RSPhysics001Cert
plain-language theorem explainer
Certificate bundle for Physics Module 001 (QCD running coupling): the domain cost vanishes on the diagonal, is nonnegative for positive mass and energy arguments, and the module's canonical threshold is strictly positive. Anyone citing the structural RS_PASS for alpha_s(M_Z) = J(phi) uses this record. It is a pure structure definition; inhabitants are built by wiring three sibling lemmas.
Claim. A certificate for RS Physics Module 001 is a record of three facts: (i) for every nonzero real $r$, the domain cost at equal arguments satisfies $C(r,r)=0$; (ii) for all positive reals $m,e>0$, $C(m,e)\ge 0$; (iii) the canonical threshold $T$ of the module obeys $T>0$.
background
Physics RS Module 1 packages the Recognition Science claim that the QCD running coupling at the Z pole equals the J-cost of the golden ratio: $\alpha_s(M_Z)=J(\varphi)\approx 0.11803$, matching PDG $0.1180$. Status is structural theorem (zero sorry, zero axiom).
The domain cost $C$ is the module-local cost on mass/energy pairs, built from the global Recognition cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$). Upstream, ObserverForcing records that every recognition event has nonnegative cost via Jcost_nonneg on the positive state. The canonical threshold is the module's positive cutoff used to gate the structural pass.
This structure does not compute $\alpha_s$; it only freezes the three cost/threshold side-conditions the module treats as the certificate interface.
proof idea
No proof body: the declaration is a structure (record type). Its three fields are Prop-valued obligations. Downstream, the definition cert inhabits it in one shot by assigning domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos to the three fields. cert_inhabited then wraps that witness as Nonempty.
why it matters
Gives the typed certificate that Module 001 must discharge before claiming RS_PASS on $\alpha_s(M_Z)=J(\varphi)$. Downstream cert and cert_inhabited are the only inhabitants exposed in-module; any later physics audit that requires a nonempty certificate for this module goes through them.
In the broader framework this sits under the J-uniqueness landmark (T5): the same $J$ that is forced by the Recognition Composition Law is evaluated at $\varphi$ (T6 fixed point) to produce the coupling. The structure itself does not re-prove T5–T6; it only locks the cost nonnegativity, diagonal vanishing, and positive threshold that the module treats as structural hygiene for the QCD claim.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.