IndisputableMonolith.Verification.EulerLagrangeCert
Verification module that packages a certificate for the Euler-Lagrange stationarity of the recognition cost J on positive reals. Physicists checking that the RS action principle is variationally consistent would land here. The module is thin: it imports the Cost layer and exposes a named certificate object rather than a long derivation.
claimA verification certificate that the recognition cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) satisfies the Euler-Lagrange stationarity condition associated to the Recognition Composition Law on $\mathbb{R}_{>0}$.
background
Recognition Science fixes a unique nonnegative cost $J$ on positive reals by the Recognition Composition Law and normalization at the fixed point $x=1$. In the Cost layer, $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$, is the T5 unique solution of that functional equation.
Euler-Lagrange structure enters when one treats $J$ as a Lagrangian density on a discrete or continuous recognition path and asks for stationary trajectories. The verification domain collects machine-checkable certificates that such variational identities hold, rather than re-deriving uniqueness of $J$.
This module sits downstream of IndisputableMonolith.Cost and upstream of any audit that needs a named EL certificate object. No further physics constants (phi-ladder, eight-tick, $D=3$) are introduced here.
proof idea
Definition and certificate packaging module, not a multi-step derivation. It imports Mathlib and the Cost API, then exposes the sibling certificate EulerLagrangeCert as the audit-facing object. Any nontrivial identities are expected to be discharged in Cost or by short wrappers around existing $J$-lemmas; this file organizes the claim for verification consumers.
why it matters in Recognition Science
Gives the Verification layer a single named handle for Euler-Lagrange consistency of the RS cost, so downstream audits need not re-open the Cost development. In the forcing chain, that cost is the T5 object $J$; stationarity of the associated action is the bridge from the functional equation (RCL) to a variational reading of recognition dynamics.
No used_by edges are recorded yet, so the module is presently a leaf certificate rather than an input to a larger proved theorem. Its place is organizational: close the variational checklist item next to other Cost-derived certificates without enlarging the foundation.
scope and limits
- Does not re-prove uniqueness of J (T5); that lives in the Cost and forcing-chain layers.
- Does not derive mass formulae, alpha bounds, or dimensional forcing (T6-T8).
- Does not supply continuum field equations beyond the J-cost EL certificate.
- Does not record downstream consumers; used_by is currently empty.