Enter, Big06-A-v2, 2026, pp. 1-99
Big06-A: Dark matter as a declared non-electromagnetic gravitational stress-energy reading
Ngo Cheung
25 October 2026 (version 1; version 0 is the user's draft of request t340u)
Abstract
Question. A gravitationally inferred stress-energy contribution, a vanishing electromagnetic current, and a microscopic dark-matter model are different mathematical objects. This sheet treats ``dark matter'' only as a declaration on a prescribed classical host: a stress-energy contribution that enters the Einstein-form source (or its weak-field Poisson reduction) and carries vanishing electromagnetic current. It proves what follows from that declaration, and what does not. Each statement is either a local result with its hypotheses printed here, or a documentary separation tagged ``source record; not a premise''. Version 1 is built on the user's draft Big06-A-v0 (request t340u), merged with the plan of request t338u. Every draft and plan item is re-proved or corrected (Appendices –).
Findings. (1) Exact non-electromagnetic constraint (Theorem). Write 𝒥 = (−g)^{−1/2}δS/δA. On a convex domain of potentials, 𝒥 ≡ 0 holds off shell exactly when the action does not change under compactly supported changes of A. This identical reading excludes every photon coupling: charge and millicharge, dipole-type, anapole, kinetic mixing, and also Rayleigh-type f(ψ)F² and axion-like φFF̃ terms. The weaker background reading, 𝒥 = 0 at A = 0, excludes exactly the terms linear in A. An explicit polarizability term has zero current on every F = 0 history but a nonzero second variation (Proposition). The plan's criterion ``operators that would produce observable photon scattering'' is replaced by this exact criterion. A vanishing current does not force an internal Noether current to vanish. (2) Conservation and transport (Propositions –). For a diffeomorphism- and gauge-invariant action, ∇ₐT^a_b = F_ba𝒥^a on shell, and ∇ₐ𝒥^a = 0 follows from gauge invariance. Under the identical reading, the dark slot is separately conserved in every Maxwell background, and separately conserved positive dust moves on geodesics. Finite moments do not determine a phase-space distribution. Positive dust, positive kinetic matter and Maxwell stress obey the null energy condition. On the bare de Sitter host, no split whose slots all obey the null energy condition has a dust dark slot. (3) Inverse problems (Propositions,,). With all other sources supplied, the dark tensor is the unique residual χ⁻¹(𝒢 + Λg) − T^known. Without them, every conserved tensor is admissible. Under spherical symmetry and circular test orbits, ρ_dm = (4πGr²)⁻¹ d(rv_c²)/dr − ρ_b. The reconstruction is Lipschitz in (v_c², (v_c²)^{′}, ρ_b), but it is discontinuous under uniform errors in v_c² alone, and every ρ_b ∈ [0, ρₜₒₜ] gives the same rotation curve. (4) Cosmological slot (Proposition,, Corollary). A separately conserved dust slot dilutes as a⁻³. On a prescribed host, 4πGρₜₒₜ = 3/2H² is an identity of the metric, not Friedmann dynamics. Linear dust perturbations respond to the total potential. On the Einstein–de Sitter comparison host the growing mode is ∝ a for every baryon fraction. (5) Non-selection (Propositions –). The vanishing-current declaration constrains only a current coordinate. Its intersection with the ordinary/dark/vacuum split fibre is the whole fibre. Fractions therefore have exact minimax risk equal to half their range, indicators 1/2 (1 for deterministic answers), and ratios infinite. Adaptive protocols with common kernels change nothing. A constant volume term, equivalently the shift (T_dm, T_{Λ}) ↦ (T_dm + λg, T_{Λ} − λg), preserves every current declaration and the null energy condition, so vacuum normalization stays open. Splits become identifiable under the stated counter-conditions: a supplied known source, a fixed dust-plus-vacuum form, or a current-sensitive family. Smooth futures that share a de Sitter past and a positive dust history differ in event-horizon existence. (6) Interfaces. The destructive Enter instrument and the Shredder floor 1 − 1/d are access theorems. They certify neither photon non-interaction nor gravitational coupling (Proposition, Theorem).
Why it matters. The sheet separates three things the phrase ``dark matter'' usually fuses: a gravitational source term, a non-electromagnetic constraint, and a microscopic or dynamical model. It states exactly which extra hypothesis each stronger conclusion needs. Observational comparisons (rotation curves, lensing, clusters, power spectrum) appear only as external steps of a declared workflow (Section). None is performed, and no observational claim is made. No priority is claimed for standard fluid, perturbation, Poisson or entanglement-breaking results.
Audit. Every cited label was re-read from the site records. Four labels named in the plan are absent from them: Big01-A thm:vacfibre and cor:P5, which exist only in an unregistered local version, and I-B-VP def:current and prop:well-defined. Registered substitutes are cited instead. Plan corrections PC-1 to PC-22, draft verdicts G0-1 to G0-30 with draft corrections DC-1 to DC-14, and a reconciliation of the draft's ``Debugging the proposed plan'' table are in the appendices. Checks DM-1 to DM-20 are executed with negative controls and pass (Appendix).
Version 2 (insertion-only). Version 2 keeps every line of version 1 and adds the following new findings. (7) The current declaration and the split survive every split-independent test, including under reading R1 itself. Moving a conserved current between the dark and ordinary slots changes no geometric or total-current datum, so whether j_dm ≡ 0 holds has minimax error 1/2 (Proposition). An explicit potential-free dust action realizes every member of the ordinary/dark fraction family on dust-total hosts, so the indicator risk 1/2 and the infinite ratio risk hold under R1, not only for accounts (Proposition). Which accounts a supplied action class realizes is decided by a projection criterion (Proposition), and R1 at one fixed metric does not make the stress independent of the potential (Proposition). (8) Adaptive experiments. Blindness to the current coordinate alone does not hide the split; a complete common process does, also with stopping rules, and approximately common processes keep a risk floor Δ/2(1 − ε^{*}) (Proposition, Theorems and). A current-sensitive supply is a data refinement, not a derivation (Corollary). The dark slot and Big02-A dark capacity are independent coordinates in both directions (Proposition). (9) Vacuum shift. The shift (T_dm, T_{Λ}) ↦ (T_dm + λg, T_{Λ} − λg) keeps every current and the null energy condition but not dust form or positivity: the weak condition holds exactly for −ρ_{Λ} ≤ λ ≤ inf ε_dm, the dominant one for λ ≤ 1/2inf ε_dm. The vacuum density then has the finite risk 1/2(ρ_{Λ} + inf ε_dm) (Proposition, Corollary). Rank is not supply: box risks hold at every rank, and a supplied value of an injective map identifies its coefficients (Theorem). The coefficient orbit is complete and independent of the split (Proposition). (10) Conditional forecasts and joint consistency relations (Sections and), each with a rejection condition and with hypotheses outside the ledger.
• The departure of a³ε_dm from a constant equals the integrated exchange.
• Two separately conserved geodesic dusts: a(θ₁ − θ₂) is constant and the contrast difference freezes; a compensated mode with zero total density and momentum leaves every linear potential zero.
• Expansion and growth must reconstruct one matter density: Ω̂ₘ(a) is constant along every solution, whatever the baryon fraction.
• For geodesic tracers the lensing–motion residual equals k²(Ψ − Φ) with no field equation, so L = 2V_g is equivalent to zero gravitational slip.
• Shell masses are stable with a sharp bound, local densities are not, and nonnegativity of the residual density is a shell inequality.
• On a flat matter-plus-vacuum branch the event-horizon radius tends to c/H_{∞}, every linear growth solution freezes, and the background mass inside the horizon tends to zero; bound systems are not claimed to dissolve.
• A positive dust slot inside a null-saturated total, such as the de Sitter past, forces a null-energy defect in the remainder that no vacuum shift can remove. (11) Errata (insertion-only). χεₜₒₜ = 3H² − Λ and χpₜₒₜ = −(2Ḣ + 3H²) + Λ replace the Λ-free forms of Proposition (b) and finding (4) (E2-1); energy-condition conclusions need χ > 0 (E2-2); a deterministic rule of a random transcript can have error 1/2, not 1 (E2-3). Why it matters: the new relations tie independently reconstructed quantities together, so one specified model must fit expansion, growth, lensing, motions and shell masses jointly, while the attribution, current and vacuum ambiguities are shown to survive every split-independent test. Verdicts and a crosswalk are in the appendices. Version status: the Big03-A site record is still version 2, and its version-3 labels are not cited. The Big01-A site record is now version 7 and contains thm:vacfibre and cor:P5 (PC2-3, PC2-9). Because material is inserted, version-1 section numbers after Section 7 shift, and version-1 reference ranges such as that of finding (5) now also span the inserted version-2 sections. par Standing ledger (verbatim): Host A and Host B stay separate prescribed backgrounds. Host C is not created. O.1 stays NOT_CLOSED. XV-B is not opened. XVI-N is not discharged. Gate 0 and ST4b stay NOT_SUPPLIED. No quantum coefficient is admitted. 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. Citations identify source records only; no proof transfer.
Bibliographic record
- Title
- Big06-A: Dark matter as a declared non-electromagnetic gravitational stress-energy reading
- Author
- Cheung, Ngo
- Publication date
- 2026
- Institution
- Enter
- Report number
- Big06-A-v2
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 99
- Language
- en