This paper develops the obstruction-theoretic grammar of Variable-Signature QuantumTheory and Obstruction Theory. Its central convention is claim-theoretic rather thanontological: a variable-signature quantum family supports a channel-relative nonclassicalityclaim only when a licensed, schema-generated recovery problem has a soundly certifiedfailure of recovery from a declared comparison class under declared admissibility constraints.The paper refines the earlier coarse triviality tuple into a partial annotated obstructionpacket whose components record decision values, invariant values, statuses, and certificatesfor representation, transition, algebraic, dynamical, operational, geometric, topological,stochastic, and compression channels. The construction separates recoverability decisions frominvariant obstruction values and proof certificates, introduces partial domains to distinguishzero obstruction from undefined, deferred, or out-of-scope channels, formalizes structurallicensing through licensed recovery schemas and proof ledgers, formalizes certification throughsound verifiers, and adds a claim-export gate separating invariant detection from certifiednonclassicality claims. The resulting framework is deliberately conservative: moving sectors,moving metrics, and moving projections are not automatically nonclassical; they support arelative nonclassicality claim only when a structurally licensed recovery problem has soundlyverified no-recovery in a specified channel and passes the declared claim-export standard
This paper develops the obstruction-theoretic grammar of Variable-Signature QuantumTheory and Obstruction Theory. Its central convention is claim-theoretic rather thanontological: a variable-signature quantum family supports a channel-relative nonclassicalityclaim only when a licensed, schema-generated recovery problem has a soundly certifiedfailure of recovery from a declared comparison class under declared admissibility constraints.The paper refines the earlier coarse triviality tuple into a partial annotated obstructionpacket whose components record decision values, invariant values, statuses, and certificatesfor representation, transition, algebraic, dynamical, operational, geometric, topological,stochastic, and compression channels. The construction separates recoverability decisions frominvariant obstruction values and proof certificates, introduces partial domains to distinguishzero obstruction from undefined, deferred, or out-of-scope channels, formalizes structurallicensing through licensed recovery schemas and proof ledgers, formalizes certification throughsound verifiers, and adds a claim-export gate separating invariant detection from certifiednonclassicality claims. The resulting framework is deliberately conservative: moving sectors,moving metrics, and moving projections are not automatically nonclassical; they support arelative nonclassicality claim only when a structurally licensed recovery problem has soundlyverified no-recovery in a specified channel and passes the declared claim-export standard
This paper develops the state and measurement layer of Variable-Signature QuantumTheory and Obstruction Theory. Building on the projected positive-sector geometry ofVSQT-OT I, it defines θ-density states, algebraic states, effects, finite POVMs, sectorcompletely positive maps, and passive instruments intrinsically on the physical sectorEθ = Pθ K, Hphys,θ = (Eθ , ⟨·, ·⟩θ ).The paper fixes the sector trace, θ-adjoint, sector positivity, amplified sector positivity, andsector complete-positivity conventions, carefully distinguishing abstract sector endomorphismsfrom ambient representatives. Its main results prove compact convexity of the θ-density statespace, Born positivity and normalization for sector POVMs, selected-operator-system states,algebraic-state restriction and positive-extension representation on finite-dimensional ∗θ-subalgebras, finite-dimensional sector Kraus representation, Schrödinger–Heisenberg dualityfor sector CP maps, passive instrument normalization, instrument-to-POVM compatibility,admissibility of stratumwise measurement families, pointwise equivalence with ordinary finite-dimensional measurement theory, a fixed-Hilbert obstruction criterion, and fixed-Hilbertrecovery in Born-pair, POVM, and instrument forms. The paper does not introduce a newfinite-dimensional Born rule or a new Kraus theorem; pointwise sector measurement theoryis ordinary finite-dimensional measurement theory written in the sector Hilbert structure.Its contribution is the sector-typed formulation of the standard measurement calculus overvariable positive sectors, together with a first obstruction class detecting when the sectorbundle cannot be globally reduced to one admissible fixed-Hilbert carrier, gauge-compatibleadmissibility rules for stratumwise families, a formal admissible comparison class, and arecovery criterion identifying when that layer is ordinary after compatible admissible transport.The selected-observable layer is an operator-system layer by default, with selected ∗θ -algebrastreated as a stronger optional case. All operator-algebraic statements are finite-dimensionaland algebraic; no infinite-dimensional, unbounded, spectral, semigroup, or von Neumannalgebraic operator theory is claimed. The paper is finite-dimensional, stratumwise, andnon-dynamical. Active inter-sector instruments are isolated as a separate typed skeletonrather than claimed as a consequence of passive measurement calculus
This paper develops the deterministic finite-dimensional dynamical layer of Variable-Signature Quantum Theory and Obstruction Theory. The regular finite-dimensional theoryis deliberately conservative: it does not produce a new local Schrödinger law. Its main roleis to calibrate VSQT-OT by proving fixed-Hilbert recoverability on regular intervals and byisolating nontriviality into typed admissibility, comparison, global, or transition constraints.Along a regular path t 7 → θ(t) inside a fixed-rank positive stratum, the physical Hilbert spaceis the moving sectorEt = PtK, Ht = (Et, ⟨·, ·⟩t),where the restricted sector metric gt induced by an ambient representativê Gt is positive.The paper defines sector-metric compatible connections, sector Schrödinger equations, sector-unitary propagators, covariant von Neumann dynamics, and transported state/effect/POVMcalculus. Its main results show that compatible dynamics preserves sector membership, sectornorm, density states, effects, POVMs, and Born probabilities in transported or Heisenbergform. It also proves that every regular finite-dimensional interval satisfying the standing C1hypotheses admits a C1 fixed-Hilbert reduction. In any such reduction,Dt = St∇tS−1t = ∂t + Γt, Γ†t = −Γt,and the effective Hamiltonian is˜Ht = SthtS−1t − iΓt.Thus regular pathwise finite-dimensional dynamics over intervals is mathematically fixed-Hilbert ordinary at the intrinsic dynamical level, relative to the unrestricted admissibletrivialization class and the full finite-dimensional time-dependent Hermitian Hamiltoniancomparison class. Nontriviality can be claimed only after declaring admissibility, comparison-class, global, or transition constraints
This paper develops the measurement-stability layer of the VSQT-OT program. Build-ing on the finite-dimensional sector measurement calculus of VSQT-OT II and the typedobstruction methodology of VSQT-OT IV, it studies finite-outcome passive measurementsover declared signature regimes and local measurement charts. The signature parameter θ isnot derived endogenously here; it is treated as a declared static parameter, prescribed path,external control, measurable stochastic index, or deferred measurement-updated/endogenousvariable. Once a pre-regular chart, typed local sector gauge, compatible finite measurementrecord, comparison channel, and the relevant quantitative measurement-control certificatesare supplied, the paper proves quantitative stability estimates for Born probabilities, fi-nite POVM distributions, passive instrument probabilities, and passive posterior stateson positive-probability domains. It also proves transported standard recovery, same-rankrecovery under exact recovery gauges, perturbative recovery under declared dual positivequasi-transport, and channelwise finite commuting-POVM statistical convergence by distance-to-class certificates. The concrete commuting-POVM comparison class is given an explicitfinite stochastic representation through joint spectral decomposition; it is not identified withjoint measurability, noncontextuality, instrument-level classical simulation, or a global classi-cal phase-space limit. Finally, it proves that rank loss, positivity-gap closure, failed algebratransport, and zero-probability conditioning obstruct unconditional full all-state/all-effectmeasurement-statistics continuation unless transition data are declared; zero-probability con-ditioning obstructs uniform posterior-stability estimates based only on unnormalized-outputcontrol, even for ordinary finite two-outcome CPTP instruments
This paper develops the first mathematical layer of Variable-Signature Quantum Theoryand Obstruction Theory in the finite-dimensional deterministic setting. The core object is asignature-parameterized family inside a fixed ambient complex vector space K ∼=CN, withparameter space Θ, Hermitian forms Gθ, representing matrices Bθ, physical-sector projectionsPθ, and selected sector-preserving observable algebras Aθ. The Paper-I projected physicalsector is not the positive cone of Gθ, but the selected positive sector HIphys,θ = (Eθ,Gθ|Eθ).On each regular stratum Θα, projected positive sectors form finite-rank Hilbert bundles. Thepaper distinguishes ambient comparability from smooth fixed-Hilbert triviality and introducesfirst-layer coarse obstructions to sector representation, transition extension, and conditionalselected-algebra transport. The algebra obstruction is evaluated only after a smooth globalsector representation exists. On regular strata, the paper recovers the ordinary fixed-Hilbertform of finite-dimensional sector kinematics, numerical Born state-effect pairings under sectorrepresentation, and selected algebra transport when the relevant algebra obstruction alsovanishes. This first paper proves mathematical nontriviality criteria for sector geometry; itdoes not claim empirical deviation from standard quantum mechanics or a full operationalreconstruction of measurement theory.
This document develops a mathematical framework for two-boundary quantum admissibility. Given initial and final wave fields ψi = √ρieiSi/ℏ and ϕf= √ρfeiSf /ℏ , one asks when an intermediate local support structure is compatible with both boundary data. The central diagnostic is the log-transport defect of a power-family density ρα = ρα i ρ1−α f transported by the symmetrized velocity v = (ui + uf)/2. 1This defect decomposes exactly into an irreducible geometric obstruction, a density- reweighting channel, and one-boundary continuity residuals. The scalar parameter α is thereby recognized not as the whole theory, but as the first repair test. Scalar reweighting removes only a one-dimensional projection of the geometric-mean defect. Promoting α to a local field α(x,t) converts the repair problem into a first-order transport equation, and failure of that equation gives a stronger form of obstruction. Finally, nodal winding mismatch TC = m 2πℏ C(ui−uf)·dℓ= ni(C)−nf(C) is invariant under scalar and local density reweightings, yielding a non-reweightable topological obstruction. The framework lifts naturally to configuration space, where physical event succession is described by admissible support connectivity rather than by ordinary- space path continuity. Classical trajectories arise as narrow projected support tubes, while quantum nonclassicality appears when admissible connectivity persists in configuration space although its ordinary-space projection is broad, thresholded, or coarse-grained. Coherence dynamics and holonomy sectors are discussed as extensions of the admissibility hierarchy rather than as part of its theorem-level core.
We formulate a contract-relative operational test for whether future-predictive quantum information lies in the post-processing capability of any admissible present description. For a declared present-access family and typed recovery policy, the present-completion deficiency is the diamond-norm distance between the predictive target and a contract-defined present-capability set. The underlying quantum channel deficiency and sufficiency mathematics is established prior art; the contribution is its use inside a path-sensitive black-hole reconstruction framework that separates coarse endpoint from rich present state, claim descent from witness descent, information insufficiency from failure of joint realizability, and local recovery from path monodromy. We prove access- and target-monotonicity under explicit typed closure conditions, give a pairwise distinguishability-gap lower bound, state a finite-dimensional attainment criterion for finitely many compact channel-type components, and derive the single/joint/virtual access hierarchy. The exact [[5,1,3]] perfect-code benchmark covers all ten three-retained and all ten two-retained access channels: every three-retained channel is exactly reversible, every two-retained channel is completely depolarizing with optimal diamond-norm deficiency 3/2 to the logical identity, and a Bell-projector discrimination certificate attains that lower bound. Two distinct three-retained erasure histories have the same coarse endpoint but different rich-present channels while both remain exactly recoverable, providing a direct null calibration of the endpoint split. The Bousso-Kaya-Lin-Shahbazi-Moghaddam finite-region family is retained as a coarse-data insufficiency benchmark, not yet as a no-present-completion theorem. The next black-hole-specific target is a certified positive fiberwise completion lower bound surviving operational quotienting, joint-realizability controls, and declared contract variations.