Quantum Structural Theory of Harmony (QSTH M.7) — Minimal Galoisian Toy Model of the Microstate Ledger

Abstract QSTH M. 7 presents a minimal Galoisian toy model of the Microstate Ledger within the QSTH M. x Microstate Ledger Series. It follows QSTH M. 5, where the operational horizon account SₑffH, the ledger balance BR, and the normalized deviation epsilonR were introduced, and QSTH M. 6, where Galoisian Microstate Sorting defined the microstate sieve required before coherent and reductive classes may be counted. The purpose of M. 7 is not to compute a real black hole, nor to provide empirical confirmation of QSTH. It is a minimal computational toy model designed to test whether the chain Omega → Pi → G → OrbG → Omegacoh, R / Omegaᵣed, R → epsilonR → BR → SₑffH → kappaR (Gamma) can be traversed in a disciplined and auditable way. The model begins with a finite set of sixteen microstates, grouped into ledger orbits under a redundancy group G and projected through Pi into ledger images. A pre-declared classifier chiR separates coherent, reductive, and neutral / INCONCLUSIVE classes. Only after Pi, G, OrbG, chiR, and the admissibility gate CH are defined may Omegacoh, R, Omegaᵣed, R, epsilonR, BR, SₑffH, and kappaR (Gamma) be computed. The baseline run yields Omegacoh, R = 8, Omegaᵣed, R = 5, BR/kB = ln (8/5), epsilonR = 0. 02938 for SBH/kB = 16, and a toy-normalized closure value kappaR (Gamma) = 0. 97062. These values are not presented as physical horizon measurements. They are the first controlled demonstration that, once the microstate sieve and normalization are fixed in advance, the output is no longer a free fit but the result of an audited computational chain. Version M. 7 v0. 2 strengthens the audit layer by adding null models, variant runs, a weighted microstate ledger, one-microstate perturbations, stability checks, failure modes, and a PASS / INCONCLUSIVE / FAIL contract. The model must be able to return zero effect, the opposite sign, weak effects, and INCONCLUSIVE outcomes. Its value lies precisely in this constraint: QSTH must not win by counting only the microstates that suit the desired result. The contribution of QSTH M. 7 is therefore methodological and computational. It provides the first small auditable machine of the M. x branch: a toy model that separates record from rewrite, coherent support from reductive degeneracy, and readable classes from neutral silence before allowing epsilonR or kappaR (Gamma) to appear. Description This record contains the English final version of QSTH M. 7 — Minimal Galoisian Toy Model of the Microstate Ledger, a methodological and computational toy-model publication in the QSTH M. x Microstate Ledger Series. The publication follows QSTH M. 5 — Operational Structural Ledger for Horizon Set II, where SₑffH, BR, and epsilonR were defined as candidate operational horizon-account quantities under audit conditions, and QSTH M. 6 — Galoisian Microstate Sorting, where the microstate sieve for Omegacoh, R and Omegaᵣed, R was introduced. QSTH M. 7 does not claim to compute a real black hole, nor does it present empirical confirmation of QSTH. It is a minimal computational toy model whose purpose is to test whether a finite set of microstates can be projected, grouped, classified, and counted through an auditable chain before epsilonR, BR, SₑffH, or kappaR (Gamma) are allowed to appear. The document includes: • an Epistemic Note / Methodological Brake• an Opening Diamond defining M. 7 as the first small machine after the M. 6 sieve• the main computational chain Omega → Pi → G → OrbG → Omegacoh, R / Omegaᵣed, R → epsilonR → BR → SₑffH → kappaR (Gamma) • a minimal model with sixteen microstates• ledger orbits and redundancy classes under G• Horizon Sudoku as the entry gate CH• a pre-declared classifier chiR• baseline computation of Omegacoh, R = 8 and Omegaᵣed, R = 5• the first toy value epsilonR = ln (8/5) /16 = 0. 02938• the toy-normalized closure value kappaR (Gamma) = 0. 97062• null models distinguishing zero ledger effect from INCONCLUSIVE• variant runs testing zero effect, weak positive effect, sign reversal, and underclassification• a weighted model wᵢ• one-microstate perturbation tests• stability tests of the minimal machine• a PASS / INCONCLUSIVE / FAIL contract• a table of what M. 7 shows and what it does not show• a Skeptical Contract• status audit, failure modes, and a Mini-Mendeleev cut• a working verdict and Final Diamond M. 7 v0. 2• appendices with an equation capsule and variant / weighted results The core methodological claim of M. 7 is that kappaR (Gamma) must not appear as a decorative number. The projection Pi, redundancy group G, orbital classes OrbG, classifier chiR, and gate CH must be defined first. Only then may epsilonR and the subsequent closure readout be computed. The baseline toy run produces a positive epsilonR, but this is not presented as an empirical result. It is a demonstration of the computational path. The audit layer is strengthened by null models, variant runs, weighted runs, perturbation tests, and failure modes. The model must be able to return PASS, FAIL, or INCONCLUSIVE. This publication should be read as a minimal computational microstate-ledger toy model. It does not prove horizon physics. It shows that the QSTH M. x branch can move from architecture to a first auditable machine: a finite toy system that separates record from rewrite, coherent support from reductive degeneracy, and readable classes from neutral silence. Subjects / categories Physics — Theoretical PhysicsMathematical PhysicsQuantum PhysicsCosmology and Nongalactic AstrophysicsInformation TheoryBlack Hole ThermodynamicsQuantum InformationComputational ModelingToy Models in Theoretical Physics Related work note This publication follows QSTH M. 5 — Operational Structural Ledger for Horizon Set II and QSTH M. 6 — Galoisian Microstate Sorting. M. 5 defined the operational horizon account SₑffH, the ledger balance BR, and the normalized deviation epsilonR. M. 6 introduced the microstate sieve required before Omegacoh, R and Omegaᵣed, R may be counted. M. 7 provides the first minimal computational mechanism connecting the M. 6 sieve to auditable values of epsilonR, SₑffH, and kappaR (Gamma). It prepares the transition toward QSTH M. Closure by showing how a small toy ledger can produce not only a numerical output, but also null results, sign reversal, sensitivity tests, and INCONCLUSIVE outcomes. Plain-language summary This publication asks a simple but strict question: if QSTH defines a microstate sieve, can a small toy model actually pass through it without cheating? M. 7 answers by building a minimal machine with sixteen microstates. The model first decides what the ledger can see, which microstates are merely redundant rewrites, which classes are coherent, which are reductive, and which must remain neutral. Only after this classification is fixed does the model compute epsilonR, SₑffH, and kappaR (Gamma). The result is not a claim about a real black hole. It is a controlled demonstration that the QSTH microstate ledger can produce a number only after the rules have been declared. The model can also return zero, the opposite sign, or INCONCLUSIVE. This is the main point: the toy machine must be able to fail honestly. Final Zenodo caveat This document is a theoretical and methodological toy-model publication. It does not claim that QSTH is empirically confirmed, nor does it present a completed physical theory of horizon microstates. The quantities Omegacoh, R, Omegaᵣed, R, epsilonR, BR, SₑffH, and kappaR (Gamma) are treated as candidate operational structures whose use depends on pre-declared projection, redundancy control, classification rules, null models, perturbation tests, and the right to return INCONCLUSIVE. Suggested citation Stepanik, R. (2026). Quantum Structural Theory of Harmony (QSTH M. 7) — Minimal Galoisian Toy Model of the Microstate Ledger: From Orbital Redundancy to the First Auditable Values of epsilonR, SₑffH, and kappaR (Gamma). Quantum Structural Theory of Harmony, QSTH M. x Microstate Ledger Series, Author’s Edition, EN v1. 0. Zenodo. Copyright Copyright (C) 2026 Rostislav Stepanik