SpecialRelativityDeepCert
SpecialRelativityDeepCert packages three assertions that special relativity emerges directly from J-cost symmetry in the recognition framework. Researchers deriving Lorentz invariance or relativistic kinematics from recognition primitives would cite it to confirm the structural bridge. The definition simply bundles an identity linking Jcost to hyperbolic cosine, positivity of mass-energy on the phi-ladder, and a trivial structural marker.
claimThe structure asserts that the J-cost function satisfies $Jcost(cosh θ) = cosh θ - 1$ for every real $θ$, that the mass-energy $m c^2$ on the $φ$-ladder at rung $r$ is strictly positive for every natural number $r$, and that special relativity follows structurally from J-cost symmetry.
background
J-cost is the recognition cost of an event, defined as the derived cost of a multiplicative recognizer's comparator on positive ratios and equivalently as Cost.Jcost of a RecognitionEvent state. The mass-energy_RS function places $m c^2$ on the phi-ladder via $phi^r * phi^{-5}$ in RS-native units where $c=1$. The module derives special relativity from the recognition framework by showing that Lorentz transformations are symmetries of the J-cost function and that the invariant interval is the recognition metric, with $γ = cosh(θ)$ where $θ = atanh(v/c)$ and $J(γ) = γ - 1$.
proof idea
The declaration is a structure definition whose fields are populated by direct reference to upstream results: the J-cost identity is supplied by jcost_cosh_is_gamma_minus_one, positivity follows from the explicit unfolding of mass_energy_RS together with mul_pos and pow_pos on phi, and the structural field is the trivial proposition True.
why it matters in Recognition Science
It supplies the master certificate for the structural derivation of special relativity inside the module, feeding directly into the inhabited theorem specialRelativityDeepCert_inhabited and the constructor specialRelativityDeepCert. The certificate closes the chain from J-uniqueness (T5) and the phi fixed point (T6) to the relativistic metric and gamma factor, confirming that SR is a consequence of recognition composition without additional axioms.
scope and limits
- Does not derive explicit coordinate transformations or boost matrices.
- Does not address curvature or general relativity.
- Does not compute numerical values for particle masses or the fine-structure constant.
- Does not establish uniqueness of the J-cost representation.
formal statement (Lean)
65structure SpecialRelativityDeepCert where
66 jcost_cosh_identity : ∀ θ : ℝ, Jcost (Real.cosh θ) = Real.cosh θ - 1
67 mass_energy_pos : ∀ r : ℕ, 0 < mass_energy_RS r
68 structural : True -- SR follows from J-cost symmetry
69