IndisputableMonolith.Physics.LorentzSymmetryFromRecognition
This module establishes that the J-cost is symmetric under boost inversion, deriving Lorentz symmetry from recognition principles. It supplies the types, counts, and certificates needed to link frame costs to relativistic invariance. Physicists reconstructing special relativity from the Recognition Composition Law would cite these results. The module builds on the Cost import to verify the symmetry by direct comparison of boosted and inverted frames.
claimLorentz symmetry: $J$ is invariant under boost inversion, so the recognition cost computed for a boost with velocity parameter $v$ equals the cost for the inverted boost with parameter $-v$.
background
Recognition Science starts from the J-cost satisfying the Recognition Composition Law and the forcing chain that yields three spatial dimensions. This module sits in the Physics domain and imports the Cost module to access the definition $J(x) = (x + x^{-1})/2 - 1$. It introduces LorentzTransformType as the structure for admissible boosts, together with auxiliary functions that count transforms and compute moving-frame costs relative to a rest-frame equilibrium.
proof idea
The module is organized as a sequence of definitions followed by targeted theorems. It first declares the Lorentz transform type and the moving-frame cost function, then applies the algebraic properties of J imported from Cost to establish invariance under sign flip of the boost parameter. The central theorem lorentz_symmetry is obtained by direct substitution and simplification using the RCL identity.
why it matters in Recognition Science
The module supplies the missing link that turns the abstract J-cost into a concrete relativistic symmetry, feeding the larger program that recovers the Lorentz group and the alpha band from recognition alone. It aligns with the T5–T8 steps of the forcing chain by showing that boost inversion leaves the cost unchanged, thereby supporting the emergence of D = 3 spatial dimensions and the observed constants.
scope and limits
- Does not construct the full Lorentz group or its Lie algebra.
- Does not treat acceleration, curved spacetime, or general covariance.
- Does not incorporate quantum fields or gauge interactions.
- Does not produce numerical predictions or compare with experiment.