IndisputableMonolith.Astrophysics.RS_AST_Structural_004
Structural certificate module for RS astrophysics claim 004: a nonnegative domain cost built from the RS cost functional, a positive canonical threshold, and an inhabited certificate packing both. Astrophysicists working the RS mass/structure ladder would cite the certificate when discharging structural side conditions. The module is mostly definitions plus elementary positivity and equality lemmas.
claimDefine a domain cost $C$ from the RS cost functional, prove $C \ge 0$ and an evaluation identity, fix a canonical threshold $\theta > 0$, and package $(C,\theta)$ into an inhabited structural certificate for RS-AST-004.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain and obeying the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Astrophysics domain and introduces a domain-level cost (a lift of $J$ to whatever domain objects the 004 claim uses), records that the cost is nonnegative, and names a positive canonical threshold against which structural comparisons are made. The certificate type bundles those ingredients so downstream astrophysics lemmas can assume a single inhabited pack rather than re-proving positivity each time.
proof idea
Definition-heavy module. domainCost is introduced from the Cost layer; domainCost_at_eq is an evaluation/unfolding identity; domainCost_nonneg is the standard nonnegativity inheritance from $J\ge 0$. canonicalThreshold is a fixed positive real (canonicalThreshold_pos). RSASTStructural004Cert (with cert and cert_inhabited) is a structure packing cost and threshold, discharged by constructing a default inhabitant. No deep tactic proof; elementary algebraic and positivity facts.
why it matters in Recognition Science
Gives the Astrophysics lane a reusable structural certificate for claim RS-AST-004 so later mass-ladder or structure theorems can cite one inhabited object instead of ad-hoc cost and threshold hypotheses. It ties local astrophysical bookkeeping to the global J-cost and the RS tick from Constants/Cost, consistent with the phi-ladder mass formula and the forcing-chain cost uniqueness (T5). No downstream used_by edges are recorded yet; the module is an interface pack waiting on parent astrophysics theorems.
scope and limits
- Does not derive observational astrophysics predictions or fit data.
- Does not prove the full RS-AST-004 physics claim, only structural cost/threshold packaging.
- Does not fix numerical rung, gap, or mass-yardstick values.
- Does not establish uniqueness of the canonical threshold beyond positivity.
- Does not connect to eight-tick, D=3, or alpha-band results.