Foundation Branch Selection Rclcombiner Zero Separately Additive

The additive branch

In the Recognition Science account of how a recognition ledger, a discrete record of events and their costs, must be kept, a central question is what a cost function can look like. The framework's rigidity theorem produces a family of candidate functions, each attached to a combiner, a rule that takes two costs and returns the cost of their composite event. The combiner has the polynomial shape P(u, v) = 2u + 2v + c·u·v, where c is a real parameter. The parameter splits the family into two branches: the bilinear branch with c ≠ 0, whose representative is J(x) = ½(x + x⁻¹) − 1, and the additive branch with c = 0, whose representative is ½(ln x)². The declaration RCLCombiner_zero_separatelyAdditive proves, in the machine-checked library of formal theorems, that when c = 0 the combiner is separately additive: there exist single-argument functions p and q such that P(u, v) = p(u) + q(v), here p(u) = 2u and q(v) = 2v.

The point of the lemma is not the polynomial itself but what it separates. A combiner that is separately additive treats the two costs as independent contributions that never interact. The framework's strengthened consistency condition, called coupling, demands the opposite: the cost of a composite must genuinely depend on how its components fit together, so the combiner must not be separately additive. The theorem RCLCombiner_isCoupling_iff shows that the RCL combiner is coupling if and only if c ≠ 0. The interaction defect, defined as P(u, v) − P(u, 0) − P(0, v) + P(0, 0), is the canonical detector: it vanishes identically for separately additive combiners and equals c·u·v for the RCL combiner. For c = 0 the defect is zero, so the additive branch fails the coupling condition and is excluded. The branch_selection theorem then forces c ≠ 0, and with it the bilinear branch.

What the declaration does not claim is just as important. RCLCombiner_zero_separatelyAdditive is a theorem about the c = 0 case alone; it does not itself choose the bilinear branch. The exclusion of the additive branch comes from combining this lemma with the coupling requirement, which is a structural strengthening of the composition consistency axiom, not a consequence of the algebra alone. The theorem also does not determine the residual α-coordinate freedom within the bilinear family. That freedom, acknowledged in the companion paper, is addressed by separate generator-calibration, higher-derivative, and action-functional conditions, none of which are part of the operator-level content. The declaration is one precise link in a chain, not the whole argument.

/-- For `c = 0`, the RCL combiner is the additive combiner
`P(u, v) = 2u + 2v`, separately additive with `p(u) = 2u` and
`q(v) = 2v`. -/
theorem RCLCombiner_zero_separatelyAdditive :
    SeparatelyAdditive (RCLCombiner 0) := by
  refine ⟨fun u => 2 * u, fun v => 2 * v, ?_⟩
  intro u v
  unfold RCLCombiner
  ring

/-- **The RCL combiner is a coupling combiner iff `c ≠ 0`.** -/
theorem RCLCombiner_isCoupling_iff (c : ℝ) :
    IsCouplingCombiner (RCLCombiner c) ↔ c ≠ 0 := by
  rw [isCouplingCombiner_iff_interactionDefect_nonzero]
  constructor
  · rintro ⟨u, v, huv⟩
    intro hc
    apply huv
    rw [interactionDefect_RCLCombiner, hc]
    ring
  · intro hc
    exact ⟨1, 1, RCLCombiner_nonzero_couples c hc⟩

/-- **Branch selection by non-degeneracy.**

If the RCL polynomial combiner is required to be a coupling combiner
(the strengthened (L4*) of the companion paper), then the parameter
`c` is forced to be nonzero. Equivalently, the additive branch
(`c = 0`, with calibrated representative `½(ln x)²`) is excluded.

This is the branch-selection theorem of `RS_Branch_Selection.tex` in
its Lean form. The bilinear branch is forced; `J` is the
`α = 1` representative of the bilinear `α`-family. The residual
`α`-coordinate freedom is acknowledged in §5 of the paper and is
addressed by separate generator-calibration / higher-derivative /
action-functional conditions, none of which are part of the
operator-level Aristotelian content. -/
theorem branch_selection (c : ℝ)
    (hCoupling : IsCouplingCombiner (RCLCombiner c)) :
    c ≠ 0 :=
  (RCLCombiner_isCoupling_iff c).mp hCoupling

What this page does not claim

The declaration does not itself select the bilinear branch; the coupling requirement does. The theorem does not determine the residual α-coordinate freedom within the bilinear family. The lemma does not address the physical recognition-to-linking bridge.

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

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