IndisputableMonolith.Verification.JcostAxiomsCert
Verification certificate module for the J-cost axioms used throughout Recognition Science. It packages the algebraic identities that uniquely characterize J(x)=(x+x^{-1})/2-1 (equivalently cosh(log x)-1) so downstream forcing and mass-ladder arguments can cite a single checked bundle. The module is thin: it imports Cost and re-exports or asserts the axiom set as a certifiable unit.
claimA verification certificate that the cost $J:(0,\infty)\to[0,\infty)$ satisfies the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$, normalization $J(1)=0$, and the uniqueness/forcing properties that pin $J(x)=(x+x^{-1})/2-1$.
background
Recognition Science derives physics from a single functional equation on a dimensionless cost $J$. The canonical form forced at T5 is $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. This $J$ is nonnegative, vanishes only at $1$, and obeys the Recognition Composition Law (RCL):
$J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$.
The parent Cost module defines $J$ and proves the elementary identities. Verification modules in this layer do not re-derive those identities; they package them as a named certificate so that later stages (forcing chain, constants, mass ladder) can depend on a single audited interface rather than scattered lemmas.
proof idea
This is a certificate/aggregation module, not a deep proof development. It imports IndisputableMonolith.Cost (and Mathlib) and exposes the J-cost axiom bundle under the JcostAxiomsCert name so consumers obtain a single dependency edge. Any internal proofs are thin wrappers or re-exports of the Cost-layer lemmas that establish RCL, normalization, and uniqueness of $J$.
why it matters in Recognition Science
Downstream forcing (T5 J-uniqueness through T8 dimension) and every mass or coupling formula that sits on the phi-ladder presuppose that $J$ is the unique cost obeying RCL. A dedicated axioms certificate keeps that hypothesis explicit and machine-checkable: if the Cost layer changes, this module is the single place verification tooling re-checks. No used_by edges are recorded yet; the module is infrastructure for the Verification domain rather than a leaf theorem cited by name in physics results.
scope and limits
- Does not re-prove uniqueness of J from first principles; that lives in Cost / the forcing chain.
- Does not derive physical constants, the phi fixed point, or the eight-tick structure.
- Does not assert numerical bounds on alpha or mass rungs.
- Does not introduce new axioms beyond those already stated for J in Cost.