IndisputableMonolith.Materials.RS_Matl_Module_003
Materials module defining a domain cost functional and a positive canonical threshold for RS materials analysis. Supplies nonnegativity of the cost, an evaluation identity, and an inhabited certificate bundle RSMatl003Cert. Cited by materials-layer arguments that need a fixed cost yardstick and threshold. Structure is definitional plus short positivity and equality lemmas.
claimIntroduce a domain cost $C$ on the materials side, prove $C \ge 0$ and an evaluation identity at equality cases, fix a canonical threshold $\theta > 0$, and package these into an inhabited certificate record for module 003.
background
Recognition Science materials layers sit on the global cost $J$ from the Cost import and the RS-native constants (including the tick $\tau_0 = 1$). The cost calculus supplies the unique $J(x) = (x + x^{-1})/2 - 1$ forced by the Recognition Composition Law; materials modules specialize that calculus to domain-level quantities rather than particle rungs.
This module introduces a domain cost (a nonnegative real-valued functional on the materials domain) together with a canonical threshold intended as a fixed comparison scale. Sibling lemmas record nonnegativity, an on-equality evaluation identity, and positivity of the threshold. The certificate record bundles those facts for downstream discharge.
Local setting is the Materials domain of the monolith: short, self-contained cost-and-threshold scaffolding rather than a full forcing-chain step (T5–T8 live upstream in Foundation).
proof idea
Definition module with thin lemmas. domainCost is introduced as a def; domainCost_nonneg and domainCost_at_eq are short algebraic or rewriting arguments off the Cost import. canonicalThreshold is a positive constant def; canonicalThreshold_pos is a one-line positivity check. RSMatl003Cert and cert package the above into a structure; cert_inhabited builds a witness by assembling those lemmas. No deep tactic scripts; mostly definitional packaging.
why it matters in Recognition Science
Gives the materials stack a named cost yardstick and threshold certificate so later materials claims can cite a single inhabited record instead of re-proving nonnegativity and positivity. Does not itself close a forcing-chain landmark (T5 J-uniqueness, T6 $\phi$, T7 eight-tick, T8 $D=3$ remain in Foundation). used_by is empty in the current graph, so this module is a leaf supplier: parents will be materials theorems that import RS_Matl_Module_003 and project out cert. Touches scaffolding closure only insofar as the inhabited certificate removes local sorries at the cost/threshold interface.
scope and limits
- Does not derive domainCost from the Recognition Composition Law or prove J-uniqueness.
- Does not fix numerical materials constants (masses, gaps, or alpha-band values).
- Does not prove existence of physical materials phases or phase transitions.
- Does not feed Foundation forcing steps T5–T8 or the eight-tick octave.
- Does not claim downstream consumers; used_by is currently empty.