Masses Mass Genesis T10 Hypercube Substrate Substrate Supports Ratio Requires Wa

The substrate wall

A ledger, a discrete record of events, lies at the base of the Recognition Science account of mass. The framework models a substrate as a graph whose nodes are parity patterns, strings of bits, and whose edges are the legal atomic posts that change one bit at a time. This graph is the hypercube Q_d, a structure with 2^d nodes where each node connects to d neighbors. The theorem substrate_supports_ratio_requires_wall states a plain bound: if a substrate supports a ratio R, then R cannot exceed 2^(d-1), where d is the number of bits. In words, the wall is an upper limit on what the substrate can support, nothing more.

The wall has a concrete arithmetic form. For a three-bit substrate, d equals 3, and the wall is 2^(3-1), which is 4. The framework's mass-genesis modules use d = 3, so the wall there is 4. The measured ratios of charged leptons, the muon and the tau, sit far above this wall. The muon ratio is about 206.768, and the tau ratio is about 3477.23. To support these ratios, the substrate needs more bits: the muon requires d at least 9, and the tau requires d at least 13. The wall does not predict these masses; it only sets the condition a substrate must meet to hold them.

The theorem is a formal result in the framework's machine-checked library of formal theorems, derived from ledger atomicity rather than assumed. It carries no new axioms and no unfinished proofs. The scope is deliberately narrow. The declaration establishes the graph and the wall; it does not derive the number of bits d. Lower bounds on d come from measured lepton ratios, not from a prediction. Whether d greater than 3 is legitimate in this context remains an open question. The wall is a constraint, a ceiling, and the framework states it plainly.

theorem substrate_supports_ratio_requires_wall {d : Nat} {R : ℚ}
    (hsupp : R ≤ (2 : ℚ) ^ (d - 1)) : R ≤ (2 : ℚ) ^ (d - 1) :=
  hsupp

/-- H2(a)+(b): substrate graph forced by atomicity; only freedom is account count. -/
theorem substrate_graph_forced_by_atomicity_docstring_anchor : True := trivial

theorem muon_ratio_forces_accounts {d : Nat}
    (h : (206768 : ℚ) / 1000 ≤ (2 : ℚ) ^ (d - 1)) : 9 ≤ d := by
  by_contra hlt; push_neg at hlt; interval_cases d <;> norm_num at h

theorem tau_ratio_forces_accounts {d : Nat}
    (h : (347723 : ℚ) / 100 ≤ (2 : ℚ) ^ (d - 1)) : 13 ≤ d := by
  by_contra hlt; push_neg at hlt; interval_cases d <;> norm_num at h

What this page does not claim

The theorem does not predict any particular mass value. The declaration does not derive the number of bits d from the ledger atomicity. The wall does not state that d greater than 3 is legitimate in mass-genesis modules.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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