Encyclopedia Masses Masses Mass Genesis T10 Creation Deposit Forcing Deposit Quantum Intended Eq Phi

ARTICLE 5 claims 5 theorems

Masses Mass Genesis T10 Creation Deposit Forcing Deposit Quantum Intended Eq Phi

A machine-checked theorem ties the mass scale to a geometric normalization, but the deepest origin of the scale remains an open conjecture.

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The deposit quantum identity

The declaration depositQuantum is part of a framework that models physical quantities as entries in a discrete ledger of events. The theorem depositQuantum_intended_eq_phi42_mul_prod_sq_postingNorm states a precise identity: for a certain intended amplitude, the deposit quantum equals phi to the power 42 multiplied by the product of the squared per-posting normalization factors across the three axes. In plain terms, the framework proves that the quantum of a creation event is the golden ratio raised to the 42nd power times a geometric factor that comes from normalizing each of three spatial directions.

The geometric factor is not arbitrary. The framework proves that each per-posting normalization is forced by unitarity to be 1 divided by the square root of 2. The product of three such factors, squared, is exactly one eighth. This one-eighth is the same number that appears in the banked relation for the square of the intended amplitude, which is phi to the 42nd power divided by 8. The identity therefore shows that the off-field factor, the part not built from phi alone, is entirely determined by the three-dimensional normalization structure.

The theorem also proves that a simple additive composition law for deposits fails. If a set of payable amounts is closed under addition, then accepting the intended amplitude forces accepting four times that amplitude, a doubled-scale decoy. The multiplicative composition, where the normalization of a composite deposit is the product of the per-posting normalizations, is the one that survives. The framework also proves that the intended amplitude belongs to the allowed rung-quantum deposit class, while twice that amplitude does not, because no power of phi equals twice phi to the 21st.

What the identity does not claim is just as important. The selection principle that a genesis creation deposit posts every cube axis once is not derived from a dynamics; it is a conjecture. The phi to the 21st remainder, half of the 42 in the mass exponent, is not explained by this theorem. That remainder carries what the framework calls a tainted electroweak rung, and the taint is not lifted. Whether the tainted candidate is the true absolute scale remains undecided.

THEOREM depositQuantum_intended_eq_phi42_mul_prod_sq_postingNorm · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean
depositQuantum_intended_eq_phi42_mul_prod_sq_postingNorm · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean:244 
/-- **The C1 deposit law multiplies the per-axis normalizations (THEOREM).**
The C1 seed's own deposit quantum, evaluated at the intended amplitude, is
the rung-42 phi scalar times the product over the three cube axes of the
SQUARED per-posting transport normalization. -/
theorem depositQuantum_intended_eq_phi42_mul_prod_sq_postingNorm {i : Sector}
    (hi : sectorRung i = 0) :
    depositQuantum intendedGapOneFactorAmplitude i =
      Constants.phi ^ 42 * ∏ _a : Axis, postingNorm ^ 2 := by
  have hsq : intendedGapOneFactorAmplitude ^ 2 = Constants.phi ^ 42 / 8 := by
    simpa [intendedGapOneFactorAmplitude] using gapOne_factorAmplitude_sq_eq_phi42_div_eight
  show (if sectorRung i = 0 then postingEventLoad intendedGapOneFactorAmplitude else 0) = _
  rw [if_pos hi]
  show intendedGapOneFactorAmplitude ^ 2 = _
  calc intendedGapOneFactorAmplitude ^ 2 = Constants.phi ^ 42 / 8 := hsq
    _ = Constants.phi ^ 42 * ((1 : ℝ) / 2) ^ 3 := by ring
    _ = Constants.phi ^ 42 * ∏ _a : Axis, postingNorm ^ 2 := by
        rw [Finset.prod_const, Finset.card_univ, Fintype.card_fin, postingNorm_sq]
THEOREM postingNorm_unique · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean
postingNorm_unique · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean:126 
/-- **Unitarity forces the normalization (THEOREM).** The unnormalized
quarter factor `1 + a` has squared norm `2`, so `c * (1 + a)` is a unit
quaternion iff `c^2 * 2 = 1`; the unique positive solution is
`c = 1 / sqrt2`. -/
theorem postingNorm_unique {c : ℝ} (hc : 0 < c) (hunit : c ^ 2 * 2 = 1) :
    c = postingNorm := by
  have hc2 : c ^ 2 = 1 / 2 := by linarith
  have h : c = Real.sqrt (1 / 2) := by
    have h1 : c = Real.sqrt (c ^ 2) := (Real.sqrt_sq hc.le).symm
    rw [hc2] at h1
    exact h1
  calc c = Real.sqrt (1 / 2) := h
    _ = Real.sqrt 1 / Real.sqrt 2 := by
        rw [Real.sqrt_div (by norm_num : (0 : ℝ) ≤ 1)]
    _ = 1 / Real.sqrt 2 := by rw [Real.sqrt_one]
    _ = postingNorm := rfl
THEOREM creationDepositNorm_sq_eq_one_div_eight · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean
creationDepositNorm_sq_eq_one_div_eight · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean:194 
/-- The squared deposit normalization is the banked eight-tick per-tick
share `1/8`: the same `1/8` that enters `f^2 = phi^42/8` through the banked
`f^2 = predictedMass/16` (the two-phase factor `2` cancels the mass law's
two-occupied-phases factor, leaving the per-tick share `8 = 2^3`). -/
theorem creationDepositNorm_sq_eq_one_div_eight :
    creationDepositNorm ^ 2 = 1 / 8 := by
  have h : creationDepositNorm ^ 2 = (postingNorm ^ 2) ^ 3 := by
    rw [creationDepositNorm_eq_pow]; ring
  rw [h, postingNorm_sq]
  norm_num
THEOREM additive_composition_gate_killed · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean
additive_composition_gate_killed · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean:340 
/-- **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
THEOREM intended_mem_rungQuantumDeposit · two_mul_intended_ne_phi_pow_mul_creationDepositNorm · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean
intended_mem_rungQuantumDeposit · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean:310 
/-- The rung-quantum refinement accepts the intended amplitude at `k = 21`:
the deposit's rational-phi scalar is the pure rung transport ratio
`phi^21`. -/
theorem intended_mem_rungQuantumDeposit :
    ∃ k : ℕ, intendedGapOneFactorAmplitude = Constants.phi ^ k * creationDepositNorm :=
  ⟨21, intended_eq_phi21_mul_creationDepositNorm⟩
two_mul_intended_ne_phi_pow_mul_creationDepositNorm · IndisputableMonolith/Masses/MassGenesis/T10CreationDepositForcing.lean:317 
/-- **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⟩

What this page does not claim

The theorem does not derive the phi to the 21st power remainder from first principles. 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 establish that 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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