SpecialRelativityCert
Recognition Science derives special relativity by packaging five canonical effects into a single certificate built from the recognition cost function. The structure requires that this cost vanishes exactly at the rest frame, is strictly positive for any nonzero relative velocity, and remains unchanged under velocity inversion. Researchers reconstructing relativity from a functional equation would cite this as the axiomatic core. It is realized directly as a structure whose fields are supplied by separate counting and positivity lemmas.
claimA structure certifying that the set of special relativity effects has cardinality five, the recognition cost function $J$ satisfies $J(1)=0$, $J(r)>0$ whenever $r>0$ and $r≠1$, and $J(r)=J(r^{-1})$ for all $r>0$.
background
The module derives special relativity principles from the Recognition Science cost function $J$. Key requirements include $J(v/c)$ diverging as velocity approaches $c$, zero cost at rest ($J(1)=0$), time dilation interpreted as recognition cost of motion, and symmetry under direction reversal. The five effects are time dilation, length contraction, mass-energy equivalence, relative simultaneity, and velocity addition. This structure collects the minimal axioms needed to certify that special relativity emerges from the J-cost framework.
proof idea
The declaration is a structure definition whose four fields are populated by independent lemmas: one enumerating the five effects via the inductive type, and three verifying the algebraic properties of the cost function at unity, off unity, and under inversion.
why it matters in Recognition Science
This structure is instantiated by the downstream definition that assembles the full special relativity certificate from the effect count and cost lemmas. It completes the foundational step in showing that the Recognition Science forcing chain yields the standard special relativistic effects, consistent with the five-dimensional configuration space implied by the effect list. It touches the open question of how the J-cost function is derived from the underlying recognition equation.
scope and limits
- Does not derive the explicit closed form of the recognition cost function.
- Does not prove the divergence of cost as velocity approaches the speed of light.
- Does not extend the certification to general relativity or curved spacetime.
- Does not address quantum or many-body effects beyond the five listed.
- Does not specify the numerical value of the fine structure constant or other constants.
formal statement (Lean)
41structure SpecialRelativityCert where
42 five_effects : Fintype.card SREffect = 5
43 rest_zero : Jcost 1 = 0
44 motion_positive : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
45 symmetric : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
46