IndisputableMonolith.Mathematics.RS_MTH_Structural_007
Structural mathematics module that packages a nonnegative domain cost, a positive canonical threshold, and an inhabited certificate for RS claim MTH-Structural-007. A mathematician checking the cost-threshold interface would cite it. The module is mostly definitions plus elementary positivity and equality lemmas, closed by a certificate inhabitant.
claimIntroduce a domain cost $C$ (nonnegative), a canonical threshold $\theta>0$, and a certificate asserting the structural package for RS mathematics item 007: $C$ agrees with its pointwise evaluation, $C\ge 0$, and $\theta>0$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ from the Cost layer (T5 uniqueness). Constants supplies the RS-native tick $\tau_0=1$. This module sits in the Mathematics domain and does not re-derive $J$; it defines a domain-level cost functional and a fixed positive threshold used as a structural gate.
Sibling objects are: domainCost and its pointwise agreement and nonnegativity facts; canonicalThreshold with a positivity lemma; and a certificate type RSMTHStructural007Cert with an inhabitant. Imports are Mathlib, Constants, and Cost only.
The local setting is a thin structural wrapper: cost nonnegativity and a positive cutoff, not a full forcing-chain step (T0–T8).
proof idea
Definition-heavy module. Domain cost is introduced, then shown equal to its evaluation form and nonnegative. The canonical threshold is defined and proved positive. A certificate record bundles those facts; inhabitation is by assembling the lemmas. No deep tactic proof or external theorem chain beyond Cost/Constants.
why it matters in Recognition Science
Gives a named, certifiable cost-and-threshold interface for mathematics claim Structural-007 inside the RS monolith. Downstream use is not yet wired in the graph (used_by empty), so the module is a leaf package rather than a forcing-chain link. It keeps the Cost layer’s nonnegativity discipline available under a stable certificate name for later structural or numerical gates. It does not itself force $\varphi$, the eight-tick octave, or $D=3$.
scope and limits
- Does not derive or uniquify the J-cost; imports Cost only.
- Does not prove any T0–T8 forcing step or RCL identity.
- Does not fix numerical constants beyond a positive threshold.
- Does not supply downstream consumers; used_by is empty.
- Does not address mass ladder, alpha band, or physical units.