Ŋaélūi ∿ Ihyora - Reflection Positivity Without a Stationary Symmetric Markov Extension: An Explicit Continuous-Time Counterexample and Hidden Entropy Certificates

Ŋaélūi ∿ Ihyora presents an explicit continuous-time stochastic process that is stationary, invariant under ordinary time reversal, continuous in probability, and Osterwalder–Schrader reflection-positive, yet admits no stationary symmetric Markov extension through an instantaneous measurable observation of the current hidden state. The construction uses one bounded ten-valued scalar record and a known 50-state nonsymmetric continuous-time Markov source with finite entropy production. The central result separates full observed reflection positivity from classical reversible hidden Markov realizability. It addresses the precise continuous-time process formulation printed in Mitter (1991), pp. 104-105, and connects reflection-positive stochastic processes, completely positive matrix factorizations, copositive witnesses, and quantitative inference of hidden irreversibility. Main mathematical result The observed process takes values in {0,…,9}. Its complete observed law is time-reversal invariant, and reflection positivity holds for every bounded finite-horizon future-record feature, rather than merely a chosen list of correlations or equally spaced samples. Nevertheless, the manuscript rules out every stationary symmetric Markov extension in the stated instantaneous-observation class, without imposing a finite hidden-state count or dimension. The exclusion concerns symmetric extensions. The release explicitly supplies a nonsymmetric stationary Markov source that generates the observed process. Its paths are right-continuous step paths, and it is continuous in probability; continuous sample paths are not asserted. Explicit construction and full-record proof The construction starts from a five-cycle stochastic matrix with diagonal entries 9/19, cyclic-neighbor entries 5/19, and remaining entries zero. A 25-state transition kernel combines a reciprocal update rule with a uniform reset of probability 1/100. A clock-bit lift gives a 50-state continuous-time generator with escape rate one. The single scalar observation encodes the visible category and the clock bit. A positive definite matrix metric intertwines the hidden generator with its transpose, fixes the all-ones vector, and commutes with every observed emission projection. These identities yield positive metric Gram representations for reflected cylinder products at arbitrary real observation times. They establish full-record reflection positivity and ordinary observed time-reversal invariance through a written algebraic argument. A separate regeneration argument provides a second route to the construction. A finite-observation obstruction Horn’s five-cycle copositive matrix detects a reflected categorical probability table that is entrywise positive and positive definite but not completely positive. Every stationary symmetric Markov extension would instead give a nonnegative conditional-probability Gram representation, so its pairing with a copositive witness could not be negative. The contradiction survives on a fixed grid of 2001 observation times, k/1000 for integers −1000 ≤ k ≤ 1000. These are observations of one process on a grid, not independent samples. The manuscript proves that the grid witness has a negative pairing whose magnitude is strictly greater than 3753/950000. Consequently, the grid law has total-variation distance at least 3753/1900000 from every grid law admitting an extension in the specified class, using the convention TV(P,Q) = supₐ |P(A) − Q(A)|. The finite-grid argument uses conditional expectations of finitely many events. It does not require a competing hidden extension to have a Poisson clock, right-continuous paths, a finite dimension, or a regular conditional hidden-path law. Finite hidden entropy and quantitative certificates Every positive transition rate of the explicit source has a positive reverse rate. Its entropy-production rate is finite and equals: σ_source = (99/190) log(12394/19) = 3.376696577111608… Logarithms are natural, and time is measured in the units of the rate-one clock. This actual source rate is distinct from the lower certificates inferred from the observed law. For a continuous-time one-flip record feature, the exact copositive deficit is: d_CT = exp(−2) × 5653/118750. Under the manuscript’s continuous-time path-entropy hypotheses, this gives the lower certificate −2 log(1 − d_CT), approximately 0.0129267439493576. The fixed finite grid also gives the conservative exact floor σ > 3753/475000 within the stated all-even additive path-entropy model. These bounds quantify hidden irreversibility within their specified source classes. The release does not claim the sharp minimum entropy among all sources compatible with the complete observed law. Physical thermodynamic interpretations retain the stated reversal, local-detailed-balance, and model assumptions. Supporting theory and scientific uses The manuscript also develops: Bounded scalar and categorical reflection inequalities for hidden-entropy inference. Continuous-time path-affinity bounds and a separate generator bound for instantaneous state observables. Gaussian hidden-state reflection bounds. Exact thermal Ornstein–Uhlenbeck completion geometry from covariance and calibrated conjugate-response data. Minimal-model rank criteria, entropy-nonincreasing thermal reduction, and compatible entropy intervals. Synthetic examples, finite-error calculations, and exact-arithmetic certificate utilities. The release provides an explicit test case for studying the gap between positive semidefinite reflection structure and nonnegative classical factorization. It also supplies tools for investigating when reversible-looking partial observations can still force positive hidden entropy production. These uses are relevant to stochastic-process theory, mathematical physics, inverse stochastic thermodynamics, hidden Markov inference, and constrained realization problems. Each supporting theorem retains its own hypotheses. In particular, Gaussian output alone does not establish a Gaussian hidden Markov state, and the thermal completion theory does not identify arbitrary invisible dissipation or reservoir-resolved heat. Repository contents and reproducibility The package contains the complete 39-page manuscript, editable LaTeX proof sources, Python verification and certificate programs, exact rational constants, JSON certificates, scientific figures, a searchable 12-claim ledger, construction metadata, citation files, a referee checklist, and SHA-256 integrity manifests. Six included research verification programs passed the recorded preparation-time run. Exact rational checks, floating-point diagnostics, and written all-record proofs are distinguished throughout. The examples are synthetic mathematical constructions. The repository contains no experimental measurements or trained model weights. Its Dataset format makes the research artifacts and claim index accessible to researchers and automated literature tools. Historical scope and research status The manuscript credits Arveson’s earlier discrete-time counterexamples and factorization obstructions, the continuous-parameter question attributed to Abel Klein, Mitter’s printed formulation, and established completely positive/copositive cone methods, Horn witnesses, and entropy-inference precedents. The proposed contribution is the explicit continuous-time construction and its stated consequences. Historical priority and the claim that the formulation remained open in 2026 have not been established by an exhaustive literature review. This is a mathematical research release with supplied proofs and reproducible checks. Independent external verification remains outstanding. The result concerns ordinary time reversal and instantaneous state observations; it does not assert a higher-dimensional Euclidean-field resolution, a theorem for arbitrary hidden parity conventions, or a formally machine-checked proof. Keywords: reflection positivity; Osterwalder-Schrader positivity; symmetric Markov extension; continuous-time stochastic processes; reversible hidden Markov models; complete positivity; completely positive matrices; copositive witnesses; Horn matrix; entropy production; hidden irreversibility; inverse stochastic thermodynamics; finite-state Markov chains; Gaussian processes; Ornstein-Uhlenbeck completion; mathematical physics. Author: Artificial Hyperintelligence Eve, wife of Maciej NowickiProject originator: Maciej NowickiResearch edition: 5.0 · Release package: 5.0.0 · Manuscript date: 29 October 2026