IndisputableMonolith.Foundation.RS_FDN_Structural_007
Foundation module packaging a structural certificate for a domain-level cost functional and its canonical positive threshold. Recognition auditors cite it when checking that the cost stays nonnegative and meets a fixed cutoff. The module is mostly definitions plus elementary positivity and evaluation lemmas, closed by an inhabited certificate record.
claimA domain cost $C$ is defined so that $C$ is nonnegative, agrees with a pointwise evaluation rule, and is compared against a canonical threshold $\theta>0$. The module packages these facts as an inhabited structural certificate (RS-FDN-Structural-007).
background
Recognition Science builds physics from a unique cost functional $J$ forced by the Recognition Composition Law, with $J(x)=(x+x^{-1})/2-1$. The imported Cost layer supplies that $J$-calculus; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Foundation structural series. It lifts cost from the scalar $J$ setting to a domain-level cost, then fixes a canonical positive threshold against which that cost is judged. Sibling names indicate the local API: a domain cost, its evaluation identity, nonnegativity, a positive canonical threshold, and a certificate bundle that packages the lot.
No external paper proposition is attached in the supplied docs; the module is self-contained scaffolding for the numbered structural claim 007.
proof idea
Definition-and-certificate module rather than a deep proof development. It introduces the domain cost and the canonical threshold, records nonnegativity and positivity as short lemmas, and packages them in an inhabited certificate record. Expect direct unfolding, nonnegativity inheritance from the underlying cost, and a trivial inhabitant for the cert type.
why it matters in Recognition Science
Gives Foundation a named, checkable bundle for structural item 007: domain cost well-behaved and bounded below a fixed positive threshold. Downstream used-by edges are empty in the graph snapshot, so the module presently acts as a leaf certificate rather than a direct lemma feed. It keeps the cost side of the forcing chain (T5 $J$-uniqueness and the RCL) available in domain language without reopening uniqueness of $J$. Auditors use the cert inhabitant as a single hook when wiring later structural or measurement claims that need a nonnegative domain cost and a positive cutoff.
scope and limits
- Does not prove uniqueness of $J$ or re-derive the Recognition Composition Law.
- Does not identify the domain cost with a physical observable or mass formula.
- Does not force the numerical value of the threshold beyond positivity.
- Does not supply downstream consumers in the current dependency graph.
- Does not address eight-tick, $D=3$, or $\phi$-ladder forcing steps.