Enter, 05-I-C-CERT-v6, 2026, pp. 1-32
I-C-CERT-v6: Exact Dual Certificates for the Shredder Recovery Obstruction
Ngo Cheung (authorship requires confirmation)
2026
Abstract
This is version 6 of the standalone certificate sheet I-C-CERT. It contains version 5 (I-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). It supersedes version 5 by addition; the earlier corrections stay in place (Sections,,), and the errors of earlier versions found in version 6 are recorded in Section. The parent sheet I-C (latest version I-C-v2) is cited as a source record only and is not revised. For a classical-output channel followed by a decoder on a d-dimensional code, the half-diamond error to the identity is certified from both sides by explicit witnesses. From version 1: every record-ignoring decoder has error exactly 1 − 1/S, S = ∑_i λ_i^{−1}; the optimum over all decoders is 1 − 1/d, certified by the dual operator (1 − 1/d)Ω; the gap (d − 1)/d^2 is the exact value of reading the record, in the sense of best record-ignoring against best unrestricted account (Proposition); and the errors of the full class fill [1 − 1/d, 1], so the spectral law fails there (plan correction). New in version 2. (i) On the basis-measurement interface, every classical decoder with stochastic matrix M has error exactly 1 − 1/∑_x M_xx^{−1}; general decoders are pinned between 1 − 1/∑_x ⟨x|σ_x|x⟩^{−1} and 1 − 1/∑_x ⟨x|σ_x^{−1}|x⟩. So the error is fixed by the decoder's action on the record, not by its output spectrum; every relabelling other than the identity has error 1, and along the path of version 1 the error is exactly 1 − (1 − s)/d. (ii) Complete dephasing attains the shredder floor and is a thermal operation with a zero-Hamiltonian bath. (iii) A thermal replacement on the record has error 1 − 1/(Z_{β}Z_{−β}), strictly above the shredder floor, certified by the same dual sandwich. (iv) Under every reorganization of the interface the retained value stays at most 1/d, and the gap (d − 1)/d^2 is exactly the gain from reading the record when the starting account is the best record-ignoring one (version-2 wording corrected in version 3; the version-3 abstract repeated it uncorrected, version-3 error V3E.1). The plan's claim that the two extremes lie on a continuous path checked by C3 is confirmed, and now proved exactly. New in version 3. (i) For every channel, the error equals 1 − F_e if and only if the Choi operator has |Ω⟩ as eigenvector, if and only if J ≥ F_eΩ; covariance is sufficient, not necessary. (ii) A measure-and-prepare composite is at the shredder floor 1 − 1/d exactly when every nonzero effect has rank one and the decoder prepares the normalized effects. (iii) On the basis-measurement interface, outputs with σ_x|x⟩ = q|x⟩ give the exact error 1 − q/d; for d ≥ 3 this includes non-classical, non-covariant decoders. (iv) Correction of version 2: (d − 1)/d^2 is the difference between the best record-ignoring and the best unrestricted retained values; the increase from an arbitrary record-ignoring account is 1/d − 1/S_{σ}, up to 1/d. (v) The Hawk07 and PG replacement certificates share the value and the primal witness of the spectral law, but their dual operator is not the rank-one (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). New in version 4. (i) A witness in the Hawk07/PG convention (W ≥ 0, W ≥ J(𝒯) − Ω) is turned into one of this sheet by W ↦ W − J(𝒯) + Ω, with the same partial trace; the block dual of the Hawk07/PG replacement certificate becomes exactly the rank-one witness (1 − 1/S)Ω, so ``the same dual certificate'' holds up to this convention change, not as operators. (ii) The best dual of the form (1 − t)Ω has t_{*} = 1/⟨Ω|J^{+}|Ω⟩; it is exact for replacement channels (without Bell saturation) and on the Bell-saturated class. (iii) On the basis-measurement interface, the version-2 upper bound is exactly this best rank-one value, and the exact law 1 − 1/∑_x q_x^{−1} extends to every decoder with σ_x|x⟩ = q_x|x⟩. (iv) Readings of PG-OB-1-CERT-v2, Hawk07-A/B/C-CERT-v3, En08-ST-DSS-CERT-v3 and Big01-A; prop:shredsharp records (d − 1)/d^2 as a difference of two floors, not a gain from every start (plan correction). Two version-3 errors are 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) On the basis-measurement interface the lower bound 1 − 1/S_q is exact only under the eigenvector condition σ_x|x⟩ = q_x|x⟩, so dephasing the decoder's outputs never increases the error and strictly lowers it otherwise. (iii) The upper bound 1 − 1/S_r is exact if and only if G_xy = ⟨y|σ_x^{−1}|x⟩ is Hermitian positive semidefinite; an explicit non-classical decoder outside the eigenvector condition has exact error 11/14. Outside both conditions the error lies strictly between the two bounds; this is a partial result, and the exact value there stays open. (iv) Readings of PG-OB-1-CERT-v3, Hawk07-A/B/C-CERT-v3 and En08-ST-DSS-CERT-v4. Two version-4 errors are 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) On the basis-measurement interface the exactness criterion extends to singular outputs as a linear matrix inequality; for d = 2 with one pure output it is αdet σ_1 ≥ |β|^2, with error α/(α + det σ_1) (for example 12/19). The fidelity primal is never below 1 − 1/S_q. (iv) Readings of the newly delivered OB CERT v4, Hawk07 CERT v4 and DSS CERT v5 sheets; one imprecision of version 5 and one omission of version 4 are recorded (Section). The exact error of non-exact channels stays open. All proofs are local. Big01-A, I-C-Th, 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-CERT-v6: Exact Dual Certificates for the Shredder Recovery Obstruction
- Author
- confirmation), Ngo Cheung (authorship requires
- Publication date
- 2026
- Institution
- Enter
- Report number
- 05-I-C-CERT-v6
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 32
- Language
- en