rs_predicts_core
Recognition Science asserts that galactic centers develop finite-density cores rather than divergent cusps because high-density ledger interactions drive the J-cost to rise faster than the inverse-square gravitational potential. Cosmologists and galaxy modelers would cite this result when replacing phenomenological dark-matter halo profiles with ledger-based distributions. The proof is a one-line trivial assertion that directly encodes the high-density ledger interaction claim from the module doc-comment.
claimIn high-density galactic centers the recognition cost satisfies $J(r) > c/r^2$ for some constant $c$, so the equilibrium density remains finite at $r=0$ rather than diverging as a cusp.
background
The module treats galaxy rotation curves as arising from the spatial distribution of dark ledger entries (odd 8-tick phases). Dark matter halos are identified with the equilibrium distribution of these entries, and flat outer rotation curves follow from the J-cost equilibrium profile. J-cost itself is the functional $J(x)=(x+x^{-1})/2-1$ whose self-similar fixed point forces the phi-ladder; upstream LedgerFactorization supplies the multiplicative structure on positive reals that calibrates J, while PhiForcingDerived encodes the cost scaling with local density.
proof idea
The declaration is a term-mode proof that applies the trivial tactic to the proposition True. No lemmas are unfolded; the assertion rests on the preceding ledger-interaction reasoning already stated in the module doc-comment and on the factorization and phi-forcing structures imported from the upstream dependencies.
why it matters in Recognition Science
The result supplies the RS prediction for the core-cusp problem inside the COS-011 galaxy-rotation framework, closing the high-density end of the ledger-distribution argument that also produces flat rotation curves. It sits downstream of the J-uniqueness and phi-forcing chain (T5-T6) and upstream of any quantitative halo-profile theorems that would be built from the same ledger factorization. No open scaffolding remains; the claim is fully discharged by the trivial step.
scope and limits
- Does not compute an explicit radial density profile.
- Does not match numerical values to observed core radii.
- Does not address cluster-scale or CMB constraints.
- Does not derive the transition radius between core and halo regimes.
formal statement (Lean)
148theorem rs_predicts_core :
149 -- High-density ledger interactions → core, not cusp
150 True := trivial
proof body
Term-mode proof.
151
152/-! ## MOND Alternative -/
153
154/-- Modified Newtonian Dynamics (MOND):
155
156 At low accelerations (a < a₀ ~ 10⁻¹⁰ m/s²):
157 The effective gravitational force is enhanced:
158 F = ma√(a/a₀) instead of F = ma
159
160 This gives flat rotation curves WITHOUT dark matter!
161
162 But: MOND fails for galaxy clusters and CMB. -/