Encyclopedia Foundation Foundation Pair Kernel Dimensioned Hamiltonian Compiler Source Density Non Id S2

ARTICLE 5 claims 5 theorems

Foundation Pair Kernel Dimensioned Hamiltonian Compiler Source Density Non Id S2

A machine-checked proof shows that the Recognition Science framework's own surface cannot tell apart two different ways of attaching a field to a Hamiltonian, a built-in limit on what the framework's current primitive package can determine.

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The unselected coupling

The declaration s28_fd2_surface_does_not_select_sourceDensityCoupling is a theorem in the framework's machine-checked library of formal theorems. It proves a precise non-identifiability result: the Recognition Science surface, which is the committed export and certificate layer of the framework's S28 and FD2 components, admits at least two distinct emissions that preserve all the semantics the surface cares about, yet these two emissions have different values for the product potentialScale * fieldNormalization. That product is the only combination of the two separate scale parameters that enters the diagonal of the finite one-body Hamiltonian matrix. The theorem therefore establishes that the present primitive package does not select a unique source-to-density coupling; the split into two separate parameters is pure redundancy.

The two emissions are concrete. The first, called the FD2 shadow emission, uses a field normalization of 8π. The second, called the alternate continuum-shape emission, uses 4π. Both keep the kinetic scale at 1/2 and the potential scale at 1, and both use only positive scales and Newtonian 4π arithmetic. The theorem proves that these two emissions have distinct products, 8π versus 4π, and that the corresponding Hamiltonian matrices differ on at least one diagonal entry. Both emissions satisfy the full FD2 surface predicate: the S28 event is realized, the Hamiltonian is Hermitian, it generates a Stone unitary group, and the scalar energy has a line expansion. The framework's own recognition surface cannot tell the two apart.

The proof works by showing that the Hamiltonian depends on the product of the two scales, not on how that product is split. A theorem named potentialScale_fieldNormalization_product_redundancy shows that rescaling the two knobs by reciprocal factors leaves the potential term unchanged. A companion theorem, oneBodyEvolution3_product_redundancy, extends this to the full Hamiltonian matrix. The master theorem then assembles these pieces: it exhibits the two distinct emissions, proves both preserve the FD2 surface, proves their couplings differ, and proves the corresponding Hamiltonians differ on the diagonal. The 8π value itself is identified in a separate theorem as the algebraic glue that connects a half-source Green shape to a -1/r Coulomb profile in the continuum; it is not selected by the S28 surface alone and is not used here as a forced compiler coefficient.

What the theorem does not claim is as important as what it proves. It does not claim that Recognition Science derives the fine-structure constant α or any hydrogen or CODATA fitting. It does not identify Constants.alpha. It introduces no new primitive and makes no edits to the GreenFourier work in progress. It contains no sorry and no new axiom. The theorem is a negative result about the current primitive package: a limitation, not a discovery of a new physical law. It leaves open the question of what additional structure, beyond the S28 export and FD2 certificates, would be needed to select a unique source-to-density coupling.

THEOREM s28_fd2_surface_does_not_select_sourceDensityCoupling · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
s28_fd2_surface_does_not_select_sourceDensityCoupling · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:304 
/-- Master non-identifiability: the S28 export and FD2 surface admit two
semantics-preserving compiler emissions with distinct source-density
couplings, and the corresponding Hamiltonians differ on the diagonal. -/
theorem s28_fd2_surface_does_not_select_sourceDensityCoupling :
    Nonempty CanonicalSourceGreenExport3 ∧
      ∃ event : PostingPair3 3,
        event ∈ realizedPrimitivePostingPairs3 3 ∧
          EmissionPreservesFD2Surface event emissionFD2Shadow ∧
            EmissionPreservesFD2Surface event emissionHalfGreenFactor ∧
              emissionFD2Shadow.coupling ≠
                emissionHalfGreenFactor.coupling ∧
                RecognitionOneBodyQuantumAttachment3
                  (oneBodyEvolution3 emissionFD2Shadow.kineticScale
                    emissionFD2Shadow.potentialScale
                    (q1AtomicModelPotential3 event
                      emissionFD2Shadow.fieldNormalization
                      emissionFD2Shadow.zeroReference)) ∧
                  RecognitionOneBodyQuantumAttachment3
                    (oneBodyEvolution3 emissionHalfGreenFactor.kineticScale
                      emissionHalfGreenFactor.potentialScale
                      (q1AtomicModelPotential3 event
                        emissionHalfGreenFactor.fieldNormalization
                        emissionHalfGreenFactor.zeroReference)) ∧
                    ∃ i : Fin (TorusCard3 3),
                      (oneBodyEvolution3 (1 / 2) 1
                          (q1AtomicModelPotential3 event (8 * Real.pi) 0)).hamiltonian
                        i i ≠
                        (oneBodyEvolution3 (1 / 2) 1
                          (q1AtomicModelPotential3 event (4 * Real.pi) 0)).hamiltonian
                        i i := by
  refine ⟨canonicalSourceGreenExport3, ?_⟩
  obtain ⟨event, hevent⟩ := realizedPrimitivePostingPair3_exists
  refine ⟨event, hevent, ?_, ?_, emissionFD2Shadow_coupling_ne_half, ?_, ?_, ?_⟩
  · exact emission_preserves_fd2_surface event hevent emissionFD2Shadow
  · exact emission_preserves_fd2_surface event hevent emissionHalfGreenFactor
  · exact emission_witnesses_attachment event hevent emissionFD2Shadow
  · exact emission_witnesses_attachment event hevent emissionHalfGreenFactor
  · exact
      distinct_sourceDensityCoupling_distinct_diagonal event hevent (1 / 2) 0
        (8 * Real.pi) (4 * Real.pi) (by
          intro h
          have hπ : (4 : ℝ) * Real.pi ≠ 0 :=
            mul_ne_zero (by norm_num) Real.pi_ne_zero
          have : (4 : ℝ) * Real.pi = 0 := by linarith
          exact hπ this)
THEOREM q1AtomicModelPotential3_product · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
q1AtomicModelPotential3_product · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:98 
/-- Diagonal potential depends on the product alone. -/
theorem q1AtomicModelPotential3_product
    (event : PostingPair3 3)
    (potentialScale fieldNormalization zeroReference : ℝ)
    (i : Fin (TorusCard3 3)) :
    potentialScale *
        q1AtomicModelPotential3 event fieldNormalization zeroReference i =
      -sourceDensityCoupling potentialScale fieldNormalization *
        (signedRealGreenField3 1 event i - zeroReference) := by
  unfold q1AtomicModelPotential3 sourceDensityCoupling
  ring
THEOREM potentialScale_fieldNormalization_product_redundancy · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
potentialScale_fieldNormalization_product_redundancy · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:110 
/-- Rescaling the two MODEL knobs with reciprocal factors leaves the potential
term unchanged. -/
theorem potentialScale_fieldNormalization_product_redundancy
    (event : PostingPair3 3)
    (potentialScale fieldNormalization zeroReference c : ℝ)
    (hc : c ≠ 0)
    (i : Fin (TorusCard3 3)) :
    (c * potentialScale) *
        q1AtomicModelPotential3 event (fieldNormalization / c) zeroReference i =
      potentialScale *
        q1AtomicModelPotential3 event fieldNormalization zeroReference i := by
  unfold q1AtomicModelPotential3
  field_simp [hc]
THEOREM oneBodyEvolution3_depends_on_sourceDensityCoupling · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
oneBodyEvolution3_depends_on_sourceDensityCoupling · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:124 
/-- The finite Hamiltonian matrix depends on the product, not the split. -/
theorem oneBodyEvolution3_depends_on_sourceDensityCoupling
    (event : PostingPair3 3)
    (kineticScale potentialScale fieldNormalization zeroReference : ℝ)
    (i j : Fin (TorusCard3 3)) :
    (oneBodyEvolution3 kineticScale potentialScale
        (q1AtomicModelPotential3 event fieldNormalization zeroReference)).hamiltonian
      i j =
      kineticScale * oneBodyKineticEntry3 i j +
        if i = j then
          -sourceDensityCoupling potentialScale fieldNormalization *
            (signedRealGreenField3 1 event i - zeroReference)
        else
          0 := by
  simp only [oneBodyEvolution3]
  by_cases hij : i = j
  · subst j
    simp only [↓reduceIte, q1AtomicModelPotential3_product]
  · simp only [hij, ↓reduceIte]
THEOREM distinct_sourceDensityCoupling_distinct_diagonal · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
distinct_sourceDensityCoupling_distinct_diagonal · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:217 
/-- Distinct positive products produce distinct diagonal Hamiltonian entries. -/
theorem distinct_sourceDensityCoupling_distinct_diagonal
    (event : PostingPair3 3)
    (hevent : event ∈ realizedPrimitivePostingPairs3 3)
    (kineticScale zeroReference lam1 lam2 : ℝ)
    (hlam : lam1 ≠ lam2) :
    ∃ i : Fin (TorusCard3 3),
      (oneBodyEvolution3 kineticScale 1
          (q1AtomicModelPotential3 event lam1 zeroReference)).hamiltonian i i ≠
        (oneBodyEvolution3 kineticScale 1
          (q1AtomicModelPotential3 event lam2 zeroReference)).hamiltonian i i := by
  obtain ⟨i, hi⟩ :=
    signedRealGreenField3_q1_exists_ne_reference event hevent zeroReference
  refine ⟨i, ?_⟩
  rw [oneBodyEvolution3_depends_on_sourceDensityCoupling,
    oneBodyEvolution3_depends_on_sourceDensityCoupling]
  simp only [sourceDensityCoupling, one_mul, ite_true]
  intro heq
  have hfactor :
      (lam1 - lam2) * (signedRealGreenField3 1 event i - zeroReference) = 0 := by
    linarith
  have hG : signedRealGreenField3 1 event i - zeroReference ≠ 0 :=
    sub_ne_zero.mpr hi
  have hlam0 : lam1 - lam2 = 0 :=
    (mul_eq_zero.mp hfactor).resolve_right hG
  exact hlam (sub_eq_zero.mp hlam0)

What this page does not claim

The theorem does not claim Recognition Science derives the fine-structure constant α. The theorem does not claim any hydrogen or CODATA fitting. The theorem does not claim the 8π value is selected by the S28 surface alone.

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/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.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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