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Finite matrix formal inverse and gauge-covariant kernel — Examples

1 · Prerequisites

2 · Summary

These examples give a rational off-shell Dirac matrix check, the exact transverse angular coefficient for a fixed vector, and a bounded-energy coherent family without a strong Fock limit. The last result is a counterexample in the declared free transverse-mode model, not a construction of an interacting charged QED sector. Draft QFT-MG01 supplier carriers are present, and their page inventory lists those items; the prerequisite records retain the exact mathematical hypotheses, with independent review and native readiness recorded separately.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

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 →

Exact rational Dirac Ward matrix example

Example

For any $μ > 0$ , use $h = diag (1, - 1, - 1, - 1)$ , $p = (0, 0, 0, 0)$ , and $k = (2 μ, 0, 0, 0)$ . Both $p$ and $p + k$ are off shell. The exact rational matrices are $S_{0} (p) = - \frac{1}{μ} I_{4}, S_{0} (p + k) = \frac{2 β + I_{4}}{3 μ}, Γ (k) = 2 μ β .$

Facts & Assumptions

Given: $μ > 0$ , $p = 0$ , $k = (2 μ, 0, 0, 0)$ , and the Dirac matrices in Off-shell free Dirac Ward telescoping and tree currents.

[F1]

Off-shell free Dirac Ward telescoping and tree currents: when both $h (p, p) - μ^{2}$ and $h (p + k, p + k) - μ^{2}$ are nonzero, $S_{0} (p) = (Γ (p) + μ I_{4}) / (h (p, p) - μ^{2})$ and $S_{0} (p + k) Γ (k) S_{0} (p) = S_{0} (p) - S_{0} (p + k)$ .

[F2]

Dirac Clifford matrices and energy projections: $β = diag (I_{2}, - I_{2})$ , so $β^{2} = I_{4}$ ; the item fixes $Γ (p)$ with the effective matrix $h = - η$ .

[A1]

$μ > 0$ has inverse-length units; the momenta $p = 0$ and $k = (2 μ, 0, 0, 0)$ use the same inverse-length coordinate convention as Off-shell free Dirac Ward telescoping and tree currents.

Verification

Given: [F1], [F2], and the numerical momenta and positive parameter in the Example.

1.1A1F1algebra

Here $h (p, p) - μ^{2} = - μ^{2}$ and $h (p + k, p + k) - μ^{2} = 3 μ^{2}$ , so both rational inverses are defined. Since $Γ (p) = 0$ and $Γ (k) = 2 μ β$ , the inverse formula in [F1] gives the displayed $S_{0} (p)$ and $S_{0} (p + k)$ .

2.1F1F2step 1.1algebra∎

Using $β^{2} = I_{4}$ from [F2], $S_{0} (p + k) Γ (k) S_{0} (p) = - \frac{(2 β + I_{4}) 2 β}{3 μ} = - \frac{4 I_{4} + 2 β}{3 μ} = S_{0} (p) - S_{0} (p + k) .$ Thus this off-shell rational instance satisfies the exact matrix Ward identity.

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 →

Fixed-vector transverse angular coefficient $8 π / 3$

Example

Let $e \in R^{3}$ be a unit vector and $P_{T} (n) = I_{3} - n n^{⊤}$ for $n \in S^{2}$ . Then the angular coefficient in the coherent infrared family with $b (n) = λ P_{T} (n) e$ , $λ \in R ∖ {0}$ , is $\int_{S^{2}} ∣ b (n) ∣^{2} d Ω = \frac{8 π λ^{2}}{3} .$

Facts & Assumptions

Given: The unit vector $e$ , the scalar $λ \neq 0$ , and the transverse projection and angular integral in Logarithmic coherent infrared tail: finite energy, no Fock limit.

[F1]

Logarithmic coherent infrared tail: finite energy, no Fock limit uses $P_{T} (n) = I_{3} - n n^{⊤}$ and $C_{b} = \int_{S^{2}} ∣ b (n) ∣^{2} d Ω$ for a measurable transverse $b \in L^{2} (S^{2}; C^{3})$ .

[A1]

$e \in R^{3}$ is a unit vector and $λ \in R ∖ {0}$ , as stated in the Example.

[F2]

Agreement with the existing polar sphere measure: the Euclidean chart measure on $S^{2}$ equals polar surface measure and is invariant under orthogonal transformations.

[F3]

Surface integration on compact C1 hypersurfaces: surface integration is the Euclidean chart integral with density $\sqrt{\det (D X^{T} D X)}$ .

[F4]

A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ : coordinate hyperplanes and singletons have Lebesgue measure zero, hence chart seams and poles have zero area when their coordinate parameter sets are null.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume the same Countable Choice carried by the coherent-tail and surface-measure suppliers; no stronger choice principle is used.

Verification

Given: [A1]–[A2], [F1]–[F4] and the specified $b (n) = λ P_{T} (n) e$ .

1.1A1A2F1F2F3F4algebra

Since [A1] gives $∣ e ∣ = 1$ and $∣ n ∣ = 1$ , $∣ P_{T} (n) e ∣^{2} = e^{⊤} (I_{3} - n n^{⊤}) e = 1 - (n \cdot e)^{2}$ . Rotational invariance in [F2] permits taking $e = (0, 0, 1)$ . Use the chart $X (u, ϕ) = (\sqrt{1 - u^{2}} \cos ϕ, \sqrt{1 - u^{2}} \sin ϕ, u)$ for $- 1 < u < 1$ and $0 < ϕ < 2 π$ . Its tangent vectors are orthogonal with squared lengths $(1 - u^{2})^{- 1}$ and $1 - u^{2}$ , so [F3] gives area density $1$ . The omitted azimuthal seam is a coordinate line in sphere graph charts, and each pole is a singleton in its graph chart; these parameter sets are null by [F4], and their chart integrals vanish. Thus with $u = n_{3}$ the angular integral is $2 π \int_{- 1}^{1} (1 - u^{2}) d u$ .

2.1A1A2F1step 1.1algebra∎

The integral equals $2 π [u - u^{3} / 3]_{- 1}^{1} = 8 π / 3$ . Multiplying by $λ^{2}$ yields $C_{b} = 8 π λ^{2} / 3 > 0$ , which is the coefficient required by [F1].

CounterexampleConstruction: 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 →

Bounded free-photon energy need not yield a Fock-vector limit

Statement refuted

Every normalized coherent family in the transverse free-photon Fock space with uniformly bounded expectation of $H_{p h} = ℏ c d Γ (∣ k ∣)$ converges strongly in that Fock space as its infrared cutoff $ϵ ↓ 0$ .

Construction

Take a unit vector $e \in R^{3}$ , set $b (n) = P_{T} (n) e$ for $n \in S^{2}$ , fix $κ > 0$ , and use $f_{ϵ} (k) = 1_{{ϵ < ∣ k ∣ < κ}} ∣ k ∣^{- 3 / 2} b (\hat{k})$ . This gives the explicit transverse coherent family from Logarithmic coherent infrared tail: finite energy, no Fock limit.

Facts & Assumptions

Given: The planned transverse free-photon space and the exact coherent-family result of Logarithmic coherent infrared tail: finite energy, no Fock limit.

[F1]

Logarithmic coherent infrared tail: finite energy, no Fock limit: for any transverse $b$ with $C_{b} = \int_{S^{2}} ∣ b ∣^{2} d Ω > 0$ , the vectors $C (f_{ϵ})$ have norm one, energy expectation $ℏ c C_{b} (κ - ϵ)$ , weak limit zero, and no strong Fock-space limit.

[A1]

$e \in R^{3}$ is a unit vector, $κ > 0$ , and $ℏ, c > 0$ ; cutoffs satisfy $0 < ϵ < κ$ . These are the chosen model parameters in the Counterexample.

[F2]

Agreement with the existing polar sphere measure: the Euclidean chart measure on $S^{2}$ equals polar surface measure and is invariant under orthogonal transformations.

[F3]

Surface integration on compact C1 hypersurfaces: surface integration is the Euclidean chart integral with density $\sqrt{\det (D X^{T} D X)}$ .

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume the same Countable Choice carried by the coherent-tail and surface-measure suppliers; no stronger choice principle is used.

Counterexample

Given: [A1]–[A2], [F1]–[F3], and the profile in the Counterexample.

1.1A1A2F1F2F3algebra

For $b (n) = P_{T} (n) e$ , $∣ b (n) ∣^{2} = 1 - (n \cdot e)^{2}$ . By [F2], rotate $e$ to the third coordinate axis. The regular sphere chart $X (y, z) = (\sqrt{1 - y^{2} - z^{2}}, y, z)$ on $∣ y ∣, ∣ z ∣ < 1 / 4$ lies in the belt $∣ n \cdot e ∣ < 1 / 2$ and has area density $(1 - y^{2} - z^{2})^{- 1 / 2} \geq 1$ by [F3]. Its parameter rectangle has area $1 / 4$ , so this belt has positive surface measure. On the belt $∣ b (n) ∣^{2} \geq 3 / 4$ , hence $C_{b} > 0$ ; also $b$ is continuous, square-integrable on the finite sphere measure, and transverse.

2.1A1A2F1step 1.1algebra∎

Apply [F1] to this $b$ . It gives unit vectors with $0 \leq ⟨ C (f_{ϵ}), H_{p h} C (f_{ϵ}) ⟩ \leq ℏ c C_{b} κ < \infty$ , yet no strong Fock-space limit. This contradicts the asserted implication and is the required counterexample.

Finite matrix formal inverse and gauge-covariant kernel — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Exact rational Dirac Ward matrix example

Example

Facts & Assumptions

Verification

Fixed-vector transverse angular coefficient 8π/38π/3

Example

Facts & Assumptions

Verification

Bounded free-photon energy need not yield a Fock-vector limit

Statement refuted

Construction

Facts & Assumptions

Counterexample

Sources

Fixed-vector transverse angular coefficient $8 π / 3$