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

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Dimensioned Hamiltonian Compiler Source Density Non Id

A machine-checked proof shows that a quantum model's core coupling is not uniquely determined by the physics it reproduces, forcing a choice the framework cannot yet make.

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The compiler's blind spot

In Recognition Science, a ledger is a discrete record of events, and a Hamiltonian is the matrix that dictates how a quantum state evolves in time. The module named foundation pair kernel dimensioned hamiltonian compiler source density non id addresses a specific question: when the framework builds a one-body quantum model from a Green field, does the physics uniquely determine the strength of the source-to-density coupling? The answer, proved in the machine-checked library of formal theorems, is no.

The model attaches a potential to each point of a torus by the formula V_i = -fieldNormalization * (G_i - zeroReference), where G is a Green field and fieldNormalization is a scaling factor. An independent potentialScale multiplies this potential. The module proves that the finite matrix and the scalar energy see only the product sourceDensityCoupling = potentialScale * fieldNormalization. The two factors themselves are pure redundancy: rescaling one by a nonzero constant and the other by its reciprocal leaves every Hamiltonian entry unchanged, as shown by the theorem potentialScale_fieldNormalization_product_redundancy.

The central result, s28_fd2_surface_does_not_select_sourceDensityCoupling, constructs two distinct emissions that preserve the entire committed Recognition surface: the S28 event, the Hermitian property, the Stone unitary group, and the scalar line expansion. One emission uses fieldNormalization = 8π, the other 4π, with the same kinetic scale and potential scale. Both satisfy the attachment predicate, yet they produce different diagonal Hamiltonian entries. The theorem states plainly that the present primitive package does not select the source-to-density coupling.

The 8π factor is not forced by the S28 Green field alone. A separate theorem, continuum_eight_pi_is_shape_glue_not_s28_selection, shows that 8π is algebraic glue connecting a half-source Green shape to a -1/r Coulomb profile in the continuum. It is a shape choice, not a derived constant. The module makes no hydrogen or CODATA fit, no identification of the fine-structure constant, and introduces no new primitive. It closes with a precise statement of what remains open: the split between potentialScale and fieldNormalization is unconstrained by the physics the framework currently recognizes.

What this changes is the epistemic status of any Hamiltonian built this way. A reader can no longer assume that the framework's one-body quantum model pins down its coupling strength. The proof turns a silent modeling assumption into a named gap, and that gap is now a target for future work rather than an invisible premise.

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 oneBodyEvolution3_product_redundancy · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
oneBodyEvolution3_product_redundancy · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:144 
/-- Reciprocal rescaling of the two knobs yields identical matrix entries. -/
theorem oneBodyEvolution3_product_redundancy
    (event : PostingPair3 3)
    (kineticScale potentialScale fieldNormalization zeroReference c : ℝ)
    (hc : c ≠ 0)
    (i j : Fin (TorusCard3 3)) :
    (oneBodyEvolution3 kineticScale (c * potentialScale)
        (q1AtomicModelPotential3 event (fieldNormalization / c)
          zeroReference)).hamiltonian i j =
      (oneBodyEvolution3 kineticScale potentialScale
        (q1AtomicModelPotential3 event fieldNormalization
          zeroReference)).hamiltonian i j := by
  rw [oneBodyEvolution3_depends_on_sourceDensityCoupling,
    oneBodyEvolution3_depends_on_sourceDensityCoupling]
  unfold sourceDensityCoupling
  field_simp [hc]
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 continuum_eight_pi_is_shape_glue_not_s28_selection · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
continuum_eight_pi_is_shape_glue_not_s28_selection · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:350 
/-- Continuum `8π` is the algebraic glue from half-source Green shape to a
`-1/r` Coulomb profile.  It is not selected by S28 alone and is not used here
as a forced compiler coefficient. -/
theorem continuum_eight_pi_is_shape_glue_not_s28_selection :
    (8 * Real.pi) * (1 / (8 * Real.pi)) = 1 ∧
      emissionFD2Shadow.fieldNormalization = 8 * Real.pi := by
  constructor
  · have hπ : Real.pi ≠ 0 := Real.pi_ne_zero
    field_simp [hπ]
  · rfl

What this page does not claim

The module does not claim that the two emissions produce identical physics; they differ on the diagonal. The module does not claim that 8π is a derived constant; it is a shape glue choice. The module does not claim that the framework cannot ever select the coupling, only that the current surface does not.

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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