newton_second_law_one_statement
plain-language theorem explainer
Newton's second law is recovered as the Euler-Lagrange equation of the small-strain J-action. A physicist embedding classical mechanics into Recognition Science would cite this domain certificate when verifying the variational origin of F = ma. The proof is a one-line term that supplies the inhabited witness constructed from the quadratic-limit theorems.
Claim. There exists a structure $C$ such that $C$.el_iff_newton asserts that for any mass $m$, potential $V$, trajectory $γ$, and time $t$, the standard Euler-Lagrange equation of the quadratic J-action vanishes if and only if $m γ̈(t) = -V'(γ(t))$, $C$.quadratic_taylor_bound supplies the bound $|J(1+ε) - ε²/2| ≤ ε²/10$ for $|ε| ≤ 1/10$, and $C$.hessian_one states that the second derivative of $J$ at 1 equals 1.
background
The module packages Newton's second law as a corollary of the J-action variational principle into a single domain certificate. NewtonSecondLawCert is the structure whose three fields record the EL-Newton equivalence, the quadratic Taylor bound on Jcost, and the Hessian calibration J''(1) = 1. These clauses are theorems already established in the QuadraticLimit module and are simply assembled here.
proof idea
The proof is the one-line term that constructs the witness by applying newtonSecondLawCert, which in turn pulls the equivalence from QuadraticLimit.newton_second_law, the Taylor bound from Jcost_taylor_quadratic, and the Hessian from Jcost_quadratic_leading_coeff.
why it matters
This declaration supplies the single-statement domain certificate required by the master UnifiedReality chain. It closes the variational derivation of Newton's second law as a corollary of the J-action in the small-strain regime, as outlined in the paper companion RS_Least_Action.tex §Newton's Second Law as a Corollary. The result relies on the quadratic Taylor expansion and Hessian calibration of Jcost at the ground state.
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