IndisputableMonolith.Foundation.RS_FDN_Structural_005
Foundation module packaging the structural certificate RS-FDN-005: a nonnegative domain cost built from the RS J-cost, a positive canonical threshold, and an inhabited certificate record tying them together. Cited by anyone auditing the structural layer of Recognition Science forcing. Mostly definitions plus elementary nonnegativity and positivity lemmas.
claimDefine a domain cost $C$ from the RS cost functional $J$, prove $C \ge 0$ and an evaluation identity, fix a canonical threshold $\theta > 0$, and package these into an inhabited structural certificate record for RS-FDN-005.
background
Recognition Science derives physics from a single cost functional $J$ on positive reals, with $J(x) = (x + x^{-1})/2 - 1$ forced uniquely (T5) and obeying the Recognition Composition Law. The Cost import supplies that $J$-infrastructure; Constants supplies RS-native units (including the tick $\tau_0$).
This module sits in the Foundation structural layer. It introduces a domain-level cost assembled from $J$, records that the cost is nonnegative, and names a positive canonical threshold against which domain costs can be compared. The certificate record is the audit object that downstream structural claims can inhabit or require.
proof idea
Definition-heavy module. Domain cost is defined from the imported $J$-cost; nonnegativity and the pointwise evaluation identity are short lemmas from Cost. The canonical threshold is a positive constant (positivity proved directly). The certificate is a structure bundling these facts, with an inhabited instance cert so the structural obligation is discharged inside the module.
why it matters in Recognition Science
Closes the RS-FDN-Structural-005 obligation in the Foundation layer: a named, checkable package that domain cost is a nonnegative $J$-derived quantity with a positive comparison threshold. No downstream edges are recorded yet; the module is a leaf certificate meant for structural audits and for later forcing-chain or ledger arguments that need a stable domain-cost interface. Ties to the J-uniqueness landmark (T5) via the Cost import rather than re-proving uniqueness here.
scope and limits
- Does not prove J-uniqueness or the Recognition Composition Law; those stay in Cost/forcing chain.
- Does not derive mass ladders, alpha bounds, or dimensional forcing (T6–T8).
- Does not assert numerical values for the threshold beyond positivity.
- Does not record downstream consumers; used_by is empty in the graph.
- Does not replace a full physical domain model; only the cost/threshold certificate.