Masses Mass Genesis T10 Channel Block Window Pricing Wall Electroweak Rung0 Pred

The gap-one witness

The golden ratio φ, about 1.618, is the number whose square is itself plus one. It appears throughout geometry and number theory. In the Recognition Science framework, φ also appears in a mass formula. The declaration electroweak_rung0_predictedMass_div_eight_eq_phi42_div_four states that a certain predicted mass, divided by eight, equals φ raised to the 42nd power, divided by four. Written out: predictedMass / 8 = φ^42 / 4. This is a proved theorem in the framework's machine-checked library of formal theorems.

The mass in question belongs to the electroweak sector at its zeroth rung. The framework models particle masses as values on a ladder of φ powers. This particular identity is a witness: a concrete example that tests a proposed pricing law. The law under test would say that the energy of an emitted photon window equals some channel-block energy. A channel-block energy is a price the framework assigns to posting events, and every such price is at most one, because each is φ to a negative power.

The witness fails that law. The intended window energy is φ^42 / 4, which is greater than one. The framework proves this in the theorem one_lt_gapOne_factorWindowEnergy. Since every channel-block price is at most one, no channel-block price can equal the witness window energy. The theorem gapOne_factorWindowEnergy_ne_channelBlockEnergy states this inequality directly. The framework therefore proves that the proposed pricing law is false: channelBlockWindowPricingLaw_wall is a theorem stating the negation of that law. This is a wall, not a hard constraint. The law fails on the physics it was meant to accept, which marks it as malformed.

The declaration also connects to a counting obstruction. A summing account would need to pay the window from coherence quanta. The required event weight on the neutral electroweak seed equals φ^47 / 4. The framework proves this weight is irrational, so it cannot be a count of events. The theorem electroweak_neutral_requiredEventWeight_not_nat states this. This means no integer number of events can produce the required weight.

What the declaration does not claim is important. It does not claim that the predicted mass is the measured electroweak boson mass. It does not claim that the pricing law is a valid physical law; in fact, the framework proves it is not. It does not claim that the mass value itself is a count of events. The declaration is a stepping stone in a proof that a tempting pricing move is malformed. It establishes a numerical identity and supports a wall against a specific law. It does not establish the mass spectrum itself.

private theorem electroweak_rung0_predictedMass_div_eight_eq_phi42_div_four :
    MassLaw.predict_mass Anchor.Sector.Electroweak 0 0 / 8 =
      phi ^ (42 : ℕ) / 4 := by
  unfold MassLaw.predict_mass
  rw [MassLaw.gap_zero_neutral]
  simp only [Anchor.yardstick, Anchor.E_coh, Anchor.B_pow_Electroweak_eq,
    YardstickPremiseFreeCert.r0_ew_certified_eq_55]
  have hphi_ne : phi ≠ 0 := phi_ne_zero
  have h2 : (2 : ℝ) ^ (1 : ℤ) = (2 : ℝ) := by norm_num
  rw [h2]
  have hexp : (((0 : ℤ) : ℝ) - 8 + (0 : ℝ)) = (-8 : ℝ) := by norm_num
  rw [hexp]
  have hrpow : phi ^ (-8 : ℝ) = phi ^ (-(8 : ℤ)) := by
    simp
  rw [hrpow]
  have hpow :
      phi ^ (-(5 : ℤ)) * phi ^ (55 : ℤ) * phi ^ (-(8 : ℤ)) =
        phi ^ (42 : ℤ) := by
    rw [← zpow_add₀ hphi_ne, ← zpow_add₀ hphi_ne]
    norm_num
  conv_lhs =>
    rw [show
      (2 : ℝ) * phi ^ (-(5 : ℤ)) * phi ^ (55 : ℤ) * phi ^ (-(8 : ℤ)) =
        (2 : ℝ) *
          (phi ^ (-(5 : ℤ)) * phi ^ (55 : ℤ) * phi ^ (-(8 : ℤ))) by ring]
  rw [hpow]
  have hnat : phi ^ (42 : ℤ) = phi ^ (42 : ℕ) := by
    simp [← zpow_natCast]
  rw [hnat]
  ring

/-- Honest inequality from `1 < phi`: the intended window energy exceeds one. -/
theorem one_lt_gapOne_factorWindowEnergy :
    (1 : ℝ) < phi ^ (42 : ℕ) / 4 := by
  have hgt : (4 : ℝ) < phi ^ (42 : ℕ) := phi_pow_fortyTwo_gt_four
  have hpos : (0 : ℝ) < 4 := by norm_num
  exact (one_lt_div hpos).2 (by linarith)

/-- No channel-block energy can equal the intended gap-one window energy:
every price is `≤ 1`, while that window is `phi^42/4 > 1`. -/
theorem gapOne_factorWindowEnergy_ne_channelBlockEnergy
    {N : ℕ} [NeZero N]
    (channelEnergy : PostingEventChannelEnergy3 N)
    (event : RealizedPostingEvent3 N) (n : ℕ) :
    eightTickWindowEnergy
        (originEmittedPhotonStateScaled canonicalQ3SettledLedgerOctave
          (primitivePositiveStationaryFactorAmplitude
            (worldlinePattern gapOneTwoPhaseMode))
          gapOne_factorAmplitude_pos).window ≠
      channelEnergy.blockEnergy event n := by
  intro heq
  have hwin := gapOne_factorWindowEnergy_eq_phi42_div_four
  have hle := channelBlockEnergy_le_one channelEnergy event n
  have hgt := one_lt_gapOne_factorWindowEnergy
  linarith

/-- On the neutral Electroweak seed the required weight is `phi^47/4`. -/
theorem electroweak_neutral_requiredEventWeight_eq_phi47_div_four :
    requiredChannelBlockEventWeight Anchor.Sector.Electroweak 0 0 =
      phi ^ (47 : ℕ) / 4 := by
  unfold requiredChannelBlockEventWeight
  rw [Anchor.B_pow_Electroweak_eq, YardstickPremiseFreeCert.r0_ew_certified_eq_55,
    MassLaw.gap_zero_neutral]
  have h2 : (2 : ℝ) ^ ((1 : ℤ) - 3) = (1 : ℝ) / 4 := by
    norm_num
  rw [h2]
  have hexp : ((55 : ℤ) : ℝ) + ((0 : ℤ) : ℝ) - 8 + (0 : ℝ) = (47 : ℝ) := by
    norm_num
  rw [hexp]
  -- `phi ^ (47 : ℝ)` is the real power; reduce to Nat power.
  have h47 : (47 : ℝ) = (↑(47 : ℕ) : ℝ) := by norm_cast
  rw [h47, ← Real.rpow_natCast]
  ring

/-- The required Electroweak event weight is not a natural number. -/
theorem electroweak_neutral_requiredEventWeight_not_nat (N : ℕ) :
    requiredChannelBlockEventWeight Anchor.Sector.Electroweak 0 0 ≠
      (N : ℝ) := by
  rw [electroweak_neutral_requiredEventWeight_eq_phi47_div_four]
  exact phi47_div_four_ne_nat N

What this page does not claim

The declaration does not claim the predicted mass equals any measured particle mass. The declaration does not claim the channel-block pricing law is valid; the framework proves it is not. The declaration does not claim the mass value is a count of 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/MassGenesis/T10ChannelBlockWindowPricingWall.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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