Enter, 06-I-C-Th-CERT-v6, 2026, pp. 1-23
I-C-Th-CERT-v6: A Coefficient Certificate for the Thermal Bill
Ngo Cheung (authorship requires confirmation)
2026
Abstract
This is version 6 of the standalone certificate sheet I-C-Th-CERT. It contains version 5 (I-C-Th-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-Th (latest version I-C-Th-v1) is cited as a source record only and is not revised. The thermal coefficient is the inverse temperature β of the reservoir preparation, a supplied input. On the parent's spectator witness, with the visible channel, the spectator Hamiltonians and the energy-conserving implementation all fixed, the designated-reservoir energy gain Q_L(β) = ε/(1 + e^{−βε}) is strictly increasing, hence injective, so no rule of admitted data alone reproduces the named heat for two values of β. The plan's generic phrase ``infinite worst-case error'' is false for the energy gain: the best admitted-data rule has worst-case error exactly ε/4, and every finite reservoir has bounded energy gain. It is true for the bill in entropy units, βQ_L, which is unbounded. Injectivity holds on the witness family, not for every implementation: the identity implementation has Q_L ≡ 0. Both points are recorded as plan corrections. No temperature is derived and no quantum coefficient is admitted; the cone and its deficit stay inputs. New in version 2: (i) the finite balance βQ_B = ΔS_M + I + D holds for every unitary and every β, so it selects nothing (vacuity); (ii) with the admitted channel data and β fixed, the identity and the swap give bills 0 and ε/(1 + e^{−βε}), so no rule from those data is correct on the pair (two-point minimax error exactly half the swap bill); (iii) the bill and β are interchangeable data, so a named temperature enters only as a supplied input, and the bill is not a horizon temperature; (iv) the bill is invariant under H_B ↦ H_B + cI while absolute reservoir energies shift, so it is a relative energy and not a vacuum-energy calculation; (v) for every fixed finite implementation, either the bill is constant in β or no admitted-data rule reproduces it, and the best worst-case error is exactly half its range, at most the spectral width. New in version 3. (i) Over all energy-conserving spectator unitaries the bill is s ε/(1 + e^{−βε}) with s ∈ [0, 1] arbitrary; at fixed β the exact minimax error is half the swap bill, with U and β both omitted it is ε/2, and on that larger family the bill no longer determines β: injectivity belongs to the specified witness. (ii) The energy bill has finite worst-case error and βQ infinite. (iii) Under H_B → H_B + cI, an absolute reservoir energy has exact two-point minimax error |c_1 − c_2|/2 on the shift-invariant reduced datum and infinite worst-case error over c ∈ ℝ. (iv) Documentary readings of VC-OB-1, Big01-A and Red06-G-DSS-CERT-v2, and a correction: the finite balance is an entropy identity, not a Bell/Choi sandwich. No version-2 statement was found to be false. New in version 4. (i) A radius lemma: a constant rule has finite worst-case error exactly when the target range is bounded, and then the error is half the range. (ii) Of the named quantities compared, Q_L and the error 1 − 1/(2 + 2cosh βΔ) of a thermal replacement are injective and bounded (finite error), while βQ_L and the de Sitter labels T^{(varkappa)}, S^{(varkappa)} are unbounded (infinite error); so Red06's thm:RC-coef has the form of the βQ_L statement, not of the Q_L statement (plan correction). (iii) The curvature tensors displayed in VC-OB-1-CERT-v2 have nonzero divergence in that sheet's own convention (exact computer-assisted check), so ``enlarges the fibre by curvature counterterms'' describes the intended tensors only (plan correction); this matches Hawk07 CERT v3's correction of the identical display. (iv) The cross-reference ``the hosts and the named quantities differ'' is false inside the de Sitter group (DS-OB-1-CERT, Red06-G-DSS-CERT and Big01-A share DSS_H). One version-3 error is recorded. New in version 5. (i) The corrected curvature-squared tensor E^{(1)} is conserved; the version-2 display satisfies ∇^{μ} ^{(1)}H_{μν} = 4R_{νλ}∇^{λ}R; and E^{(1)}, E^{(2)} and both displayed tensors vanish on Einstein regions. (ii) Of the three parts of the enlarged-fibre proposition of VC-OB-1-CERT-v2: the negative part (a) survives through the c_0-subfamily; (b) survives; and (c) survives qualitatively, with the values −360 and −120 (computer-assisted). The conservation of E^{(2)} stays partial. (iii) Readings of VC-OB-1-CERT-v3, DS-OB-1-CERT-v3, Red06-G-DSS-CERT-v4 and Hawk07-F-CERT-v3. No version-4 error was found (Section). New in version 6. (i) E^{(2)} is divergence-free for every metric (hand proof), so every member T + c_0g + c_1E^{(1)} + c_2E^{(2)} of the corrected fibre is conserved, and the version-5 proviso is removed. (ii) On the radiation metric η^2(− dη^2 + dx^2), which is not Einstein, E^{(1)} = 0 and (computer-assisted) E^{(2)} = 0: the Einstein condition is sufficient, not necessary, for the curvature-squared directions to drop out. (iii) Readings of VC-OB-1-CERT-v4, Hawk07-F-CERT-v4, DS-OB-1-CERT-v4 and DSS CERT v5; one version-4 reading is qualified (Section). (iii) The version-2 displays read in the two curvature-sign conventions give, respectively, the relations of Theorem (b) and the corrected tensors themselves, so the version-5 relation and the opposite-convention reading of VC-OB-1-CERT-v4 are both correct (Proposition). All proofs are local; the parent, Big01-A VC-OB-1, VC-OB-1-CERT-v2 and Red06-G-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-Th-CERT-v6: A Coefficient Certificate for the Thermal Bill
- Author
- confirmation), Ngo Cheung (authorship requires
- Publication date
- 2026
- Institution
- Enter
- Report number
- 06-I-C-Th-CERT-v6
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 23
- Language
- en