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 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 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 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 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
Paper 3 of a four-paper program on generalized quantum measurement, realization geometry, and informational invariance. Generalized quantum measurements admit multiple purification and dilation realizations of the same operational state–measurement data. This paper studies the geometric structure induced by that representation freedom and develops a transport theory on the resulting realization space. On each finite-dimensional regular fixed-rank state–POVM stratum, framed purification amplitudes and framed Naimark realizations are shown to assemble into a smooth principal bundle whose fibers are precisely compact auxiliary gauge orbits. The paper establishes complete-POVM robustness results controlling states, labeled measurement architectures, fixed coarse events, canonical normalization, and smooth operational paths. A framed Hilbert–Schmidt response geometry is then introduced on the realization bundle. The associated minimum-response construction yields a smooth gauge-equivariant principal connection and an induced positive response metric on the operational base manifold. This provides a geometrically distinguished transport law and establishes relative canonicity within the framed response geometry. The manuscript further introduces Quantum Informational Identity (QII) and Dynamical Quantum Informational Identity (DQII) as explicit foundational identity assumptions. Exact gauge transformations are shown to preserve the generated informational profile of a realization, while smooth gauge paths preserve the corresponding dynamical profile history. Under QII and DQII, all quantum-informationally admissible principal connections determine the same represented quantum-physical transport content. Relative to the framed Hilbert–Schmidt response geometry, the minimum-response connection is the unique minimum-response representative of that common content. The paper refers to this combination of: uniqueness of represented quantum-physical transport content, and uniqueness of the framed minimum-response representative, as absolute canonicity within the generated realization framework. The results are restricted to finite-dimensional generated realizations on regular fixed-rank strata with fixed outcome labels. The paper does not claim that the framed response geometry is uniquely derived from informational principles, nor does it extend the transport theory across singular or rank-changing loci.
We develop a systematic framework for quantum mechanics on a finite graph-resolved class of second-countable, T1, non-Hausdorff one-manifolds. The central problem is that local differential expressions do not by themselves determine the quantum theory: non-Hausdorff incidence, bundle transport, smooth extension, and analytic completion can impose additional global constraints on the operator domain. We formulate these constraints using transported jets, Whitney realisability, Sobolev traces, and formal return holonomy, obtaining exact closure results for finite smooth graph resolutions.
At first Sobolev order, scalar dynamics reduce to a weighted quantum graph whose edge weights arise from a marked resolving presentation, leading naturally to weighted Kirchhoff Hamiltonians. Multiple-origin lines and circles, branching junctions, finite trees, and split-and-rejoin geometries then yield explicit deficiency indices, reflection and transmission laws, interference conditions, compact dark states, and embedded cavity modes.
For finite-rank Hermitian bundles, unitary gluing selects a transmitting fixed subspace, so scattering becomes projection onto the invariant sector of the subgroup generated by the gluing matrices. Compact-group representations therefore turn non-Hausdorff incidence into an invariant-sector selection mechanism; connected compact semisimple groups require at most two relative gluing matrices for full invariant completion, with explicit SU(3) examples.
The resulting theory separates topological non-Hausdorff data from the analytic and representation-theoretic structures that remain detectable by quantum dynamics.