Foundation Pair Kernel Scale Bearing Self Dual Posting Law Scale Bearing Self Du

The dilation response

In Recognition Science, a posting is a discrete record of an event, and the framework's central object is the cost of recognizing that event. The declaration scaleBearingSelfDualPostingAction27_hasDerivAt_dilation proves a precise fact about that cost: if you take a posting and smoothly vary its scale, the rate at which the cost changes is given by a specific formula. Concretely, the theorem states that the derivative of the action with respect to the dilation parameter equals (1 - d⁻²) / 2, where d is the positive scale factor. This is not an approximation or a numerical fit; it is a proved identity in the machine-checked library of formal theorems.

The formula itself is the content. The action is the sum of two terms: the cost function J(d) plus a Weyl field cost that does not depend on the scale. The theorem shows that the scale-dependence lives entirely in the J term, and that its derivative has the simple closed form above. This matters because the derivative vanishes exactly when d = 1, and the second derivative is positive there. The framework then proves that the ground state, the configuration of minimal cost, occurs precisely at unit dilation. The derivative theorem is the analytical hinge: it converts the question of where the cost is minimized into a statement about a single real variable, and the answer is that the scale must be one.

The theorem does not claim that the scale-bearing law was forced by earlier Recognition premises. The law itself is a foundational model, a definitional choice authorized on 2026-07-20, and the properties proved from it are theorems. The declaration also does not claim that any empirical constant or external unit conversion is involved; the native extent is measured in the framework's own voxel standard, and conversion to meters remains outside the law. What the theorem does establish is internal and exact: given the model, the derivative of the action with respect to dilation is the stated function, and that function's unique zero at unit scale is what selects the ground state.

theorem scaleBearingSelfDualPostingAction27_hasDerivAt_dilation
    (event : RealizedPostingEvent3 3)
    (dilation : PostingDilationCoord)
    (ψ : WeylEventAmplitude27) :
    HasDerivAt
      (fun d : ℝ =>
        Cost.Jcost d + selfDualPostingWeylCost27 event ψ)
      (scaleBearingPostingDilationResponse27 dilation.1)
      dilation.1 := by
  simpa [scaleBearingPostingDilationResponse27] using
    (Cost.hasDerivAt_Jcost dilation.1 dilation.2.ne').add_const
      (selfDualPostingWeylCost27 event ψ)

/-- Any joint ground pair has the unique native dilation. -/
theorem jointPostingGroundState27_dilation_eq_unit
    (admissible : WeylEventAmplitude27 → Prop)
    (event : RealizedPostingEvent3 3)
    (dilation : PostingDilationCoord)
    (ψ : WeylEventAmplitude27)
    (hground :
      IsJointPostingGroundState27
        admissible event dilation ψ) :
    dilation = unitPostingDilation := by
  have hle := hground.2 unitPostingDilation ψ hground.1
  have hnonneg := Cost.Jcost_nonneg dilation.2
  have hzero : Cost.Jcost dilation.1 = 0 := by
    unfold scaleBearingSelfDualPostingAction27 at hle
    simp only [unitPostingDilation, Cost.Jcost_unit0, zero_add] at hle
    exact le_antisymm (by linarith) hnonneg
  have hdilation :
      dilation.1 = 1 :=
    (Cost.Jcost_eq_zero_iff dilation.1 dilation.2).mp hzero
  exact Subtype.ext hdilation

/-- FOUNDATIONAL MODEL: one stable dilation cost plus the one-weight Weyl
sector.  No numerical coefficient or length field is accepted. -/
def scaleBearingSelfDualPostingAction27
    (event : RealizedPostingEvent3 3)
    (dilation : PostingDilationCoord)
    (ψ : WeylEventAmplitude27) : ℝ :=
  Cost.Jcost dilation.1 + selfDualPostingWeylCost27 event ψ

What this page does not claim

The scale-bearing law was forced by earlier Recognition premises. The theorem provides an external conversion from the native voxel standard to meters. The derivative formula holds for non-positive scale factors, which the framework excludes by definition.

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

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