Physics Mass Residue No Go

The no-go separation

In the Standard Model of particle physics, the mass of a charged lepton (electron, muon, tau) receives small corrections from quantum effects. These corrections, called mass-anomalous-dimension integrals, are dimensionless numbers. For leptons, a literal integration over any reasonable interval produces a small residue, on the order of 10⁻² to 10⁻¹. This is the kind of number a physicist might expect to add to a base value to get a final mass.

Recognition Science, a framework that derives physical structure from a forced cost function, produces its own closed-form band value for charged leptons at an anchor point. This value, called gap 1332, is a discrete record of a recognition event: a specific number in the framework's ledger of events. The framework's library, a machine-checked collection of formal theorems, proves that this band value is greater than 13.953. The gap is a large number, not a small correction.

The no-go result is a pure arithmetic separation. The framework's library proves that if any number x has absolute value at most 0.1, then the absolute difference between x and gap 1332 is greater than 10. Since gap 1332 is provably above 13.953, a small number near zero is simply far away from it. No physical interpretation is needed to see the conflict: a small residue cannot equal a large band value.

This matters because the framework's anchor identity requires an experimental residue f to equal gap 1332. If someone tried to interpret f as a small Standard Model correction, the identity would fail. The library proves this failure directly: for any small x, the distance to gap 1332 exceeds 10, so x cannot match gap 1332 within a tolerance of 1e-6, or any small tolerance. The theorem does not assume a particular renormalization group kernel; it is a statement about numbers, not about physics details.

In plain language, the module closes a loophole. It shows that the framework's large band value cannot be faked by a small Standard Model correction. The separation is not a matter of approximation or tuning; it is a proved inequality. A reader can now see that the framework's anchor value stands apart from small perturbative corrections by a wide margin, and any attempt to bridge that gap with a small residue is mathematically impossible.

theorem gap_1332_gt_13_953 : (13.953 : ℝ) < gap 1332 :=
  (gap_1332_bounds).1

theorem abs_sub_gap1332_gt_ten {x : ℝ} (hx : |x| ≤ (0.1 : ℝ)) :
    (10 : ℝ) < |x - gap 1332| := by
  have hx' : (-0.1 : ℝ) ≤ x ∧ x ≤ (0.1 : ℝ) := abs_le.mp hx
  have hx_upper : x ≤ (0.1 : ℝ) := hx'.2
  have hg : (13.953 : ℝ) < gap 1332 := gap_1332_gt_13_953

  -- Bound y := x - gap 1332 by a strict negative number.
  have hy : x - gap 1332 < (-10 : ℝ) := by
    have h1 : x - gap 1332 < (0.1 : ℝ) - (13.953 : ℝ) := by
      nlinarith [hx_upper, hg]
    have hnum : (0.1 : ℝ) - (13.953 : ℝ) < (-10 : ℝ) := by norm_num
    exact lt_trans h1 hnum
  have hy_neg : x - gap 1332 < (0 : ℝ) := lt_trans hy (by norm_num)

  have hpos : (10 : ℝ) < -(x - gap 1332) := by linarith
  simpa [abs_of_neg hy_neg] using hpos

theorem not_within_micro_tolerance {x : ℝ} (hx : |x| ≤ (0.1 : ℝ)) :
    ¬ (|x - gap 1332| < (1e-6 : ℝ)) := by
  intro h
  have h10 : (10 : ℝ) < |x - gap 1332| := abs_sub_gap1332_gt_ten (x := x) hx
  have h_lt10 : |x - gap 1332| < (10 : ℝ) := lt_trans h (by norm_num)
  exact (lt_irrefl (10 : ℝ)) (lt_trans h10 h_lt10)

theorem small_x_ne_gap1332 {x : ℝ} (hx : |x| ≤ (0.1 : ℝ)) :
    x ≠ gap 1332 := by
  intro h
  have h10 : (10 : ℝ) < |x - gap 1332| := abs_sub_gap1332_gt_ten (x := x) hx
  have h10' : (10 : ℝ) < (0 : ℝ) := by simpa [h] using h10
  have hcontra : ¬ ((10 : ℝ) < (0 : ℝ)) := by norm_num
  exact hcontra h10'

What this page does not claim

This does not claim that Standard Model corrections are the only possible source of a residue. This does not claim that the framework's mass predictions match experiment. This does not claim that gap 1332 is derived from the Standard Model.

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

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