Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Constructed regulated polynomial-model evolution

Physical scope

Global Schrödinger evolution at one fixed finite regulator, with the form-selected unbounded Hamiltonian and its exact domain. No continuum or empirical claim.

Statement

Assume Countable Choice. Use one fixed finite-regulator data set from Adopted finite-regulator polynomial model, and let $h$ be the self-adjoint form realization supplied by Self-adjoint unbounded polynomial interaction at fixed regulator, with domain $D (h)$ and eigenpairs $(ψ_{n}, λ_{n})$ . Let $E_{*} > 0$ joules and set $H_{p h y s} : = E_{*} h$ on $D (h)$ . For physical time $τ$ in seconds define

$U_{p h y s} (τ) : = U (E_{*} τ / ℏ),$

where $U (s)$ is the dimensionless-parameter group of Global unitary evolution at fixed finite regulator. Then $U_{p h y s}$ is a strongly continuous unitary group, $U_{p h y s} (0) = I$ , $U_{p h y s} (τ_{1} + τ_{2}) = U_{p h y s} (τ_{1}) U_{p h y s} (τ_{2})$ , and $∥ U_{p h y s} (τ) ψ ∥ = ∥ ψ ∥$ for every $ψ \in H_{N}$ . Its norm derivative exists exactly for $ψ \in D (h)$ , and then

$\frac{d}{d τ} U_{p h y s} (τ) ψ = - \frac{i}{ℏ} H_{p h y s} U_{p h y s} (τ) ψ .$

The result is a conditional deduction for the selected finite-regulator model. It is not an empirical observation or a continuum-limit theorem.

Facts & Assumptions

Given: Countable Choice; a fixed finite-regulator model; $E_{*} > 0$ joules; $ℏ > 0$ in joule-seconds; the operator and domain of the mathematical finite-regulator construction.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function. The mathematical QFT-IR01 spectral construction assumes this principle.

[A2]

Adopted finite-regulator polynomial model: the regulator and polynomial data are fixed, and the physical Hamiltonian is $E_{*} h$ when the form realization $h$ is actually constructed; $E_{*}$ is a positive energy scale.

[A3]

Physical interpretation of a supplied constructed quantum model: only after a selected model has an actually constructed self-adjoint Hamiltonian and unitary evolution, adopt the Schrödinger convention $U (t) = \exp (- i t \hat{H} / ℏ)$ . This postulate itself constructs neither operator nor group.

[F1]

Self-adjoint unbounded polynomial interaction at fixed regulator: under Countable Choice and the finite-mode coercivity hypotheses, $h$ is the specified self-adjoint form realization, with complete eigenfamily $(ψ_{n})$ , real eigenvalues $λ_{n}$ , and $D (h) = {\sum_{n} c_{n} ψ_{n} : \sum_{n} λ_{n}^{2} ∣ c_{n} ∣^{2} < \infty}$ .

[F2]

Global unitary evolution at fixed finite regulator: under Countable Choice, $U (s) = \sum_{n} e^{- i s λ_{n}} ⟨ ψ, ψ_{n} ⟩ ψ_{n}$ is a strongly continuous unitary group for dimensionless $s \in R$ ; its norm derivative exists exactly on $D (h)$ and equals $- i h U (s) ψ$ there.

Proof

technique · direct

1.1A1A2F1algebra

Under Countable Choice [A1], the hypotheses of [F1] hold, so $h$ is self-adjoint on $D (h)$ . Multiplication by the positive real scalar $E_{*}$ preserves self-adjointness and the domain, so $H_{p h y s} = E_{*} h$ is self-adjoint on $D (h)$ with eigenvalues $E_{*} λ_{n}$ . The units in [A2] make $E_{*} τ / ℏ$ dimensionless.

2.1A1F2step 1.1algebra

Put $s = E_{*} τ / ℏ$ . Since this is a real linear change of parameter, Countable Choice [A1] discharges the hypothesis of [F2], which gives unitarity, strong continuity, the identity at zero, and the group law for $U_{p h y s} (τ) = U (s)$ . Unitarity also gives $∥ U_{p h y s} (τ) ψ ∥ = ∥ ψ ∥$ .

3.1A1F1F2step 1.1step 2.1algebra

If $ψ \in D (h)$ , the chain rule and [F2] give $\frac{d}{d τ} U (E_{*} τ / ℏ) ψ = - i (E_{*} / ℏ) h U (E_{*} τ / ℏ) ψ = - i H_{p h y s} U_{p h y s} (τ) ψ / ℏ$ . Conversely, because $E_{*} / ℏ > 0$ , the change $s = E_{*} τ / ℏ$ is invertible, so norm differentiability in $τ$ is equivalent to differentiability in $s$ ; [F2] makes its domain exactly $D (h)$ .

4.1A1A3F1F2step 1.1step 2.1∎

[A3] supplies the physical reading of this already constructed self-adjoint Hamiltonian and group as the selected model's Schrödinger evolution. It adds no existence claim beyond [F1] and [F2], and no empirical premise is used.

Constructed regulated polynomial-model evolution

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources