Foundation Pair Kernel Physical Unit Map S12 Native Action And Exact Jconjugate

The source of a fixed action

In classical mechanics, a source is the quantity that appears in a variational principle: when you vary a field to find a stationary action, the source is the conjugate variable that couples to the field. The Recognition Science framework carries this idea into its discrete ledger, a record of recognition events where each event has a cost. The declaration ledger nativeAction_and_exactJConjugate_force_source proves that, under the framework's assumptions, the source magnitude attached to a primitive posting event is forced to be a specific constant, not a parameter that can be tuned.

The framework models the cost of a recognition event with the function J(ε) = cosh ε − 1. Its variational conjugate, the source, is sinh ε. The theorem establishes that when the action is at its canonical native drop, the source equals sinh of that drop, a positive number less than 1. The proof is machine-checked in the framework's library of formal theorems, with no unproved axioms beyond the standard logical ones. The result is not a numerical approximation; it is an exact equality derived from the definitions.

The theorem does not claim that this source is a physical charge, an electromagnetic coupling, or any SI-measured quantity. The docstring states plainly: "No electromagnetic, length, or SI interpretation is asserted." The identification of this exact-J conjugate with a physical Gauss source remains an explicit semantic arrow, a choice the framework makes, not a consequence it proves. The theorem also distinguishes this source from other candidate numbers, such as the reciprocal of the action quantum, which is proved not to be the variational source.

What the theorem does establish is internal consistency: within the framework, if a field carries the native action and a source is defined as the exact-J conjugate, then that source is uniquely determined. This is a structural result about the framework's own mathematics. It shows that the framework does not leave the source as a free input; the structure of the cost function forces it. This is a step toward the framework's goal of deriving physical constants from the single principle of a forced recognition cost, though the bridge from this mathematical source to any measured physical quantity is not part of this theorem.

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

/-- Exact one-edge J action in log-field coordinates. -/
def exactOneEdgeJAction (ε : ℝ) : ℝ :=
  Real.cosh ε - 1

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 source is not identified with any electromagnetic charge or measured physical quantity. The theorem does not prove that the framework's source matches any experimental value. The bridge from this mathematical source to a physical interpretation is not established by this theorem.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: