Foundation Pair Kernel Affine Weyl Event Action

The affine Weyl carrier

In Recognition Science, the ledger (a discrete record of recognition events) assigns to each event two basic quadratic costs: a shift cost, which measures how far the event's amplitude spreads across sites, and a clock cost, which measures how the amplitude is distributed around a chosen center. The affine Weyl event action (a model action built on this ledger) introduces one positive real coordinate, the dilation, and lets it rescale the two costs reciprocally: multiplying the dilation by a factor divides the shift cost by the square of that factor, and multiplies the clock cost by the square of that factor.

Several properties of this action are established. The action is always nonnegative. At dilation equal to one, it recovers the previous equal-weight action exactly. The first variation of the action with respect to the dilation coordinate is computed exactly, and the action is stationary in that coordinate precisely when the fourth power of the dilation balances the two costs: dilation to the fourth times the clock cost equals the shift cost. When the clock cost is positive, this balance condition has a unique positive solution for the dilation, so the model action does not retain a free shift-to-clock ratio. A relative length response is also defined by multiplying the on-shell dilation by a fixed mesh scale, and this response is positive.

In Recognition Science, this construction builds a reversible primary carrier for events without changing the foundations. The action is a model, not a forced result: the framework does not force the physical Fourier exchange or an absolute event length. The variational, balance, uniqueness, and relative-chart properties are proved as theorems. Recognition-forced Fourier exchange, dimensioned physical length, SI attachment, and molecular identity remain open targets. The construction contains no unproved assumptions and no empirical constants.

theorem affineWeylEventAction27_nonneg
    (event : RealizedPostingEvent3 3)
    (dilation : ℝ)
    (ψ : WeylEventAmplitude27) :
    0 ≤ affineWeylEventAction27 event dilation ψ := by
  unfold affineWeylEventAction27
  exact add_nonneg
    (mul_nonneg (sq_nonneg _)
      (finiteWeylShiftOccupationCost27_nonneg ψ))
    (mul_nonneg (sq_nonneg _)
      (finiteWeylClockOccupationCost27_nonneg _ ψ))

/-- Unit dilation recovers the previous equal-weight MODEL action. -/
theorem affineWeylEventAction27_at_one
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27) :
    affineWeylEventAction27 event 1 ψ =
      finiteWeylEventAction27
        (realizedPostingWeylClockCenter27 event) ψ := by
  calc
    affineWeylEventAction27 event 1 ψ =
        finiteWeylTwoWeightAction27
          (realizedPostingWeylClockCenter27 event) 1 1 ψ := by
      simp [affineWeylEventAction27,
        finiteWeylTwoWeightAction27]
    _ = finiteWeylEventAction27
          (realizedPostingWeylClockCenter27 event) ψ :=
      finiteWeylTwoWeightAction27_one_one _ ψ

/-- Stationarity is exactly reciprocal balance of the two quadratic costs. -/
theorem affineWeylDilationCritical27_iff_balance
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27)
    (dilation : EventDilationCoord) :
    AffineWeylDilationCritical27 event ψ dilation ↔
      dilation.1 ^ 4 *
          finiteWeylClockOccupationCost27
            (realizedPostingWeylClockCenter27 event) ψ =
        finiteWeylShiftOccupationCost27 ψ := by
  unfold AffineWeylDilationCritical27
  unfold affineWeylDilationResponse27
  have hdilation : dilation.1 ≠ 0 := ne_of_gt dilation.2
  constructor <;> intro h
  · field_simp [hdilation] at h
    nlinarith
  · field_simp [hdilation]
    nlinarith

/-- When the clock cost is positive, the positive on-shell dilation is unique.
Thus this MODEL action does not retain a free shift/clock ratio. -/
theorem affineWeylDilationCritical27_unique_of_clock_pos
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27)
    (hclock :
      0 < finiteWeylClockOccupationCost27
        (realizedPostingWeylClockCenter27 event) ψ)
    (dilation₁ dilation₂ : EventDilationCoord)
    (hcritical₁ :
      AffineWeylDilationCritical27 event ψ dilation₁)
    (hcritical₂ :
      AffineWeylDilationCritical27 event ψ dilation₂) :
    dilation₁ = dilation₂ := by
  have hbalance₁ :=
    (affineWeylDilationCritical27_iff_balance
      event ψ dilation₁).mp hcritical₁
  have hbalance₂ :=
    (affineWeylDilationCritical27_iff_balance
      event ψ dilation₂).mp hcritical₂
  have hpow : dilation₁.1 ^ 4 = dilation₂.1 ^ 4 := by
    nlinarith
  apply Subtype.ext
  exact
    (pow_left_inj₀ dilation₁.2.le dilation₂.2.le
      (by norm_num : 4 ≠ 0)).mp hpow

theorem affineWeylRelativeLengthResponse27_pos
    (N : ℕ)
    (hN : 0 < N)
    (dilation : EventDilationCoord) :
    0 < affineWeylRelativeLengthResponse27 N dilation :=
  mul_pos dilation.2 (selfDualWeylMeshScale_pos N hN)

What this page does not claim

This construction does not force physical Fourier exchange or an absolute event length. The affine Weyl event action is a model, not a consequence of the recognition forcing chain. The relative length response is dimensionless until a physical event-length law is supplied.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: