Theorem · Chain Position 70 of 346

[[070_T8.1_SIGN-INVARIANCE-THEOREM|SIGN INVARIANCE]] THEOREM

** For any self-generated unitary Û: [σ̂, Û] = 0.

Connections

Assumes

  • None

Enables

  • None
Objections & Responses
Objection: "This assumes unitarity—what if evolution isn't unitary?"
"Real moral change might involve non-unitary processes from the start."
Response

Self-operations are unitary by definition. Non-unitary evolution requires coupling to external systems (Lindblad dynamics). If you claim internal non-unitarity, you've either (a) expanded the system to include external input (contradicting "self-generated") or (b) violated quantum mechanics. The theorem applies within standard quantum formalism.

Objection: "Commutators can be zero accidentally"
"Just because [σ̂, Û] = 0 doesn't mean σ is preserved. Eigenvalues could still change."
Response

When two Hermitian operators commute, they share eigenstates. Unitary evolution preserves the eigenvalue associated with each eigenstate. If σ̂ and Û commute, and |−⟩ is a σ̂ eigenstate with eigenvalue −1, then after Û acts, the state remains in the −1 eigenspace. This is not accidental—it's the spectral theorem.

Objection: "What if the Hamiltonian is unbounded?"
"Unbounded operators have domain issues. The theorem might not apply."
Response

For physically realistic Hamiltonians (bounded below, finite energy), the generated unitaries are well-defined. The sign operator acts on a 2-dimensional space (compact); domain issues don't arise for σ̂. The theorem holds for all relevant cases.

Objection: "This seems circular"
"You defined σ̂ to commute with self-operations. That's not a theorem, it's a stipulation."
Response

No. The theorem derives from: (1) Self-operations are generated by the system's own Hamiltonian H. (2) σ̂ represents a symmetry of the system (moral orientation is intrinsic). (3) Symmetries commute with Hamiltonians (Noether). (4) Therefore [σ̂, Û] = 0 for Û = exp(−iHt). The commutation is derived, not stipulated.

Objection: "People's moral orientation isn't a quantum observable"
"You can't apply QM to morality."
Response

The soul has quantum structure ([[087_E10.1_Soul-Field-Equation|E10.1]]). The sign operator acts on the moral Hilbert space, which is part of the soul's full Hilbert space. Theophysics extends quantum formalism to include moral degrees of freedom. If you reject this extension, you must provide an alternative—but the quantum structure is the most rigorous available.

Physics Layer

Symmetry and Conservation

Noether's Theorem: Every continuous symmetry corresponds to a conserved quantity.

For sign:

  • Symmetry: σ̂ → σ̂ under self-evolution (sign is intrinsic)
  • Conservation: Sign eigenvalue is preserved under self-dynamics

Physical parallel: Charge conservation follows from U(1) symmetry. Sign conservation follows from Z₂ moral symmetry.

Mathematical Layer

Formal Proof

Theorem ([[070_T8.1_Sign-Invariance-Theorem|T8.1]]): For any self-generated unitary Û, [σ̂, Û] = 0.

Proof:

1. Let Û be self-generated: Û = exp(−iHt/ℏ) for system Hamiltonian H

2. σ̂ is a symmetry: [σ̂, H] = 0 (sign is conserved under internal dynamics)

3. Baker-Campbell-Hausdorff: [σ̂, exp(−iHt)] = Σ (−it)ⁿ/n! [σ̂, Hⁿ]

4. [σ̂, Hⁿ] = 0 for all n (since [σ̂, H] = 0)

5. Therefore: [σ̂, Û] = 0 ∎

Defeat Conditions

To Falsify This

  1. **Find a self-generated Û where [σ̂, Û] ≠ 0** — Demonstrate a unitary operation internally generated that doesn't commute with sign
  2. **Show sign is not a symmetry of self-dynamics** — Prove the individual's Hamiltonian doesn't respect sign conservation
  3. **Demonstrate sign-flip through internal evolution** — Show Û|−⟩ = |+⟩ for some self-generated Û
  4. **Break the unitarity-symmetry connection** — Find unitary operators that change eigenvalues of commuting observables