Masses Baseline Derivation Minimal Complete Coefficients

The completeness condition

In the Recognition Science framework, the masses of elementary particles are not free parameters. They are placed on a ladder of integers, each rung derived from the geometry of a three-dimensional cube. The declaration minimal_complete_coefficients is a theorem in the framework's machine-checked library of formal theorems. It states a simple inequality: for any two positive integers a and b, their sum is at least 2. The proof is a single line of arithmetic, but the statement carries weight because it is a completeness condition for the framework's particle ladder.

The framework models the universe as maintaining a discrete record of events, a ledger. The cost of a transition between two states is forced by a proved formula, J(x) = (x + 1/x)/2 - 1, which is zero only when the transition ratio x equals 1. A transition with ratio 1 is operationally indistinguishable from no transition, so it is excluded from the physical event count. The theorem minimal_complete_coefficients is a companion to this: it says that when the framework's counting variables a and b are both at least 1, their sum cannot be less than 2. In plain terms, the ledger cannot record a single event as a fraction; every recorded transition counts as at least one whole unit.

The theorem is proved in the framework's library, in the module IndisputableMonolith/Masses/BaselineDerivation.lean. It is a formal consequence of the definitions of the counting variables, not an assumption added to the framework. The library proves it with the standard arithmetic tactic, which means the statement is checked by the machine. The theorem is part of a larger derivation that upgrades what were previously boundary assumptions into derived results, all from the single input that the spatial dimension D equals 3. The octave offset, the neutrino baseline, the lepton and quark baselines, and the generation ordering all trace back to cube geometry at D = 3.

What the theorem does not claim is equally important. It does not say that the framework derives the actual measured masses of particles. The scaling law that places particle masses on a phi-power ladder is proved, but the comparison against measured PDG values is an empirical check, not a theorem. The theorem minimal_complete_coefficients is a structural statement about the ledger's counting consistency, not a prediction of any specific mass. It also does not claim that the framework derives the fine-structure constant alpha; that remains an open target. The theorem is a piece of the framework's internal architecture, not a bridge to experimental data.

/-- The minimal complete solution is (a, b) = (1, 1). -/
theorem minimal_complete_coefficients :
    ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → a + b ≥ 2 :=
  fun a b ha hb => by omega

/-- The minimal complete solution is (a, b) = (1, 1). -/
theorem minimal_complete_coefficients :
    ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → a + b ≥ 2 :=
  fun a b ha hb => by omega

/-- J(x) = 0 ⟹ x = 1 (for x > 0). -/
theorem J_eq_zero_imp_one (x : ℝ) (hx : 0 < x) (hJ : J x = 0) : x = 1 := by
  unfold J at hJ
  have hsum : x + x⁻¹ = 2 := by linarith
  have hxx : x * x⁻¹ = 1 := mul_inv_cancel₀ (ne_of_gt hx)
  have hinv : x⁻¹ = 2 - x := by linarith
  have : x * x⁻¹ = x * (2 - x) := by rw [hinv]
  rw [hxx] at this
  nlinarith [this]

/-- At D = 3: T_min = 8. -/
theorem T_min_at_D3 : T_min D = 8 := by native_decide

What this page does not claim

The theorem does not derive the actual measured masses of particles from the framework. The theorem does not claim that the framework derives the fine-structure constant alpha. The theorem does not state that the ledger can record fractional events.

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/BaselineDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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