Layer 3 · the derivation · Galaxy rotation curves

Galaxy rotation curves — Branch-A Poisson → SPARC fit

Modified Poisson equation from F01 in the weak-field limit, regularised at the Newton-limit acceleration. Tested against 175 SPARC galaxies (F120).

Result — demonstrated, not derived

(1+f)-modified Poisson reproduces 175 SPARC galaxies with median RMS 13.8 km/s; on the Q=1 sub-sample (99 galaxies) ISST beats MOND.

The enhancement mechanism (the factor 4/3 from F01) is first-principles. The interpolation S(a) that smoothly connects the Newton limit to the ISST limit is phenomenological — two parameters fit, not derived.

F120 full SPARC 175: ISST median RMS 13.8 km/s · MOND 12.7 km/s · ISST wins on Q=1 sub-sample (99 galaxies, ISST 12.1 vs MOND 13.3)

Step 1

Start from F01 in the weak-field, static-source limit

nab l a^{2} P hi; =; 4 p i, G_{t e x t e f f}, t f r a c 43, (1 + f_{p}) (1 + f_{s}), r h o_{b}

Take the F01 action, expand the field equations to leading order in Φ/c² for a static, spherically symmetric source. The conservation identity for the Ψ-equation produces the factor 4/3 in front of (1+f) ρ_b. The correction has been confirmed unchanged by the F130b path-ζ framework.

Frameworks consulted

ALLOWED·F01 modified Poisson
DENIED·Standard Poisson[matter_coupling_universal=false]

Step 2

Read off circular velocity

V^{2} (R) = R, f r a c d P hi d R; =; V_{t e x t ba r}^{2} (R) c d o t f r a c 43, (1 + f_{p}) (1 + f_{s} (R))

For a galaxy whose baryonic mass distribution is known from photometry plus a stellar mass-to-light ratio, the standard prediction V_bar²(R) is enhanced by (4/3)(1+f_p)(1+f_s). With f_p = 5.664 (the universal primordial enhancement from A05/F59), the asymptotic flat-curve prediction is set; the radial profile of f_s(R) is what couples the local processing history of the stellar/gas content to the rotation curve.

Frameworks consulted

ALLOWED·Baryonic Tully-Fisher relation
WARNING·MOND interpolation[different mechanism, similar phenomenology]

Step 3

Newton-limit regulator (phenomenological)

V^{2} (R) = V_{t e x t ba r}^{2} (R), B i g 1 + bi g [E (R) - 1 bi g], S! bi g (ba r a (R) bi g) B i g

E (R) = t f r a c 43, (1 + f_{p}, (a_{0} / ba r a)^{q}), q u a d S (x) = f r a c 1 1 + (x / a_{t e x t cr i t})^{p}

To match the Newton-limit (no enhancement at high acceleration, e.g. inside the solar system) and the ISST limit (full enhancement at low acceleration, e.g. galaxy outskirts), the v8 formula introduces a sigmoidal regulator S(x) with a_crit ≈ cH₀/(2π) ≈ 1.0 × 10⁻¹⁰ m/s² (the only natural acceleration scale in the cosmology). Two parameters are fit to the SPARC ensemble: a₀ = 0.0075 × 10⁻¹⁰ m/s² (full 175 optimum) and q = 0.45. This is the honest gap: deriving a₀ and q from the underlying structural-index physics is the open problem F120 explicitly disclaims.

Frameworks consulted

WARNING·v8 phenomenological regulator[two free parameters: a_0, q]
DENIED·First-principles closure[open — full f_s(R, T) profile not yet derived]

Step 4

Apply to SPARC 175

t e x t R M S_{t e x t m e d ian}^{t e x t I S S T} = 13.8 t e x t k m / s, q u a d t e x t R M S_{t e x t m e d ian}^{t e x t M O N D} = 12.7 t e x t k m / s

On the full SPARC 175-galaxy benchmark, the v8 ISST formula is competitive with MOND: ISST wins on 86 of 175 galaxies; on the high-quality Q=1 sub-sample (99 galaxies), ISST gives 12.1 km/s vs MOND 13.3 km/s. On the 14-galaxy 2019 Lelli benchmark, ISST 10.2 vs MOND 12.4, with ISST winning 10/14. The Baryonic Tully-Fisher slope from this fit is 1.15 (literature: 1.0 ± 0.1).

Frameworks consulted

ALLOWED·SPARC photometric inputs
ALLOWED·Stellar mass-to-light fit

Step 5

What this denies, and what it doesn't

The rotation-curve fit does not require any non-baryonic mass component. The same (1+f) coupling that drives this fit is the one that the engine surfaces as denyingNFW-halo and WIMP frameworks. ISST and MOND are observationally close here but distinguishable on cluster scales (where ISST's structural-index dependence matters and MOND's pure acceleration scale doesn't).

Frameworks consulted

DENIED·NFW dark-matter halo[dark_matter_halo_ontology=false]
DENIED·WIMP cold dark matter[dark_matter_particles_exist=false]
DENIED·Warm dark matter[dark_matter_particles_exist=false]
WARNING·MOND (a₀ tuning)[phenomenologically similar — distinguishable on cluster scales]