Masses Channel Cost Channel Cost Independent Without Premises

The independence result

A recognition event, in this framework, is a discrete record of a distinction being made. The cost of recognition is forced by a proved functional equation, and particle masses are priced on a ladder of powers of the golden ratio. The declaration channel_cost_independent_without_premises is a theorem in the framework's machine-checked library of formal theorems. It states a precise fact about a base rule that assigns a coefficient to each active channel class of a fermion: even if you drop the two named modeling premises that normally justify the coefficient, the structural form of the rule survives.

The two premises are the per-degree-of-freedom pricing rule (one rung per configuration degree of freedom) and the channel-as-distinction count (one gauge channel is one distinction, and a distinction has exactly two sides). Under both, the coefficient is forced to be 2. The theorem shows that at a deformation where the coefficient becomes 4 instead of 2, every banked structure of the table still survives, so the independence holds. But the observable predictions differ: the predicted total at the lepton level for the electron under the deformed rule is not equal to the prediction under the base rule, for any integer offset Z.

What the theorem does not claim is just as important. It does not say the two premises are false, nor that the coefficient 2 is wrong. It says the structural form of the rule does not depend on those particular premises, while the numerical predictions do. The independence is about the shape of the rule, not about its output. The theorem also does not claim that any deformation is physically realized; it only establishes a formal independence and a formal difference in predictions.

The result matters because it separates two questions that are easy to conflate. One question is whether the base rule has a certain shape; the other is whether that shape is justified by specific modeling choices. The theorem shows the shape is stable under dropping those choices, while the predictions are not. That is a sharp, checkable statement about where the framework's mass formulas get their structure and where they get their numbers.

/-- **Without the premises, the coefficient is independent.** At deformation `k = 2`
(coefficient 4 instead of 2) every banked structure of the table survives
(`RungBaseBoundary.shiftedRungs_factors`), yet observable predictions change
(`RungBaseBoundary.baseRule_predictions_differ`). -/
theorem channel_cost_independent_without_premises (Z : ℤ) :
    (∀ f : Fermion,
      shiftedRungs 2 f
        = (2 + 2) * activeChannelClasses f + Integers.tau (genOf f).val
          + (if f = Fermion.nu3 then 2 else 0))
    ∧ predictAt .Lepton (deformedTotal 2 Fermion.e) Z
        ≠ predictAt .Lepton (deformedTotal 0 Fermion.e) Z :=
  ⟨shiftedRungs_factors 2, baseRule_predictions_differ 2 (by norm_num) Z⟩

/-- **Without the premises, the coefficient is independent.** At deformation `k = 2`
(coefficient 4 instead of 2) every banked structure of the table survives
(`RungBaseBoundary.shiftedRungs_factors`), yet observable predictions change
(`RungBaseBoundary.baseRule_predictions_differ`). -/
theorem channel_cost_independent_without_premises (Z : ℤ) :
    (∀ f : Fermion,
      shiftedRungs 2 f
        = (2 + 2) * activeChannelClasses f + Integers.tau (genOf f).val
          + (if f = Fermion.nu3 then 2 else 0))
    ∧ predictAt .Lepton (deformedTotal 2 Fermion.e) Z
        ≠ predictAt .Lepton (deformedTotal 0 Fermion.e) Z :=
  ⟨shiftedRungs_factors 2, baseRule_predictions_differ 2 (by norm_num) Z⟩

/-- **Without the premises, the coefficient is independent.** At deformation `k = 2`
(coefficient 4 instead of 2) every banked structure of the table survives
(`RungBaseBoundary.shiftedRungs_factors`), yet observable predictions change
(`RungBaseBoundary.baseRule_predictions_differ`). -/
theorem channel_cost_independent_without_premises (Z : ℤ) :
    (∀ f : Fermion,
      shiftedRungs 2 f
        = (2 + 2) * activeChannelClasses f + Integers.tau (genOf f).val
          + (if f = Fermion.nu3 then 2 else 0))
    ∧ predictAt .Lepton (deformedTotal 2 Fermion.e) Z
        ≠ predictAt .Lepton (deformedTotal 0 Fermion.e) Z :=
  ⟨shiftedRungs_factors 2, baseRule_predictions_differ 2 (by norm_num) Z⟩

What this page does not claim

The theorem does not claim the two premises are false or that the coefficient 2 is wrong. The theorem does not claim any deformation is physically realized. The theorem does not claim the structural independence extends to the numerical predictions.

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

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Derived articles

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