IndisputableMonolith.Verification.EulerLagrangeCert
Verification module packaging a certificate that the Recognition cost functional satisfies its Euler-Lagrange stationarity condition. Physicists checking the variational origin of the J-cost cite it when auditing the forcing chain. The module is thin: it imports the Cost layer and exposes a named certificate object rather than a long derivation.
claimA verification certificate asserting that the Recognition cost $J(x)=(x+x^{-1})/2-1$ is stationary under the Euler-Lagrange equation associated to the Recognition Composition Law, i.e. the first variation of the action built from $J$ vanishes on admissible paths.
background
Recognition Science fixes a unique nonnegative cost $J$ on ratios by the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$, with closed form $J(x)=\cosh(\log x)-1$. The Cost module supplies that functional and its elementary identities.
In a variational reading, physical trajectories extremize an action whose Lagrangian density is built from $J$. The Euler-Lagrange equation is the corresponding stationarity condition. This module lives in the Verification domain: it does not re-derive $J$, but records a machine-checkable certificate that the EL identity holds for the RS cost.
The only import beyond Mathlib is IndisputableMonolith.Cost, so all analytic content is inherited from that layer.
proof idea
Definition and certificate module, not a multi-step derivation. It wires the Cost API into a named Euler-Lagrange certificate object (sibling EulerLagrangeCert) that downstream checkers can reference. Any nontrivial identities are expected to reduce to algebraic consequences of the closed form of $J$ already proved in Cost; this file mainly packages and names that fact for the verification graph.
why it matters in Recognition Science
Gives the verification graph a single hook for the claim that RS dynamics are variationally consistent with $J$. That matters for the forcing chain: once $J$ is unique (T5) and $\varphi$ is fixed (T6), one still wants a clean certificate that the same $J$ is the Lagrangian density whose EL equation reproduces the intended discrete/continuous dynamics (eight-tick structure, $D=3$).
No downstream consumers are listed yet (used_by is empty), so the module is presently a leaf certificate: available for audit pages and for future parents that assemble a full variational derivation of the mass ladder or the alpha band. It does not itself close T5-T8; it only certifies the EL side of the cost story.
scope and limits
- Does not re-prove uniqueness of $J$ (that is T5 in the forcing chain).
- Does not derive the mass ladder, alpha band, or dimension $D=3$.
- Does not supply a full field-theoretic action principle beyond the named certificate.
- Does not list downstream theorems; currently a verification leaf.
- Does not address discrete eight-tick dynamics except insofar as Cost already does.