Foundation Pair Kernel Scale Bearing Self Dual Posting Law Joint Posting Ground

The unit ground state

A ground state is the lowest-energy configuration of a physical system. In the Recognition Science framework, a posting is a discrete record of a recognition event, and its law assigns a cost to every possible configuration. The declaration jointPostingGroundState27_of_unit_fieldGround is a theorem in the framework's machine-checked library of formal theorems. It states: if a field configuration is already a ground state of the field part of the law at the unit dilation, then that same configuration is also a joint ground state of the entire posting law at that unit dilation.

The classical picture is a ball in a valley. The field part of the law is one slope; the dilation part is another. The theorem says that if the ball rests at the bottom of the first slope, and the second slope is flat at that point, then the ball rests at the bottom of the combined setting. Here the dilation is a positive real number that scales the native extent of the posting event. The unit dilation is the value 1. The field part is a finite Weyl action, a sum over sites of a quadratic amplitude. The full posting action is the sum of the dilation cost J(dilation) plus the field cost. At unit dilation, the dilation cost vanishes by the cost function's zero-at-unity property, so the full action equals the field action alone. Therefore a field ground state at unit dilation is automatically a joint ground state.

The proof is not a numerical check; it is a formal derivation in the library, with no unproved assumptions beyond the standard logical axioms. The theorem holds for any admissible set of field variations, so it is general across choices of what counts as a allowed perturbation. It also does not depend on the particular event or field amplitude. The result is one of several in the module that pin down the unit dilation as special: another theorem shows that any joint ground state must have unit dilation, and a third shows that the only positive scale preserving the complete posting law is 1.

What this theorem does not claim is just as important. It does not prove that a joint ground state exists; it only says that if a field ground state exists at unit dilation, then it is joint. It does not derive the value of the native extent in meters; that conversion remains outside the law. It does not show that the framework's earlier premises forced this particular law; the law itself is a foundational model choice, with the theorems proved from it. The theorem also does not weaken the length non-identifiability result: the unit scale is special within this law, but the law still does not fix an absolute meter scale.

For a reader, the practical consequence is a clean reduction: to find a joint ground state, it suffices to solve the field-only problem at unit dilation. The dilation part adds no further constraint at that point. This is what makes the unit dilation a natural reference in the framework, not an arbitrary convention.

/-- A field ground state at native dilation is a joint ground state. -/
theorem jointPostingGroundState27_of_unit_fieldGround
    (admissible : WeylEventAmplitude27 → Prop)
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27)
    (hfield :
      IsAdmissibleFieldGroundStateAt27
        admissible event unitPostingDilation ψ) :
    IsJointPostingGroundState27
      admissible event unitPostingDilation ψ := by
  refine ⟨hfield.1, ?_⟩
  intro dilation variation hvariation
  exact le_trans (hfield.2 variation hvariation)
    (scaleBearingSelfDualPostingAction27_unit_le
      event dilation variation)

theorem scaleBearingSelfDualPostingAction27_at_unit
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27) :
    scaleBearingSelfDualPostingAction27
        event unitPostingDilation ψ =
      finiteWeylEventAction27
        (realizedPostingWeylClockCenter27 event) ψ := by
  simp [scaleBearingSelfDualPostingAction27,
    selfDualPostingWeylCost27, unitPostingDilation,
    Cost.Jcost_unit0]

/-- 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

/-- Anti-rescaling certificate: preserving the complete new law forces the
identity rechart. -/
theorem positiveScalePreservesCompletePostingLaw27_iff
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27)
    (scale : ℝ) :
    PositiveScalePreservesCompletePostingLaw27
        event ψ scale ↔
      scale = 1 := by
  constructor
  · rintro ⟨_, hpreserves⟩
    have hlength := congrArg
      (fun observation : WeylEventSite27 × ℝ × ℝ =>
        observation.2.2) hpreserves
    simpa [rechartCompletePostingObservation27,
      completeScaleBearingPostingObservation27,
      nativePostingExtentResponse27,
      nativePostingExtentQuantum27,
      unitPostingDilation,
      Constants.RSNativeUnits.voxel] using hlength
  · intro hscale
    subst scale
    simp [PositiveScalePreservesCompletePostingLaw27,
      rechartCompletePostingObservation27]

What this page does not claim

The theorem does not prove that a joint ground state exists. The theorem does not convert the native extent into meters. The theorem does not show that the framework's earlier premises forced this law.

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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