RSCOSStructural001Cert
plain-language theorem explainer
A certificate structure packaging three structural cost axioms for the first RS cosmology module: diagonal vanishing of the domain cost, non-negativity for positive mass/energy arguments, and positivity of the canonical threshold. Cosmology and forcing-chain auditors cite it as the typed interface that the module's concrete certificate must inhabit. It is a pure structure definition; no proof body.
Claim. A certificate consists of three facts: (i) for every nonzero real $r$, the domain cost of the pair $(r,r)$ is zero; (ii) for all positive reals $m,e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
Module RS_COS_Structural_001 records a structural cosmology prediction of Recognition Science: the J-cost $J(x)=\frac12(x+1/x)-1$, the golden ratio $\varphi$ as self-similar fixed point, and $D=3$ forced (status: structural theorem, zero sorry, zero axiom).
The domain cost is the local cost functional on pairs of positive reals used in this cosmology layer; the certificate demands it vanish on the diagonal (matched arguments) and stay nonnegative off it. The canonical threshold is the positive cutoff against which structural comparisons are made.
Upstream, ObserverForcing already proves that every recognition-event cost is nonnegative via $J$-cost nonnegativity. The present structure lifts that nonnegativity pattern, together with diagonal vanishing and threshold positivity, into an explicit cosmology certificate type.
proof idea
No proof: this is a structure declaration. The three fields are Prop-valued requirements (diagonal vanishing of domain cost, nonnegativity on positive arguments, positivity of the canonical threshold). Inhabitation is discharged later by the concrete cert value, which fills the fields with domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos.
why it matters
This certificate is the typed gate for Cosmology RS Structural Module 1. Downstream, cert builds a concrete inhabitant by wiring the three sibling lemmas, and cert_inhabited records Nonempty of the structure, closing the module's structural claim.
In the broader framework it sits under the forcing chain landmarks T5 (J-uniqueness), T6 ($\varphi$ fixed point), and T8 ($D=3$), packaging the cost axioms those landmarks induce for cosmology-scale statements. It does not itself derive $J$, $\varphi$, or $D=3$; it only standardizes the cost/threshold interface those results feed.
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