Masses Channel Distinction B22 Counts Two Sided Axis As One

The B-22 counting precedent

The declaration b22_counts_two_sided_axis_as_one establishes a counting convention for degrees of freedom in the Recognition Science framework. The theorem itself is the arithmetic identity 3 + 1 + 1 = 5. In plain language, it says that when the framework's coherence exponent counts three spatial dimensions, one temporal dimension, and one balance symmetry, the two-sided balance symmetry contributes exactly one axis, not two. The companion theorem sides_as_dofs_breaks_b22 makes the alternative explicit: counting the two sides of the balance as two separate degrees of freedom would give 3 + 1 + 2 = 6, which is not 5, so that reading is refuted by the precedent.

The declaration sits inside a larger argument about how the framework counts the channels through which a species couples. A Boolean channel predicate is read as a two-sided distinction, and the framework proves that the number of active channels equals the number of distinctions affirmed. Each channel carries two banked axes: presence and orientation. For charge, a nonzero value is exactly a value with a sign. For color, the up/down orientation is a genuine two-sided datum because the two equivariant coupling tables are provably distinct. The B-22 precedent then fixes the unit of counting: the degree of freedom is the distinction axis, not the side.

The arithmetic identity is kernel-checked, but the identification of a degree of freedom with a distinction axis is a model. The framework states this plainly: the axes and the precedent are proved, while the identification is a definitional choice. A structure called ChannelDistinctionModel packages the choice that each channel carries exactly two axes, and a theorem shows that under minimal per-degree-of-freedom pricing this yields a base coefficient of 2. The boundary is also marked: one axis per channel and three axes per channel both meet every banked clause of the table, and both change observable predictions, so the choice of two axes is not forced by the proved theorems alone.

What the declaration does not claim is that the balance symmetry is one-dimensional in any physical sense, or that the counting convention is derived from deeper principles. It claims only that within the framework's own precedent, the two-sided balance counts as one degree of freedom. The proof is a reflexivity check on integer arithmetic, not a statement about the physical world. The model that identifies a degree of freedom with a distinction axis is a choice, and the framework says so explicitly.

/-- **THEOREM (the B-22 precedent: one two-sided axis counts as ONE dof).**
The coherence exponent counts 3 spatial + 1 temporal + 1 balance = 5; the
two-sided balance symmetry contributes one axis. -/
theorem b22_counts_two_sided_axis_as_one : (3 : ℤ) + 1 + 1 = 5 := rfl

/-- **THEOREM (the sides-as-dofs reading is refuted by the precedent).**
Counting the two-sided balance as two dofs would give exponent 6, not the
banked 5. So the dof unit is the distinction axis, not the side. -/
theorem sides_as_dofs_breaks_b22 : (3 : ℤ) + 1 + 2 ≠ 5 := by decide

/-- **THEOREM (the charge channel carries two distinctions).** A nonzero
charge is exactly a charge with a sign: presence and orientation are the two
distinction axes of the charge channel. -/
theorem charge_channel_two_distinctions (q : ℤ) :
    q ≠ 0 ↔ (q < 0 ∨ 0 < q) :=
  ⟨fun h => lt_or_gt_of_ne h, fun h => by rcases h with h1 | h1 <;> omega⟩

/-- The remaining model: a configuration dof is a distinction AXIS (B-22's
own counting), and each coupled gauge channel carries exactly the two banked
axes of Part II. The axes and the precedent are kernel-checked; the
identification is the model. -/
structure ChannelDistinctionModel where
  distinctionsPerChannel : ℕ
  distinctionsPerChannel_eq_two : distinctionsPerChannel = 2

/-- **THEOREM (alternative axis counts are observably distinct).** One axis
per channel (coefficient 1) and three axes per channel (coefficient 3) both
meet every banked clause of the table, and both change observable
predictions. -/
theorem channelDistinction_boundary (Z : ℤ) :
    predictAt .Lepton (deformedTotal (-1) Fermion.e) Z
      ≠ predictAt .Lepton (deformedTotal 0 Fermion.e) Z
    ∧ predictAt .Lepton (deformedTotal 1 Fermion.e) Z
      ≠ predictAt .Lepton (deformedTotal 0 Fermion.e) Z :=
  ⟨baseRule_predictions_differ (-1) (by norm_num) Z,
    baseRule_predictions_differ 1 (by norm_num) Z⟩

What this page does not claim

The balance symmetry is one-dimensional in any physical sense. The counting convention is derived from deeper principles within the framework. The choice of two axes per channel is forced by the proved theorems.

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/ChannelDistinction.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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