Enter, 08-I-C-Ex2-CERT-v6, 2026, pp. 1-28
I-C-Ex2-CERT-v6: Exact Dual Certificates for the Access Law 1 − k/d
Ngo Cheung (authorship requires confirmation)
2026
Abstract
This is version 6 of the standalone certificate sheet I-C-Ex2-CERT. It contains version 5 (I-C-Ex2-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-C-Ex2 is cited as a source record only and is not revised. For a cut whose accessible output is a classical record plus quantum blocks of dimension at most k, followed by any decoder on a d-dimensional code, the entanglement fidelity is at most k/d and the half-diamond error at least 1 − k/d. Here the optimum 1 − k/d is certified on both sides: the Bell vector is the primal witness and (1 − k/d)Ω is a written dual witness for the attaining channel T_{d, k}. For the parent's finite memory model (d = qk, classical label of dimension q, quantum memory of dimension k), the dual witness (1 − 1/q)Ω certifies the recovery error 1 − k/d on both sides. For every object of the class the error is at least 1 − F_e ≥ 1 − k/d, and it equals 1 − F_e exactly on the covariant class. Exact recovery forces k = d. The plan's statement that the error of every such channel equals the formula is false: the errors of the class fill [1 − k/d, 1]. This is recorded as a plan correction. New in version 2: the twirl of any channel is covariant, so its error is exactly 1 − F_e by the written dual, and the twirl is a replacement channel exactly when F_e = 1/d^2, where the spectral law gives the same value; every replacement channel twirls to ℛ_{I/d}. The class is closed under unitary pre-encoding, decoder changes, block-local refinement and coin mixing, so 1 − k/d is a hard floor (retained value at most k/d) along every reorganization. The commutant step of the attainer rests on a local Schur–Weyl argument: the OpenAI catalogue, which the plan proposed as a source record for it, contains no Choi-operator or isotropic-channel entries (plan correction). New in version 3. (i) For every channel, the error equals 1 − F_e if and only if J|Ω⟩ = dF_e|Ω⟩, if and only if J ≥ F_eΩ; covariance is sufficient, not necessary. (ii) The composites of the class are exactly the channels with a Kraus family of rank at most k, and such a channel is at the floor 1 − k/d exactly when it has the form X ↦ ∑_i λ_iP_iXP_i with rank-k projections and ∑_i λ_iP_i = I; each has the certificate (Φ_d, (1 − k/d)Ω), with no covariance assumed. (iii) The memory-model attainer and the covariant attainer T_{d, k} are two members of this class. (iv) The Hawk07 and PG replacement certificates are comparisons whose dual operator is not of the form (1 − t)Ω (plan correction; qualified in version 4, since PG-OB-1-CERT-v2 also has the rank-one witness and the block dual corresponds to it under a sign change). No version-2 statement was found to be false. New in version 4. (i) A witness in the Hawk07/PG convention is turned into one of this sheet by W ↦ W − J(𝒯) + Ω, with the same partial trace; the Hawk07/PG block dual of a replacement channel becomes exactly the rank-one witness (1 − 1/S)Ω. (ii) The best dual of the form (1 − t)Ω has t_{*} = 1/⟨Ω|J^{+}|Ω⟩, exact for replacement channels and on the Bell-saturated class. (iii) Enlarging the accessible block from k to k^{′} lowers the floor by (k^{′} − k)/d; scaling the code and the block by the same factor leaves it unchanged; so thm:kdreorg is closure under the local moves of Definition, not under Big01-A's def:reorg (plan correction). (iv) Readings of PG-OB-1-CERT-v2, Hawk07-A/B/C-CERT-v3, En08-ST-DSS-CERT-v3 and Big01-A. One version-3 error is recorded. New in version 5. (i) For every channel, the best rank-one dual (1 − t_{*})Ω is exact if and only if ker (J − t_{*}Ω) contains vec (R) for a state R; for full-rank J, if and only if the matrix of J^{−1}|Ω⟩ is Hermitian positive semidefinite. Exactness is invariant under unitary conjugation. (ii) Pinching onto m orthogonal blocks of arbitrary sizes has error exactly 1 − 1/m, certified by the rank-one dual; for unequal sizes this channel is neither covariant nor Bell-saturated, and its error lies strictly above the floor 1 − k/d. (iii) Readings of PG-OB-1-CERT-v3, Hawk07-A/B/C-CERT-v3 and En08-ST-DSS-CERT-v4. One version-4 error is recorded (Section). New in version 6. (i) The fidelity primal L_F, a convex program over states, satisfies 1 − F_e ≤ L_F ≤ ε ≤ 1 − t_{*}, and the best rank-one dual is exact if and only if L_F = 1 − t_{*}. (ii) Non-exactness is generic: the channels whose matrix M_{𝒯} is not Hermitian form an open dense set of full measure, on which ε < 1 − t_{*} strictly. A depolarized unitary pAd_U +(1 − p)ℛ_{I/d} is exact if and only if Tr U = 0 or U is a phase times a Hermitian unitary. (iii) The mixtures p id + (1 − p) ·(pinching onto m blocks of any sizes) have error exactly (1 − p)(1 − 1/m), with a witness independent of p. (iv) Readings of the newly delivered OB CERT v4, Hawk07 CERT v4 and DSS CERT v5 sheets; one omission of version 4 is recorded (Section). The exact error of non-exact channels stays open. All proofs are local; the parent and Big01-A, Hawk07-A/B/C-CERT, PG-OB-1-CERT and En08-ST-DSS-CERT-v2 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-C-Ex2-CERT-v6: Exact Dual Certificates for the Access Law 1 − k/d
- Author
- confirmation), Ngo Cheung (authorship requires
- Publication date
- 2026
- Institution
- Enter
- Report number
- 08-I-C-Ex2-CERT-v6
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 28
- Language
- en