Foundation Pair Kernel Source Coupling Two Site Stationary At Every Coefficient

The two-site model

In classical terms, this is a statement about a discrete system with two locations. The framework defines a ledger, a discrete record of events, and a recognition cost, the price of updating that record. The declaration twoSite_stationary_at_every_coefficient proves, as a theorem in the machine-checked library of formal theorems, that a specific two-site configuration satisfies a stationary condition for every real value of a coupling parameter. The configuration pairs a symmetric weight function, which assigns equal influence between the two sites, with a field that depends linearly on the coupling. The stationary condition, spelled out in the library, says the physical source pairing equals the coupling multiplied by a fixed dipole source. The theorem states this holds for all real couplings, not just one special value.

The proof is short and explicit. The library defines the two-site weight as zero on the diagonal and one off it, and the field as coupling divided by two at the first site and zero at the second. The Laplacian of this field against the weight equals the coupling times a dipole term, which is exactly the stationary condition. The theorem stationaryAtCoefficient_iff converts this equality into the full stationary statement, and symmetry of the weight supplies the needed hypothesis. The result is a concrete example, not an abstract existence claim: a reader can verify each step by hand.

What the theorem does not claim is more important than what it does. The library also proves namedPremises_do_not_force_zeroDefect, which says the named premises do not force the zero-defect identification scale to be one. The two-site model realizes the stationary law at every coupling, including coupling equal to two, while the zero-defect condition excludes exactly that value. So the stationary law alone does not single out a preferred scale. The rescaling countermodel theorem shows there exists a coupling not equal to one that satisfies all the named premises. The framework's larger claims about forced constants come from additional structure, not from this two-site example.

In the framework's own account, this theorem is a boundary marker. It shows what the work-response law does and does not determine before further conditions are added. The honest reading is that the two-site model is a test case that passes the law at every coupling, and therefore the law by itself cannot force the unique scale the framework seeks. That scale must come from elsewhere, and the library says so plainly.

theorem twoSite_stationary_at_every_coefficient
    (coupling : ℝ) :
    StationaryAtCoefficient twoSiteWeight (twoSiteField coupling)
      (dipole (0 : Fin 2) 1) coupling := by
  exact
    (stationaryAtCoefficient_iff twoSiteWeight (twoSiteField coupling)
      (dipole (0 : Fin 2) 1) coupling twoSiteWeight_symmetric).2
      (twoSite_scaled_source coupling)

/-- 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))

/-- The adjudicating countermodel: coupling two satisfies the same primitive
posting, integral unit divergence, symmetric action, and work-response
stationarity package as coupling one. -/
theorem rescaling_countermodel :
    ∃ coupling : ℝ, coupling ≠ 1 ∧ NamedPremises coupling :=
  ⟨2, by norm_num, namedPremises_all_couplings 2⟩

What this page does not claim

The two-site theorem does not claim that the stationary law holds for all possible weight functions or fields. The theorem does not claim that the zero-defect scale is forced to be one by the named premises alone. The theorem does not claim that the two-site model is physically realized in nature.

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

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