mass_energy_RS
plain-language theorem explainer
The mass-energy equivalence in Recognition Science places the rest energy of a particle at rung r on the phi-ladder as the product of phi to the power r and phi to the negative fifth power. Researchers deriving special relativity from the J-cost symmetry would cite this when establishing the structural mass-energy relation with c set to 1. The definition is introduced directly as an algebraic scaling expression without intermediate lemmas.
Claim. The mass-energy $m c^2$ at rung $r$ equals $phi^r · phi^{-5}$ in recognition units.
background
In the Recognition Science treatment of special relativity, the J-cost function supplies the Lorentz factor via $gamma = 1 + J(e^theta)$ where $J(x) = (x + x^{-1})/2 - 1$. The module sets $c = 1$ and derives the invariant interval from symmetries of this cost function, with energies scaled along the phi-ladder. The constant $phi^{-5}$ matches the native unit for $hbar$, so the expression aligns with the mass formula that places rest energies at discrete rungs.
proof idea
This is a direct definition that writes the mass-energy as the product of two powers of phi. No lemmas or tactics are invoked; the expression stands alone as the base term for later statements.
why it matters
The definition supplies the mass-energy term required by SpecialRelativityDeepCert, whose fields certify that special relativity follows from J-cost symmetry with no additional axioms. It implements the relation $m c^2 = phi^r · E_coh$ at rung r and connects to the framework landmarks T6 (phi as fixed point) and the phi-ladder scaling. The downstream positivity result and the master certificate both unfold this expression directly.
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