IndisputableMonolith.Physics.CouplingRunning3_FromJCost
Module packaging a certificate that three-scale coupling running is controlled by the J-cost on a positive domain. It defines a domain cost, a canonical positive threshold, and an inhabited CouplingRun3Cert record. Physicists tracking RS renormalization or three-generation structure would cite the certificate. Content is definitional plus nonnegativity and positivity lemmas, not a deep existence proof.
claimOn a positive domain one has a cost $C$ built from the RS $J$-cost, with $C\ge 0$ and a canonical threshold $\theta>0$. A certificate record asserts the three-scale coupling-running constraints implied by that cost; the module supplies an inhabited instance of the certificate.
background
Recognition Science forces the unique symmetric cost $J(x)=(x+x^{-1})/2-1$ (T5) and the Recognition Composition Law. Couplings in RS are not free parameters: their scale dependence is read off from $J$ on multiplicative ratios of scales, with the golden ratio $\phi$ fixing the self-similar ladder (T6).
This module sits in the physics layer and imports only Constants (RS time quantum $\tau_0$) and Cost (the $J$-cost infrastructure). It introduces a domain-restricted cost functional, equality at a reference point, nonnegativity, and a strictly positive canonical threshold used as the running cutoff or matching scale.
The three in the name aligns with the forced spatial dimension $D=3$ (T8) and with three-family structure in the broader RS coupling story; the module itself only packages the cost-side certificate, not the full Standard Model beta functions.
proof idea
Definition module with thin lemmas. domainCost and canonicalThreshold are defs; domainCost_at_eq and domainCost_nonneg record algebraic identities and $J\ge 0$ on the positive reals; canonicalThreshold_pos is a positivity fact. CouplingRun3Cert is a structure bundling those ingredients; cert builds an instance and cert_inhabited shows the type is nonempty. No multi-step analytic argument beyond cost nonnegativity and threshold positivity.
why it matters in Recognition Science
Gives a named, inhabitable certificate that three-scale coupling running is downstream of $J$-cost rather than an extra RG postulate. That matches the RS program: once $J$ is forced (T5) and $\phi$ and $D=3$ are fixed (T6, T8), coupling thresholds should be cost-derived.
No downstream consumers are wired in the graph yet (used_by empty), so the module is a leaf certificate ready for a parent theorem on running couplings, $\alpha$ matching, or three-generation mass/coupling ladders. It closes a scaffolding gap between pure cost theory and physics-facing running statements without claiming a full beta-function derivation.
scope and limits
- Does not derive Standard Model beta functions or loop coefficients.
- Does not prove uniqueness of the three-scale scheme beyond the certificate fields.
- Does not compute numerical values of $\alpha$ or gauge couplings.
- Does not connect to eight-tick dynamics or mass-ladder rungs.
- Does not supply downstream theorems; the use graph is currently empty.