Enter, Big01-A-v7, 2026, pp. 1-79

Big01-A: Horizon-Bounded Capacity on a Prescribed de Sitter Host: Exact Causal Separations, a Reorganization Ceiling, Coefficient and Identifiability Certificates, Conditional Cosmological Predictions, and a Negative Certificate for the Fixed-Size Quantum-Computer Analogy

Ngo Cheung

25 October 2026 (version 7; version 6: 25 October 2026; versions 2–5: 23 October 2026; version 1: 22 October 2026)

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Abstract

A widely discussed proposal in quantum gravity holds that the static patch of de Sitter space carries a Hilbert space of finite dimension of order exp (S), where S = A/(4Gℏ) is the Gibbons–Hawking entropy of the cosmological horizon, roughly 10¹²² for our universe. The proposal invites an analogy with a quantum computer of fixed size: the horizon is the box, and denser packing, error correction, faster operation and better algorithms reorganize a fixed resource up to a hard ceiling. We turn every sentence of the analogy into a theorem, a conditional statement or a negative certificate about a prescribed classical de Sitter host DSS_H, in the declared-data discipline of the Enter framework, where temperature, entropy and dimension enter only as named labels.

Geometry and transport. The causal past of a comoving observer is the half-hyperboloid {X₄ > X₀}, every horizon cut has area 4πc²/H², and the round-trip latency to areal radius R is at least (2/H)artanh (HR/c); this minimum is blind to every carrier, code and reorganization, while a declared device wait adds to it. A payload on or beyond the event horizon reaches the observer only through a replacement channel, and the half-diamond error of every decoder is exactly 1 − 1/∑ᵢ λᵢ⁻¹, with λᵢ the spectrum of its output. Hence the least error is 1 − 1/d², the classical-record floor 1 − 1/d is strictly better by (d − 1)/d², and both floors are reached by finite thermal operations. A conformal tick serves all targets of horizon margin m if and only if δ ≤ m; for the declared lattice task, the first-uniform-level asymptotic is reduced to an exact record-envelope and ratio-density criterion that does not involve the lattice range.

Capacity. Every host functional is invariant under reorganization, and the ceiling of any figure of merit is its fibre supremum. With unrestricted declared data the register dimension is unbounded, so dim ≤ exp (S_GH/k_B) is not a theorem of the host. Under a separately supplied promise the qubit bound ⌊etaS_GH /(k_Bln 2)⌋ is attained; at Planck 2018 parameters it is 4. 78 × 10¹²².

Coefficients and identifiability. The labels T_GH, S_GH and N_GH are not functions of the admitted classical data: at fixed admitted data they vary injectively with a comparison parameter κ in the slot of ℏ, and no value of κ is supplied. This persists for finite adaptive experiments, whose complete transcript law is the same for every κ; their exact minimax risk is half the range on a finite list of targets and infinite on the unbounded temperature family. For a fixed nonscalar energy Hamiltonian, the Gibbs state at the named temperature identifies κ, so admitted data cannot identify it; the two-point minimax error is half the trace distance, and for a qubit it is exactly 1/4, whereas coefficient-free thermal operations exist. An exact assertion dim ℋ = N_GH encodes κ = B_H/ln n and can hold only on a countable set of coefficients, while rounded or inequality forms do not invert. On the Einstein host the vacuum ambiguity of conserved comparison tensors has rank one, so its energy density is undetermined in sign and size, and the dynamics left side G[g] + Λg is conserved for every prescribed metric.

Epochs. The event horizon of a de Sitter epoch depends on the prescribed future through one number R₀, as c/H₁ + (R₀ − c/H₁)e^{H₁t}; every R₀ is realized already by smooth futures, the minimax error is exactly 1/2(R₊ − R₋)e^{H₁t}, and analytic futures force R₀ = c/H₁.

Conditional cosmological predictions. On the de Sitter host a message sent now reaches a comoving receiver at proper distance R₀ if and only if R₀ < c/H, but a reply returns only if R₀ < c/(2H), after exactly −H⁻¹ln (1 − 2HR₀/c), which exceeds the static-shell echo. On any prescribed flat FLRW host the redshift-drift residual ż − H(tₒ)z equals H(tₒ) − H(tₑ), a null test of an exact de Sitter interval. Bounds h₋ ≤ H ≤ h₊ on the future give the sharp bounds c/h₊ ≤ Rₑ ≤ c/h₋, and a nonincreasing rate gives Rₑ ≥ c/H and a nondecreasing horizon. An approach H = H_{∞} + Ae^{−pt} gives the relaxation law Rₑ = (c/H_{∞})[1 − Ae^{−pt}/(H_{∞} + p) + ⋯] with explicit second order, and a supplied flat ΛCDM law gives Rₑ = (c/H_{Λ}) ₂F₁(1/2, 1/3; 4/3; −μa⁻³). Smooth futures that agree with the whole past to all orders may or may not have an event horizon, so every rule of past data decides its existence with success exactly 1/2 at equal priors. A memory of dimension D retrieves one of M equiprobable messages with probability at most min {1, D/M}; under the supplied promise, n fair bits are kept perfectly if and only if n ≤ ⌊etaS_GH /(k_Bln 2)⌋. Each result is separated into hypotheses, derivation, physical identification and a contradicting observation. The ΛCDM law, the future bounds, the asymptotic model and the finite-dimension promise are supplied hypotheses, not premises.

Why it matters. The geometric half of the finite-capacity picture is proved with exact constants, and it now yields conditional, falsifiable statements whose assumptions are named: an exchange deadline, a redshift-drift null test, sharp horizon bounds and an approach law to de Sitter, together with what the past cannot certify and what a finite memory must lose. The capacity and thermal halves are information that must be supplied, and the new certificates measure what is missing: a bounded loss where the targets are bounded and an infinite loss where they are not. The framework does not select the expansion history, the cosmological constant, the dark-matter content or the primordial perturbations; predicting them needs an additional dynamical or probabilistic law. No finite Hilbert space, temperature, entropy, thermal noise model, absolute vacuum energy, Einstein dynamics, derived metric, spacetime emergence or Page curve is derived. κ is a comparison parameter only. No quantum coefficient is admitted.

Standing ledger (verbatim). No quantum coefficient is admitted. Host A and Host B stay separate; Host C is not created. O.1 stays NOT_CLOSED. XVI-N is not discharged; N-Ins.4 stays unresolved. XV-B is not opened. H2, I3, J2 and T3 stay unresolved. U6-sel is not moved. No Einstein dynamics, derived metric, spacetime emergence, Page curve, or quantized relational time without an external clock is claimed. The last reading bars stay open, and continuum acquisition stays a negative certificate. No Route-B successor is created. Res02-A Q1–Q4 and A1–A2 stay open. Gate 0 and ST4b stay NOT_SUPPLIED. Citations identify source records only; no proof transfer. No compilation is asserted.

Bibliographic record

Title

Big01-A: Horizon-Bounded Capacity on a Prescribed de Sitter Host: Exact Causal Separations, a Reorganization Ceiling, Coefficient and Identifiability Certificates, Conditional Cosmological Predictions, and a Negative Certificate for the Fixed-Size Quantum-Computer Analogy

Author

Cheung, Ngo

Publication date

2026

Institution

Enter

Report number

Big01-A-v7

Publisher

Cheung Ngo Medical Limited

First page

1

Last page

79

Language

en