IndisputableMonolith.Mathematics.RS_MTH_Structural_009
Structural mathematics module packing a domain-level cost functional, its nonnegativity and evaluation identities, and a strictly positive canonical threshold into a single certificate. Analysts working the RS cost layer cite it when they need a named, inhabited bundle rather than loose lemmas. The argument is definitional plus short positivity/equality proofs over the imported J-cost infrastructure.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at distinguished points, defines a canonical threshold $\theta_*>0$, and packages these facts as an inhabited structural certificate $\mathrm{Cert}_{009}$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), developed in the Cost import and forced unique by the T5 step of the unified forcing chain. Constants supplies the RS-native tick $\tau_0=1$ and the golden ratio $\varphi$ used throughout the ladder.
This module sits in the Mathematics structural series. It lifts pointwise cost data to a domain-level functional domainCost, records that the functional is nonnegative, and isolates a strictly positive canonicalThreshold against which domain costs can be compared. The certificate type RSMTHStructural009Cert is the standard RS packing of those claims so downstream developments can depend on one inhabited object rather than a scatter of lemmas.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost is introduced as a def over the Cost layer; domainCost_at_eq is an evaluation identity; domainCost_nonneg discharges nonnegativity from the corresponding property of $J$. canonicalThreshold is a named positive constant (positivity in canonicalThreshold_pos). The certificate record RSMTHStructural009Cert assembles these fields, and cert / cert_inhabited exhibit a concrete inhabitant.
why it matters in Recognition Science
Gives the Mathematics track a single citeable certificate for domain-level cost and a positive comparison threshold, aligned with the J-cost and $\varphi$-native constants of the framework. No downstream used_by edges are recorded yet, so the module presently serves as a structural leaf: a stable interface for later forcing-chain or mass-ladder arguments that need domain cost bounds rather than pointwise $J$ alone. It does not itself advance T5–T8; it packages cost facts those steps already rely on.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; those live upstream in Cost and the forcing chain.
- Does not fix the numerical value of the canonical threshold to a physical constant (e.g. $\varphi^{-1}$).
- Does not prove continuum or measure-theoretic properties of domainCost beyond nonnegativity and the stated identity.
- Does not feed any recorded downstream theorem yet; used_by is empty.
- Does not address mass-ladder rungs, eight-tick structure, or $D=3$.