Masses Mass Genesis T10 Creation Deposit Forcing Two Mul Intended Ne Phi Pow Mul

The doubled-scale gate

The declaration two_mul_intended_ne_phi_pow_mul_creationDepositNorm is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states, in plain terms: there is no whole number k such that twice the intended gap-one factor amplitude equals phi raised to the k-th power times the creation deposit norm. The intended amplitude is the value the framework banks for a specific mass-generation step, written as phi^21 times the creation deposit norm. The declaration proves that doubling that amplitude cannot be expressed in the same form, because it would require phi^(m-21) to equal 2, and the framework has already proved that 2 is not a power of phi.

This result is the gate that rejects a doubled-scale decoy. In the framework's picture, a creation deposit is a discrete record of events, and its normalization is forced by unitarity: each posting along a cube axis carries a factor of 1/sqrt(2), and the full deposit across three axes carries (1/sqrt(2))^3 = sqrt(2)/4. The intended amplitude fits this shape exactly. The doubled candidate, 2 * phi^21 * creationDepositNorm, also lies in the phi-field, so the rejection does not come from the field alone. It comes from the rung-quantum shape: the scalar must be a pure phi power, and 2 is not one. The theorem is kernel-checked with no sorry and no new axioms.

The declaration does not claim that the phi^21 remainder is explained. That exponent carries the tainted electroweak rung, and the taint is not lifted. It does not claim that a genesis creation deposit must post every cube axis once; that selection principle is a conjecture, though the minimal total-parity-flip event is proved to have length exactly three. It also does not claim that the additive composition law is dead in general; it shows that an additive-over-axes deposit law accepts a quadruple-scale decoy, so that composition fails the gate here.

What the theorem changes is the boundary of what is forced. The transport product fixes only the off-field factor, sqrt(2)/4, and every phi-field scalar times that factor is off-field. The rejection of the doubled candidate is therefore a precise, local result: it closes one decoy path, while leaving the selection principle and the phi^21 remainder as open targets for the framework.

/-- **Gate discrimination (THEOREM).** The rung-quantum law rejects the
doubled decoy: `2 f = phi^m * creationDepositNorm` would force
`phi^(m-21) = 2`, excluded by the banked `two_not_zpow_phi` (the diagonal
ratio `2` is no integer power of `phi`). -/
theorem two_mul_intended_ne_phi_pow_mul_creationDepositNorm :
    ¬ ∃ k : ℕ, 2 * intendedGapOneFactorAmplitude =
      Constants.phi ^ k * creationDepositNorm := by
  rintro ⟨k, hk⟩
  rw [intended_eq_phi21_mul_creationDepositNorm, creationDepositNorm_eq_pow] at hk
  have hpn : postingNorm ≠ 0 := ne_of_gt postingNorm_pos
  have hk2 : (2 * Constants.phi ^ 21) * postingNorm ^ 3 =
      Constants.phi ^ k * postingNorm ^ 3 := by
    linear_combination hk
  have h1 : 2 * Constants.phi ^ 21 = Constants.phi ^ k :=
    mul_right_cancel₀ (pow_ne_zero 3 hpn) hk2
  have hφ : Constants.phi ≠ 0 := Constants.phi_ne_zero
  have hz : Constants.phi ^ ((k : ℤ) - (21 : ℕ)) = 2 := by
    rw [zpow_sub₀ hφ, zpow_natCast, zpow_natCast, ← h1]
    rw [div_eq_iff (pow_ne_zero 21 hφ)]
  exact two_not_zpow_phi ⟨(k : ℤ) - (21 : ℕ), hz⟩

/-- The intended gap-one factor amplitude is the rung-21 phi scalar times
the full-dimension creation-deposit normalization. -/
theorem intended_eq_phi21_mul_creationDepositNorm :
    intendedGapOneFactorAmplitude = Constants.phi ^ 21 * creationDepositNorm := by
  rw [intended_eq_tainted, creationDepositNorm_eq_sqrt2_div_four]
  calc taintedAmplitude = Constants.phi ^ 21 * Real.sqrt 2 / 4 := rfl
    _ = Constants.phi ^ 21 * (Real.sqrt 2 / 4) := by ring

/-- 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

/-- **The additive-over-axes composition is gate-killed (banked).** Any
deposit law whose payable loads compose additively accepts the quadruple of
every accepted load, hence cannot accept the gap-one load while rejecting
the doubled-scale decoy. This instantiates the banked
`additiveGenesis_accepts_gapOne_decoy`; it is not re-proved here. -/
theorem additive_composition_gate_killed {Payable : ℝ → Prop}
    (hadd : ∀ {x y : ℝ}, Payable x → Payable y → Payable (x + y))
    (hintended : Payable (Constants.phi ^ 42 / 4)) :
    Payable (4 * (Constants.phi ^ 42 / 4)) :=
  additiveGenesis_accepts_gapOne_decoy hadd hintended

What this page does not claim

The theorem does not explain the phi^21 exponent or lift the taint from the electroweak rung. The theorem does not prove that a genesis creation deposit must post every cube axis once; that selection principle remains a conjecture. The theorem does not show that additive composition is universally invalid, only that it fails this gate by accepting a quadruple-scale decoy.

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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