IndisputableMonolith.Astrophysics.RS_AST_Structural_002
Structural certificate module for RS astrophysics claim AST-002. It packages a nonnegative domain cost built from the RS J-cost, a strictly positive canonical threshold, and an inhabited certificate record tying those facts together. Astrophysicists working the RS mass/ladder or galactic-structure chain would cite the certificate when discharging the structural side conditions of AST-002. The module is mostly definitions plus short positivity and equality 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 for astrophysics claim AST-002.
background
Recognition Science measures mismatch with the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced at T5 of the unified forcing chain and obeying the Recognition Composition Law. The Cost import supplies that $J$; Constants supplies the RS-native tick $\tau_0 = 1$.
In the astrophysics layer, structural claims need a domain-level cost (aggregate mismatch on a spatial or mass domain) and a fixed positive threshold against which that cost is compared. This module introduces those two objects for claim AST-002 and wraps them in a certificate record so downstream astrophysics theorems can assume a single inhabited witness rather than re-proving nonnegativity and positivity locally.
The setting is purely structural: no dynamical evolution, no observational fit, only the algebraic skeleton required before ladder or eight-tick arguments are applied.
proof idea
Definition module with short supporting lemmas. domainCost is built by composing or evaluating the imported $J$-cost on a domain; domainCost_at_eq is an evaluation identity; domainCost_nonneg follows from nonnegativity of $J$. canonicalThreshold is a positive RS-native constant; canonicalThreshold_pos is the corresponding positivity proof. RSASTStructural002Cert (and the cert / cert_inhabited pair) assembles those facts into an inhabited structure. No deep tactic scripts; the argument is packaging plus inheritance from Cost.
why it matters in Recognition Science
AST-002 sits in the astrophysics domain of the RS mirror. Structural certificates of this form are the gate that later mass-ladder, galactic, or threshold arguments must pass: without a nonnegative domain cost and a positive canonical cut, those claims cannot be stated cleanly in Lean.
used_by is presently empty, so this module is a leaf certificate rather than an intermediate lemma. It still earns its place by closing the structural side of AST-002 in the same style as other RS_*Cert packages: one inhabited record, zero sorry, ready for a parent theorem to require cert as a hypothesis. Framework landmarks touched only indirectly are T5 ($J$-uniqueness) via the Cost import and the phi-ladder mass formula once a parent applies the certificate to rung/gap data.
scope and limits
- Does not derive observational astrophysics or fit data.
- Does not prove dynamical stability or evolution equations.
- Does not instantiate the phi-ladder mass formula or rung gaps.
- Does not claim a unique numerical threshold beyond positivity.
- Does not discharge any parent theorem; used_by is empty.