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

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 S_eff^H, the ledger balance B_R, and the normalized deviation epsilon_R 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 → Orb_G → Omega_coh,R / Omega_red,R → epsilon_R → B_R → S_eff^H → kappa_R(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 chi_R separates coherent, reductive, and neutral / INCONCLUSIVE classes. Only after Pi, G, Orb_G, chi_R, and the admissibility gate C_H are defined may Omega_coh,R, Omega_red,R, epsilon_R, B_R, S_eff^H, and kappa_R(Gamma) be computed. The baseline run yields Omega_coh,R = 8, Omega_red,R = 5, B_R/k_B = ln(8/5), epsilon_R = 0.02938 for S_BH/k_B = 16, and a toy-normalized closure value kappa_R(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 epsilon_R or kappa_R(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 S_eff^H, B_R, and epsilon_R were defined as candidate operational horizon-account quantities under audit conditions, and QSTH M.6 — Galoisian Microstate Sorting, where the microstate sieve for Omega_coh,R and Omega_red,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 epsilon_R, B_R, S_eff^H, or kappa_R(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 → Orb_G → Omega_coh,R / Omega_red,R → epsilon_R → B_R → S_eff^H → kappa_R(Gamma)• a minimal model with sixteen microstates• ledger orbits and redundancy classes under G• Horizon Sudoku as the entry gate C_H• a pre-declared classifier chi_R• baseline computation of Omega_coh,R = 8 and Omega_red,R = 5• the first toy value epsilon_R = ln(8/5)/16 = 0.02938• the toy-normalized closure value kappa_R(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 w_i• 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 kappa_R(Gamma) must not appear as a decorative number. The projection Pi, redundancy group G, orbital classes Orb_G, classifier chi_R, and gate C_H must be defined first. Only then may epsilon_R and the subsequent closure readout be computed. The baseline toy run produces a positive epsilon_R, 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 S_eff^H, the ledger balance B_R, and the normalized deviation epsilon_R. M.6 introduced the microstate sieve required before Omega_coh,R and Omega_red,R may be counted. M.7 provides the first minimal computational mechanism connecting the M.6 sieve to auditable values of epsilon_R, S_eff^H, and kappa_R(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 epsilon_R, S_eff^H, and kappa_R(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 Omega_coh,R, Omega_red,R, epsilon_R, B_R, S_eff^H, and kappa_R(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 epsilon_R, S_eff^H, and kappa_R(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