Enter, I-B-G-Q-B-v7, 2026, pp. 1-70

I-B-G-Q-B: one interacting product, still not an admission

Ngo Cheung (authorship requires confirmation)

Version 6 — 22 October 2026

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Abstract

This new sheet takes the compatibility criterion, the finite part and the three version-5 typing propositions of I-B-G-Q as inputs and proves two things. First, for one declared interaction (a massless complex scalar coupled to an external U(1) field on auxiliary Euclidean ℝ⁴) we construct the renormalized vacuum-polarization product Π^{(ν)} = c₀(∂∂ − δΔ)u_{ν}, c₀ = −1/(48π⁴). We prove that it extends the off-diagonal product, satisfies the interacting Ward identity exactly, is covariant, and is unique up to the scale ν under the stated degree bound. For it, descent, Ward identity and normalization hold together, while scale independence fails. We prove that it is not the inherited coefficient: its singular support is a point, whereas the inherited datum is singular along a codimension-two set. Second, under the stated typing we prove that the inherited coefficient is not determined by the inputs: an injective, identically conserved worldsheet family Θ(𝒟^{′}(W)) preserves every typed datum, and the contractible incidence closure supplies no monodromy constraint. We also prove that the typing alone decides neither success nor failure of the actual-image condition, and we record a conditional support separator. Its hypothesis, a nonzero defect of the actual lift, is not established. No quantum coefficient is admitted, and the certificate stays RG-open.

Version 2 adds: the exact running of Π^{(ν)} in position and momentum space, ν∂_{ν}Π^{(ν)} = −1/(24π²)(∂∂ − δΔ)δ₀ and Π̂^{(ν)} = 1/(48π²)(δk² − kk)(log (k²/ν²) − c_{⋆}/π²) (Theorem); a renormalized product on regular balls of the Euclidean section of the prescribed cone for deficits α = 1/N, by images, with exact Ward identity, a seagull term forced by the Ward identity, N-independent running, and coincidence value (α⁻² − 1)/(48π²r²) (Theorem); the precise status of the Lorentzian and general-deficit transfer (Theorem); and the exact dimension of the translation-invariant free family, 25, 45, 71 for ω = 4, 5, 6 (3, 3, 5 with covariance), which equals the number of real normalization data that are necessary and sufficient (Theorem).

Version 3 corrects the v2 singularity hypothesis (T0), which the benchmark |x_{⊥}|⁻² does not satisfy, to a non-removable-smooth-singularity hypothesis under which the non-admission theorem holds. It proves that the Euclidean product and all its proved properties do not determine a flat Lorentzian product, even modulo local counterterms (a two-parameter family with identical Ward identity, scaling and running and non-local differences), and writes one explicit declared Lorentzian datum; its identification with the time-ordered product is not proved. It proves that finite images realize the cone exactly for α = 1/N, that the typed inputs determine neither the value nor the sign of the actual-lift detecting functional, and that in every degree all 3m² + 5m + 3 normalization slots are left unfilled by every subset of the typed inputs. No quantum coefficient is admitted; the certificate remains RG-open. Version 4 identifies the declared Feynman datum as i times the unitary Epstein–Glaser time-ordered product, unique up to the scale, with unitarity as exactly the extra datum; constructs the Euclidean cone product for every deficit by a closed-form mode sum; and evaluates the defect of a naive lift on the declared model (local, of size (α⁻² − 1)/(24π²r²), removable). Version 5 constructs the Lorentzian local cone remainder for every deficit by analytic continuation of the closed form (real, symmetric, a bisolution, equal to the image sum at α = 1/N) and the Ward-fixed seagull 2iH_{α}^L(x, x), without a state or time-ordered identification; evaluates the defect for a whole class of lifts; shows that the running does not place the product outside the Oai282-A Ward certificate; and counts the normalization slots. Version 6 determines the global singular structure of the continued Lorentzian remainder (image-cone poles with residue 1/sinh u_{*}, a diffracted pole whose residue vanishes iff α = 1/N, conjugate branches beyond the diffracted front) and turns it into a method obstruction: the seagull s = 2iH is global on the diagonal, but globalizing off it requires a boundary-value prescription that the inputs do not supply. It shows that the defect class of every lift agreeing off the diagonal is zero, with undetermined representative.

No quantum coefficient is admitted. No compilation is asserted. Cited sheets are documentary, never premises.

Bibliographic record

Title

I-B-G-Q-B: one interacting product, still not an admission

Author

confirmation), Ngo Cheung (authorship requires

Publication date

2026

Institution

Enter

Report number

I-B-G-Q-B-v7

Publisher

Cheung Ngo Medical Limited

First page

1

Last page

70

Language

en