Big02-A: Dark-Capacity Certificates on the Prescribed Static Patch: Redundancy, Complete Causal-Access Loss, Junk Sectors, and Future-Dependent Horizons

Cheung, Ngo

Enter, Big02-A-v7, 2026, pp. 1-121

Big02-A: Dark-Capacity Certificates on the Prescribed Static Patch: Redundancy, Complete Causal-Access Loss, Junk Sectors, and Future-Dependent Horizons

Ngo Cheung

25 October 2026 (version 7; versions 1–6: 23–24 October 2026)

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Abstract

Big01-A turned the fixed-size quantum-computer analogy for the de Sitter static patch into theorems about a prescribed classical host DSS_H. In that analogy one may call ``dark'' any part of a finite, horizon-bounded register that constrains the usable computation while lying outside the controllable, readable logical subspace. The analogy names four such parts: error-correction redundancy, horizon leakage, junk sectors, and information that has left the patch. This sheet asks which sentences about these four parts are theorems, and proves the ones that are. Each statement here is either a local theorem proved from hypotheses stated in the sheet, or a documentary separation that depends on a source record and is tagged ``source record; not a premise''. Temperature, entropy and dimension enter only as named labels with an external coefficient slot κ in the place of ℏ.

Four main theorems. (I) Redundancy. Suppose a code on n declared q-dimensional sites corrects the erasure of any δ − 1 sites (a Knill–Laflamme condition). Then the reference decouples from every set of δ − 1 sites, and ln K ≤ (n − 2(δ − 1))ln q. The same bound holds for the logical factor of a subsystem code. Under a supplied capacity promise P_{η} this becomes ln K ≤ etaS_GH /k_B − 2(δ − 1)ln q. Without the promise, K is unbounded at fixed distance. No code improves on a replacement channel. (II) Complete causal-access loss. A source with u = X₄ − X₀ ≤ 0 reaches a comoving observer only through a replacement channel. The least error over located circuits is exactly (1 − 1/d²) 1[p ∉ J⁻(γ)]. This error is independent of κ, it is not a function of placement-blind data, and the same temperature label is compatible with errors 0, 1 − 1/d and 1 − 1/d². (III) Junk. An orthogonal complement forces no error at all. Flagged loss costs exactly p(1 − 1/k²). A flagged m-dimensional quantum cut costs exactly 1 − m/d. Scrambling the complement costs exactly 1 − 1/(1 + (d − k)²), which is strictly below the horizon floor. The cut floor lies below the horizon floor by (md − 1)/d². (IV) Information that has left the patch. On smooth futures that share the entire past, the junction radius R₀ takes every value in (0, ∞). A fixed past event can be dark for one future and visible for another, so any rule of past data has worst-case error at least 1/2(1 − 1/d²) on the least error. The area proxies, the late label N^{(κ)}(K) and Θₜₜ = 3(K² − H₁²) all differ across these futures.

Tools. A primal–dual sandwich lemma: if J(T) ≥ t ΩΩ^{†} then ε(T) ≤ 1 − t, and every input gives a lower bound. The sandwich is exact on replacement channels, block pinchings, isotropic channels and the flagged-cut attainers. It is not exact on every channel: amplitude damping has t^{*} = 0 but error below 1. The sheet also has a fibre lemma, a coefficient lemma, a monotone null coordinate and an ancestor lemma.

A background section presents the dark-matter analogy as structural motivation only. Every physical sentence in it is marked heuristic and not claimed, and points to its finite-channel certificate where one exists. A user draft (t272u) was checked claim by claim and its verified results are adopted. No finite Hilbert space, temperature, entropy, particle species, clustering mechanism, Einstein dynamics or Page curve is derived. No quantum coefficient is admitted.

Version 2 (plan t273u; user draft t275u). Version 2 keeps every statement of version 1 unchanged. By addition only, it adds a robustness subsection after each main theorem, a section of results integrated from the user's t275u draft, a cross-cutting section, checks DC-17 to DC-31, an updated plan audit, a verdict appendix for the draft, and new documentary edges. I-R: a code that corrects every error of weight ≤ t has distance ≥ 2t + 1, so ln K ≤ (n − 4t)ln q; the form (n − 2t)ln q is the bound for t erasures. Approximate correctability gives (1 − ε_A − ε_B)ln K ≤ ln dim ℋ_C + 1/2(g(ε_A) + g(ε_B)) with g(ε) = (1 + ε)h(ε/(1 + ε)), which costs a dimension-dependent term 2εln K as well as O(εln (1/ε)). II-R: if the receiver channel is within half-diamond distance ε of a replacement, the error is at least 1 − 1/d² − ε, and this is tight. Per-observer darkness does not imply joint darkness. A cut with L classical labels and an m-dimensional register has error at least 1 − min {m/d, Lm²/d²}, attained with a shared seed when m ∣ d. Every error in [0, 1 − 1/d²] is realized by a thermal operation at every β > 0. III-R: a residual state τ gives installed error 1 − 1/(1 + Tr τ⁻¹), with an exact entropy-constrained envelope, but optimized recovery has error 1 − 1/(1 + j²) whatever the entropy. Amplitude damping has error exactly γ. IV-R: explicit moduli for R₀ hold under tail hypotheses, and local regularity alone gives none. Quasi-analytic futures are rigid, while Gevrey (s > 1), C^{k, α}, Sobolev and C^{∞} futures realize every R₀. A null-type condition Ḣ ≤ 0 restricts R₀ to [c/H₁, ∞) without removing the dark-event floor. Junction radii are affinely tied to later ones, and the status-flip region has an exact 4-volume.

Version 3 (plans t276u and t277u; user drafts t279u). Version 3 keeps every statement of version 2 unchanged. By addition only, it adds an early-universe comparison host with a thermodynamic interface (Section), packaging results and further expansions (Section), two corrections to version-2 wording (Section), checks DC-32 to DC-43, plan audits, a verdict appendix for the drafts, and documentary edges. Early host: a radiation-type prefix a = √(1 + 2H₀t) is prescribed, not derived. Its causal relation, arrival times and deadlines are explicit. Radiation labels Tᵣ ∝ κ^{3/4}/a and Sᵣ ∝ κ^{−3/4} are not functions of the admitted data, but an exact calibrated label reveals κ. With the terminal rate H₁ fixed, the settled late radius is c/H₁, and the early-junction radius ranges over (0, ∞), or over (c/H₀, c/H₁) under Ḣ ≤ 0; a settling deadline shrinks these ranges further. The status-flip four-volume has a closed polynomial form. Early-source loss is a replacement floor independent of every label. A late Gibbons–Hawking label is not determined by early data unless H₁ and κ are supplied, and G + Λg conservation is vacuous on a prescribed host. Packaging: the declared-data monoid acts uniformly only through its spectator lift. The seeded (L, m) optimum is 1 − Fₘₐₓ, with explicit sign-correct witnesses. It equals 1 − min {m/d, Lm²/d²} exactly when Lm ≤ d or when I_d is a weighted sum of L rank-m projectors, and fails for (3, 2, 2). A mean-energy promise supplies no dimension. The cutoff-and-reset error is exactly (q + √(4q − 3q²))/2. The information defect is additive, and its allowance keeps a (ε_A + ε_B)ln K term.

Version 4 (request t280u; derived agenda). Version 4 keeps every statement of version 3 unchanged. No plan was attached to the request, so the agenda was derived from the open, partial and refuted items of version 3 (Appendix). By addition only, version 4 adds:

• (L, m) values. A Ky Fan branch bound gives the exact variational formula d²Fₘₐₓ = max ∑ₓ ‖√(Eₓ)‖₍ₘ₎² over L-outcome POVMs. For two labels the closed form is d²Fₘₐₓ = 2r² + 2√(2)rk + k², so e_seed(3, 2, 2) = (6 − 2√(2))/9 exactly.

• Energy floor. The energy-constrained replacement floor is exactly (ν + √(4ν − 3ν²))/2 for ν = E/Δ ≤ 1/3 in any number of modes. For one mode it is 3ν(1 − ν) on [1/3, 2 − √(22)/3], with an explicit dual certificate, and it is bracketed beyond. Optimized decoding does not improve cutoff-and-reset at q ≤ 1/3.

• Coding cost. A flag code shows that the coefficient ε_A + ε_B of ln K is tight to first order for the information defect, which replaces the refuted dimension-free cost; the dimension form stays open.

• Early host. Power-law prefixes give general-w labels with Jacobian −3w/(1 + w) and a flip four-volume for every exponent. Rate-bounded futures give the early-junction range (B, R̅) with R̅ = B[γρ^{1 − 1/γ} − 1]/(γ − 1), and decelerating futures have no event horizon.

• Checks and re-check. Checks DC-44 to DC-53 pass, and the re-check of version 3 found no erratum.

Version 5 (plan t283u; user draft t285u). Version 5 keeps every statement of version 4 unchanged and adds, by insertion only:

• (L, m) decompositions. Rank conditions, complement and Naimark moves give an unconditional classification for three labels: I_d is a weighted sum of three rank-m projectors iff d/m ∈ {1, 3/2, 2, 3}. For every L, a finite Horn–Klyachko linear program decides the question and a finite second-order-cone program gives Fₘₐₓ; both rest on a cited spectral theorem.

• Energy. For M modes, the optimized replacement error is exactly min_{u, v > 0, A_M < 1} (u + νv)/(1 − A_M(u, v)²) at every energy, with matching primal and dual certificates. Vacuum replacement is optimal up to ν = M²/(M² + M + 1). Cutoff-and-reset recovery is a certified limit of finite SDPs.

• Coding. A Bell-twirled approximate Singleton bound sharpens the version-2 allowance. The version-4 allocation code is small only in Bell form: its half-diamond error is at least 1/2. The dimension form remains open.

• Geometry. Simultaneous minimax for several junctions, a window-plus-tail modulus with sharp tail constant, an exact radiation–de Sitter benchmark volume 341479π/80640, and a classification of κ-free label combinations.

• Checks. DC-54 to DC-69 are executed and pass. The re-check of version 4 found no erratum.

Version 6 (plan t287u, items 3 and 6). Version 6 keeps every statement of version 5 unchanged and adds, by insertion only:

• Closed (L, m) values. A meeting-subspace inequality and a two-parameter certificate give the seeded optimum below m/d in closed form, without the cited Horn–Klyachko theorem: for every L in the band (L − 1)m ≤ d ≤ Lm and in the co-band (L − 1)d ≤ Lm, and for every d when L = 3. The version-5 numerical surds 5 + 2√(6), 7 + 2√(6), 8 + 4√(3) and 13 + 4√(3) become theorems, and Q_{7, 4, 5} = 18 + 12√(2).

• Early labels. For any supplied system Hamiltonian, the identity, eigenbasis dephasing and Gibbs replacement are thermal operations at the label temperature, with recovery errors 0, 1 − 1/d and 1 − 1/d² that do not depend on κ. ℛ_{I/d} is thermal only for scalar Hamiltonians. Otherwise the undecoded Gibbs-replacement error depends on the label, with exact minimax error 1/(2d²).

• Checks. DC-70 to DC-78 are executed and pass. The re-check of version 5 found no erratum.

Version 7 (plans t333u and t334u; user draft t335u). Version 7 keeps every statement of version 6 unchanged and adds, by insertion only:

• Split-fibre non-selection (plan t333u, user draft t335u). Every dark-capacity certificate of this sheet factors through the map that forgets the ordinary/dark/vacuum split of the total source, so all of them, and every adaptive rule built from them, are constant on each fixed-account split fibre. On explicit fibres with nonnegative energy densities, every such rule has exact minimax error 1/2 on a dominance indicator (1 for deterministic yes/no answers), 2/3 on naming the dominant component, and infinite error on the dark-to-ordinary ratio and on the absolute vacuum term (Theorem, Proposition). The certificate package depends on the declared account, while R₀ and the access floor are genuine functionals of the full prescribed future (Proposition). This matters because it shows, from this sheet's own definitions, that operational darkness cannot be read as dark matter or vacuum energy, and it says precisely in what sense dark capacity is ``not a host functional''.

• Horizons. Without an energy condition R₀ ranges over (0, ∞]; under Ḣ ≤ 0 it ranges over [c/H₁, ∞], which is exactly the Big05-A range, so the two records agree and neither strengthens the other. Past geometry, past splits, past experiments and any finite ledger of causal statuses leave R₀ in an interval of positive length, except in one boundary case; the drift identity Ṙₑ = HRₑ − c selects no horizon (Propositions,, Corollary).

• Coefficient blindness and rank. No coefficient-blind adaptive experiment, even with quantum memory and stopping, estimates κ or a label with bounded worst-case error, while one exact label value identifies κ; the injectivity of Theorem is unchanged. A constant recovery floor or split independence does not make a process coefficient-blind, and full rank of a tensor ambiguity selects no parameter (Theorem, Propositions,).

• Conditional cosmological predictions. On every flat FLRW future the least access error is a_d1[r ≥ χₑ(t)]. A declared band ε₋ ≤ −Ḣ/H² ≤ ε₊ < 1 gives c/(H(1 − ε₋)) ≤ Rₑ ≤ c/(H(1 − ε₊)), sharp, together with certified visible and dark regions. Each comoving source crosses to dark exactly once, at a computable time, and the accessible population decays at rate 3c/Rₑ. A declared capacity promise gives a critical radius √((ln K + 2rln q)/C_{κ}) that grows with √(κ). An optional, supplied holographic closure (Li's model, outside the ledger) forces w_X = −1/3 − (2√(Ω_X))/(3b), a constant diagnostic ℬ ≡ b, and three futures: de Sitter (b = 1), power law (b > 1) and a finite-time rip (b < 1). Redshift drift and H(z) are the observables. These results matter because they turn the non-identification theorems into stated interval predictions and falsifiers under declared assumptions, without fits and without a value of κ (Section, Table).

• Audit. All three inputs (t333u, t334u, t335u) receive per-item verdicts. A plan pre-check claiming that the critical-radius bound should use n − r is rejected: n − 2r is the quantum Singleton bound and is attained. All cited labels were re-read from the site records, and every mismatch is recorded (Appendices –). Checks DC-79 to DC-90 are executed and pass. The re-check of version 6 found no erratum. 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: Big02-A: Dark-Capacity Certificates on the Prescribed Static Patch: Redundancy, Complete Causal-Access Loss, Junk Sectors, and Future-Dependent Horizons
Author: Cheung, Ngo
Publication date: 2026
Institution: Enter
Report number: Big02-A-v7
Publisher: Cheung Ngo Medical Limited
First page: 1
Last page: 121
Language: en