IndisputableMonolith.Foundation.RS_FDN_Structural_006
Foundation certificate module for RS-FDN-Structural-006: a nonnegative domain cost built from the RS cost functional, together with a strictly positive canonical threshold. Auditors of the structural forcing layer cite it when checking that cost-based cutoffs are well-posed. The module is mostly definitions plus short nonnegativity and positivity lemmas over the imported Cost and Constants layers.
claimOn the Recognition Science cost structure, define a domain cost $C$ (pulled from the standard $J$-cost), prove $C \ge 0$ pointwise, and fix a canonical threshold $T > 0$. Package these facts as the inhabited structural certificate RS-FDN-Structural-006.
background
Recognition Science measures mismatch with the $J$-cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced unique by the Recognition Composition Law and the T5 step of the unified forcing chain. The Cost import supplies that functional and its elementary calculus; Constants supplies the RS-native tick $\tau_0 = 1$.
This module sits in the Foundation structural series. It does not re-derive $J$; it specializes cost to a domain-level quantity used as a cutoff or acceptance score, and names a positive canonical threshold against which that score is compared.
Sibling objects are the domain cost itself, its evaluation identity, nonnegativity, the canonical threshold and its positivity, and the certificate record that bundles them for downstream structural checks.
proof idea
Definition-and-certificate module, not a deep derivation. Domain cost is introduced as a thin wrapper on the imported cost functional; equality-at-a-point is definitional. Nonnegativity is inherited from the known $J \ge 0$ (or the corresponding Cost lemma) by rewriting. The canonical threshold is a fixed positive RS constant; positivity is a one-line arithmetic fact. The certificate type and its inhabited instance simply package those fields so later modules can demand a single cert hypothesis rather than a scatter of lemmas.
why it matters in Recognition Science
Structural foundation certificates pin the bookkeeping that later forcing and phenomenology modules assume without re-proving. FDN-Structural-006 guarantees that any domain-level cost gate is nonnegative and compared to a strictly positive threshold, so acceptance/rejection cuts cannot collapse to the zero functional or to a non-positive barrier.
No downstream used_by edges are recorded on this page yet; the natural consumers are later Foundation or phenomenology modules that quantify over an RSFDNStructural006Cert (or the bare nonnegativity and threshold facts). In the broader RS chain this supports clean use of cost cutoffs beside T5 $J$-uniqueness and the phi-ladder mass and coupling bookkeeping, without smuggling sign or scale ambiguities into those arguments.
scope and limits
- Does not re-prove uniqueness of $J$ or the Recognition Composition Law.
- Does not derive numerical values of couplings, masses, or $\alpha$.
- Does not force dimension $D=3$ or the eight-tick octave.
- Does not assert a specific physical domain; only the abstract cost/threshold package.
- Does not discharge phenomenology certificates beyond the structural bundle.