Masses Mass Genesis T10 Creation Deposit Forcing Count Axis Eq One Of Length Thr

The three-step deposit

A cube with three axes, and a path that moves along its edges. Each move along an axis flips a parity bit for that axis. The theorem countAxis_eq_one_of_length_three_all_flipBits states: if a path has length exactly three and flips every axis's bit an odd number of times, then it must have moved along each axis exactly once. A length-three path that flips all three bits cannot, for instance, move along the x-axis twice and the y-axis once, because that would leave the z-axis bit unflipped. The proof is a direct counting argument: the total length is the sum of the per-axis counts, each count must be at least one, and the sum of three numbers each at least one equals three only when each is exactly one.

The result matters because it identifies the minimal event that touches every axis: length three, one move per axis. In the framework's model of mass generation, this minimal event carries a normalization factor. Each single-axis move contributes a factor of 1 divided by the square root of 2, forced by the requirement that the move be a unit quaternion. The product over three axes is therefore (1/√2)³, which equals √2/4. This number, √2/4, is the off-field factor appearing in the predicted amplitude for a gap-one mass, alongside a factor of the golden ratio raised to the 21st power.

What the theorem does not claim is equally precise. It does not say that the length-three, all-flip path is the physically realized creation event; that identification is a conjecture. It does not explain why the exponent 21 appears, nor does it resolve whether the tainted candidate amplitude is the true absolute scale. The theorem proves a combinatorial fact about walks on a three-axis cube, and that fact supplies one factor in a larger calculation. The rest of the story, including the meaning of the 21 and the choice of the minimal path as the physical one, remains open.

/-- **Exactness (THEOREM).** A minimal total-flip deposit (length `3`) posts
each axis exactly once: three odd counts summing to `3` are all `1`. -/
theorem countAxis_eq_one_of_length_three_all_flipBits {w : Walk} (hw : w.length = 3)
    (h : ∀ a : Axis, flipBits w a = true) (a : Axis) : countAxis a w = 1 := by
  have hodd : ∀ a : Axis, Odd (countAxis a w) :=
    fun a => (flipBits_true_iff_odd w a).mp (h a)
  have hsum : countAxis 0 w + countAxis 1 w + countAxis 2 w = 3 := by
    have hlen := length_eq_sum_countAxis w
    rw [hw] at hlen
    rw [Fin.sum_univ_three] at hlen
    exact hlen.symm
  obtain ⟨k0, hk0⟩ := hodd 0
  obtain ⟨k1, hk1⟩ := hodd 1
  obtain ⟨k2, hk2⟩ := hodd 2
  have ha : a = 0 ∨ a = 1 ∨ a = 2 := by fin_cases a <;> decide
  rcases ha with rfl | rfl | rfl <;> omega

/-- The full-dimension deposit normalization is exactly the off-field factor
`sqrt2 / 4`. -/
theorem creationDepositNorm_eq_sqrt2_div_four :
    creationDepositNorm = Real.sqrt 2 / 4 := by
  rw [creationDepositNorm_eq_pow]
  exact inv_sqrt_two_pow_three

/-- **Minimality (THEOREM).** Any deposit flipping every parity bit posts at
least `3 = dim` times: each axis count is odd, hence at least one. -/
theorem three_le_length_of_all_flipBits {w : Walk} (h : ∀ a : Axis, flipBits w a = true) :
    3 ≤ w.length := by
  rw [length_eq_sum_countAxis]
  have h1 : ∀ a : Axis, 1 ≤ countAxis a w := by
    intro a
    obtain ⟨k, hk⟩ := (flipBits_true_iff_odd w a).mp (h a)
    omega
  calc (3 : ℕ) = ∑ _a : Axis, 1 := by decide
    _ ≤ ∑ a : Axis, countAxis a w := Finset.sum_le_sum fun a _ => h1 a

What this page does not claim

The theorem does not establish that the length-three all-flip path is the physically realized creation event. The theorem does not explain the origin of the phi^21 factor in the amplitude. The theorem does not resolve whether the tainted candidate is the true absolute mass scale.

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/T10CreationDepositForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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