VPT3 – Admissibility Geometry and Invariance of Linear and Quadratic Virtual Probability Readouts Cone Positivity, Score-Visible Nonnegativity, Readout-Kernel Fibers, Null-Set Geometry, and Declared Interference Diagnostics
Virtual Probability Theory represents observable finite probability laws as normalized readoutsof latent virtual contribution objects. VPT1 [1] established the declared readout architectureand the admissible-score-to-probability theorem. VPT2 [2] classified the readout fibers andobservable quotient structures induced by aggregation, scoring, normalization, and observation.This paper develops the third foundational layer: the admissibility geometry and invariancetheory of fixed declared linear and quadratic score mechanisms.The linear regime is modeled on a finite-dimensional ordered real vector space (V, K), whereK is a closed convex pointed cone and each declared score functional belongs to the dual cone,ℓO,y ∈ K∗.The paper distinguishes cone-positive aggregate domains, visible nonnegative-score domains,and fully admissible domains. It proves that cone-positive aggregate domains are inverse-imagecone domains, develops visible-cone gap theory for nonseparating score channels, formalizesconstrained reachability of visible gaps, identifies hidden cone-violation cancellation, and recoversordinary finite source probability and its deterministic contextwise pushforwards as the positivescalar-linear sector.The quadratic regime is modeled on real or complex finite-dimensional mode spaces withfixed positive semidefinite forms GO,y . The paper proves the null-channel characterizationsquad,AO (y) = 0 ⇐⇒ AO,y A ∈ ker GO,y ,and describes the admissible domain as the complement of contextwise null sets. It then studiesreadout-kernel invariance, real and complex scale invariance, global phase invisibility, localrelative-phase diagnostics, declared component interference, and partition-coarsening behavior.The contribution is not new cone duality or new quadratic-form theory in isolation. Thecontribution is a regime-specific admissibility and invariance calculus for fixed VPT readoutmechanisms. On the nonempty admissible domain, observable probabilities remain ordinaryfinite probability laws throughout.
Virtual Probability Theory represents observable finite probability laws as normalized readoutsof latent virtual contribution objects. VPT1 [1] established the declared readout architectureand the admissible-score-to-probability theorem. VPT2 [2] classified the readout fibers andobservable quotient structures induced by aggregation, scoring, normalization, and observation.This paper develops the third foundational layer: the admissibility geometry and invariancetheory of fixed declared linear and quadratic score mechanisms.The linear regime is modeled on a finite-dimensional ordered real vector space (V, K), whereK is a closed convex pointed cone and each declared score functional belongs to the dual cone,ℓO,y ∈ K∗.The paper distinguishes cone-positive aggregate domains, visible nonnegative-score domains,and fully admissible domains. It proves that cone-positive aggregate domains are inverse-imagecone domains, develops visible-cone gap theory for nonseparating score channels, formalizesconstrained reachability of visible gaps, identifies hidden cone-violation cancellation, and recoversordinary finite source probability and its deterministic contextwise pushforwards as the positivescalar-linear sector.The quadratic regime is modeled on real or complex finite-dimensional mode spaces withfixed positive semidefinite forms GO,y . The paper proves the null-channel characterizationsquad,AO (y) = 0 ⇐⇒ AO,y A ∈ ker GO,y ,and describes the admissible domain as the complement of contextwise null sets. It then studiesreadout-kernel invariance, real and complex scale invariance, global phase invisibility, localrelative-phase diagnostics, declared component interference, and partition-coarsening behavior.The contribution is not new cone duality or new quadratic-form theory in isolation. Thecontribution is a regime-specific admissibility and invariance calculus for fixed VPT readoutmechanisms. On the nonempty admissible domain, observable probabilities remain ordinaryfinite probability laws throughout.
This paper uses the standard triangular odd-cycle anticorrelation obstruction as a finitebenchmark for Virtual Probability Theory. The obstruction itself is classical: three binary paircontexts can be locally overlap-compatible while admitting no positive global gluing, becauseperfect anticorrelation around an odd cycle violates the parity constraint. The contributionis therefore not a new contextuality obstruction, but a complete VPT fiber-and-certificateaudit of this minimal benchmark.On the declared triangular source architecture, the paper proves the benchmark member-ship profileP△ /∈ C+△, P△ ∈ Clin±△ , P△ ∈ CQ2,R△ ,and also proves the fixed-source nonnegative scalar Q2 obstruction and the same-sourcecoherent-versus-incoherent scalar-square control separation. It characterizes the signed-linearfiber, the architecture-relative hidden-negativity minimum, the exact coherent scalar Q2-fiber for the perfect target, the closed noisy scalar realization profile, and a finite-unionaggregate-cone description for noisy targets.All conclusions are finite, architecture-relative, readout-relative, and algebraic. ScalarQ2 is used as a score-generating parametrization, not as a new observable probabilitycalculus. The finite-code certificate is context-outcome-address-free and target-entry-value-free, but explicitly benchmark-pattern-aware; its code-length field is a syntactic audit, not aKolmogorov-complexity, minimum-description-length, or compression theorem
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
We study how much of the Hilbert-space and Born-rule structure of quantum theory can be obtained from operational primitives that involve records but not probabilities. The setting is a regime in which coherent alternatives recombine without leaving a stable record. We obtain three results of different kinds. Geometry. From composable contributions, record-inert reversible controls and amplitude scalings, under named conditions, we obtain a real inner-product event geometry. The existence of the inner product is equivalent to bounded mixing and is assumed in that form. In a worked interferometer with one mode per channel, the metric is not an input: metric-free port swaps and phase flips fix the form up to scale, and boundedness only forces the other controls to preserve it. Its uniqueness up to scale, the orthogonality of distinct outcomes, and the representation of events by orthogonal projections are derived. Obstruction. This geometry, even with full symmetry about the preparation and exact dyadic standards, does not select the quadratic weight: the preparation mixed with the maximally mixed state survives. The structural operations exhibited here certify only values on which the quadratic weight equals the counting weight. On restricted event families where Gleason’s theorem is unavailable, a calibration order oriented by the quadratic magnitude selects it: given exact standards, a state is the quadratic one exactly when it respects all such brackets. Whether such an orientation can be earned is the open problem. Given such brackets and refinement only to a finite depth K, every admissible state lies within 2^-K of the quadratic weight, without the Archimedean step. On the full projection lattice, Gleason’s theorem with null support, or sphere-transitive symmetry with null support, gives other conditional routes. Bridge. A predictive probability kernel is introduced separately. Its identification with the real quadratic weight ||P_A ψ||² / ||ψ||² is conditional on kernel-admission hypotheses, one of which is equivalent to the Born value; the complex form needs phase invariance in addition. The hypotheses select a class of contexts, and none is shown to be forced. For each we state the physical feature it expresses and a sufficient condition where one is known, and we work out one apparatus in which the geometry-stage hypotheses hold together. For all but one we give a context in which it fails (for the founding conditions, one that leaves the regime); for the remaining one, TCC-1, we give an illustration. This paper is the first in a series built on a common framework, Structural Ledger Consistency (in preparation).
Resolution Geometry: A Kernel–Stabilizer Geometry of Observability Abstract Resolution Geometry (RG) is a protocol-explicit geometry of observability constructed from the joint observation law of a declared family of experimental protocols. The primitive object is not a covariance decomposition, a dynamical model, or a metric space, but a complete observation system together with the operational rules that determine which distinctions are admissible. Equality of observation laws induces a universal observational equivalence relation whose kernel groupoid represents exact observational ignorance. The resulting quotient provides the canonical domain on which all experiment-internal statements must be defined. Within regular classical statistical experiments, local distinguishability is represented by the Fisher-Rao metric through the standard information-geometric characterization under sufficient-Markov invariance. Admissible observational sectors are selected by a uniquely characterized spectral projector satisfying idempotence, metric self-adjointness, information compatibility, and threshold consistency. Observable sectors transport naturally under experiment isomorphisms and assemble into compatible observable atlases through identity and cocycle laws. The mature structure of RG is organized by two objects: the kernel groupoid of the joint observation law and the effective symmetry group induced by the stabilizer of that law. The kernel identifies observational equivalence, exact ignorance, and quotient structure. The stabilizer determines an effective symmetry group acting faithfully on the observational quotient. Together they provide the organizing principle from which observable geometry is assembled. A central result is a natural-descent theorem. Observable invariants, Fisher geometry, cone fields, clock readouts, transport structures, and distance protocols are shown to arise as pullbacks from law space and therefore descend automatically to the observational quotient. On regular strata, exact ignorance is represented by the kernel groupoid while infinitesimal ignorance is represented by the radical of the Fisher tensor. Representative invariance, kernel basicness, and projectability emerge as consequences of naturality rather than independent assumptions. The framework separates several logically distinct layers. The quotient layer characterizes what an experiment can and cannot distinguish. The metric layer quantifies local distinguishability. The admissibility layer determines which distinguishable directions survive stability, conditioning, nuisance closure, and transport requirements. The symmetry layer determines which higher-level geometric structures are forced. Entropy appears only downstream as a derived quantity associated with observational quotients and unresolved sectors, recovering familiar entropy constructions as special cases rather than primitives. Higher-level physical structure emerges through symmetry content. Lorentzian symmetry forces the quadratic causal cone and conformal class on regular strata. Equal clock kernels generate a duration groupoid whose transition maps satisfy a forced cocycle law and become affine under compatible time flows. Euclidean symmetries determine unique two-point invariants and metric proportionality classes. Counted channels remove dilation freedom and provide the anchor necessary for operational distance identification. Physical distance therefore appears not as a primitive notion but as a derived consequence of observational equivalence, symmetry structure, and experimentally anchored protocols. A covariance realization provides a concrete computational model of the general framework. Schur-complement identities describe observable geometry in the presence of nuisance structure; resolved-null coupling characterizes estimator improvement; canonical correlations provide observable-sector diagnostics; and transport admits gauge-theoretic and globalization interpretations. These realizations supply practical tools for inverse problems, statistical estimation, information geometry, experiment design, and automated sector identification, while remaining consequences of the general theory rather than its axioms. The geometric picture that emerges is not a single smooth manifold. Observable geometry naturally occupies a Whitney-stratified orbit-space bundle whose regular strata support Fisher-Riemannian path geometry, transport, holonomy, and monodromy, while singular strata encode degeneracy structure, admissible distinguishability, kernel transitions, orbit-type changes, and unresolved directions. The resulting assembly combines observational quotients, information geometry, kernel groupoids, symmetry reduction, cocycle descent, and stratified geometry into a unified framework for partial observability. The constituent mathematical ingredients—statistical experiment theory, Fisher geometry, observational quotients, groupoids, symmetry reduction, spectral projectors, Schur complements, canonical correlations, cocycle theory, and stratified geometry—are classical. The contribution of Resolution Geometry is their characterization, integration, and assembly into a single kernel-stabilizer geometry of observability. Scope and non-claims are explicit: the framework does not derive admissibility protocols from first principles, does not identify observability with ontology, does not derive physical laws from information alone, and does not reduce physics to statistics. Its purpose is to characterize, for a declared experiment, the geometry of what can be distinguished, transported, synchronized, measured, and consistently inferred. Repository The static, inferential, classificatory, and dynamical results are facets of a single object and are linked by exact identities; a result from any one layer rests on the invariants of the others. Readers building on any component are asked to cite the framework as a whole. Released under CC BY-NC-ND 4.0, [DOI]; the results are offered in the ordinary scientific manner, with the single request that their common origin be acknowledged.
CPT-Coherence / Mirror-Mind Theory formulates physical law as branch-local descent from a parent-boundary coherence structure. The fundamental datum consists of a parent Hilbert space ℋ6, a closed semibounded parent action 𝒜6, a self-adjoint parent operator 𝒦6, a boundary trace system γ∂, an antiunitary CPT reflection ΘCPT, a signed boundary Green form 𝒲∂, and a selected reducing branch projector Πours. Observable law is obtained through trace quotient, passive Schur compression, flux-null boundary selection, reflection-positive reconstruction, and stable observable descent. This paper establishes the apparent-transport sector of that theory. Apparent superluminal readout is not local faster-than-light motion. It is the branch-local situation in which the operational arrival time between fixed endpoint events is reduced relative to the background comparison time, R[p, q] = Tbr[p, q] / Tbg[p, q], 0 < R[p, q] < 1, while local propagation remains causal with respect to the effective branch cone. The corresponding background-inferred velocity vapp = Lbg[p, q] / Tbr[p, q] = c / R[p, q] may exceed c only as a comparison readout; no branch signal leaves its local causal cone. The transport mechanism is generated by the frozen half-line parent sector X+ = Σ4 × ℝ+, with fourth-order coherence-depth operator: 𝒦τ = α∂τ4 - ∂τ2 + μ, α > 0, μ ≥ 0. The form-domain traces u(0) and ∂τu(0) define the accessible boundary variables, while the conjugate flux traces ∂τ2u(0) and ∂τ3u(0) enter through the Green form and the Dirichlet-to-Neumann sector. This separation produces a fourth-order boundary phase space, a Lagrangian residual law, a positive boundary-loading block, and the falsifiable impedance scaling ||N∂(λ)|| = C|λ|3/4(1 + o(1)) along admissible large-resolvent rays. The same residual-square mechanism co-generates the trace-access metric MQ and the hidden-visible Schur coupling, so the transport response is not inserted as a free metric profile. The gravitational sector is governed by conservative branch descent. A branch geometry carries a gravitational interpretation only when the coherence-depth exchange current vanishes, Jτ = 0, and the projected four-dimensional Einstein equation Gμν(4) = κ4Teffμν is accompanied by conservation, ∇μTeffμν = 0, together with null, weak, averaged-null, chronology, boundary-power, and flux-null ledgers. The theory therefore permits apparent superluminal readout only through causal parent-boundary response, never through unaccounted metric insertion or local cone violation. The primary empirical signature is not macroscopic transport. It is the correlated boundary-response triad 𝓞triad = (ΔT, Δφ, ΔSspec), measured under frozen endpoint, clock, phase, spectral, and boundary controls, together with the fourth-order impedance exponent 3/4 distinguishing the parent-boundary sector from ordinary second-order propagation. Apparent transport is thereby reduced to a precise operator-boundary question: a lawful branch response exists exactly when parent-generated impedance, Schur compression, effective principal symbol, arrival-time response, causal admissibility, and conservative energy accounting close simultaneously. Keywords: CPT-Coherence; Mirror-Mind Theory; apparent superluminal readout; parent-boundary theory; boundary impedance; Dirichlet-to-Neumann map; fourth-order operator; flux-null states; Schur compression; conservative branch gravity; effective metric; arrival-time response; causal admissibility; energy conditions.
This paper develops a reconstruction of probability that does not begin by assuming a probability space, a probability simplex, or the Born rule. Instead, it asks a prior foundational question: under what physical and mathematical conditions are a set of alternatives entitled to carry positive, additive probabilities at all? The construction begins with lawful histories, future interfaces, and retained records. It distinguishes coherent alternatives from exclusive empirical events, preventing multiple interfering routes from being mistaken for separate probabilistic outcomes. A future-sufficient quotient identifies distinctions that no admitted future test can detect, while a record-qualification step determines when alternatives have become genuine empirical events. Dagger-compatible comparison and positive record factorization then produce a positive comparison kernel, from which normalized event probabilities arise only after the relevant record sectors are certified as exclusive. The framework is extended beyond individual measurements. Compatible finite record refinements generate a global probability measure under explicit realizability conditions; stochastic kernels provide a compositional calculus for carrying probabilistic information through unresolved future structure; and locally classical measurement contexts are assembled into a single positive state on a generally non-Boolean effect structure. This allows globally coherent probability to exist even when no context-independent deterministic assignment of outcomes exists. On a reflection-positive Hilbert-space branch, admissible future contexts are represented by positive effects and positive normalized states by density operators. The Born state-effect law then appears as the finite-dimensional representation of a probability structure that has already been earned through record qualification, positivity, exclusivity, refinement, and contextual consistency rather than introduced as a primitive postulate. The paper also separates two mechanisms that are often conflated: quadratic Born-type weighting arises from dagger self-comparison, whereas equal probabilities for symmetric alternatives arise from symmetry of the outcome structure. A further state-effect bridge analysis classifies residual coordinate freedom and identifies an exceptional rank-two adjugate transformation with Lorentzian signature, which is removed when positive self-dual identification is required. The central interpretation is that probability can function as a lawful form of global completion: it can carry future-relevant law through unresolved structure even when deterministic uniqueness is unavailable. The paper does not derive a mechanism selecting one actual outcome and does not claim that every physical sector satisfies the stated hypotheses. Instead, it provides an explicit dependency-controlled architecture connecting lawful completion, reflection positivity, event qualification, contextuality, stochastic continuation, Hilbert reconstruction, and the Born rule.