Quantum Structural Theory of Harmony (QSTH M.1C) — Alpha-I-Dim: Horizon Microstates, Transition Indices and Relational Geometry

QSTH M.1C — Alpha-I-Dim: Horizon Microstates, Transition Indices and Relational Geometry Accompanied by QSTH M.1D — Adversarial Audit of the STOP Verdict This publication presents a consolidated mathematical and physical audit within the Quantum Structural Theory of Harmony (QSTH) research programme, focusing on the relationship between horizon microstates, geometric invariants, spectral transition indices, and the information required to define physically meaningful horizon responses. The central question is: Which information must be retained for a quantity to characterize the physical object itself, rather than merely the chosen representation, deformation, or measurement prescription? The investigation develops two independent analytical branches. First Lock — Horizon Microstates and Entropy The First Lock examines mechanisms capable of supporting localized interface states and area-scaling state capacity. It distinguishes the mathematical availability of localized degrees of freedom from the independent derivation of physical horizon entropy. A scalar–fermion mechanism class and a specified Schwarzschild/Hartle–Hawking reference diagnostic are examined. The audited fixed-parameter realization retains a mass-dependent critical window and does not establish universal, mass-independent horizon microphysics. The result is a conditional state-capacity construction with explicit physical limitations, rather than a derivation of universal horizon entropy. Second Lock — Geometric Invariants and Transition Indices The Second Lock develops a sequence of geometric and spectral audits involving rotational information, marginally outer trapped surfaces (MOTS), stability operators, eigenvalue criticality, and the parity of admissible real Fredholm operator paths. The K9 control analysis provides an explicit example in which the same smooth spherical horizon worldtube admits different parity results under different deformation prescriptions. This demonstrates that a geometric horizon history does not, by itself, determine every response-dependent quantity. The relevant deformation, foliation, operator path, and admissibility conditions must remain part of the mathematical specification. The targeted connection audit further distinguishes principal spectral criticality, geometric projection behaviour, and genuine fold conditions. Relational Geometry and Information Sufficiency A common mathematical criterion is identified across the audited constructions: a target quantity can be reconstructed from a retained readout precisely when it is constant on the fibres of that readout. This criterion unifies the logic of information sufficiency without identifying the distinct physical mechanisms. Here, relational geometry refers to the explicit relationship between a geometric object and the slices, normals, deformations, and readouts used to characterize it. It does not introduce an additional spacetime dimension or a new gravitational field equation. Supplementary M.1D — Adversarial Audit The accompanying QSTH M.1D document critically re-examines earlier conclusions concerning intrinsic physical response selection. It clarifies that failure of a response-dependent quantity to factorize through an unrestricted prescription space does not imply the nonexistence of a mathematically valid section. A canonical area-normalized null response channel can be constructed from the matter structure of the strictly accreting Vaidya control model. However, geometric uniqueness within this channel does not establish its physically privileged role. The audit therefore corrects overly strong no-go interpretations while retaining the unresolved status of the physical selection problem. Principal Outcome and Scientific Scope The consolidated results establish specific model-level calculations, geometric and spectral relations, reproducible control examples, and explicit limitations on their physical interpretation. They do not establish a universal horizon invariant, independently derived microscopic horizon entropy, or a complete intrinsic physical response-selection mechanism. The investigation preserves the distinction between mathematical identities, conditional model results, candidate physical interpretations, and independently testable physical claims. The principal contribution is an auditable framework for distinguishing the definition of a physical object from the prescriptions required to characterize its response. The accompanying reproducibility materials document the numerical and symbolic checks within their stated domains. Computational reproducibility does not constitute empirical confirmation or independent peer review. QSTH M.1C concludes a defined analytical stage. QSTH M.1D remains a supplementary corrective audit; its broader physical selection programme is frozen pending independently justified input or a new valid derivation. Research status: Completed scoped mathematical-physical audit; universal physical closure not established. The M.1C reproducibility archive retains earlier source materials for provenance; the authoritative manuscript for this release is M.1C English Final v1.2.