Foundation Pair Kernel Physical Unit Map S12 Mapped Ledger Action And Scale Law

The canonical drop

The declaration mappedLedgerAction_and_scaleLaw_select_canonicalDrop is a theorem in the framework's machine-checked library of formal theorems. It states a precise consequence: if a field on a torus carries a mapped ledger action at some scale, and that scale obeys the primitive posting action scale law, then for any realized primitive posting pair with distinct endpoints, the absolute difference of the field values equals a specific constant called the native action canonical drop. In plainer words, once you fix the unit map that turns ledger cost into action, the theory forces the size of the smallest possible nonzero field difference.

The theorem's content is conditional. It does not assert that any such field exists, nor that the scale law holds. It says: if you have both, then the drop is forced. The scale law itself is a separate definition, PrimitivePostingActionScaleLaw, which requires that the unit map sends the ledger cost of a unit posting step to the constant hbar. The theorem proves that this scale, if it exists, is unique: it must equal primitivePostingActionScaleCandidate, which is hbar divided by the ledger cost of a unit step. The identity scale, where one ledger unit maps to one action unit, is explicitly shown to fail this law.

The canonical drop itself is not a free parameter. The library proves it is positive, and its square equals hbar times (hbar plus 2). Its value is approximately 0.236, which is less than 1. The theorem also distinguishes this drop from another candidate, the reciprocal of hbar, which equals phi to the fifth power and is greater than 1. The two are proved distinct. This distinction matters because the reciprocal assignment satisfies a different dual relation, but it is not the variational source selected by the exact one-edge action.

What the theorem does not claim is as important as what it proves. The docstring is explicit: no electromagnetic, length, or SI interpretation is asserted. The linear unit scale and the identification of the exact-J conjugate with a physical Gauss source remain explicit semantic arrows. The theorem is a statement about a formal ledger and a field on a torus, not about any particular physical system. It establishes a mathematical constraint, not an empirical fact.

In Recognition Science, this theorem is a step in a chain. It isolates the remaining linear scale between the ledger and the action, and it shows that the canonical drop is selected by the same variational principle that defines the source. The consequence is that the framework's internal constants are not chosen by hand at this stage; they are forced by the structure of the cost function and the scale law. What remains open is the physical interpretation of these formal objects, which the framework does not assert.

theorem mappedLedgerAction_and_scaleLaw_select_canonicalDrop
    {N : ℕ} [NeZero N]
    {scale : ℝ}
    {field : Fin (TorusCard3 N) → ℝ}
    (hfield : RealizedPostingFieldCarriesMappedLedgerAction3 scale field)
    (hscale : PrimitivePostingActionScaleLaw scale)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N)
    (hne : event.1 ≠ event.2) :
    |field event.1 - field event.2| =
      nativeActionCanonicalDrop :=
  realizedPostingFieldCarriesNativeAction_selects_canonicalDrop
    (mappedLedgerAction_and_scaleLaw_imply_nativeAction hfield hscale)
    hevent hne

/-- Exact OPEN scale law required to map one integer posting cost to the native
action quantum. -/
def PrimitivePostingActionScaleLaw (scale : ℝ) : Prop :=
  ledgerToActionUnitMap scale (Cost.Jlog 1) = Constants.hbar

theorem primitivePostingActionScaleLaw_iff
    (scale : ℝ) :
    PrimitivePostingActionScaleLaw scale ↔
      scale = primitivePostingActionScaleCandidate := by
  unfold PrimitivePostingActionScaleLaw ledgerToActionUnitMap
    primitivePostingActionScaleCandidate
  constructor
  · intro h
    apply (eq_div_iff (ne_of_gt ledgerJlogUnitCost_pos)).mpr
    simpa [mul_comm] using h
  · intro h
    subst scale
    field_simp [ne_of_gt ledgerJlogUnitCost_pos]

/-- Identity conversion is ruled out by S11. -/
theorem identityUnitMap_fails_nativeAction :
    ¬ PrimitivePostingActionScaleLaw 1 := by
  intro h
  unfold PrimitivePostingActionScaleLaw ledgerToActionUnitMap at h
  exact ledgerJlogUnitCost_ne_constants_hbar (by simpa using h)

theorem nativeExactJConjugateSource_sq :
    nativeExactJConjugateSource ^ 2 =
      Constants.hbar * (Constants.hbar + 2) := by
  unfold nativeExactJConjugateSource exactJVariationalSource
  have hpyth := Real.sinh_sq nativeActionCanonicalDrop
  rw [nativeActionCanonicalDrop_cosh] at hpyth
  nlinarith

theorem nativeExactJConjugateSource_ne_inv_hbar :
    nativeExactJConjugateSource ≠ Constants.hbar⁻¹ := by
  rw [← nativeActionQuantum_eq_constants_hbar,
    nativeActionQuantum_eq_inv_phi_pow_five, inv_inv,
    ← nativeActionQuantumInv_eq_phi_pow_five]
  exact nativeExactJConjugateSource_ne_nativeActionQuantumInv

What this page does not claim

No electromagnetic, length, or SI interpretation is asserted for the canonical drop. The theorem does not assert that any field satisfying the conditions exists. The canonical drop is not claimed to be the same as the reciprocal of hbar.

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/PairKernelPhysicalUnitMapS12.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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