Masses Channel Cost Base Rule Of Channel Cost Premises

The base rule's two premises

The Recognition Science framework models physical structure through a discrete record of recognition events, called a ledger: a bookkeeping system where each entry records a distinction the universe participates in. One of its library's theorems, base_rule_of_channelCostPremises, establishes a precise conditional statement: if two modeling premises hold, then the base cost coefficient is forced to be 2. The first premise is that each configuration degree of freedom costs exactly one rung on the cost ladder. The second is that each gauge channel, a distinction a species participates in, has exactly two sides. The theorem proves that under these two premises, the coefficient multiplying the number of active channel classes is exactly 2, for every fermion.

The theorem is a conditional, not an unconditional derivation. It does not prove that the premises are true; it proves that if they are true, the coefficient follows. The two premises are definitional choices, tagged as MODEL in the framework's own accounting. The theorem itself is a formal consequence, tagged THEOREM, verified in the machine-checked library. The structure ChannelCostPremises packages the two premises as a single object: a channel has two sides, and each degree of freedom costs one rung.

The library also proves the contrapositive in a useful sense. Without the premises, the coefficient is not forced. At a deformation where the coefficient is 4 instead of 2, every banked structure of the table survives, but observable predictions change. This is a separate theorem, channel_cost_independent_without_premises, which shows that the coefficient's value is genuinely load-bearing: it is not a harmless bookkeeping choice. The two premises are what close the base rule, and the theorem shows exactly what hangs on them.

What the theorem does not claim is just as important as what it proves. It does not claim that the two premises are physically necessary. It does not claim that the coefficient 2 is the only possible value. It does not claim that the premises are derived from the framework's forcing chain; they are modeling inputs, not outputs. The theorem's role is narrower and cleaner: given the premises, the coefficient is 2; without them, it is free. That is the whole content, and the library is careful to say so.

/-- **Under the two named premises the base rule is forced.** The coefficient is
`rungPerDof * dofOfChannel = 1 * 2 = 2`, closing
`RungBaseBoundary.channel_cost_premise_closes_base`. -/
theorem base_rule_of_channelCostPremises (P : ChannelCostPremises) :
    ∀ f : Fermion,
      (P.rungPerDof * P.dofOfChannel : ℕ) * activeChannelClasses f
        = 2 * activeChannelClasses f := by
  intro f
  rw [P.rungPerDof_eq_one, P.dof_eq_two_sides]
  norm_num

/-- The two modeling premises under which the base-rule coefficient is derived,
in the pattern of B-22 (`Foundation/GapDerivation.lean`): the per-dof pricing rule
(one rung per configuration degree of freedom, the same rule that prices `E_coh`),
and the channel-as-distinction count (one gauge channel is one distinction the
species participates in, and a distinction has exactly two sides,
`Foundation/BoolFromLogic.lean`). -/
structure ChannelCostPremises where
  dofOfChannel : ℕ
  rungPerDof : ℕ
  dof_eq_two_sides : dofOfChannel = 2
  rungPerDof_eq_one : rungPerDof = 1

/-- **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 prove the two premises are physically true. The theorem does not show the coefficient 2 is the only possible value. The theorem does not derive the premises from the framework's forcing chain.

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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