Foundation Pair Kernel Physical Posting Semantics S11 Reciprocal Torsor Satisfie

The reciprocal source rule

In the Recognition Science framework, a ledger is a discrete record of posting events, each event pairing two spatial points with a numerical action. The declaration reciprocalTorsor_satisfies_explicitActionDual is a proved theorem in the framework's machine-checked library of formal theorems. It states: for any positive real number q, if a posting's action is q and its source magnitude is q⁻¹, then the source magnitude is the explicit dual of the action. The dual map is simply reciprocal: the dual of an action a is 1/a. The theorem proves that this reciprocal assignment satisfies the framework's definition of an action-source dual pair.

The theorem is conditional on the source magnitude being positive and on the action being the reciprocal of that magnitude. It does not claim that any particular action value is forced. The framework's own constants, such as the native action quantum, are not selected by this theorem. The theorem only establishes the consistency of the reciprocal assignment as a dual pair. A separate theorem in the same module, fieldNativeAction_and_explicitDual_force_phi_pow_five, shows that if a field carries the native action and the source is its explicit dual, then the source magnitude equals φ⁵. That selection is a different result, not part of the reciprocal torsor theorem itself.

The theorem also does not claim that the reciprocal assignment is unique, nor that it is the only possible dual pair. The framework's definition of an explicit action dual requires the source magnitude to be the reciprocal of the posting action for every realized posting. The theorem proves that the constant reciprocal assignment satisfies this definition. It does not prove that every dual pair must be constant, nor that the reciprocal map is the only dual map. The involutive property of the reciprocal map, that applying it twice returns the original action, is a separate proved theorem in the same module.

What the theorem contributes is a precise boundary: reciprocity between action and source is a valid dual pairing, but it is not an absolute scale selector. The framework's own constants, such as the native action quantum, are not selected by this theorem. The theorem only establishes the consistency of the reciprocal assignment as a dual pair. A separate theorem in the same module, fieldNativeAction_and_explicitDual_force_phi_pow_five, shows that if a field carries the native action and the source is its explicit dual, then the source magnitude equals φ⁵. That selection is a different result, not part of the reciprocal torsor theorem itself.

/-- Explicit torsor countermodel. Involution and action-source duality admit
every positive reciprocal pair; they do not select the native action. -/
theorem reciprocalTorsor_satisfies_explicitActionDual
    {N : ℕ} [NeZero N]
    (q : ℝ) (hq : 0 < q) :
    PostingSourceIsExplicitActionDual3
      (fun _ : PostingPair3 N => q⁻¹)
      (fun _ : PostingPair3 N => q) := by
  rw [postingSourceIsExplicitActionDual_iff]
  exact reciprocalAssignments_satisfy_sourceActionDuality q hq

theorem actionSourceDual_eq_inv (action : ℝ) :
    actionSourceDual action = action⁻¹ := by
  rw [actionSourceDual, eulerJActionUnit_eq_one, one_div]

/-- Consequently, the two explicit field/map arrows force the no-fit magnitude
`phi^5` for every realized event. -/
theorem fieldNativeAction_and_explicitDual_force_phi_pow_five
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    {field : Fin (TorusCard3 N) → ℝ}
    {sourceMagnitude : PostingPair3 N → ℝ}
    (hfield : RealizedPostingFieldCarriesNativeAction3 field)
    (hdual :
      PostingSourceIsExplicitActionDual3
        (postingExactActionFromField3 field) sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    sourceMagnitude event = Constants.phi ^ (5 : ℕ) :=
  physicalPostingAttachment_forces_phi_pow_five
    (fieldNativeAction_and_explicitDual_imply_attachment
      hN hfield hdual)
    hevent

What this page does not claim

The theorem does not claim that the reciprocal assignment is the only possible dual pair. The theorem does not claim that any particular action value, such as the native action quantum, is forced by reciprocity alone.

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/PairKernelPhysicalPostingSemanticsS11.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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