IndisputableMonolith.Foundation.RS_FDN_Structural_008
Foundation module that packages a domain cost functional, its nonnegativity and reference-point identity, and a strictly positive canonical threshold into the structural certificate RS-FDN-008. Auditors of the Recognition forcing chain cite it when a certified nonnegative cost-on-domain with fixed cutoff is required. Content is definitional plus elementary positivity and inhabitation lemmas over Cost and Constants.
claimIntroduces a domain cost $C$ with $C \ge 0$, an identity $C$ at a reference evaluation, and a canonical threshold $\theta > 0$, assembled as the inhabited structural certificate RS-FDN-Structural-008.
background
Recognition Science measures mismatch by a nonnegative cost built from the unique $J$-functional forced in the T5 step of the unified forcing chain. The Cost import supplies that cost infrastructure; Constants supplies the RS-native time quantum $\tau_0 = 1$ tick used as the discrete unit of recognition.
This module sits in the Foundation structural layer. It specializes the ambient cost to a domain cost (a real-valued functional on the working domain), records that the cost is nonnegative, and fixes a canonical positive threshold against which domain values may be compared. The threshold is the structural cutoff used by later certificate consumers, not a derived physical constant such as the Berry threshold $\phi^{-1}$.
Sibling declarations name the pieces: the domain cost itself, its evaluation identity, nonnegativity, the canonical threshold and its positivity, and the certificate record together with an inhabitation proof.
proof idea
Definition-and-certificate module rather than a deep theorem development. Domain cost and canonical threshold are introduced as definitions; nonnegativity and positivity are discharged by elementary real inequalities inherited from the Cost layer. The certificate record bundles those facts; inhabitation is a one-line constructor application showing the bundle is realized.
why it matters in Recognition Science
Closes structural item RS-FDN-008 in the Foundation layer: a reusable, Lean-certified package of nonnegative domain cost plus positive cutoff. Downstream used-by edges are empty in the current graph, so the module presently acts as a leaf certificate that later forcing or measurement developments can import without re-proving cost positivity. It does not itself advance T5–T8 (J-uniqueness, $\phi$, eight-tick octave, $D=3$), but supplies the cost-side scaffolding those steps assume when they quantify recognition mismatch on a domain.
scope and limits
- Does not derive the unique J-cost; imports Cost.
- Does not identify the canonical threshold with $\phi^{-1}$ or any physical constant.
- Does not prove forcing-chain steps T5–T8.
- Does not supply mass-ladder or $\alpha$ numerics.
- Does not currently feed named downstream theorems in the graph.