I-B-G-Q-C: The Ward-Fixed Cone Product Globalized under a Declared Wick-Rotation Boundary Value, with Its Representative, and the Support Bridge for the Same Kernel

confirmation), Ngo Cheung (authorship requires

Enter, I-B-G-Q-C-v2, 2026, pp. 1-79

I-B-G-Q-C: The Ward-Fixed Cone Product Globalized under a Declared Wick-Rotation Boundary Value, with Its Representative, and the Support Bridge for the Same Kernel

Ngo Cheung (authorship requires confirmation)

2026

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Abstract

This is version 9 of I-B-G-Q-C (book edition t427u)-v2 (changes in Sections,,,, and); it answers request t253u. It answers Plan 4 about the vacuum-polarization sheets I-B-G-Q and I-B-G-Q-B, and it revises no other sheet.

I-B-G-Q-B cor:Q-globalobs shows that the Ward-fixed Lorentzian cone product of I-B-G-Q-B thm:Q-lorentzalpha does not globalize by analytic continuation, because a boundary-value prescription is missing. Here such a prescription is declared as new data (Declaration). The cone Green function G_{α} of I-B-G-Q-B def:Q-Galpha is continued to Lorentzian pairs as the boundary value from Im w > 0, which is the limit along the Wick rotation. I-B-VP-2 lem:boundaryvalue supplies the boundary values.

The following is proved for 0 < α ≤ 1:

1. As a function of the invariant w, G_{α} is holomorphic off (−∞, −1] and finitely many real poles. The boundary value G_{α}^bv is a global distribution on off-axis pairs and solves the wave equation off the diagonal (Theorem).

2. Its singular part is explicit. On each image cone it is exactly the flat Feynman image term 1/(4π²(σₖ + i0)): the residue in w is 1 for every α at generic θ (v3 correction: 2 at θ = 0, where two images coincide; Proposition). On the diffracted front it is (w + 1 + i0)^{−1/2} times an analytic coefficient. This is an inverse-square-root singularity, not a pole in w, except at shadow angles, where a pole m/2(w + 1)⁻¹ is added (v3 correction, Proposition (b)); the source's description is corrected (Section). The prescription selects one of the two conjugate branches beyond the front.

3. The off-diagonal product built from G_{α}^bv glues with the regular-ball product into a globally conserved product. Its Ward seagull, which is the computed representative of the lift, is i(α⁻² − 1)/(24π²r²) η_{μν}δ(x − x^{′}). (v3 correction) The conjugate prescription forces −R_{α}, not R_{α} (Proposition). It is unique up to the conserved diagonal-supported tensors of I-B-G-Q-B thm:Q-actualdefect (Theorem).

4. Tested against I-B-G-Q-B thm:Q-notinherited, the representative is separated from the inherited codimension-two datum. For the global product, the singular-support test is inconclusive under (T0) alone; it separates for the benchmark datum (Proposition).

5. Support bridge: yes. For the same-prescription kernel, supp Im G_{α}^bv lies in the closed completion relation J = {σ ≤ 0} (v3 wording, Remark). So both support orderings give fgIm G_{α}^bv = 0, with no sphere–ball geometry needed (Theorem). Still, no time-ordered product is constructed, and the IX-E trace certificate is not verified. New in version 2. Plan t233u asked for a negative theorem: that the declared prescription yields no state and no time-ordered product for α ∉ {1/N}. That is false, and the correct statement is proved. G_{α} is the Euclidean kernel of the Friedrichs Laplacian (Lemma). For every 0 < α ≤ 1, G_{α}^bv is the time-ordered two-point function of the static ground state ω_{α} of the Friedrichs dynamics, which is positive and has unitary time evolution (Theorem). The opposite prescription gives the anti-time-ordered function. Off the diagonal, the global product is the ground-state time-ordered bilinear (Corollary). The separation from the inherited codimension-two datum is extended to every (T0) datum whose singular support contains no hypersurface piece (Proposition). Composite time-ordered products, inherited identification, the IX-E-3 certificate and RG stay open. New in version 3 (Section).

1. Two-current time-ordered product (partial). For the complex free field in ω_{α}, T(jₐj_b^{′}) is constructed as a form on a dense domain of smooth finite-particle vectors (Theorem). It satisfies causal factorization off the diagonal, and an operator Ward identity, with total seagull 2iη_abδ(x − x^{′})(mathopen: φ^{*}φmathclose: +h), h = (α⁻² − 1)/(48π²r²). Its c-number part is v2's seagull, and its operator coefficient is forced.

2. Corrections of v2: the residue is 2 at θ = 0; shadow angles produce a pole at w = −1 (Proposition); the conjugate prescription forces −R_{α} (Proposition); and J is the closed completion relation (Remark). The distribution-level existence results all stand. New in version 4 (Section).

1. Exact shadow set (Proposition). Shadow angles are θ ≡ π mod 2πα. They exist iff 1/α is not an odd integer; they are then ±θₛ, and the multiplicity is 2 exactly for α = 1/N, N even, θ = 0. So α = 1/N with N odd, including the flat case, has no shadows.

2. Ward-compatible diagonal terms (Theorem). A diagonal term ∑_{β} d_ab^{β}(x^{′})∂^{β}δ(x − x^{′}), with operator-valued coefficients, preserves the Ward identity iff its symbol has the Koszul form ξ^cΛ_cab, Λ antisymmetric in c, a. Every order-zero term is excluded, for every tensor structure; at order two, Q(∂ₐ∂_b − η_abBox)δ is allowed for every Q, but satisfies the Ward identity in both variables only for constant Q. This settles the v3 open row on operator-valued diagonal terms, relative to the Ward identity.

3. Audit of v3: no mathematical error; one imprecision (V3-N1: ``shadows occur for α = 1/N'' holds only for even N). New in version 5 (Section).

1. Two-sided classification (Theorem). A polynomial 2-tensor symbol transverse in both indices is ξ^cξ^dR_cadb with R antisymmetric in each pair (degree ≥ 2), or ξ^cΘ_cab with Θ a constant 3-form (degree 1). This classifies the diagonal terms with point-independent coefficients that preserve both Ward identities; the Lorentz-covariant c-number ones are (∂ₐ∂_b − η_abBox)S(Box)δ, with ⌊p/2⌋ of them under the order bound p. For point-dependent coefficients the leading symbol is classified.

2. Symmetry (Proposition). The two-current product is symmetric, and then satisfies both Ward identities, iff the local product of I-B-G-Q-B is symmetric; that is not established here.

3. Higher products stay open; the missing input is wavefront information on G_{α}^bv (Remark). Audit of v4: no error. New in version 6 (Section).

1. Symmetry (Theorem). The local product T_{α} of I-B-G-Q-B is symmetric, by a direct computation from its defining formula. Hence the global product and the two-current product T(jₐj_b^{′}) are symmetric and satisfy the Ward identity in both arguments.

2. Point-dependent coefficients (Theorem). The diagonal terms compatible with both identities are exactly ξ^c(ξ^d − ∂_{x^{′}}^d)R_cadb + ξ^cΘ_cab with Θ a divergence-free 3-form; the proof uses a second Koszul complex for ξ − ∂_{x^{′}} (Lemma).

3. Three or more currents stay open (Remark). Audit of v5: no error; one scope note (V5-N1). New exact checks Q22, Q23. New in version 7 (Section).

1. Wavefront positivity (Lemma): off the diagonal every covector of WF(G_{α}^bv) is ξ_w dw + ξ_{θ} dθ with ξ_w > 0, by analyticity in the angle. Pairing with the time-scaling field gives a product criterion for every graph of contractions (Lemma).

2. Three-current product; partial (Theorem): defined on all of (M^{∘})³, symmetric, time ordered and causally factorized; Furry cancellation of the c-number part; Ward identity off the total diagonal. The contact Ward identity and n ≥ 4 stay open (Remark).

3. Audit of v6: no error; one scope note (V6-N1). New checks Q24, Q25 (exact) and Q26 (numerical corroboration). New in version 8 (Section).

1. Contact Ward identity (Theorem): ∂_{x₁}^aT₃ = 0 on all of (M^{∘})³. No diagonal term is needed and no new coefficient enters. The diagonal freedom of T₃ is not classified.

2. Four currents; partial (Proposition): defined, with the Ward identity, off the total diagonal. The only obstruction is the C-even cycles (squares, seagull loops, seagull bubbles) at the total diagonal.

3. Non-inheritance for every (T0) datum whose singular support has no hypersurface piece (Proposition; documentary).

4. Audit of v7: no error in proofs or statements; two scope errors in Remark (V7-N1, V7-E1). New checks Q27 (exact, combinatorial) and Q28 (exact). Higher time-ordered products, the full Epstein–Glaser axioms, inherited identification, the IX-E-3 certificate and RG stay open.

The plan's Oai282 wording agrees with I-B-G-Q-B thm:Q-282; the plan misattributes three v6 results to that theorem (Remark). The certificate stays RG-open. No quantum coefficient is admitted. Host C is not created; Host A and Host B stay separate. O.1 stays NOT_CLOSED. XVI-N is not discharged (N-Ins.4 unresolved), and there is no Route-B successor. XV-B is not opened. H2, I3 and T3 stay unresolved; J2 stays Unresolved. U6-sel is not moved. Res02-A Q1–Q4 and A1–A2 stay open. The last reading bars (quantized observable, no external clock) stay open. No Einstein dynamics, derived metric, spacetime emergence, external-clock removal or Page curve is claimed. Gate 0 stays NOT_SUPPLIED, and thm:J is not substituted. Continuum acquisition stays a negative certificate. No compilation is asserted. Citations identify source records only; no proof transfer. Authorship requires confirmation. Nothing is pushed or registered.

Bibliographic record

Title: I-B-G-Q-C: The Ward-Fixed Cone Product Globalized under a Declared Wick-Rotation Boundary Value, with Its Representative, and the Support Bridge for the Same Kernel
Author: confirmation), Ngo Cheung (authorship requires
Publication date: 2026
Institution: Enter
Report number: I-B-G-Q-C-v2
Publisher: Cheung Ngo Medical Limited
First page: 1
Last page: 79
Language: en