ledgerShadowProperties
Ledger shadows in Recognition Science cosmology are characterized by gravitational coupling through J-cost, electromagnetic invisibility, weak self-interaction, and cold phase-locking to the ledger. Cosmologists modeling structure formation and CMB spectra would cite this list to connect RS mechanisms to observed dark matter dominance. The definition is a direct enumeration grounded in the 8-tick phase structure and cost functions.
claimThe properties of ledger shadows are the list: gravitational coupling of J-cost to geometry, absence of electromagnetic interaction, weak self-interaction satisfying σ/m < 1 cm²/g, and cold phase-locking to the ledger.
background
In the Cosmology.DarkMatter module, dark matter is the σ=0, Z≠0 phantom sector viewed at temporal resolution as odd-phase orbits of the 8-tick parity cycle. J-cost is the recognition cost J(x) = (x + x^{-1})/2 - 1 that couples ledger entries to geometry. The 8-tick phase is defined as kπ/4 for k = 0 to 7 and is periodic with period 2π. Upstream results include the phase definition from EightTick, which supplies the parity cycle, and the cost definition from ObserverForcing, which states that the cost of any recognition event is its J-cost.
proof idea
This is a direct definition that enumerates the four properties as a List String. It draws on the phase definition from EightTick and the cost definitions from MultiplicativeRecognizerL4 and ObserverForcing to justify the descriptions of gravitating, non-luminous, collisionless, and cold behavior.
why it matters in Recognition Science
This definition supports the COS-010 view of dark matter as non-luminous ledger configurations, aligning with the eight-tick octave and the forcing chain steps that fix D = 3. It feeds explanations of galaxy rotation curves, cluster dynamics, and the Ω_dm/Ω_b ratio. The module notes consistency with J(φ)-suppressed signals at 1.79 GeV while leaving open the full unification across projections in PhantomSectorStratification.
scope and limits
- Does not derive the numerical value of Ω_dm ≈ 0.27.
- Does not prove the phase-mismatch suppression for null detections.
- Does not specify mass formulas or candidate spectra.
- Does not address unification with other phantom projections.
formal statement (Lean)
187def ledgerShadowProperties : List String := [
proof body
Definition body.
188 "Gravitates (J-cost → geometry)",
189 "No electromagnetic interaction",
190 "Weak self-interaction (collisionless)",
191 "Cold (phase-locked to ledger)"
192]
193
194/-- Self-interaction of dark matter:
195
196 Ledger shadows can interact with each other.
197 But cross-section is small: σ/m < 1 cm²/g (cluster limits).
198
199 In RS: Odd-phase × odd-phase → even-phase is suppressed. -/