summary
Recognition Science treats dark matter as odd-phase ledger entries within the eight-tick cycle, producing gravitational J-cost without photon coupling and a ratio near phi cubed plus one. Cosmologists and particle physicists working in the RS framework cite this definition as the compact statement of the ledger-shadow model. The definition is a direct list of five explanatory strings with no lemmas or computation.
claimThe summary is the list of strings encoding the RS dark-matter claims: eight-tick phases separate visible and dark sectors, dark matter consists of odd-phase ledger entries, such entries carry J-cost but emit no photons, the ratio satisfies $Ω_{dm}/Ω_b ≈ φ^3 + 1$, and detection is suppressed by phase mismatch.
background
The module sits inside the COS-010 ledger-shadow account of the phantom sector at temporal resolution. It rests on the eight-tick phase definition phase(k) = k π/4 for k : Fin 8, whose periodicity is 2π, and on the fundamental time quantum tick = 1 (denoted τ₀). Recognition events incur J-cost, defined as the derived cost of the multiplicative-recognizer comparator or equivalently Cost.Jcost of the event state. The local setting is the σ=0, Z≠0 phantom sector realized as odd-phase orbits of the eight-tick parity cycle, one of five projections of the same object.
proof idea
The definition is a direct literal list of five strings. No lemmas are invoked and no tactics are applied; the body simply enumerates the high-level assertions drawn from the eight-tick structure and J-cost mechanism.
why it matters in Recognition Science
This definition supplies the top-level statement for the COS-010 module and therefore anchors the ledger-shadow explanation of the observed 5:1 ratio. It directly instantiates the eight-tick octave (T7) and J-uniqueness (T5) inside cosmology, and the module document records consistency with a J(φ)-suppressed 1.79 GeV signal. The entry leaves open the unification across the five projections referenced in Cosmology.PhantomSectorStratification.
scope and limits
- Does not derive the numerical value of the ratio from the Recognition Composition Law.
- Does not specify the mass spectrum or particle content of the dark sector.
- Does not compute interaction cross-sections beyond the phase-mismatch statement.
- Does not address the spatial or consciousness-level projections of the same sector.
formal statement (Lean)
270def summary : List String := [
proof body
Definition body.
271 "8-tick gives visible and dark sectors",
272 "Dark matter = odd-phase ledger",
273 "Gravitates (J-cost) but doesn't shine",
274 "Ratio Ω_dm/Ω_b ≈ φ³+1 ≈ 5.2",
275 "Detection suppressed by phase mismatch"
276]
277
278/-! ## Falsification Criteria -/
279
280/-- The derivation would be falsified if:
281 1. Dark matter doesn't exist (MOND works fully)
282 2. DM ratio very different from φ³+1
283 3. DM couples strongly to photons -/