equivalence_ratio_unity_structural
plain-language theorem explainer
The declaration proves that the ratio of inertial to gravitational mass equals one whenever the two masses are equal and the gravitational mass is nonzero. Researchers deriving the equivalence principle from a single cost function would cite this to confirm the structural identity. The proof is a direct term-mode application of the elementary ratio lemma after introducing the mass variables and hypotheses.
Claim. For all real numbers $m_1, m_2$, if $m_2 ≠ 0$ and $m_1 = m_2$, then $m_1 / m_2 = 1$.
background
In Recognition Science both inertial and gravitational masses are extracted from the single cost function J(x) = ½(x + x⁻¹) − 1. Inertial mass is the quadratic coefficient J''(1) = 1; gravitational mass is the integrated J-cost defect. Module G-003 therefore treats any mass theory that pulls both quantities from this unique J as necessarily producing equal values. The upstream definition equivalence_ratio_unity encodes precisely the statement that the ratio is unity under equality (with nonzero gravitational mass). The lemma ratio_one_when_equal supplies the algebraic identity that equal nonzero reals have ratio one.
proof idea
The proof is a one-line term-mode wrapper. It introduces the inertial mass m_i, gravitational mass m_g, the nonzero hypothesis, and the equality hypothesis, then applies the lemma ratio_one_when_equal directly to these inputs.
why it matters
This theorem closes the structural half of the G-003 derivation. It supplies the definition equivalence_ratio_unity that is invoked by rs_equivalence_ratio to obtain the concrete result for Jcost_mass_theory. Within the framework it realizes the T5 uniqueness of J: because both mass functionals are built from the same cost function, their equality is a tautology rather than an empirical fact. No open scaffolding remains at this node.
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