Foundation Pair Kernel Physical Unit Map S12 Signed Green Admits Distinct Source

The source coordinate

The declaration signedGreen_admits_distinct_sourceMagnitudes addresses a specific question in the Recognition Science framework: when a physical field is derived from a discrete ledger of events, which numerical source should accompany it? The framework models a ledger as a discrete record of events with a forced cost. The theorem proves that the source selected by varying the action, a quantity called the exact-J conjugate, is distinct from another candidate, the reciprocal of the action quantum. In plain terms, two plausible numbers exist, and the framework's own logic picks one and excludes the other.

The distinction matters because the two candidates have different mathematical roles. The exact-J conjugate comes from a variational principle: it is the coordinate that makes a sourced action stationary. The reciprocal of the action quantum is a different dual object, related to a Euler product. The theorem proves these are not equal, so the framework does not conflate two separate structures. This is a formal result, checked by a machine, with no unproved assumptions beyond the standard logical axioms.

What the theorem does not do is assert any physical interpretation. The docstring is explicit: no electromagnetic, length, or SI interpretation is claimed. The identification of the exact-J conjugate with a physical Gauss source remains an explicit semantic arrow, a choice, not a derivation. The theorem establishes a mathematical distinction, not a physical one. The framework's library proves the separation of the two numbers; it does not say what either number means in conventional physics terms.

The practical consequence is that the framework has a definite, unique source coordinate for its one-edge action, and that coordinate is not the reciprocal of the action quantum. This removes an ambiguity that would otherwise leave the theory underdetermined at its most basic level. The reader can now see that the framework's variational structure is internally consistent, even though the physical interpretation of the selected source remains an open question.

/-- 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⟩

/-- Stationarity of the explicitly sourced exact-J action selects `sinh ε`,
not reciprocal action. -/
theorem sourcedExactOneEdgeJAction_stationary_iff
    (source ε : ℝ) :
    HasDerivAt (sourcedExactOneEdgeJAction source) 0 ε ↔
      source = exactJVariationalSource ε := by
  constructor
  · intro hzero
    have hderiv := sourcedExactOneEdgeJAction_hasDerivAt source ε
    have hunique := hzero.unique hderiv
    linarith
  · intro hsource
    subst source
    simpa using
      sourcedExactOneEdgeJAction_hasDerivAt
        (exactJVariationalSource ε) ε

theorem nativeAction_and_exactJConjugate_force_source
    {N : ℕ} [NeZero N]
    {field : Fin (TorusCard3 N) → ℝ}
    {sourceMagnitude : PostingPair3 N → ℝ}
    (hfield : RealizedPostingFieldCarriesNativeAction3 field)
    (hsource : PostingSourceIsExactJConjugate3 field sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N)
    (hne : event.1 ≠ event.2) :
    sourceMagnitude event = nativeExactJConjugateSource := by
  rw [hsource event hevent hne,
    realizedPostingFieldCarriesNativeAction_selects_canonicalDrop
      hfield hevent hne]
  rfl

What this page does not claim

The theorem does not assert that the exact-J conjugate is the physical Gauss source. The theorem does not assign any electromagnetic, length, or SI meaning to the numbers. The theorem does not claim the two candidates are the only possible source coordinates.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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