IndisputableMonolith.Mathematics.RS_MTH_Structural_002
Structural certificate module for a domain-restricted cost and its canonical positivity threshold in Recognition Science units. It packages nonnegativity of the domain cost, a positive threshold cut, and an inhabited certificate record used by downstream structural audits. The argument is definitional plus short positivity and equality lemmas over the imported J-cost.
claimOn the RS cost $J$, define a domain-restricted cost $C_D$ with $C_D\ge 0$, a canonical threshold $\theta>0$, and an inhabited structural certificate asserting these facts in RS-native units ($c=1$, tick $\tau_0=1$).
background
Recognition Science builds physics from a unique cost functional $J$ fixed by the Recognition Composition Law and the forcing chain (T5: $J(x)=(x+x^{-1})/2-1$). The Cost import supplies that $J$-calculus; Constants fixes the RS time quantum $\tau_0=1$ tick and related native units.
This module sits in the Mathematics structural layer. It introduces a domain cost (restriction or evaluation of the cost on a designated domain), records that the cost is nonnegative there, and names a canonical positive threshold against which structural comparisons are made. Sibling definitions assemble these facts into a certificate type RSMTHStructural002Cert with an inhabited instance, so later audits can demand a single package rather than ad hoc lemmas.
proof idea
Definition module with short supporting lemmas, not a deep derivation. Domain cost is defined from the imported Cost layer; an evaluation-equality lemma pins the definition at concrete points. Nonnegativity follows from the standard $J\ge 0$ (or domain restriction thereof). The canonical threshold is a positive constant in RS units; positivity is a one-line arithmetic fact. The certificate record bundles these propositions and is shown inhabited by supplying the proved fields.
why it matters in Recognition Science
Gives a reusable structural certificate for domain cost nonnegativity and a positive threshold cut, aligned with the T5 $J$-uniqueness landmark and RS-native units from Constants. No downstream used-by edges are recorded on the page yet; the intended consumers are later Mathematics or audit modules that require an inhabited structural cert rather than raw Cost lemmas. Closes a small packaging gap so structural claims can cite one named certificate instead of re-proving $C_D\ge 0$ and $\theta>0$ at each call site.
scope and limits
- Does not derive uniqueness of $J$ or re-prove the Recognition Composition Law.
- Does not fix numerical particle masses, $\alpha$, or rung placements on the $\phi$-ladder.
- Does not assert physical measurement protocols beyond RS-native units.
- Does not supply downstream consumers; used_by is currently empty.
- Does not replace the full Cost API; only packages domain cost and threshold facts.