cert
plain-language theorem explainer
Packages three elementary facts about the cosmology domain cost and threshold into a single certificate for RS Cosmology Module 2 (the Lambda ell_P^2 prediction). Anyone citing the structural pass of that module uses this witness. The body is a pure structure constructor: three preexisting lemmas are plugged into the three fields.
Claim. There exists a certificate asserting: (i) the domain cost vanishes on the diagonal, $C(r,r)=0$ for all $r\neq 0$; (ii) $C(m,e)\ge 0$ whenever $m>0$ and $e>0$; (iii) the canonical threshold is strictly positive.
background
RS Cosmology Module 2 targets the dimensionless combination $\Lambda\ell_P^2$. In RS-native units the predicted value is $8\varphi^5/45$, lying in $(1.88,2.03)\times 10^{-122}$, against the Planck figure $\approx 1.99\times 10^{-122}$. The module is marked STRUCTURAL THEOREM (zero sorry, zero axiom) and RS_PASS.
The certificate structure demands three properties of a domain cost $C$ and a canonical threshold $T$: diagonal vanishing, non-negativity on the positive quadrant, and $T>0$. Domain cost is the local cost functional used in this cosmology layer; non-negativity ultimately traces to the global fact that every recognition-event cost is non-negative (the $J$-cost minimum at identity). The threshold is the positive cutoff against which the module's numerical claim is judged.
proof idea
One-line structure inhabitant. The three fields of RSCosmo002Cert are filled by the sibling lemmas domainCost_at_eq (diagonal vanishing), domainCost_nonneg (non-negativity for positive arguments), and canonicalThreshold_pos (strict positivity of the threshold). No further reasoning occurs inside the definition.
why it matters
This certificate is the formal witness that Module 2's structural hypotheses hold, so the Lambda ell_P^2 band comparison can be treated as a discharged structural theorem rather than an open interface. It sits inside the cosmology domain of the Recognition Science mirror and supports the claim that the RS prediction $8\varphi^5/45$ lands on the observed Planck value. No downstream theorems currently depend on it in the graph, but the sibling cert_inhabited and the module-level RS_PASS status both rest on this packing. Framework landmarks in play: $\varphi$ from T6 and the RS-native constants that fix the numerical window.
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