IndisputableMonolith.Mathematics.RS_MTH_Structural_008
Structural mathematics module packaging a domain cost functional, its nonnegativity, and a strictly positive canonical threshold. Researchers cite it when a recognition-cost bound must sit above a fixed cutoff rather than at zero. The file is mostly definitions and elementary positivity lemmas, closed by an inhabited certificate record.
claimA domain cost $C$ is defined on the relevant recognition domain, satisfies $C \ge 0$ pointwise, and is paired with a canonical threshold $\theta > 0$. Equality cases of the cost at distinguished points are recorded. The package is witnessed by an inhabited certificate object.
background
The module sits in the mathematics layer of the Recognition Science monolith and imports the global constants (including the RS tick $\tau_0$) together with the Cost library. In RS, the fundamental cost is the J-functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law; domain-level costs are built from that seed.
Here the local objects are a domain cost (a real-valued functional on the structural domain under study), its evaluation identities, and a canonical threshold meant to separate the zero-cost locus from the positive-cost regime. Nonnegativity of the cost and positivity of the threshold are the elementary analytic facts the rest of the file relies on.
The certificate pattern (RSMTHStructural008Cert, cert, cert_inhabited) is the standard RS packaging: a structure bundling the named inequalities so downstream modules can assume the whole package by inhabitation rather than by repeating hypotheses.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost and canonicalThreshold are introduced as defs; domainCost_at_eq records evaluation identities; domainCost_nonneg and canonicalThreshold_pos discharge nonnegativity and strict positivity by direct appeal to the Cost library and elementary real arithmetic. The certificate structure aggregates those facts, and cert_inhabited supplies a canonical inhabitant so the package can be passed as a single hypothesis.
why it matters in Recognition Science
RS structural mathematics claims are numbered and certified so later forcing or measurement arguments can cite a single inhabited record rather than a scatter of inequalities. This module is the 008 entry: it freezes the domain-cost-plus-threshold package used whenever a recognition cost must clear a positive cutoff (Berry-type creation thresholds and related phi-ladder cutoffs sit in the same conceptual family).
No downstream edges are recorded in the current graph, so the file presently acts as a leaf certificate rather than an active dependency. Its value is archival and compositional: once a parent theorem needs a nonnegative domain cost with a fixed positive threshold, it can import this cert instead of re-proving the elementary facts. It does not itself advance the T0–T8 forcing chain; it supplies a reusable analytic side condition those chains may later consume.
scope and limits
- Does not derive the J-cost uniqueness theorem or the Recognition Composition Law.
- Does not fix numerical values of physical constants (c, hbar, G, alpha).
- Does not prove mass-ladder or eight-tick octave statements.
- Does not claim downstream consumers; the usage graph is currently empty.
- Does not replace a full Cost-module development; it only packages domain-level facts.