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| Formato: | Recurso digital |
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Zenodo
2026
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| Acceso en línea: | https://doi.org/10.5281/zenodo.18283041 |
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- <p><br>## Overview<br>This preprint presents a purely mathematical toy program in which a “universe” is modeled by a finite-dimensional real state space equipped with a complex-structure operator, and its “cosmic evolution” is driven by a matrix (operator) generator. The aim is not physical cosmology, but an operator-theoretic framework that formalizes:</p> <p>- internal (“coordinate-free”) notions of expansion-like behavior,<br>- observational equivalence and non-identifiability under one-observable chronologies,<br>- Big-Bang-like events as spectral phase transitions, and<br>- an explicit short-window certification bound producing acceleration-like inference with computable failure probability.</p> <p>## Core setup<br>- State space: a finite-dimensional real inner-product space \(V \cong \mathbb{R}^{2N}\) with a complex structure \(J\) satisfying \(J^2=-I\).<br>- Evolution: an internal chronology parameter \(s\) drives a linear flow<br> \[<br> \Psi'(s)=A(s)\Psi(s),<br> \]<br> with generator \(A(s)\in\mathrm{End}(V)\).<br>- Symmetric/antisymmetric split:<br> \[<br> A(s)=\mathrm{skew}(A(s))+\mathrm{sym}(A(s))=:K(s)+S(s),<br> \]<br> where \(S(s)\) is used to define expansion-like quantities internally (via trace/sector averages).<br>- Slice observer: a fixed decomposition \(V=V_4\oplus V_x\) and access to a one-observable chronology (e.g. a scalar series \(Q_4(s)\), optionally with derived rates).</p> <p>## What is meant by “expansion” and “Big-Bang-like events”<br>- “Expansion” is treated as an inference tag defined from internal observables of \(S(s)\) (e.g. trace/sector rates), rather than an external geometric primitive.<br>- “Big-Bang-like events” are defined as spectral phase transitions of \(A(s)\) or \(S(s)\), such as:<br> - gap collapse or sign changes in the spectrum of \(S(s)\),<br> - robust near-defectiveness / loss of diagonalizability (Jordan-type behavior),<br> - pseudospectral instability that marks structural transitions in non-normal dynamics.</p> <p>## Main results and contributions<br>1. **Observational equivalence and non-identifiability.**<br> The paper formalizes when two distinct generators are observationally equivalent with respect to one-observable chronologies, and proves non-identifiability: one-observable data generally cannot reconstruct the full generator.</p> <p>2. **Mixing indices and probabilistic structure from spectral information.**<br> The program introduces interpenetration/mixing indices (UIC/pUIC and band-averaged variants) to connect spectral “interpenetration” to slice-weight regularity, enabling within-band typicality conclusions.</p> <p>3. **RQC (Rayleigh-quotient concentration) and coupling control.**<br> A weak mixing hypothesis is formulated via concentration of sector Rayleigh quotients around trace rates, with explicit exponential tails in sector dimensions, together with an explicit coupling regime controlling cross-block leakage.</p> <p>4. **Short-window certification inequality (explicit failure probability).**<br> Under high-band mixing, trace advantage, coupling control, and a positive alignment channel linking mixing growth to acceleration-like inference, the paper derives a closed-form certification bound on a short window \(J\):<br> \[<br> \mathbb{P}\Big(\exists\,s\in J:\widehat{\alpha}_4(s)\ge a/2\Big)<br> \ \ge\<br> 1-(N+1)\Big(\pi+\Delta_{\mathrm{high}}/\tau\Big),<br> \]<br> where \(\pi\) is an explicit RQC tail bound, \(\Delta_{\mathrm{high}}\) is the high-band mixing deficit, \(\tau\) controls the tolerance window, and \(N\) is the grid size used in the union bound.</p> <p>## Scope and non-toy status (within the toy program)<br>- All statements are finite-dimensional and operator-theoretic.<br>- “Expansion/acceleration” are inference tags derived from internal data; the paper does not assert physical cosmology.<br>- The point of the program is to isolate a minimal, reproducible pipeline:<br> **engine (operator structure) → inference (slice data) → certification (explicit probability)**.</p> <p> </p>