We develop a spacetime-free reconstruction from a faithful state on a structured family of observable algebras. Compatible half-sided modular inclusions generate positive translations and, after finite homogeneous assembly, a pointed Lorentz cone. Each faithful standard split inclusion determines a canonical Petz--Gram transfer, and we isolate a faithful modular boundary-code condition with three clauses: compact discarded response, a uniquely dominant displacement sector, and complete-order recovery of product defects. Under this condition, overlap transport, and the assembly hypotheses, we prove strict tail contraction, completely bounded two-boundary factorization, exact coherently aligned M2(C) embeddings, selection of three response dimensions, and complex multiplication up to the opposite algebra. Exact finite-dimensional and type-III counterexamples show that none of the three local code clauses follows from faithfulness, split or modular nuclearity, positivity, KMS symmetry, or covariance alone. A genuinely interacting spin-1 AKLT/Haldane chain realizes the logical edge algebra without microscopic qubits, with closed-form transfer bands and completely bounded encoding distortion 1+O(3^{-N}). Relative modular two-jets define the response fields (C,u,Q); their Gaussian covariance dynamics are Lie--Poisson with modular-spectrum Casimirs, and pseudodifferential commutators yield the mass-independent Q^{ij} deformation anchor. Spatial locality, branch smoothness, and two-derivative closure remain explicit conditional inputs.
We develop a spacetime-free reconstruction from a faithful state on a structured family of observable algebras. Compatible half-sided modular inclusions generate positive translations and, after finite homogeneous assembly, a pointed Lorentz cone. Each faithful standard split inclusion determines a canonical Petz--Gram transfer, and we isolate a faithful modular boundary-code condition with three clauses: compact discarded response, a uniquely dominant displacement sector, and complete-order recovery of product defects. Under this condition, overlap transport, and the assembly hypotheses, we prove strict tail contraction, completely bounded two-boundary factorization, exact coherently aligned M2(C) embeddings, selection of three response dimensions, and complex multiplication up to the opposite algebra. Exact finite-dimensional and type-III counterexamples show that none of the three local code clauses follows from faithfulness, split or modular nuclearity, positivity, KMS symmetry, or covariance alone. A genuinely interacting spin-1 AKLT/Haldane chain realizes the logical edge algebra without microscopic qubits, with closed-form transfer bands and completely bounded encoding distortion 1+O(3^{-N}). Relative modular two-jets define the response fields (C,u,Q); their Gaussian covariance dynamics are Lie--Poisson with modular-spectrum Casimirs, and pseudodifferential commutators yield the mass-independent Q^{ij} deformation anchor. Spatial locality, branch smoothness, and two-derivative closure remain explicit conditional inputs.
This paper establishes the structural closure layer of the canonical QMS–spectral program. Working entirely within the framework of uniformly continuous quantum Markov semigroups (QMS) on von Neumann algebras, the paper proves that strict contractivity, faithful invariant states, and complementary slackness at active constraints force a series of nontrivial structural consequences. The results do not assume spacetime geometry, field equations, or particle content. They are operator-theoretic consequences of contractive fixed-point dynamics. The principal structural results include: Algebraic collapse at the active constraint surface via a quantitative pinching gap: superpositions of fixed-point and transient components are excluded exactly, not merely asymptotically. Born-exponent rigidity: modular implementability and the Jordan–von Neumann parallelogram characterization force the unique Schatten exponent p = 2 within the unitarily invariant operator norm family. Recovery of the Born probability rule via Gleason’s theorem once Hilbert geometry is enforced. Strict directedness of the non-fixed-point sector induced by spectral gap and contraction. Primitive completeness: the fixed-point algebra is completely characterized by the commutant of the GKSL generator data; no additional independent structural primitive exists. Exact constraint count equal to eight under bidual independence. Minimal admissible carrier classification: symmetric bilinear structure is uniquely compatible with the constraint budget; linear and higher-arity carriers are excluded. Sector parameter exclusion via Osterwalder–Schrader measure positivity: the deformation parameter theta must vanish in order for Euclidean positivity and positive-norm Hilbert space reconstruction to hold. The paper proves that once strict contractivity is imposed on structural data, the fixed-point algebra, probability geometry, carrier structure, and admissible sector parameters are rigidly determined. This completes the structural closure layer of the five-part canonical QMS–spectral program: Construction layer – canonical spectral bridge and OS positivity. Rigidity layer – spectral compactness, Weyl invariance, and capacity constraints. Structural closure layer – algebraic collapse, modular rigidity, primitive completeness, and sector exclusion (this paper). Universal reduction theorem – reduction from minimal relativistic QFT assumptions to the canonical framework. Universality theorem – forcing of Einstein dynamics at leading order. The present work contains no phenomenological assumptions and introduces no adjustable parameters. All results follow from operator-algebraic fixed-point structure and contractive dynamics.
Quantum theory and Lorentzian geometry both make essential use of ordered cones, but a mathematical cone isomorphism does not show that the corresponding orders have the same physical origin. This paper asks whether quantum-statistical positivity and an independently reconstructed one-sided transfer order can be represented as two aspects of a single physical transfer sector. A finite-dimensional local transfer semigroup defines a proper cone CT . A statistical state–effect duality, pulled back through a dynamically faithful process–state representation, defines a typed map ♯T : VT → VT ∗ . Witness pullback completeness fixes the dual order operationally, while causal-sharp compatibility (CSC) requires ♯T (CT ) = CT ∗ . An explicit counterexample shows that CSC does not follow from statistical self-duality, witness completeness, or cone homogeneity alone. If admissible invertible order filters act transitively on int CT , the cone is symmetric and therefore Jordan-representable. Intersecting an independent complex-matrix branch JT ≃ Hermn (C) with a quadratic causal-transfer algebraic boundary forces n = 2, giving CT ≃ Herm2(C)+≃ L4. Familiar qubit–Lorentz identities then become cross-pipeline consistency tests between independently reconstructed quantities. The construction is local, finite-dimensional, and conditional; it does not derive spacetime dynamics or gravity.
Relational quantum causal processes formulate finite operational contexts as normal positive functionals on local completely positive maps. Response differences generate an influence algebra, and its central projections define jointly readable Boolean events. We develop this starting point through a sequence of exact and controlled models. Fresh-environment unitary collision circuits produce dephasing-exchange kinetics with an exact charge-center fixed algebra, a uniform finite-step limit at fixed response order, and graph-controlled metastable Markov dynamics. An absorbing-state model exhibits a sharp transition between non-Abelian quantum memory and Boolean records. A reversal-covariant defect dynamics generates a locally finite partial order on a restricted graph family without assuming a Lyapunov time. Conditional on a certified order, a positive additive record measure, compactness, and identifiability, we prove subsequential convergence to a Lorentzian metric-measure space, finite reconstruction bounds, and uniqueness of admissible smooth limits. Complementary finite regulators provide controlled tests of modular-to-boost response, null tomography, same-update variational identities, induced quadratic gravity, and compatible common-refinement limits. These results are exact or controlled within their stated models, but they do not yet constitute a single background-independent microscopic law that jointly generates adjacency, time, volume normalization, dimension, signature, nonlinear Einstein constraints, and quantum matter. We therefore present RQCP-QG as a theorem-indexed framework that separates established mechanisms, conditional compositions, and open assumptions.
We construct a supported local spacetime adaptation of the observation-boundary contract of VII-E and the finite hybrid-channel contract of VII-G. The two contracts are proved independently within one laboratory on I-B’s prescribed conical Lorentzian product. For the observation boundary, we prove the effect-pullback criterion, its reference-stable form, the exact sharp-measurement coherence criterion, and a qubit separation of trace distance 1/2 between ideal residual targets with identical records. For hybrid evolution, we prove a complete branch representation, history-sensitive observation closure, spectator consistency, and the exact recovery–constant-complement criterion. An informative classical record consequently excludes recovery of every unknown quantum input, even when the declared residual is also available. The additional contribution is an end-to-end finite observation package that keeps these distinctions visible in supplied spacetime. We provide a compactly supported 3+1-dimensional lift of a four-face transverse adjacency exchange, an exact guarded reversible encoding, and a separately constructed qubit instrument selecting strictly ordered slices without changing the conical metric. A two-step specialization has two histories with the same terminal slice but different conditional residuals. After coarse aggregation, the record remains unchanged while two complete residual targets have the exact half-diamond gap 4(1 − 1/√2)/25. An explicit residual audit enlarges the required input observation space from the diagonal algebra to the full qubit algebra. A destructive finite clock, complete-channel approximation bounds, and a causal route for both payload and classical control complete the local Grade S construction. The package may serve as a precise testbed for distinguishing record-level claims from instrument-level claims in geometric feedback models. The channel and recovery principles themselves are established finite-dimensional facts; the claimed contribution is their supported assembly and the explicit certificates and separations. No metric emergence, gravitational dynamics, field instrument, radiative coefficient, or quantum Ward identity is constructed.
Bostelmann, Cadamuro and Minz noted that they presented no proof of convergence of finite-dimensional modular data to the continuum, and that this requires further work; the closest constructive framework, the operator-algebraic scaling-limit programme, likewise leaves modular-data convergence untreated. For half-line free fermions we prove such a theorem for compact-local wavelet schemes, and one delimiting it. Positively, for gauge-invariant quasi-free CAR states, strong exhaustion, strong embedded covariance convergence, and endpoint faithfulness imply norm convergence of local state restrictions and strong-resolvent convergence, after faithful block-completion, of the log-odds modular generators; Trotter–Kato gives local modular-automorphism convergence. We verify these inputs for admissible schemes, including the pinned Daubechies embedding along a dyadic refinement sequence and, under a multiresolution-exhaustion hypothesis, the fixed-order Cohen–Daubechies–Vial boundary model; in the pinned model, local state restrictions on smooth windows away from the endpoint converge at rate O(a^{s}), 0<s ≤ p fixed, those reaching it at rate O(a^{1/2}), the exact order of the driving defect for p ≥ 2 with nonzero endpoint value. Negatively, the locality of this convergence is sharp within a subclass (compactly supported orthonormal-translate interior generator, boundary-free pullback modulo compacts, alias-free, and satisfying the hypotheses of the abstract theorem) for the raw (un-renormalized, infrared-unmodified) fixed-cutoff comparison: a non-exact passband at an interior Fermi-window point makes both Powers–Størmer–Araki square-root defects noncompact, so that comparison supplies no same-folium global normality, whereas exact passband with trace-class boundary error does. Compact support forbids exact passband: the locality yielding local modular convergence obstructs the raw global comparison.
We derive a causal law of observer-indexed entropy retrieval: retrieved reference information grows toward the supply available to a finite observer. A recorded admission-and-readout instrument produces orthogonal operational sectors from arbitrary coherent radiation; exact diary fidelity and decoupling give the flagged family, while branch errors bound its approximate realization. An exact quantum transfer channel yields the remaining-gap law and its continuous Lindblad evolution. The channel has an explicit unitary collision realization: admitted reference information moves from radiation to memory, while the capture record remains diary-blind. Its memory marginal is the Choi state of an erasure channel, giving an intrinsic positive-coherent-information boundary at retrieved fraction r = 1/2. A strip-holomorphic modular speed limit bounds how rapidly the channel can activate, with the hyperbolic tangent selected uniquely at extremality. The integrated retrieval dose is invariant under a change of clock, making the Kerr lapse cancellation one geometric instance of a general result. At fixed supply and kinetic rate under an affine clock, the transfer and activation laws compose into a closed proper-time retrieval trajectory on the equality branch, identify that trajectory as the sharp fastest-retrieval envelope, and fix the earliest admissible retrieval horizon. In the continuum formulation, Araki relative entropy defines retrieval on restricted observable algebras, and a split Type-I factor provides the regulated density-operator representation. The experimental program follows the same causal sequence through retrospective analysis and staged intervention. Tier 0A determines whether archived analog-horizon data retain the shot-level, timing, protocol, and processing information needed to ask the retrieval question. Tier 0B then tests whether qualifying data contain observer-, protocol-, readout-, or analysis-operator-indexed correlation structure that standard pooling or filtering suppresses. A positive Tier 0B result identifies retrieval-relevant structure and motivates Tier 1; it does not by itself confirm the complete law. The prospective Tier 1 BEC program targets the held-out g²(t₁,t₂) surface, bandwidth scaling, and protocol dependence. Independent activation readout tests the modular bound and tanh rigidity. Reference-tagged supply and memory test cumulative dose, composite-hazard recovery, degraded-channel ordering, and two-memory allocation. The v2 verification artifact provides synthetic benchmarks and numerical consistency checks at bond dimensions D = 4 and D = 8 for the transfer law, inverse recovery, finite-resolution robustness, and matched adversarial alternatives. Standard RT/HRT geometry and a separate accessibility functional place the retrieval dynamics within holographic language. The retrieval law closes the observer-local path between global encoding and retained access. Information conservation, radiation encoding, causal collection, finite-observer retrieval, formal reconstruction, and comprehension are distinct physical operations. The black-hole information problem contains a proper-time access dynamics that global entropy accounting alone does not determine. Physical complexity enters through encoding, supply, clock, activation, transfer, retention, and observation. Those layers locate each source of retrieval failure instead of idealizing it away. The theory orders the encoding, supply, and retrieval clocks; distinguishes late supply from early supply through their overlap with future transfer dose; and gives an exact finite-lifetime retrieval criterion. Here rᵢ(τ) is the normalized reference information retained by observer i, eᵢ(τ) is the causally available supply, kᵢ(τ) is the composite access hazard, and τ_char is the characteristic activation scale.