Masses Mass Genesis T10 Counting Bridge Obstruction Int Combination Eq Zero

The rational independence lemma

The golden ratio φ is the positive solution to r² = r + 1, approximately 1.618. One of its less famous properties is that it is irrational: it cannot be written as a fraction of two whole numbers. The declaration int_combination_eq_zero states a sharper version of that fact: if integers a and b satisfy a·φ + b = 0, then both a and b must be zero. In other words, 1 and φ are linearly independent over the rational numbers, meaning no nonzero integer combination of them can collapse to zero.

This lemma is proved in the framework's machine-checked library of formal theorems. The proof runs through a standard route: φ = (1 + √5)/2, so the equation a·φ + b = 0 rearranges to √5 = −(2b + a)/a, which would express √5 as a rational number. Since √5 is irrational (a theorem also in the library), that is impossible unless a = 0, which then forces b = 0. The declaration is a formal theorem, not a definition or a hypothesis.

In Recognition Science, this lemma does concrete work. The framework models particle masses as values built from powers of φ, and it defines "block energies" as φ^(−n) for integer n. Every finite sum of signed multiples of such block energies reduces to the form A·φ + B with integers A and B. The lemma then says this representation is unique: two different integer pairs cannot give the same value. That uniqueness is what lets the framework prove a negative result: the specific load φ^42 / 4 cannot be written as any finite integer combination of block energies, because doing so would force 4 to divide the Fibonacci number F_41, which it does not (F_41 = 165580141 ≡ 1 mod 4).

The lemma itself does not claim anything about particle masses, about which loads are representable, or about the physical meaning of φ. It is a pure number-theoretic fact about the golden ratio. Its role is as a tool: it establishes the rational independence that makes the later obstruction theorem possible. The framework's own documentation describes this as closing a specific repair route for a counting puzzle in mass genesis, but the lemma alone says nothing about that puzzle.

/-- **`{1, phi}` is rationally independent** (integer form). -/
theorem int_combination_eq_zero {a b : ℤ}
    (h : ((a : ℝ)) * Constants.phi + ((b : ℝ)) = 0) : a = 0 ∧ b = 0 := by
  by_cases ha : a = 0
  · subst ha
    simp at h
    exact ⟨rfl, by exact_mod_cast h⟩
  · exfalso
    have hane : (a : ℝ) ≠ 0 := by exact_mod_cast ha
    have hphi : Constants.phi = (((-(b : ℚ) / (a : ℚ)) : ℚ) : ℝ) := by
      push_cast
      field_simp
      linear_combination h
    exact irrational_phi ⟨-(b : ℚ) / (a : ℚ), hphi.symm⟩

/-- **`{1, phi}` is rationally independent** (integer form). -/
theorem int_combination_eq_zero {a b : ℤ}
    (h : ((a : ℝ)) * Constants.phi + ((b : ℝ)) = 0) : a = 0 ∧ b = 0 := by
  by_cases ha : a = 0
  · subst ha
    simp at h
    exact ⟨rfl, by exact_mod_cast h⟩
  · exfalso
    have hane : (a : ℝ) ≠ 0 := by exact_mod_cast ha
    have hphi : Constants.phi = (((-(b : ℚ) / (a : ℚ)) : ℚ) : ℝ) := by
      push_cast
      field_simp
      linear_combination h
    exact irrational_phi ⟨-(b : ℚ) / (a : ℚ), hphi.symm⟩

theorem irrational_sqrt5 : Irrational (Real.sqrt 5) := by
  rw [show (5 : ℝ) = ((5 : ℕ) : ℝ) by norm_num, irrational_sqrt_natCast_iff]
  rintro ⟨m, hm⟩
  have hle : m ≤ 2 := by
    by_contra h3
    push_neg at h3
    have hsq : 3 * 3 ≤ m * m := Nat.mul_le_mul h3 h3
    omega
  interval_cases m <;> norm_num at hm

What this page does not claim

The lemma does not claim that φ is transcendental, only irrational. The lemma does not claim that any particular load is or is not representable as a block combination. The lemma does not claim anything about the physical masses of particles.

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/T10CountingBridgeObstruction.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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