IndisputableMonolith.Foundation.RS_FDN_Structural_003
Foundation module packaging the domain-cost functional and its canonical positive threshold used by structural certificate RS-FDN-003. It records nonnegativity of the cost, evaluation at equality, and an inhabited certificate bundle. Anyone citing the structural forcing chain or cost-threshold gates would import it. The file is mostly definitions plus short positivity and equality lemmas over the Cost layer.
claimDefine a domain cost $C$ (built from the RS $J$-cost), prove $C \ge 0$ and the on-equality evaluation identity, fix a canonical threshold $\theta > 0$, and package these facts as an inhabited structural certificate for RS-FDN-003.
background
Recognition Science measures mismatch with the $J$-cost from the Cost module, $J(x)=(x+x^{-1})/2-1$, which is nonnegative and vanishes only at $x=1$. Constants supplies the RS-native tick $\tau_0=1$. This module lifts that cost to a domain-level functional (domain cost) and fixes a single positive canonical threshold against which structural claims are checked.
The local setting is the Foundation structural series: short, certificate-shaped modules that pin elementary analytic facts (nonnegativity, positivity of a threshold, evaluation identities) so later forcing or ledger arguments can cite one named bundle rather than reopen the Cost API.
Upstream material is thin: only Constants and Cost are imported. No deeper forcing-chain theorems are required here; the module stays at the level of cost calculus and a threshold constant.
proof idea
Definition-heavy module. Domain cost is introduced as a thin wrapper around the Cost layer; equality-at-matching and nonnegativity are short lemmas reducing to the corresponding $J$-cost facts. The canonical threshold is a positive constant (positivity proved by a direct numeric or algebraic check). The certificate record simply bundles these fields; inhabitation is by constructing that record from the lemmas above. No multi-step tactic developments.
why it matters in Recognition Science
Gives the Foundation layer a named structural certificate (RS-FDN-003) for domain cost and its threshold, so later recognition or ledger arguments can depend on one inhabited bundle instead of ad-hoc Cost lemmas. Downstream use is not yet wired in this graph (no used_by edges), but the sibling certificate pattern matches other RS-FDN structural pins that feed forcing and measurement gates. Ties to the cost side of the framework (J-uniqueness / T5 background) without claiming any step of the T0–T8 chain itself. Closes a small scaffolding slot: threshold positivity and cost nonnegativity are discharged here rather than left sorry'd upstream.
scope and limits
- Does not prove J-uniqueness or any T5–T8 forcing step.
- Does not derive numerical values of c, hbar, G, or alpha.
- Does not define mass rungs, Berry threshold, or dream fraction.
- Does not assert physical units beyond RS-native cost and threshold.
- Does not yet appear as a dependency of any recorded downstream theorem.