IndisputableMonolith.Foundation.CPT_Theorem3FromJCost
Module packaging CPT Theorem 3 as a certificate built from the J-cost on positive reals. It defines a domain cost, a canonical positive threshold, and an inhabited certificate record tying nonnegativity and threshold positivity to the cost. Foundation authors cite it when discharging the third CPT step from the unique J functional rather than from an external axiom.
claimOn the positive reals, the domain cost is the J-cost $J(x)=(x+x^{-1})/2-1$. A canonical threshold $T>0$ is fixed, and a CPT-3 certificate asserts $J\ge 0$ together with $T>0$, witnessed by an inhabited certificate record.
background
Recognition Science forces the cost functional $J$ uniquely (forcing step T5): $J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $\cosh(\log x)-1$. The Recognition Composition Law and the self-similar fixed point $\varphi$ sit upstream; this module works only with the cost side.
The module imports RS constants (including the native tick $\tau_0$) and the Cost library. It introduces a domain cost equal to $J$ on the positive reals, records that this cost is nonnegative, and fixes a canonical positive threshold used by the CPT-3 statement. The certificate type bundles those facts so later layers can consume a single inhabited witness rather than re-proving cost inequalities.
proof idea
Definition-and-certificate module, not a deep derivation. Domain cost is identified with the standard $J$-cost; equality-at-point and nonnegativity are recorded as lemmas. The canonical threshold is a positive constant with a positivity lemma. CPT3Cert is a structure packing those properties; an explicit inhabitant cert is supplied so cert_inhabited holds by construction.
why it matters in Recognition Science
Closes the third CPT step inside the Foundation layer by deriving the needed positivity and threshold facts from J-cost rather than leaving them as free hypotheses. Feeds any downstream development that expects an inhabited CPT-3 certificate when assembling the forcing chain or cost-based selection rules. No further used-by edges are recorded yet; the module is a leaf certificate source for later CPT and selection arguments that quote T5 J-uniqueness.
scope and limits
- Does not prove J-uniqueness; assumes the Cost library's J.
- Does not derive spatial dimension, eight-tick period, or phi fixed-point facts.
- Does not connect the certificate to mass ladders, alpha, or Berry thresholds.
- Does not supply analytic bounds beyond nonnegativity and threshold positivity.
- Does not claim a used-by consumer inside the current graph.