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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Adopted finite-regulator bosonic polynomial model — Examples

1 · Prerequisites

2 · Summary

The one-mode example instantiates the finite-regulator polynomial model with a positive quartic term and applies the page's exact evolution and energy-bound results. The separate scope remark explains what the cited Euclidean distribution-extension identity does and does not establish physically. Neither item reports data or asserts a continuum construction.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

RemarkRemark: AI-adaptedProof: Not applicableverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Scope of the Euclidean distribution-extension identity

Physical scope

A scope clarification for one four-dimensional Euclidean distribution-extension identity; it asserts no measured running parameter, Lorentzian observable, or continuum field theory.

Physical status

Assume the Axiom of Choice (The Axiom of Choice) as in Exact four-dimensional Euclidean logarithmic scale anomaly. That proposition gives an exact Euclidean distribution-extension identity for the specified punctured kernel $u (x) = 1 / (16 π^{4} ∣ x ∣^{4})$ : changing its cutoff scale changes the extension by $(8 π^{2})^{- 1} \log (μ / ν) δ_{0}$ . Its statement explicitly classifies it as a Euclidean distribution identity. By itself it does not say that an experimentally measured parameter changes with energy scale, choose which extension a physical model uses, or define a measurement procedure. It constructs no Lorentzian field theory. This remark asserts neither existence nor nonexistence of a field theory obtained by taking a continuum limit and supplies no empirical claim.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Constructed one-coordinate positive-quartic example on full $L^{2}$

Physical scope

One fixed dimensionless coordinate model on full $L^{2} (R, d q; C)$ with a positive quartic potential term, form-selected Hamiltonian, global unitary evolution, and the proved energy-increment bound. No empirical data or continuum field theory.

Example

Choose $N = 1$ , $a = 1$ , $r = 1$ , and $λ = 1$ , all dimensionless. The potential is $V (q) = q^{2} / 2 + q^{4}$ , and the physical Hamiltonian is $H_{p h y s} = E_{*} h$ with $E_{*} > 0$ joules. The coordinate-space Hilbert space is the full $H_{1} = L^{2} (R, d q; C)$ of Finite-lattice polynomial Hamiltonian data, not a finite matrix truncation. The positive $q^{4}$ term is present in the potential of the full form-selected operator. This is a one-coordinate model; no field-scattering, continuum, or empirical conclusion is asserted.

Facts & Assumptions

Given: The $N = 1$ data above, Countable Choice, and the conditional results below.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function, as assumed by the QFT-IR01 mathematical suppliers.

[F1]

Constructed regulated polynomial-model evolution: for the constructed self-adjoint form realization, the physical evolution is a strongly continuous unitary group for all real seconds, with the generator domain specified in that theorem.

[F2]

Controlled quartic-model ground-energy bound: for the potential $V_{λ} (q) = r^{2} q^{2} / 2 + λ q^{4}$ , $0 \leq E_{g}^{p h y s} (λ) - E_{g}^{p h y s} (0) \leq 3 E_{*} a λ / (2 r^{2})$ .

[F3]

Finite-lattice polynomial Hamiltonian data: for $N = 1$ the coordinate space is $R$ , the Hilbert space is $L^{2} (R, d q; C)$ , and the polynomial differential expression is defined on $C_{c}^{\infty} (R; C)$ ; the definition does not assert a self-adjoint operator domain.

Verification

1.1F3algebra

The parameters satisfy $a > 0$ , $r > 0$ , and $λ > 0$ , and $V (q) = q^{2} / 2 + q^{4} \geq q^{2} / 2$ , so the coercivity hypotheses hold with $c = 1 / 2$ and $C_{0} = 0$ . The coordinate variable $q$ ranges over $R$ ; vectors in $H_{1}$ are square-integrable complex functions of $q$ , identified up to equality almost everywhere by [F3]. The characteristic functions $1_{[3 n, 3 n + 1)}$ , $n \in N$ , are nonzero pairwise orthogonal vectors, so this Hilbert space is infinite-dimensional.

2.1A1F2step 1.1algebra

By [A1], the Countable Choice hypothesis of [F2] holds. Substituting $a = r = λ = 1$ into it gives $0 \leq E_{g}^{p h y s} (1) - E_{g}^{p h y s} (0) \leq 3 E_{*} / 2$ . This is an exact bound for the selected model, not an exact value of either ground energy.

3.1A1F1step 1.1algebra∎

By [A1], the Countable Choice hypothesis of [F1] holds. Applying [F1] to the same selected model gives a global unitary group $U_{p h y s} (τ)$ for every $τ \in R$ , norm preservation for all states, and differentiability precisely on the stated Hamiltonian domain. The quartic coefficient is positive, and the Hilbert space is infinite-dimensional by step 1.1; thus this is a one-coordinate polynomial operator model on the full Hilbert space, not a finite matrix replacement. No observation or continuum extrapolation is asserted.