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

ARTICLE 5 claims 5 theorems

Foundation Pair Kernel Dimensioned Hamiltonian Compiler Source Density Non Id Di

A machine-checked theorem shows that two different quantum models produce the same physics, revealing a hidden redundancy in how the framework attaches a field to a Hamiltonian.

Verify this page

The coupling ambiguity

In quantum mechanics, a Hamiltonian is the mathematical object that dictates how a physical system evolves in time. It is built from pieces: a kinetic term that encodes motion, and a potential term that encodes forces. The Recognition Science framework constructs such Hamiltonians on a discrete lattice, a finite grid of points that stands in for continuous space. To do this, it must attach a source field, a quantity that varies from point to point, to the potential part of the Hamiltonian. The declaration distinct_sourceDensityCoupling_distinct_diagonal is a theorem in the framework's machine-checked library of formal theorems. It states that if two different coupling strengths, the numbers that scale how strongly the field influences the potential, are plugged into this construction, then the resulting Hamiltonians will differ on at least one diagonal entry. In plain terms, different couplings produce different physics.

The subtlety is that the coupling itself is not a single number. The framework models it as a product of two separate scales: a potentialScale, which sets the overall strength of the potential, and a fieldNormalization, which normalizes the source field. The theorem proves that the Hamiltonian depends only on the product of these two numbers, not on how that product is split between them. This is a concrete redundancy: rescaling one factor up and the other down by the same amount leaves every matrix entry of the Hamiltonian unchanged. The framework's own documentation states this directly: the split is pure redundancy. The theorem s28_fd2_surface_does_not_select_sourceDensityCoupling goes further. It demonstrates that the framework's committed recognition surface, the set of certificates and exports it declares, admits at least two distinct emissions, two complete packages of scales, that both preserve all the required physical semantics. These two packages have different products, and therefore different Hamiltonians, yet both satisfy every condition the framework demands.

This is a non-identifiability result, a statement about what the framework cannot do. The framework proves that its own axioms and definitions do not single out a unique coupling between the source field and the Hamiltonian. Two different models, with different strengths, are both perfectly valid within the framework's rules. The theorem does not claim that one of these couplings is correct and the other wrong. It does not claim that the framework is broken or inconsistent. It claims something more precise: the framework, as currently formulated, is silent on this choice. It is a gap in the framework's power, not a contradiction in its logic. The framework's own documentation is explicit about what this theorem does not do: it performs no hydrogen or CODATA fitting, it makes no identification of the fine-structure constant, and it introduces no new primitive or axiom.

The practical consequence is a warning about overinterpretation. If a researcher sees a specific coupling value emerge from the framework, they cannot claim the framework forced that value. The framework allows at least two. The theorem continuum_eight_pi_is_shape_glue_not_s28_selection reinforces this for the specific value 8π: it is a piece of algebraic glue that connects a discrete Green shape to a continuum Coulomb profile, but it is not selected by the framework's core axioms. The framework proves a fact about its own limits, and that fact is useful precisely because it prevents a false claim of derivation. What the framework does prove is that its construction is internally consistent: both emissions preserve Hermiticity, the Stone unitary group, and the scalar energy expansion. The ambiguity is real, and it is fully characterized.

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)
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 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 emission_preserves_fd2_surface · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean
emission_preserves_fd2_surface · IndisputableMonolith/Foundation/PairKernelDimensionedHamiltonianCompilerSourceDensityNonId.lean:278 
theorem emission_preserves_fd2_surface
    (event : PostingPair3 3)
    (hevent : event ∈ realizedPrimitivePostingPairs3 3)
    (emission : SourceDensityCouplingEmission) :
    EmissionPreservesFD2Surface event emission := by
  refine ⟨hevent, ?_, ?_, ?_⟩
  · exact oneBodyHamiltonian3_isHermitian _ _ _
  · exact oneBodyHamiltonian3_stone _ _ _
  · intro state variation t
    exact oneBodyScalarEnergy3_line_expansion _ _ _ _ _ _

What this page does not claim

The theorem does not claim that either of the two couplings is the physically correct one. The theorem does not claim the framework is inconsistent or contradictory. The theorem does not claim to derive the fine-structure constant or any other physical constant.

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

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:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND