Foundation Pair Kernel Physical Unit Map S12

The unit map

The module ledger (a discrete record of events) tracks costs, and a separate scale must convert those costs into physical actions. The unit map is the rule that does this conversion. The central result is that the only scale which works is hbar / Jlog 1, where Jlog 1 is the cost of a single posting step. This is proved in primitivePostingActionScaleLaw_iff.

The naive choice, a scale of 1, fails. The theorem identityUnitMap_fails_nativeAction shows that using the identity map does not produce the required action. This is because the cost of one posting step, Jlog 1, is not equal to the constant hbar. The correct scale is the ratio of these two quantities.

Beyond the scale, the module derives the source term that makes the action stationary. For an action of the form cosh(ε) - 1, the variational conjugate is sinh(ε). At the canonical drop, this source equals sqrt(hbar * (hbar + 2)), and the Hessian is 1 + hbar. The theorem nativeAction_and_exactJConjugate_force_source proves that this source is forced for any realized posting field.

A key distinction is drawn: the reciprocal 1/hbar = phi^5 is a different Euler-J product dual and is proved distinct from this exact-J conjugate source in nativeExactJConjugateSource_ne_inv_hbar. The module does not assert any electromagnetic, length, or SI interpretation for these quantities. The identification of the conjugate with a physical Gauss source remains an explicit semantic arrow, not a derived fact.

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

This module does not assert any electromagnetic, length, or SI interpretation for the derived quantities. The identification of the exact-J conjugate with a physical Gauss source is not proved here. The module does not derive the fine-structure constant or any other coupling constant.

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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