IndisputableMonolith.Astrophysics.RS_AST_Structural_006
Certificate module packaging a nonnegativity and positivity package for a domain-restricted recognition cost and a canonical threshold used in RS astrophysics structural claim 006. Astrophysicists working the RS mass/structure ladder would cite the inhabited certificate rather than the individual lemmas. The module is mostly definitional: cost and threshold defs, elementary positivity proofs, then a bundled cert record.
claimDefine a domain-restricted recognition cost $C_D$ built from the RS $J$-cost, prove $C_D \ge 0$ and an evaluation identity, introduce a canonical threshold $\theta_\ast > 0$, and package these facts as an inhabited structural certificate for RS astrophysics claim 006.
background
Recognition Science measures mismatch with the unique cost $J(x) = (x + x^{-1})/2 - 1$ forced by the Recognition Composition Law (T5). The Cost import supplies that $J$-calculus; Constants supplies the RS-native tick $\tau_0 = 1$ and related units.
This module sits in the Astrophysics domain and isolates a structural fragment labeled 006: a domain cost (cost restricted to an astrophysical domain of interest), its nonnegativity, a matching evaluation identity, and a strictly positive canonical threshold against which that cost is compared. The threshold is the gate that turns a pure cost inequality into a structural yes/no statement.
Sibling names indicate a small closed package: domainCost and its algebraic/positivity lemmas, canonicalThreshold with positivity, then a certificate record and an inhabitation proof that the record can be filled.
proof idea
Definition-first module, not a deep derivation. The domain cost is introduced as a def from the imported Cost layer; nonnegativity and the pointwise evaluation identity are short algebraic or nlinarith-style facts inherited from $J \ge 0$ and $J(1)=0$. The canonical threshold is a positive constant def; positivity is immediate from the RS constant stack. The certificate is a structure bundling those props, discharged by cert_inhabited (constructor application of the preceding lemmas). No multi-step forcing-chain argument lives here.
why it matters in Recognition Science
Gives a named, machine-checkable certificate object for RS astrophysics structural claim 006 so downstream astrophysics developments can depend on one inhabited record instead of re-proving cost nonnegativity and threshold positivity. Used_by is currently empty in the mirror graph, so this module is a leaf packaging step rather than an intermediate lemma in a long chain. It touches the Cost/$J$ layer (T5) and the Constants layer, and is the natural place a referee looks when auditing whether structural 006 is fully discharged or still only interface-level. No T0–T8 forcing step is completed here; the module only freezes the cost/threshold side conditions those steps need in the astrophysics setting.
scope and limits
- Does not derive $J$-uniqueness or the Recognition Composition Law; those are imported.
- Does not prove any observational astrophysics fit or mass-ladder numerical match.
- Does not fix the physical meaning of the domain beyond the cost restriction.
- Does not establish eight-tick, $D=3$, or $\phi$-ladder forcing (T6–T8).
- Does not currently feed a recorded downstream theorem in the mirror graph.