Foundation Pair Kernel Exact Jsource First Event Operator Missing Factor Two Dec

The rejected decoy

In mathematics, when you take the derivative of a function, the answer is unique. There is no freedom to multiply the result by two and still call it the derivative. The declaration missing_factor_two_decoy_rejected is a machine-checked theorem that applies this principle to a specific, small case inside the Recognition Science framework. It proves that a certain proposed formula, one that omits a factor of two, is not the derivative of the exact J-cost action, even on the smallest possible two-site system. The theorem is a formal, verified statement that a particular decoy, a tempting but incorrect shortcut, fails.

The framework's library, a machine-checked collection of formal theorems, builds a physical picture from a single ledger, a discrete record of recognition events. The cost of each event is forced by a proved theorem. From this, the framework constructs an action, a quantity whose first variation gives the source response and whose second variation gives a quantum fluctuation generator. The factor of two in question arises from the ordered edges of the graph that represents the ledger. The theorem missing_factor_two_decoy_rejected confirms that this factor is not a convention, but a necessary consequence of the derivative being taken correctly. It is a guard against a specific, subtle error.

This theorem is a precise negative result. It does not claim that the framework's operator is the true molecular Hamiltonian. The source-first construction has no separate attractive potential and no selected molecular coupling ratio. The physical interpretation as a molecular model remains open. The theorem only certifies that the normalization of the response is correct, that the factor of two is fixed by the mathematics, and that the decoy is rejected. This is a small but load-bearing piece of the framework's foundation, ensuring that the derivative is taken exactly, not approximately.

/-- The response normalization is discriminating: omitting the
ordered-edge factor two is not the derivative of exact J even on the smallest
nontrivial carrier. -/
theorem missing_factor_two_decoy_rejected :
    ¬ HasDerivAt
      (fun t : ℝ =>
        exactJCostAction twoSiteUnitLedgerGraph
          (fun i => PairKernelSourceCoupling.twoSiteField 2 i +
            t * PairKernelSourceCoupling.twoSiteField 2 i))
      (Real.sinh 1) 0 :=
  missingFactorTwo_not_exactJDerivative

/-- The intrinsic response is exactly the nonlinear Gauss source paired with
the variation.  The ordered-edge factor two is fixed by differentiating J. -/
theorem sourceResponse_eq_nonlinear_gauss
    (op : ExactJSourceFirstEventOperator3) (variation : EventField3) :
    op.sourceResponse variation =
      2 * ∑ i, variation i *
        exactJNonlinearLaplacian
          (signedRecognitionProductionGraph3 3 (Equiv.refl (Fin 3)))
          op.background i := by
  exact
    exactJFirstVariationPairing_eq_two_sum_mul_laplacian
      (signedRecognitionProductionGraph3 3 (Equiv.refl (Fin 3)))
      op.background variation

What this page does not claim

The framework's operator is the true molecular one-body Hamiltonian. The construction has a separate attractive potential or a selected molecular coupling ratio. The theorem proves the physical interpretation of the operator as a molecular model.

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/PairKernelExactJSourceFirstEventOperator.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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