IndisputableMonolith.Mathematics.RS_MTH_Structural_010
Structural mathematics module for RS claim MTH-010: it defines a domain cost on the positive reals, proves it is nonnegative and agrees with evaluation at a point, and introduces a strictly positive canonical threshold. A small certificate bundle packages these facts for downstream audit. Anyone checking the cost-threshold interface in the mathematics layer would cite it. The argument is mostly definitional with short positivity and nonnegativity lemmas.
claimOn the positive reals, a domain cost $C$ is defined so that $C(x)\ge 0$ and evaluation at a point recovers the cost value. A canonical threshold $\tau_*>0$ is fixed. These facts are packaged as an inhabited certificate for structural claim MTH-010.
background
The module sits in the mathematics layer of Recognition Science and imports the RS constants (including the native time quantum) and 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 and the T5 step of the forcing chain. Domain cost here is the local specialization of that cost language to a domain-level functional used by structural claim 010.
Sibling declarations introduce domainCost, its pointwise evaluation identity, nonnegativity, a canonicalThreshold with a positivity lemma, and a certificate type RSMTHStructural010Cert with an inhabited instance. No external physics constants beyond the cost/constants imports are required for the interface.
proof idea
Definition-first module. Domain cost is introduced as a def; a one-line evaluation identity equates the named cost at a point with direct application. Nonnegativity is a short lemma (inherited from the J-cost shape or an elementary inequality on the positive reals). The canonical threshold is a positive constant def; positivity is immediate from the construction. The certificate record bundles these facts, and cert_inhabited supplies a canonical inhabitant so downstream code can assume the bundle without rebuilding the lemmas.
why it matters in Recognition Science
Gives the mathematics layer a named, auditable handle for structural claim MTH-010: domain cost plus a positive canonical threshold, sealed as a certificate. Downstream consumers (none linked in the current graph) can import the certificate rather than re-proving nonnegativity or threshold positivity. In the broader RS stack this keeps cost-side structural hypotheses aligned with the J-cost uniqueness story (T5) and the constants/cost imports, without yet tying to mass ladders, eight-tick structure, or dimension forcing. It is infrastructure for later structural theorems that need a clean cost-threshold pair.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law.
- Does not fix numerical RS constants beyond the cost/threshold interface.
- Does not prove mass, alpha, or dimension results.
- Does not connect domain cost to the phi-ladder or eight-tick octave.
- Does not supply downstream physics theorems; used_by is empty in the graph.