Enter, Big05-A-v3, 2026, pp. 1-70
Big05-A: Is Spacetime Curvature Mainly Caused by Dark Matter and Cosmic Expansion?
Ngo Cheung
25 October 2026 (version 3; version 2: 25 October 2026; version 1: 24 October 2026)
Abstract
We test, inside the declared-data discipline of the Enter framework, the hypothesis ``if the curvature of spacetime is mainly, to the most extent, caused by the dark matter and expanding universe instead''. On a prescribed classical metric with declared cosmological label Λ and classical coupling χ ≠ 0, the total source Tₜₒₜ = χ⁻¹(G + Λg) is fixed by the metric, but its split Tₜₒₜ = T_b + T_dm + T_{Λ} into ordinary, dark and vacuum pieces is not: every dominance target (pointwise energy, norm suppression, averaged ratio) takes both truth values on one data fibre, also with every piece of positive energy, so no rule of host data decides it (worst-case indicator error 1/2, ratio error infinite). Only the combination Λ + χρ_{Λ} is fixed. Conservation makes a supplied law compatible only when it sums to Tₜₒₜ. On the bare host DSS_H every split into pieces obeying the null energy condition is of vacuum type, so the hypothesis holds there trivially and says nothing local. On Kottler (Schwarzschild–de Sitter) hosts the Ricci part of the curvature is exactly the Λ piece and the Weyl part exactly the mass label; the mass label dominates the Kepler, Kretschmann and tidal readings exactly inside (3m/Λ)^{1/3} < (18m²/Λ²)^{1/6} < (6m/Λ)^{1/3} and the Λ piece outside. Null orbit shapes are Λ-free, bound orbits need m > 0, and Λ enters the perihelion advance first through πΛa³√(1 − e²)/m. Local dark dominance is therefore either a relabelling invisible to every reading or a change of the metric labels that the readings decide. Big02-A's operational dark capacity is not a host functional and is not identified with stress-energy. Observational numbers enter only as a documentary record; the plan's H₀²r ≈ 10⁻¹⁵ m s⁻² at 1 AU is corrected to 7. 1 × 10⁻²⁵ m s⁻². The user's draft v0 is integrated item by item.
Version 2. Version 2 is an insertion-only upgrade: every result of version 1 keeps its label and text. It adds six local results, each proved from definitions printed here and checked by computer algebra or numerics with negative controls. (i) On Kottler hosts, with a declared static-observer optical contract, the finite-distance bending angle and the emitter's radar interval are exact quadratures. They are constant on split fibres, and the observer angle depends on Λ although the orbit shape does not; two angles of one ray at known radii, with known turning radius, decide (m, Λ) (Theorem). (ii) Effective rank four is C⁴-open and dense, and attained by an explicit analytic metric; on every rank-four metric the total is not of vacuum form somewhere, and the dominance targets stay non-selected (Theorem). (iii) On a prescribed McVittie comparison metric the effective total, the tides and a Kretschmann comparison function are explicit; the Ricci part couples to the mass through Ḣ/√(1 − 2m/r), and the source pressure is unbounded at r = 2m iff Ḣ ≠ 0 (Theorem). (iv) A separately prescribed halo-plus-Λ host has exact halo/central and Λ/mass crossover equations, and it is not a Kottler host (Theorem). (v) Declared clocks do not select a split or remove the external parameter (Proposition). (vi) The radiation-label coefficient orbit and the split are independent coordinates (Proposition). (vii) At the user's request t323u the sheet is organized around a central theorem: every observable built from the metric, its curvature, declared observers, clocks, references and other split-independent declarations is constant on the compatible split fibre, so ordinary/dark/vacuum attribution needs a split-coupled observation or a constitutive law (Theorem); it is supported by general identifiability theorems (quotient, convex fibre, observation kernel, two-point bounds), a rank/split separation, McVittie, halo, optical and contract-relative results, some of them partial (Appendix). The user's version-2 draft (t320u) is integrated item by item, with verdicts in Appendix. Items (c) and (d) of t315u are closed in the scoped sense stated in Section.
Version 3. Version 3 is insertion-only over version 2 and turns non-selection into auditable, conditional predictions. (i) Past geometry does not fix the future. Over smooth expanding flat hosts with a de Sitter past, whose effective totals obey the null and weak energy conditions with Λ = 0, the event-horizon radius takes exactly the values [c/H_E, ∞]. Every value is attained by a future that agrees with the past to all orders, and a horizon smaller than c/H_E forces a violation of the null energy condition. Over real-analytic futures the future is forced (Theorem). (ii) Without any evolution law, comoving redshift drift and transverse distance obey [(1 + z)H₀ − ż]D^{′} = c on flat hosts, and a z-independent curvature combination on curved ones. The drift curve is stationary exactly at acceleration transitions (Theorem). (iii) A separately declared, optional closure Ḣ = −n/2(H² − H_{∞}²), which is not a premise of the sheet, gives H² = H_{∞}² + (H₀² − H_{∞}²)(1 + z)ⁿ on all three branches and the diagnostic N(z) ≡ n, with its exact relation to Om. The acceleration transition, the full drift curve and the asymptotic horizon c/H_{∞} then follow jointly (Theorems,). At the documentary Planck-2018 benchmark, with radiation neglected, the conditional values are z_acc = 0. 6316, z_{*} = 1. 9162, Δv = +2. 160 and −5. 503 cm s⁻¹ over ten years at z = 1 and 4, and c/H_{∞} = 17. 543 Gly. (iv) On the McVittie comparison, 𝒟_K/(r − 2m) → 1/(32m⁵Ḣ²), so the slow-expansion limit is not uniform at r = 2m (Theorem). (v) Every forecast is constant on split fibres: predicting expansion does not identify its source decomposition (Proposition). This matters because each prediction now names the theorem that yields it, the supply it depends on and the observation that could falsify it.
noindentCitations identify source records only; no proof transfer. No quantum coefficient is admitted.
Standing ledger. No quantum coefficient is admitted. Host C is not created. O.1 stays NOT_CLOSED. XVI-N is not discharged. XV-B is not opened. H2, I3, and J2 stay unresolved. T3 stays unresolved. U6-sel is not moved. Gate 0 stays NOT_SUPPLIED. No compilation is asserted. No machine-checked proof is claimed. Numerical execution is not asserted. This is not metric reconstruction, not Einstein dynamics, not a Page curve, not a de Sitter state, not spacetime emergence, and not quantized relational time without an external clock.
Standing ledger of the plan (t316u), carried verbatim. No quantum coefficient is admitted. 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. H2, I3, J2 stay unresolved. T3 stays unresolved. U6-sel is not moved. Gate 0 stays NOT_SUPPLIED. ST4b stays NOT_SUPPLIED. Absolute vacuum energy stays open. No Einstein dynamics, derived metric, spacetime emergence, external-clock removal, or Page curve is claimed. Cited sheets (including OpenAI catalogue entries) are source records and are not premises. A source record is not a premise.
Bibliographic record
- Title
- Big05-A: Is Spacetime Curvature Mainly Caused by Dark Matter and Cosmic Expansion?
- Author
- Cheung, Ngo
- Publication date
- 2026
- Institution
- Enter
- Report number
- Big05-A-v3
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 70
- Language
- en