Foundation Pair Kernel Affine Weyl Event Action Affine Weyl Dilation Critical27

The unique balance point

In the Recognition Science framework, an event is described by two competing costs: a shift cost and a clock cost. These are quadratic forms, sums of squared magnitudes over a finite set of axes and sites. The framework models a ledger, a discrete record of events, by assigning such costs to each posting. A positive real parameter, the dilation, rescales these two costs in opposite directions: multiplying the dilation by a factor divides the shift cost by that factor squared, and multiplies the clock cost by the same factor squared.

The declaration affineWeylDilationCritical27_unique_of_clock_pos is a theorem in the framework's machine-checked library of formal theorems. It states that, for any event and any amplitude configuration where the clock cost is strictly positive, there is at most one positive dilation value at which the derivative of the total action with respect to the dilation is zero. In plainer terms: if the clock cost is not zero, the model has exactly one way to balance the shift and clock costs against each other. The balance condition itself is an equation, dilation^4 * clock_cost = shift_cost, and the theorem proves that this equation has at most one positive solution.

The theorem does not claim that such a balance point always exists. It only says that if one exists, it is unique. The statement requires the clock cost to be positive; if the clock cost is zero, the balance equation may have no solution or infinitely many, and the theorem says nothing about those cases. It also does not claim that the dilation value has any physical meaning: the framework explicitly leaves open whether this dilation corresponds to a measurable length or time.

The practical consequence is that the model does not contain a free parameter for the relative strength of the shift and clock costs. If the clock cost is positive, the balance point is fixed by the costs themselves. This is a structural property of the model, not an empirical claim about the physical world.

/-- 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

What this page does not claim

The theorem does not prove that a balance point exists, only that it is unique if it does. The theorem does not assign any physical dimension or SI unit to the dilation coordinate. The theorem does not claim that the shift and clock costs are derived from first principles; they are part of a MODEL action.

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)

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