Foundation Pair Kernel Exact Jnonlinear Gauss S13 Native Ordered Exact Jsource N

Two distinct scales

In the Recognition Science framework, a ledger, a discrete record of events, assigns a cost to each configuration of fields. The framework's central action, called the exact-J cost, is built from the sum over pairs of cosh(field difference) minus one. From this action, the framework derives a nonlinear version of a discrete Gauss law, relating the field's curvature to a source term. The theorem nativeOrderedExactJSource_ne_nativeActionQuantumInv proves that two specific numbers, which arise in this context, are distinct.

The first number, nativeOrderedExactJSource, is defined as twice the conjugate source, and the framework proves it equals 2 * sqrt(hbar * (hbar + 2)). The second, nativeActionQuantumInv, is the inverse of the framework's quantum of action. The theorem states that these two are not equal. The proof is direct: it shows the first is less than one, while the second is greater than one.

This inequality is a structural fact, not a numerical coincidence. It separates two roles in the framework: the source that appears in the nonlinear Gauss law, and the quantum scale that governs individual events. The theorem confirms that the framework's native source is not simply the inverse of its action quantum, a distinction that matters when interpreting the framework's constants and their relationships.

The theorem does not claim that either number has a specific physical value. It does not assert that nativeActionQuantumInv is the fine-structure constant, nor that nativeOrderedExactJSource is a measured coupling. It only establishes their inequality. The framework's library, a machine-checked collection of formal theorems, proves this without any unverified assumptions, but the physical interpretation of these numbers remains a separate question.

theorem nativeOrderedExactJSource_ne_nativeActionQuantumInv :
    nativeOrderedExactJSource ≠ nativeActionQuantumInv :=
  ne_of_lt
    (lt_trans nativeOrderedExactJSource_lt_one
      one_lt_nativeActionQuantumInv)

theorem nativeOrderedExactJSource_eq_two_sqrt :
    nativeOrderedExactJSource =
      2 * Real.sqrt
        (Constants.hbar * (Constants.hbar + 2)) := by
  rw [nativeOrderedExactJSource,
    nativeExactJConjugateSource_eq_sqrt]

theorem nativeOrderedExactJSource_ne_nativeActionQuantumInv :
    nativeOrderedExactJSource ≠ nativeActionQuantumInv :=
  ne_of_lt
    (lt_trans nativeOrderedExactJSource_lt_one
      one_lt_nativeActionQuantumInv)

What this page does not claim

The theorem does not claim that nativeActionQuantumInv is the fine-structure constant. The theorem does not assert that either number has a specific measured physical value. The theorem does not establish that the two numbers are related by any simple algebraic expression.

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/PairKernelExactJNonlinearGaussS13.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.

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