and Φ c . The remaining limitations lie beyond this regularity result. 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 a physical spacetime identification, geodesic or curvature dynamics, gravitational field equations, reachability and growth laws, or global causal reconstruction. Here “global” means global on the selected fixed-rank QII-regular domain; no extension across QII rank-changing loci is asserted. 66

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry 20 Conclusion Paper 4 Version 4.3 establishes a staged informational-to-geometric architecture and supplements the formal development with a dependency map, logical-status table, regularity discussion, and finite-dimensional worked model. QII supplies admissibility, LAS supplies canonical realization, and UPCP supplies orientation. Their synthesis yields the Quantum Horizon, which the Horizon Equilibrium Structure Theorem identifies as the UPCP informational-equilibrium separatrix and primary invariant balance manifold. The response-threshold architecture then refines positive orientation into propagation. RCSP identifies the unique response-capacity saturation class, and BET proves that this class is the nonzero null propagation boundary. HRCEL unifies these results into the organizing balance law R o (v)≶ c o A o (v), with sub-capacity propagation, exact null-critical saturation, and super-capacity non-propagation. The pointwise results do not by themselves promote the capacity assignment to a smooth field. Conditional on the separate Smooth Informational Assembly Principle and the hypotheses of the Selector Rigidity Theorem, the pointwise capacities assemble into a smooth positive field over the selected fixed-rank QII-regular domain. Under that regularity premise, the propagation formG, defect tensorD, cone fieldK + , nonzero null hypersurfaceN + , normalized boundary bundleB, threshold graph Σ J , and canonical bridge representative Φ c assemble smoothly. The resulting conditional Global Smooth HRCEL Realization establishes Strong Smooth Regular Realizability: SIA premise + Selector Rigidity hypotheses =⇒ SSR. The resulting geometry remains a smooth Lorentz-signature propagation geometry on the QII- admissible bundle, not yet a reconstruction of physical spacetime. Tangent-bundle identification, 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. [2]Luca Bombelli, Joohan Lee, David Meyer, and Rafael D. Sorkin. Space-time as a causal set. Physical Review Letters, 59(5):521–524, 1987. [3]Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Informational derivation of quantum theory. Physical Review A, 84(1):012311, 2011. [4] Lucien Hardy. Quantum theory from five reasonable axioms, 2001. [5] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of Differential Geometry, volume 1 of Interscience Tracts in Pure and Applied Mathematics. Interscience Publishers, New York, 1963. 67

Paper 4 Version 4.3Canonical Realization and Hybrid Geometry [6]Shoshichi Kobayashi and Katsumi Nomizu. Foundations of Differential Geometry, volume 2 of Interscience Tracts in Pure and Applied Mathematics. Wiley-Interscience, New York, 1969. [7]James Ladyman, Don Ross, David Spurrett, and John Collier. Every Thing Must Go: Meta- physics Naturalized. Oxford University Press, Oxford, 2007. [8]John M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, 2 edition, 2013. [9]John M. Lee. Introduction to Riemannian Manifolds, volume 176 of Graduate Texts in Mathematics. Springer International Publishing, Cham, 2 edition, 2018. [10] Juan M. Maldacena. The largeNlimit of superconformal field theories and supergravity. Advances in Theoretical and Mathematical Physics, 2(2):231–252, 1998. [11]Barrett O’Neill. Semi-Riemannian Geometry with Applications to Relativity, volume 103 of Pure and Applied Mathematics. Academic Press, New York, 1983. [12] R. Tyrrell Rockafellar. Convex Analysis, volume 28 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1970. [13] Mark Van Raamsdonk. Building up spacetime with quantum entanglement. General Relativity and Gravitation, 42(10):2323–2329, 2010. [14]John Archibald Wheeler. Information, physics, quantum: The search for links. In Wojciech H. Zurek, editor, Complexity, Entropy, and the Physics of Information, volume 8 of Santa Fe Institute Studies in the Sciences of Complexity, pages 3–28. Addison-Wesley, Redwood City, CA, 1990. [15] John Worrall. Structural realism: The best of both worlds? Dialectica, 43(1–2):99–124, 1989. 68