Enter, Big03-A-v3, 2026, pp. 1-65

Big03-A: A Supplied-Coefficient Class beside the Negative Coefficient Certificate: evaluation of the named horizon labels at a declared external coefficient, with the unsupplied class, the channel floors and dynamics vacuity unchanged

Ngo Cheung

24 October 2026 (version 1)

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Abstract

Big01-A version 5 proves that the three named horizon labels T_GH, S_GH and N_GH are not functions of the data the ledger admits: with the prescribed host, c, G, k_B, the declared data and every report held fixed, they vary injectively with the coefficient in the slot of ℏ (Big01-A thm:Hcert). A rule reproduces them for every account exactly when it receives that coefficient as a supplied input. This sheet writes down, on this sheet only, the declared class in which that input is supplied, and records what follows and what does not.

Two classes. The unsupplied class 𝒦ᵤₙ is the family κ ∈ (0, ∞) at fixed admitted data, exactly as in thm:Hcert. The supplied class 𝒦ₛᵤₚ adds one positive real κₛᵤₚ, named as an external coefficient; the sentence ``κₛᵤₚ is the SI value of ℏ'' is a documentary label whose external record is the SI/CODATA value, not a premise (Definition).

Unsupplied half (re-proved here). Theorem restates the negative certificate for 𝒦ᵤₙ and re-proves it locally: injectivity of the three labels, the fibre bound, infinite worst-case error, and the exact finite-interval error (H(κ₂ − κ₁))/(4πk_B) + |y − (H(κ₁ + κ₂))/(4πk_B)| of En08-ST-DSS-CERT thm:EC-coef(b), with the two-point prior as dual witness, its analogues for S_GH and N_GH (also against randomized rules), an exact refinement criterion, and sharp relative risks r_T = r_S = δ, r_N = tanh (Cδ/(κ₀(1 − δ²))) for a calibration of relative width δ.

Supplied half (evaluation only). Proposition is the substitution Tₛᵤₚ = κₛᵤₚH/(2πk_B), Sₛᵤₚ = πk_Bc⁵/(κₛᵤₚGH²), Nₛᵤₚ = exp (Sₛᵤₚ/k_B). At a named sample Hubble rate it evaluates to Tₛᵤₚ ≈ 2. 19591 × 10⁻³⁰K and Sₛᵤₚ/k_B ≈ 3. 31259 × 10¹²²; these agree with the supplied evaluation already printed in Big01-A thm:D(c). At the declared example H_{*} = 10⁻¹⁸s^−1 with the CODATA 2022 central G, executed rational Machin enclosures give 1. 21566 < T_{*}/10⁻³⁰K < 1. 21567 and 1. 08086 < s_{*}/10¹²³ < 1. 08087 (Proposition). It is an evaluation, not a derivation.

Channel and dynamics halves. The floors 1 − 1/d², 1 − 1/d and 1 − m/d have κ-derivative zero and are unchanged (Proposition). Already-proved thermal-operation errors are instantiated at βₑᵥₐₗ = 1/(k_BTₛᵤₚ) with exact primal and dual certificates in both Choi signs and both factor orders (Proposition). Gibbs replacement has error 1 − 1/(Z(β)Z(−β)) with Bell-ray duals (1 − f)W and γ ⊗ I − fW (Proposition); for a sample qubit gap κₛᵤₚH, βₑᵥₐₗ · κₛᵤₚH = 2π exactly and the error is (1 + e^{2π} + e^{4π})/(1 + e^{2π})². A temperature and a Hamiltonian still leave exact minimax error 1/2 for the channel error over thermal operations (Theorem). χ⁻¹(G + Λg) is weakly conserved for every C² metric, every Λ and every coupling χ ≠ 0, and the ℏ-slot does not occur in it, so the supply selects nothing (Proposition); a nonzero scalar cannot replace a missing distributional construction (Proposition).

Status. O2 stays closed negatively for the unsupplied class; the supplied form is an evaluation. O1 and O6 stay open (Remark). The slots εₛ, U and β are not filled (Remark). The audit found eleven plan corrections (Appendix); for example, the general-w statement with εₛ is thm:v4-wlabels, not thm:v3-labels, and nuADaM01-A's radiation labels share the slot κ, so they are evaluated by the supply and are not an independent slot. The user draft t295u is integrated item by item (Appendix): every mathematical item is verified, four with corrections, and its unexecuted/unhashed provenance statements are replaced by executed checks and recorded hashes. Checks Big03-01–Big03-31 pass.

Version 3 (requests t336u, t337u, t339u): new findings. Version 3 is insertion-only: every line of version 2 is kept verbatim and in order. It adds Part III, in which every result is proved locally on this sheet's host, with a witness where one is needed and a non-claim after each result. The user's findings draft t339u is the backbone, the plan t336u fixes the scope, and the prediction layer t337u is a separate section. Where two inputs overlap, only the stronger verified form is kept. (i) Adaptive non-identifiability (Theorem, Corollary, Proposition). On 𝒦ᵤₙ, every finite adaptive experiment whose complete accessible process is coefficient-blind has a κ-independent law; this covers quantum memory, stopping and postselection. So the two-point, packing (≤ 1/M), exact finite-range and relative-loss (= 1) bounds hold for κ and for the three labels, and no such procedure gives intervals of uniformly positive coverage. The supply is a data refinement: the transcript carries no information about κₛᵤₚ. Placement blindness and marginal blindness do not imply coefficient blindness; two finite counterexamples show this. (ii) Thermal identifiability and dimension (Propositions –). A calibrated Gibbs state at β_{κ} = 2π/(κH) with a non-scalar κ-free spectrum identifies κ, even through its mean energy. It is itself a coefficient-sensitive datum, whether or not an input is named κ. Finite-copy testers never discriminate two values without error. Identity, energy-conserving unitaries and eigenbasis dephasing have β-free thermal implementations on both classes. Exact equality dim ℋ = N^{(κ)} holds only on the countable set κ = C/log D, while a bound or an integer part gives an interval. (iii) Split- and vacuum-fibre non-selection (Theorem, Propositions,). Both classes are constant on independent source fibres, and the supply removes exactly the κ-factor. A binary attribution keeps minimax error 1/2, and the vacuum coordinate has box risk 1/2[(b − a) + L(e − d)]. Nonnegative witnesses exist, and full rank is not the supply of a tensor value. (iv) Relative-risk alignment (Proposition, Corollary, Propositions,). A label Aκ^p under a window of relative width δ has sharp deterministic and randomized relative risk tanh (|p|artanh δ). This gives δ for this sheet's horizon labels and tanh (3/4artanh δ) < δ for Big04-A's radiation labels, at the same δ. The exponential label exceeds δ exactly when C_K/κ₀ > (1 − δ²)artanh δ/δ. A label orbit is not a confounding orbit unless α is omitted, and a log-linear readout matrix identifies κ iff e₁ ∈ row M. (v) Operational cosmological forecasts (Section), each tagged as a framework theorem, a conditional theorem or a testable hypothesis. Coefficient supply fixes no future access: worst-case error 1/2(1 − 1/d²), with an explicit smooth NEC witness pair, both members of which accelerate. Under detector detailed balance, ln (Γ_downarrow/Γ_uparrow)/Δ is gap-independent and gives a sharp κ_eff window with the risks of (iv). The family a = (t/t₀)(1 + ln (t/t₀))^p always accelerates but has an event horizon iff p > 1. Under ε ∈ [ε₋, ε₊] there is a sharp horizon envelope c/(H(1 − ε_{∓})) with guaranteed-access and guaranteed-dark regions. Under P_{η} and K ≥ 2, a protected code requires H ≤ H_crit. The relaxation closure Ḣ = −α(H² − H_{∞}²) forces j − 1 = (2α − 3)(q + 1) and gives the drift ż = (1 + z)H₀ − H(z). (vi) Documentary separations from Big01-A, Big02-A, Big04-A, Big04-A-coeff, Big05-A, DS-OB-1-fibre, En04-A/B, I-D, Hawk05-B and the OpenAI catalogue; none is a bridge. Why it matters. ``Evaluation is not a derivation'' becomes a set of checkable statements. Adaptivity, a split declaration and a calibration window each fail to supply the slot, and the supply selects no split, no vacuum term and no future. Every quantitative forecast is the product of a declared bridge and a framework theorem, and it carries its own rejection condition. Every cited label was re-read in the site records. Plan corrections PC8-1 to PC8-22, forecast verdicts X8-1 to X8-22, draft verdicts F9-1 to F9-30 and draft corrections DC-23 to DC-28 are recorded. Among them: the six Big01-A labels named in the plan are absent from its site record (PC8-1); the detector-thermality attribution is corrected (X8-6); and the coding bound needs K ≥ 2 (X8-12). Checks Big03-61 to Big03-78 pass, and the version-1 and version-2 programs are rerun unchanged. Standing ledger (verbatim): Host A and Host B stay separate. Host C is not created. O.1 stays NOT_CLOSED. XV-B is not opened. XVI-N is not discharged. N-Ins.4 stays unresolved. H2, I3, J2, and T3 stay unresolved. U6-sel is not moved. Gate 0 stays NOT_SUPPLIED. ST4b stays NOT_SUPPLIED. No quantum coefficient is admitted. No compilation is asserted. No Einstein dynamics, derived metric, spacetime emergence, Page curve, island, vacuum energy, or quantized relational time without an external clock is claimed. A source record is not a premise. The OpenAI catalogue is not a premise. The companion does not verify those proofs.

Bibliographic record

Title

Big03-A: A Supplied-Coefficient Class beside the Negative Coefficient Certificate: evaluation of the named horizon labels at a declared external coefficient, with the unsupplied class, the channel floors and dynamics vacuity unchanged

Author

Cheung, Ngo

Publication date

2026

Institution

Enter

Report number

Big03-A-v3

Publisher

Cheung Ngo Medical Limited

First page

1

Last page

65

Language

en