Encyclopedia Masses Masses Mass Genesis T10 Inhomogeneous Field Current Selector Wall No Field Curre

ARTICLE 4 claims 4 theorems

Masses Mass Genesis T10 Inhomogeneous Field Current Selector Wall No Field Curre

A machine-checked theorem proves that no simple rule can turn a field's strength into a particle's mass without first naming the mass itself.

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The class wall

In the Recognition Science framework, a recognition event is a discrete record of the universe taking notice of one of its own configurations, and the cost of that notice is forced by a proved uniqueness theorem. The theorem named no_fieldCurrentLaw_passes_selectorGate closes a specific route in the framework's program for deriving particle masses from first principles. It concerns a class of candidate laws, each of the form a·u = C, where a is a field amplitude, u is a fixed unit magnitude, and C is a constant. The theorem proves that no such law can satisfy three admission clauses at once: its unit and constant must both be expressible as rational combinations of the golden ratio φ; it must accept one intended test amplitude; and it must reject exactly twice that amplitude.

The proof is a line of exact algebra, not a numerical search. Acceptance forces the intended amplitude to equal C/u. Since both C and u are rational-φ combinations, their ratio would be too. But the intended amplitude is not: its square is φ⁴²/8, and the representation p + qφ would force a rational square congruent to 2 modulo 4, which no perfect square is. The obstruction is arithmetic, tied to the Cassini identity for Fibonacci numbers, so the wall has real width, not the zero-width failure of a dense set that merely happens to miss a point.

The framework's machine-checked library of formal theorems also shows the gate is not vacuous. One candidate law, with unit 1 and constant equal to the intended amplitude, passes the two witness clauses and fails only the independence clause, in the strongest way: its constant is the target itself. At the intended pattern, its acceptance condition is exactly the zero-cost predicate, a modeling choice the framework labels R4. So discrimination is available, but only at the price of naming the target, which is the identification the binding decision already prices.

What the theorem does not claim is broader. It does not prove that no field-to-current law exists; it proves that none exists within the typed class of laws whose constants lie in the rational-φ field. A law whose unit reads the pattern rather than the amplitude is outside the class entirely. And the theorem does not derive the missing absolute constant; it types the target exactly. The new absolute must live outside ℚ(φ), and that derivation remains an open target, not an impossibility.

THEOREM no_fieldCurrentLaw_passes_selectorGate · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean
no_fieldCurrentLaw_passes_selectorGate · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean:376 
/-- **The class wall (THEOREM).** No field-to-current magnitude law with
target-free (φ-field) provenance accepts the intended witness: acceptance
forces `f = C / u` into the φ-field, and `f` is not there. -/
theorem no_fieldCurrentLaw_passes_selectorGate (L : FieldCurrentMagnitudeLaw) :
    ¬ PassesSelectorGate L := by
  rintro ⟨hu, hC, hacc, -⟩
  have hf : intendedGapOneFactorAmplitude = L.C / L.u :=
    (eq_div_iff L.u_ne).mpr hacc
  exact intendedGapOneFactorAmplitude_not_inQPhi (hf ▸ hC.div hu L.u_ne)
THEOREM intendedGapOneFactorAmplitude_not_inQPhi · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean
intendedGapOneFactorAmplitude_not_inQPhi · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean:283 
/-- **Exact algebra.** The intended gap-one factor amplitude `f`
(`f² = φ⁴²/8`, banked) is not of the form `p + q φ` with rational `p, q`.
The representation would force `p² + q² = F₄₁/8` and `2pq + q² = F₄₂/8`
(by rational independence of `{1, φ}` and the Fibonacci reduction), hence
`q² = 29953729/2` or `q² = 74884322/5`; in the first case `(2q)² = 59907458`
and in the second `(5q)² = 374421610`, both integers `≡ 2 mod 4`, which no
perfect square is. The elimination's discriminant is the Cassini identity
`F₄₁² + F₄₁F₄₂ − F₄₂² = 1`, equivalently `299537289² − 80 · 33489287² = 1`. -/
theorem intendedGapOneFactorAmplitude_not_inQPhi :
    ¬ InQPhi intendedGapOneFactorAmplitude := by
  rintro ⟨p, q, h⟩
  have hsq : intendedGapOneFactorAmplitude ^ 2 = Constants.phi ^ (42 : ℕ) / 8 := by
    simpa [intendedGapOneFactorAmplitude] using
      gapOne_factorAmplitude_sq_eq_phi42_div_eight
  have hexpand : intendedGapOneFactorAmplitude ^ 2 =
      ((p : ℝ) ^ 2 + (q : ℝ) ^ 2) +
        (2 * (p : ℝ) * (q : ℝ) + (q : ℝ) ^ 2) * Constants.phi := by
    rw [h]
    linear_combination ((q : ℝ) ^ 2) * Constants.phi_sq_eq
  have hpow : Constants.phi ^ (42 : ℕ) / 8 =
      (165580141 : ℝ) / 8 + (33489287 : ℝ) * Constants.phi := by
    have hf := phi_pow_fib 42
    have hf42 : Nat.fib 42 = 267914296 := by decide
    have hf43 : Nat.fib 43 = 433494437 := by decide
    rw [hf42, hf43] at hf
    have hf' : Constants.phi ^ (42 : ℕ) =
        (267914296 : ℝ) * Constants.phi + 165580141 := by
      rw [hf]
      push_cast
      ring
    linear_combination hf' / 8
  have heq := hexpand.symm.trans (hsq.trans hpow)
  have heq2 : (((p ^ 2 + q ^ 2 : ℚ) : ℝ)) +
        (((2 * p * q + q ^ 2 : ℚ) : ℝ)) * Constants.phi =
      (((165580141 / 8 : ℚ) : ℝ)) + (((33489287 : ℚ) : ℝ)) * Constants.phi := by
    push_cast
    linear_combination heq
  obtain ⟨hA, hB⟩ := rat_coeff_eq_of_phi heq2
  have hT : 5 * q ^ 4 - (299537289 / 2 : ℚ) * q ^ 2 + (33489287 : ℚ) ^ 2 = 0 := by
    have h1 : p ^ 2 = (165580141 / 8 : ℚ) - q ^ 2 := by linear_combination hA
    have h2 : 2 * p * q = (33489287 : ℚ) - q ^ 2 := by linear_combination hB
    have h3 : 4 * p ^ 2 * q ^ 2 = ((33489287 : ℚ) - q ^ 2) ^ 2 := by
      linear_combination (2 * p * q + ((33489287 : ℚ) - q ^ 2)) * h2
    rw [h1] at h3
    linear_combination -h3
  have hT' : (10 * q ^ 2 - (299537289 / 2 : ℚ)) ^ 2 = (1 / 2 : ℚ) ^ 2 := by
    have hconst : (299537289 / 2 : ℚ) ^ 2 - 20 * (33489287 : ℚ) ^ 2 = 1 / 4 := by
      norm_num
    have hsq : (10 * q ^ 2 - (299537289 / 2 : ℚ)) ^ 2 = 1 / 4 := by
      linear_combination 20 * hT + hconst
    rw [hsq]
    norm_num
  rcases sq_eq_sq_iff_eq_or_eq_neg.mp hT' with hcase | hcase
  · have hq2 : q ^ 2 = (29953729 / 2 : ℚ) := by linear_combination hcase / 10
    have h2q : (2 * q) ^ 2 = (59907458 : ℚ) := by
      have hqq : (2 * q) ^ 2 = 4 * q ^ 2 := by ring
      rw [hqq, hq2]
      norm_num
    exact not_rat_sq_of_mod_four 59907458 (by norm_num) (2 * q)
      (by exact_mod_cast h2q)
  · have hq2 : q ^ 2 = (74884322 / 5 : ℚ) := by linear_combination hcase / 10
    have h5q : (5 * q) ^ 2 = (374421610 : ℚ) := by
      have hqq : (5 * q) ^ 2 = 25 * q ^ 2 := by ring
      rw [hqq, hq2]
      norm_num
    exact not_rat_sq_of_mod_four 374421610 (by norm_num) (5 * q)
      (by exact_mod_cast h5q)
THEOREM targetConstantLaw_accepts_intended · targetConstantLaw_rejects_doubled · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean
targetConstantLaw_accepts_intended · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean:417 
theorem targetConstantLaw_accepts_intended :
    LinearMagnitudeLaw 1 intendedGapOneFactorAmplitude
      intendedGapOneFactorAmplitude := by
  show intendedGapOneFactorAmplitude * 1 = intendedGapOneFactorAmplitude
  exact mul_one _
targetConstantLaw_rejects_doubled · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean:423 
theorem targetConstantLaw_rejects_doubled :
    ¬ LinearMagnitudeLaw 1 intendedGapOneFactorAmplitude
      (2 * intendedGapOneFactorAmplitude) := by
  have hf : 0 < intendedGapOneFactorAmplitude := gapOne_factorAmplitude_pos
  intro h
  have h' : 2 * intendedGapOneFactorAmplitude * 1 =
      intendedGapOneFactorAmplitude := h
  rw [mul_one] at h'
  linarith
THEOREM targetConstantLaw_is_sigmaZero_at_intended · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean
targetConstantLaw_is_sigmaZero_at_intended · IndisputableMonolith/Masses/MassGenesis/T10InhomogeneousFieldCurrentSelectorWall.lean:433 
/-- At the intended pattern, the target-constant law's acceptance predicate
is the σ-zero equation (banked
`scalePattern_gapOne_at_factorAmplitude_sigmaZero`). -/
theorem targetConstantLaw_is_sigmaZero_at_intended :
    loadRecognitionCost intendedGapOneFactorPattern = 0 ∧
      intendedGapOneFactorAmplitude =
        primitivePositiveStationaryFactorAmplitude
          intendedGapOneFactorPattern := by
  obtain ⟨-, -, hcost, hamp⟩ := scalePattern_gapOne_at_factorAmplitude_sigmaZero
  exact ⟨hcost, hamp⟩

What this page does not claim

No field-to-current law exists at all, only none within the rational-φ class. The missing absolute constant has been derived; it remains an open target. The R4 zero-cost predicate is a proved theorem rather than a modeling choice.

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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/T10InhomogeneousFieldCurrentSelectorWall.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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