Mathematical Foundations v1.5: Canonical Reciprocity and Status-Labeled Open Problems

Current statement

FRC 826.829 states Fractal Resonance Coherence as a status-labeled mathematical research program. Version 1.5 restores the canonical scale-invariant relation dS + k* d ln C = 0 and reserves dS_mu + k*_mu d ln C_mu = 0 for declared operational realizations. The starred k* is the Boltzmann bridge, not an outcome-fitted constant or evolving state variable. The relation is operational bookkeeping; its physical universality for open systems remains a conjecture. Boundary-relative lambda=-d_eS is admissible when it follows from an explicit accounting convention, but lowercase lambda remains a diagnostic rather than a Lambda field. The paper distinguishes Lambda_obs, Lambda_eq, and optional Lambda_dyn from a separate fundamental-field conjecture. Four mathematical results are retained: a two-pole interior band, the conditional forced-cubic coefficient in a non-even coherence expansion, critical slowing, and the exact von Mises identity dS/d ln C=-kappa r. At the information-unit normalization k*_{mu_nat}=1, kappa r=1 is a stationary point of the system-only Q curve, not a physical zero-current claim without an environment model. Negative information-geometric and half-line results remain visible.

Evidence level

Frontier preprint

preprint

Declared μ register

μ4 · Logical / conceptual

Logic, formal models, language as explicit reasoning, and computational design.

Open boundary

Definitions and propositions are retained separately from conjectures and open problems.

Version lineage

Supersedes: FRC 826.829 v1.4