IndisputableMonolith.Physics.Gravitational_Fine_Structure_RS
Module defining the RS gravitational fine-structure certificate: a nonnegative domain cost, a positive canonical threshold, and an inhabited proof that the cost sits at the threshold. Physicists tracing how G and the fine-structure scale meet on the phi-ladder would cite it. The argument is definitional plus elementary nonnegativity and positivity lemmas from the Cost and Constants imports.
claimThe module introduces a domain cost $C$, proves $C \ge 0$ and an evaluation identity, fixes a canonical threshold $\theta > 0$, and packages an inhabited certificate $\mathrm{GravFineStructCert}$ asserting the gravitational fine-structure relation in RS-native units.
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 Recognition Composition Law. Constants are fixed in RS-native units: $c = 1$, $\hbar = \varphi^{-5}$, $G = \varphi^5/\pi$, with $\alpha^{-1}$ pinned inside $(137.030, 137.039)$.
This physics module sits on Constants (time quantum $\tau_0 = 1$ tick) and Cost. It specializes those primitives to a gravitational fine-structure domain cost and a canonical positive threshold against which the cost is certified.
Sibling definitions name the cost, its nonnegativity and pointwise evaluation, the threshold and its positivity, and the certificate type GravFineStructCert together with an inhabited instance.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos discharge the sign constraints needed by the certificate; domainCost_at_eq records the evaluation identity. GravFineStructCert, cert, and cert_inhabited package those facts into an inhabited certificate record. No multi-step tactic proof is required beyond the Cost/Constants infrastructure.
why it matters in Recognition Science
Gives a named, checkable certificate that the gravitational and fine-structure scales are tied through the RS cost and the phi-ladder (with $G = \varphi^5/\pi$ and the forced $\alpha$ band). Downstream consumers can assume cert_inhabited rather than re-proving nonnegativity and threshold positivity. The module has no recorded used-by edges yet; it is a leaf certificate in the physics layer, ready for coupling to mass-ladder or forcing-chain (T5–T8) results that need a gravitational fine-structure hypothesis discharged.
scope and limits
- Does not derive $G = \varphi^5/\pi$ or the $\alpha$ band from first principles here.
- Does not prove uniqueness of the domain cost beyond the stated defs.
- Does not connect the certificate to experimental bounds or SI unit conversion.
- Does not feed any recorded downstream theorem yet (used_by is empty).
- Does not address higher-order gravitational or QED loop corrections.