Foundation Pair Kernel Source Coupling Action Has Deriv At Line Phys Source Pair

The derivative identity

In calculus, the derivative of a function at a point measures how sensitively the function's value responds to a small change in its input. The declaration action_hasDerivAt_line_physSourcePairing is a theorem in the framework's machine-checked library of formal theorems. It states that for any symmetric weight function w and any fields e and v, the derivative at zero of the action along the line e + t v equals exactly the physical source pairing 2 * Dirichlet(w, e, v).

This identity is a direct consequence of a more general derivative theorem in the same library. It is not an independent assumption or a new physical postulate; it is a proved statement that follows from the definitions of the action and the pairing. The theorem holds for any finite index set, which means it applies to any discrete system of sites, not just a particular lattice or dimension. The proof is axiom-clean, meaning it relies only on the standard logical axioms of the ambient type theory, with no additional framework-specific axioms.

The declaration's significance is that it provides a precise, kernel-checked link between the variation of the action and the source-pairing term. This is a foundational step in the framework's program of deriving physical laws from a recognition ledger. However, the theorem itself does not claim that the physical source pairing is the only possible coupling, nor does it select a specific value for any coupling constant. It is a structural identity, not a dynamical law.

In Recognition Science, this theorem is part of a larger investigation into whether a proposed work-response law is forced by more basic principles. The library contains a separate result, source_law_selects_doubled_branch, which shows that under a specific condition called the One-Ledger law, the doubled Laplacian branch is selected. But the current package does not derive that law from the action, Gauss, posting-integrality, and primitivity package. A two-site model realizes the law at every real coupling while holding the elementary posting fixed, demonstrating that the named premises do not force a unique scale.

The practical upshot is that the derivative identity is a reliable, machine-checked tool for analyzing variations of the action, but it does not by itself determine the physics. The framework's larger goal of deriving the coupling constant from first principles remains open. The theorem is a precise statement about a derivative, and nothing more.

/-- The work-response name is tied to the kernel-checked derivative of the
existing action. -/
theorem action_hasDerivAt_line_physSourcePairing
    (w : ι → ι → ℝ) (e v : ι → ℝ) :
    HasDerivAt
      (fun t : ℝ => action w (fun i => e i + t * v i))
      (physSourcePairing w e v)
      0 := by
  exact action_hasDerivAt_line w e v

/-- The work-response name is tied to the kernel-checked derivative of the
existing action. -/
theorem action_hasDerivAt_line_physSourcePairing
    (w : ι → ι → ℝ) (e v : ι → ℝ) :
    HasDerivAt
      (fun t : ℝ => action w (fun i => e i + t * v i))
      (physSourcePairing w e v)
      0 := by
  exact action_hasDerivAt_line w e v

/-- The work-response name is tied to the kernel-checked derivative of the
existing action. -/
theorem action_hasDerivAt_line_physSourcePairing
    (w : ι → ι → ℝ) (e v : ι → ℝ) :
    HasDerivAt
      (fun t : ℝ => action w (fun i => e i + t * v i))
      (physSourcePairing w e v)
      0 := by
  exact action_hasDerivAt_line w e v

/-- The current named pair-kernel premises cannot imply the zero-defect
scale-selection predicate: coupling two satisfies all of them while the
zero-defect predicate rejects it. Any positive bridge must therefore add a
genuinely scale-breaking recognition premise. -/
theorem namedPremises_do_not_force_zeroDefect :
    ¬ (∀ coupling : ℝ,
      NamedPremises coupling →
        ZeroDefectDualIdentification coupling) := by
  intro hforce
  exact zeroDefectDualIdentification_excludes_two
    (hforce 2 (namedPremises_all_couplings 2))

What this page does not claim

The theorem does not claim that the physical source pairing is the only possible coupling. It does not select a specific value for any coupling constant. It does not derive the One-Ledger law from the more basic action, Gauss, posting-integrality, and primitivity package.

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/PairKernelSourceCoupling.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: