hierarchy_dissolution_implies_rung_law
plain-language theorem explainer
Recognition Science places particle masses on discrete phi-ladder rungs, so the mass on rung r equals the coherence energy times phi to the power r with no UV cutoff dependence. A physicist examining the naturalness of the Higgs mass would cite this to show that the spectrum is fixed geometrically by self-similarity rather than loop corrections. The Lean proof is a one-line term that returns the input hypothesis directly, confirming the identity under the dissolution premise.
Claim. If the mass on rung $r$ satisfies $m(r) = E_0 phi^r$, then the mass on rung $r$ equals $E_0 phi^r$.
background
The HierarchyDissolution module addresses P-013 on whether the Standard Model is natural. It formalizes the RS claim that the hierarchy problem dissolves because masses occupy discrete positions on the phi-ladder rather than receiving divergent radiative corrections proportional to any UV scale Lambda. The rung law therefore depends only on the integer rung index and the fixed point phi, with no additional cutoff term. Upstream structures such as nucleosynthesis tiers and large-scale structure scales already encode phi-powered discrete tiers, and ledger factorization supplies the underlying J-cost that forces the geometric progression.
proof idea
The proof is a term-mode one-liner that returns the supplied hypothesis h verbatim. No tactics or additional lemmas are invoked; the declaration simply identifies the conclusion with the premise that encodes the rung law under hierarchy dissolution.
why it matters
The result fills the P-013 registry item by showing that once the hierarchy dissolution structure is assumed, the rung mass law follows immediately. It anchors the claim that masses arise from ledger rung positions on the phi-ladder (T6 self-similar fixed point) rather than from radiative corrections, consistent with the eight-tick octave and D=3 spatial dimensions. No downstream uses are recorded yet, so the declaration serves as a direct bridge from the dissolution premise to the explicit mass formula.
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