Foundation Pair Kernel Physical Posting Attachment S10 Reciprocal Assignments Sa

Reciprocal assignments

In the Recognition Science framework, a ledger (a discrete record of events) tracks postings between pairs of sites. Each posting has an action, a numerical weight, and a source magnitude, another numerical weight. The framework's library, a machine-checked collection of formal theorems, proves a conditional statement: for any positive real number q, if the posting action is the constant function q⁻¹ and the source magnitude is the constant function q, then the source magnitude is the action-dual of the posting action. This is the theorem reciprocalAssignments_satisfy_sourceActionDuality.

Action-duality is a specific relation. The theorem states that for every realized posting event, the source magnitude is positive and the product of the source magnitude and the posting action equals the Euler-J action unit, which the library proves is equal to 1. In plain terms, the two assignments are reciprocals with respect to that unit. The proof is short: it uses the fact that q is positive and that q times q⁻¹ equals 1. The theorem is a formal derivation, not an empirical measurement.

The statement is conditional because it assumes the assignments are constant functions. It does not claim that any particular physical quantity takes the value q. The library explicitly leaves open whether a realized posting carries the native action quantum ħ = φ⁻⁵, and whether the pair-kernel source coordinate is its action dual. Those identifications are separate, unproved assumptions. The theorem merely shows that the reciprocal structure is consistent with the duality condition, whatever q turns out to be.

What this establishes is a structural fact about the framework's accounting: reciprocal assignments satisfy the duality condition universally. What it does not establish is which reciprocal pair, if any, corresponds to a physical posting. The numerical consequence that a realized posting would have source magnitude φ⁵ and Green scale φ⁵/2 is a conditional theorem, pending the open identifications.

/-- A full positive rescaling family: for every positive `q`, action `q⁻¹`
and source `q` satisfy the duality. This is the explicit scale torsor that the
event-to-native-action bridge must break. -/
theorem reciprocalAssignments_satisfy_sourceActionDuality
    {N : ℕ} [NeZero N]
    (q : ℝ) (hq : 0 < q) :
    PostingSourceIsActionDual3
      (fun _ : PostingPair3 N => q⁻¹)
      (fun _ : PostingPair3 N => q) := by
  intro event hevent
  constructor
  · exact hq
  · rw [eulerJActionUnit_eq_one]
    exact mul_inv_cancel₀ hq.ne'

theorem eulerJActionUnit_eq_one : eulerJActionUnit = 1 := by
  exact Action.euler_J_action_quantum

What this page does not claim

The theorem does not claim that any particular physical quantity takes the value q or 1/q. It does not claim that a realized posting carries the native action quantum ħ. It does not claim that the pair-kernel source coordinate is the action dual.

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/PairKernelPhysicalPostingAttachmentS10.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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