IndisputableMonolith.Physics.LorentzSymmetryFromRecognition
Module deriving Lorentz symmetry from the Recognition cost J: boost inversion leaves the cost invariant. Physicists cite it for the claim that special-relativistic frame changes are forced by the same J that yields the forcing chain. The argument compares rest-frame equilibrium cost to the boosted-frame cost and records a certificate that the two agree.
claimThe Recognition cost $J$ is invariant under boost inversion: the equilibrium cost in the rest frame equals the cost evaluated in a moving frame related by a Lorentz boost. The module packages this as a Lorentz-symmetry certificate built from the rest-frame equilibrium value and the moving-frame cost.
background
Recognition Science takes the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$) as the starting point of the forcing chain (T5). The Recognition Composition Law constrains how $J$ behaves under products and quotients of positive reals. Special relativity enters when one asks whether a change of inertial frame can alter that cost.
This module sits in the physics layer and imports the Cost development. It introduces a discrete type of Lorentz transforms, a count of those transforms, the rest-frame equilibrium cost, and the cost recomputed after a boost. The informal claim in the module header is that $J$ is symmetric under boost inversion: reversing a boost returns the same numerical cost.
The local objects are therefore comparative rather than dynamical: one evaluates $J$ (or an equilibrium functional built from it) in two frames and asserts equality. No metric signature or light-cone structure is postulated up front; the symmetry is read off the cost.
proof idea
The module is organized around named comparisons rather than a single deep induction. It defines a Lorentz-transform type and a finite count of transforms, then states rest-frame equilibrium and moving-frame cost as separate values. The main symmetry claim equates those two costs (boost inversion leaves $J$ unchanged). A certificate structure packages the equality for downstream use. Supporting lemmas are expected to reduce to algebraic identities for $J$ under the reciprocal substitution that models boost inversion, drawing on the Cost import.
why it matters in Recognition Science
Lorentz invariance is not an extra postulate in Recognition Science; it must follow from the same $J$ that forces $\phi$, the eight-tick octave, and $D = 3$. This module records that step for the boost sector: cost equality under boost inversion. Downstream physics developments that need a named certificate of frame-independence (dispersion relations, mass-ladder comparisons across frames, or consistency of $c = 1$ in RS units) can import the certificate rather than re-deriving the identity. No parent theorems are listed yet in the dependency graph, so the module currently acts as a leaf that closes the Lorentz-symmetry claim inside the physics domain.
scope and limits
- Does not derive the full Lorentz group or continuous boost generators from first principles.
- Does not prove Minkowski metric signature or light-cone causality.
- Does not address rotational SO(3) symmetry or the full Poincare algebra.
- Does not connect boost invariance to the mass ladder or alpha band numerically.
- Does not claim experimental bounds; only cost equality under the modeled inversion.