Masses Ladder Offset Gauge Predict Mass Factored

The observable mass law

In the Recognition Science account, particle masses sit on a ladder: each mass is a power of the golden ratio φ times a power of two, with the power of two fixed by the particle's sector. The framework's master formula, MassLaw.predict_mass, writes that exponent as a sum of several named pieces: a sector offset, a rung index, and a correction. The theorem predict_mass_factored proves, in the machine-checked library of formal theorems, that this decomposition is a rewriting, not a new model: the mass depends only on the total integer exponent, the sum of those pieces, and not on how the sum is split.

The content of the theorem is an identifiability statement. Two different ways of assigning the named pieces produce the same predicted mass exactly when their total exponents agree. The library proves the forward direction, that equal totals give equal masses, and the converse, sum_identifiable, that equal masses at a single charge force equal totals. So the total exponent is the observable content of the mass law, no more and no less. A companion theorem, rung_shift_absorbs_offset, shows the practical consequence: shifting every rung by an integer and taking that shift back out of the cycle period changes no mass anywhere.

The same library proves that the decomposition is not entirely flimsy. The power of two in the mass formula is not absorbable into the rung ladder: no nonzero power of φ is rational, so a sector's power of two cannot be traded for rungs. That makes the sector dependence a real, measurable claim. The offset gauge, by contrast, is a genuine freedom: the named offsets can be flattened away entirely, and the prediction is unchanged. The theorem does not claim that the framework derives particle masses from first principles, nor that the ladder matches measured PDG values; those are separate empirical checks, not part of this result.

theorem predict_mass_factored (s : Sector) (r Z : ℤ) :
    MassLaw.predict_mass s r Z = predictAt s (totalIntExponent s r) Z := by
  unfold MassLaw.predict_mass predictAt Anchor.yardstick Anchor.E_coh
  have hexp : ((totalIntExponent s r : ℤ) : ℝ) + MassLaw.gap_correction Z
      = ((-(5 : ℤ) : ℤ) : ℝ) + ((r0 s : ℤ) : ℝ)
        + (((r : ℤ) : ℝ) - 8 + MassLaw.gap_correction Z) := by
    simp only [totalIntExponent]
    push_cast
    ring
  rw [hexp]
  simp only [Real.rpow_add phi_pos, Real.rpow_intCast]
  ring

/-- The converse, which is what makes the previous theorems an identifiability statement
rather than a curiosity: predictions agreeing at a single charge already force the sums to
agree. So the sum is exactly the observable content, no more and no less. -/
theorem sum_identifiable (s : Sector) (e₁ e₂ Z : ℤ) (h : predictAt s e₁ Z = predictAt s e₂ Z) :
    e₁ = e₂ := by
  unfold predictAt at h
  have h2pos : (0 : ℝ) < (2 : ℝ) ^ (B_pow s) := zpow_pos (by norm_num) _
  have hphi : phi ^ ((e₁ : ℝ) + MassLaw.gap_correction Z)
      = phi ^ ((e₂ : ℝ) + MassLaw.gap_correction Z) :=
    mul_left_cancel₀ (ne_of_gt h2pos) h
  have hlog := congrArg Real.log hphi
  rw [Real.log_rpow phi_pos, Real.log_rpow phi_pos] at hlog
  have hne : Real.log phi ≠ 0 := ne_of_gt (Real.log_pos one_lt_phi)
  have : ((e₁ : ℝ)) = ((e₂ : ℝ)) := by
    have := mul_right_cancel₀ hne hlog
    linarith
  exact_mod_cast this

/-- **A different rung table predicts identically.** Shift every rung by `k` and take `k` back
out of the cycle period: no mass anywhere changes. Stated against the master formula, so it is
a claim about the model and not about the rewriting. -/
theorem rung_shift_absorbs_offset (s : Sector) (r k Z : ℤ) :
    predictAt s (totalIntExponent s (r + k) - k) Z = MassLaw.predict_mass s r Z := by
  rw [predict_mass_factored]
  congr 1
  simp only [totalIntExponent]
  ring

/-- **The power of two is not absorbable.** An integer rung shift multiplies the mass by a
power of `φ`, and no nonzero power of `φ` is rational, so the sector's power of two cannot be
traded for rungs. `B_pow` therefore makes a claim a measurement can see. -/
theorem power_of_two_not_absorbable :
    ¬ ∃ k : ℤ, (2 : ℝ) = phi ^ k := by
  rintro ⟨k, hk⟩
  have h1 : (1 : ℝ) ≤ phi := phi_ge_one
  rcases le_or_lt k 0 with hle | hpos
  · have hup : phi ^ k ≤ 1 := by
      calc phi ^ k ≤ phi ^ (0 : ℤ) := zpow_le_zpow_right₀ h1 hle
        _ = 1 := by simp
    rw [← hk] at hup
    norm_num at hup
  · rcases eq_or_lt_of_le (show (1 : ℤ) ≤ k by omega) with heq | hgt
    · rw [← heq, zpow_one] at hk
      linarith [phi_lt_two]
    · have hge : phi ^ (2 : ℤ) ≤ phi ^ k := zpow_le_zpow_right₀ h1 (by omega)
      have hsq : phi ^ (2 : ℤ) = phi + 1 := by
        rw [show (2 : ℤ) = ((2 : ℕ) : ℤ) by norm_num, zpow_natCast]
        exact phi_sq_eq
      rw [hsq, ← hk] at hge
      linarith [one_lt_phi]

What this page does not claim

The theorem does not claim that particle masses are derived from first principles without empirical input. It does not claim that the predicted masses match measured PDG values; that is a separate check. It does not claim that the named offsets are physically meaningful, only that they are unobservable.

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/LadderOffsetGauge.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: