Enter, I-B-G-Q-C-CERT-v2, 2026, pp. 1-51
I-B-G-Q-C-CERT: A Conical Coefficient Certificate for the Diagonal Slot of the Two-Current Product
Ngo Cheung (authorship requires confirmation)
2026
Abstract
This is version 7 (book edition t427u; file CERT-v2) of the standalone certificate sheet I-B-G-Q-C-CERT-v2. It contains version 5 (I-B-G-Q-C-CERT-v5, which contains versions 4, 3, 2 and 1; all stay untouched) and adds the items of request t268u (no plan was attached), superseding version 5 by addition; errors of earlier versions found in version 6 are recorded in Section. The parent sheet I-B-G-Q-C (latest version I-B-G-Q-C-v8) is cited as a source record only and is not revised.
On the prescribed cone with deficit parameter 0 < α ≤ 1, take the parent's declared boundary value and the c-number two-current products that agree with it off the diagonal, satisfy the Ward identity in both arguments, are symmetric, and differ from one another by Lorentz-covariant constant-coefficient diagonal terms of order at most two. We prove that, once one such product T₀ is given, they are exactly T₀ + c (∂ₐ∂_b − η_abBox)δ(x − x^{′}) with c ∈ ℂ: one slot. The slot maps affinely and injectively to the characteristic coefficient χ_F read by a fixed test tensor. No rule of the admitted conical data reproduces it for two slot values, and every such rule has infinite worst-case error. At order p the ⌊p/2⌋ slots map bijectively. The order-zero diagonal term is not a slot: the parent forces it to a function of α, so a certificate for it would be false. The boundary value stays declared new data. The four-current product stays off the total diagonal, and the certificate stays RG-open: in the flat Euclidean source the scale moves along the slot. The inherited characteristic coefficient is not admitted.
New in version 2. (i) In the flat Euclidean model, the free-field product Π^{(ν)} = c₀D^Eu_{ν} satisfies descent and the Ward identity, and its scale derivative is exactly −1/(24π²)D^Eδ₀; the constant is derived here as 2π²c₀ from Vol(S³) = 2π² and the declared off-diagonal datum, whose Wick derivation is taken from the source record. Every covariant conserved extension of order at most two is Π^{(ν)} for exactly one ν, and a prescribed normalization holds for exactly one member. So, in that model, scale and slot are the same datum and the RG-open status is a theorem. (ii) The slot count ⌊p/2⌋ does not depend on α. (iii) On the cone, a prescribed value of χ_F holds for exactly one member: the plan's ``descent, Ward and normalization cannot hold together'' is false as stated; what holds is ``for exactly one member, never for two''. (iv) χ_F is a named coefficient of the declared data, not the inherited one. (v) No cosmological-type contact term is admitted, so the slot is not a vacuum-energy term. (vi) Under a declared record of the parent's four-current construction, no choice inside the admitted family completes it at Δ₄; this is a negative inside the family, not a permanent no-go.
New in version 3. (i) For two slot values the exact minimax error of a rule that omits the slot is |c₁ − c₂|/2, on a real interval half its length, on the real line infinite; changing the reference product only reparametrizes the slot. (ii) A smooth order-zero contact term a(x)η_abδ(x − x^{′}) added to a Ward-satisfying product preserves the Ward identity only for a ≡ 0; this concerns differences and does not exclude an order-zero term forced inside a complete product, such as the parent's recorded representative. (iii) Under (H), (H4) and a supplied global difference, slot substitution transports extendability at Δ₄ but does not complete the recorded construction. (iv) Documentary comparisons with DS-OB-1-CERT, Red06-G-DSS-CERT, VC-OB-1-CERT, the Casimir CERT sheets, En08-ST-DSS-CERT and V-DST-DSS-CERT share only the omitted-parameter logic. No version-2 statement was found to be false.
New in version 4. (i) The slot direction D is transverse, symmetric and of pure order two, and a test-function pairing separates it from every vacuum-energy-type term ρ η_abδ(x − x^{′}) without using the Ward identity; this re-proves and sharpens the version-2 statement that the slot is not a vacuum-energy term. (ii) Readings of DS-OB-1-CERT-v2, the Casimir certificates of Hawk07-F/G/H-CERT-v3, VC-OB-1-CERT-v2 and Red06-G-DSS-CERT-v3; the plan's ``both concern differences'' is accurate only for the contact statement, and its line ``the hosts and the named quantities differ'' is false inside the de Sitter group. No version-3 error was found.
New in version 5. (i) Under a declared dilation-stability hypothesis (Hdil), the admitted family carries a unique running constant β_{α} with S_{λ}T_c = T_{c + β_{α}log λ}. The value of β_{α} is not derived, and (Hdil) is not verified for the parent. (ii) At the total diagonal Δ₄, a Ward-satisfying extension of the four-current product, if one exists, is never unique: there is an explicit conserved term of order eight. Among constant-coefficient differences at most one extension is dilation-invariant. Existence stays open. (iii) Readings of V-DST-DSS-CERT-v4, DS-OB-1-CERT-v3, VC-OB-1-CERT-v3 and Hawk07-F/G/H-CERT-v3. No version-4 error was found (Section).
New in version 6. (i) If the parent's boundary value commutes with dilations (declared hypothesis (BV)), then G_{α}^bv is homogeneous of degree −2 and B[G_{α}^bv] is dilation-invariant off the diagonal; (Hdil) then reduces to the constant-coefficient, order-two form of the diagonal remainder S_{λ}T₀ − T₀, which stays open. (ii) A declared window [c₁, c₂] for the slot value would give least worst-case error (c₂ − c₁)/2; no window is declared. (iii) Readings of the newly delivered companions. The value of β_{α} and existence at Δ₄ stay open.
That some product satisfies the admission conditions is recorded in the parent; this sheet uses it as a declared hypothesis (H), not as an imported proof. The parent and the other cited sheets are source records, not premises. 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-CERT: A Conical Coefficient Certificate for the Diagonal Slot of the Two-Current Product
- Author
- confirmation), Ngo Cheung (authorship requires
- Publication date
- 2026
- Institution
- Enter
- Report number
- I-B-G-Q-C-CERT-v2
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 51
- Language
- en