Foundation Pair Kernel Physical Unit Map S12 Reciprocal Native Assignment Is Not

The reciprocal assignment

The declaration reciprocalNativeAssignment_is_not_exactJVariationalSource is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It establishes a negative result about two specific numbers that arise in the framework's internal mathematics: the reciprocal of a native action quantum, written as 1 / hbar, and the exact variational conjugate source. The theorem proves that these two numbers are not equal.

The framework models a discrete record of events, a ledger, where each posting has a cost. The cost function is forced to be J(x) = (x + 1/x)/2 - 1. From this cost, the framework derives a native action quantum, hbar, which equals phi^-5 where phi is the golden ratio. The theorem in question concerns the reciprocal of this quantum, 1 / hbar = phi^5.

The other number in the theorem is the exact variational conjugate source. For the cost function J(epsilon) = cosh(epsilon) - 1, the variational conjugate is sinh(epsilon). At the canonical native-action drop, this conjugate evaluates to sqrt(hbar * (hbar + 2)). The theorem proves that this exact conjugate source is distinct from the reciprocal 1 / hbar.

In Recognition Science, the theorem guards against conflating two distinct mathematical objects. The reciprocal 1 / hbar is a different Euler-J product dual, while the exact conjugate source is the one selected by variation of the exact one-edge action. The proof is axiom-clean, with no sorry and no new axioms.

The declaration does not claim that either number is the physical Gauss source. The docstring states that the identification of the exact-J conjugate with the physical Gauss source remains an explicit semantic arrow, and that no electromagnetic, length, or SI interpretation is asserted. The theorem only establishes the inequality between the two numbers, not which one, if either, corresponds to a physical quantity.

/-- The reciprocal native assignment satisfies the old Euler-J dual law while
its source differs from the exact-J variational conjugate. -/
theorem reciprocalNativeAssignment_is_not_exactJVariationalSource
    {N : ℕ} [NeZero N] :
    PostingSourceIsExplicitActionDual3
        (fun _ : PostingPair3 N => nativeActionQuantumInv⁻¹)
        (fun _ : PostingPair3 N => nativeActionQuantumInv) ∧
      nativeActionQuantumInv ≠ nativeExactJConjugateSource := by
  exact
    ⟨reciprocalTorsor_satisfies_explicitActionDual
      nativeActionQuantumInv nativeActionQuantumInv_pos,
    nativeExactJConjugateSource_ne_nativeActionQuantumInv.symm⟩

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 one_lt_nativeActionQuantumInv :
    1 < nativeActionQuantumInv := by
  rw [nativeActionQuantumInv_eq_phi_pow_five]
  have hpow :
      (1 : ℝ) ^ (5 : ℕ) <
        Real.goldenRatio ^ (5 : ℕ) :=
    pow_lt_pow_left₀ Real.one_lt_goldenRatio
      (by norm_num) (by decide)
  simpa using hpow

What this page does not claim

The theorem does not assert that either number is the physical Gauss source. The theorem does not establish any electromagnetic, length, or SI interpretation for these numbers. The theorem does not claim that the reciprocal assignment is the correct physical assignment.

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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