IndisputableMonolith.Physics.ElasticScattering_FromJCost
Module packaging elastic-scattering certificates built from the Recognition Science J-cost. It defines a domain cost, proves nonnegativity and evaluation identities, fixes a positive canonical threshold, and packages an inhabited certificate type. A scattering or RS-cost theorist would cite it when tying kinematic thresholds to J. Content is mostly definitions plus short positivity and equality lemmas over the Cost and Constants imports.
claimOn a domain equipped with the RS cost $J$, define a domain cost $C$, prove $C\ge 0$ and the pointwise evaluation identity, fix a canonical threshold $\tau>0$, and assemble an inhabited elastic-scatter certificate asserting that scattering is certified once the cost clears $\tau$.
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick scale used to normalize thresholds.
This physics module lifts those primitives to an elastic-scattering setting. A domain cost is the restriction or pullback of $J$ (or a monotone transform of it) to the kinematic variables of a scatter event. The canonical threshold is a fixed positive real against which that cost is compared; positivity of the threshold is recorded so later inequalities do not collapse.
The certificate type packages the numerical comparison (cost at or above threshold) into a Prop-carrying structure that downstream physics lemmas can inhabit or discharge without re-proving nonnegativity of $J$.
proof idea
Definition-heavy module. Domain cost is introduced as a def, with a one-line evaluation identity and a nonnegativity lemma inherited from nonnegativity of $J$ on the positive reals. The canonical threshold is a positive constant (proved positive by a short arithmetic or Constants fact). ElasticScatterCert is a structure or Prop bundling the cost-threshold comparison; inhabitation is a constructor application once nonnegativity and positivity are in hand. No deep tactic scripts; the argument is assembly of Cost lemmas into a named physics certificate.
why it matters in Recognition Science
Places elastic scattering inside the J-cost ledger rather than as an external Born-level postulate. Downstream pages can assume an inhabited scatter certificate instead of re-deriving $J\ge 0$ or threshold positivity. In the broader RS chain this sits under the cost layer (T5 J-uniqueness and RCL) and feeds any kinematics that need a dimensionless barrier measured in $J$-units. The module currently has no recorded used-by edges, so it is a leaf packaging layer ready for cross-section or phase-shift developments.
scope and limits
- Does not derive differential cross sections or partial-wave amplitudes.
- Does not prove uniqueness of the canonical threshold from first principles.
- Does not treat inelastic channels, spin, or multi-particle final states.
- Does not connect the threshold numerically to alpha, G, or the mass ladder.
- Does not claim experimental fits; only the cost-certificate interface.