Spatial Displacement Theory · Paper 08 · Prepared for Submission
Galactic Dynamics from Collective Displacement: The Derived Floor a₀ = cH₀/2π, Eclipse Saturation, and an Audited Ledger of Passes and Failures
Melbourne, Australia · July 2026 · Domain 08 — Galactic Dynamics
Abstract. Galactic rotation is treated with zero halo parameters. The acceleration floor is derived, a₀ = cH₀/2π ≈ 1.04 × 10⁻¹⁰ m/s²; a deliberately wrong floor degrades the observed radial-acceleration collapse sixfold, establishing load-bearing status. Mass at galactic scale is exhibited as collective spatial displacement — a pressure gradient of the substrate — of which the koppa point-source law is the dilute limit; the root relation zd = h(𝔇) carries zero parameters. On the rotation-curve database the audited record is mixed and reported in full: the canonical three-dimensional eclipse construction FAILS real photometry (RMS 66%; an earlier success traced to a circular mock and was voided); the twin-regime construction reaches 23.8% but imports its crossover shape; the drafting-crossover pilot re-earned 8.8% on direct rerun with a 6× control margin, its shape still borrowed and labelled so. A mass-free koppa-density regression spans 693 systems at 5.9%. The streak analysis removes a +35% halo-inference bias to +0.2%. Spiral structure follows from winding arithmetic: ~62 turns per 10 Gyr forces standing pressure waves with a spoke clock of 162 Myr.
Keywords: acceleration floor · eclipse saturation · collective displacement · rotation-curve audit · winding problem · standing pressure wave
1. The derived floor
The floor is the residual convergence-depth deficit that interstellar space cannot shed — the relay ceiling's imprint (galactic depth floor zgal ≈ 3.5 × 10⁻⁷; absolute ceiling c∞ = c/(1−zgal) ≈ c + 105 m/s). Its load-bearing status is tested, not asserted: substituting a deliberately wrong floor degrades the radial-acceleration collapse by a factor 6. The floor's numerical proximity to the phenomenological value used by the modified-inertia rival is a convergence of independent origins, disclosed; the rival's interpolation shape is not derived here, and every use of a borrowed shape below is labelled.
2. Mass as collective displacement
At galactic surface densities the displacement fields of ~10¹¹ sources cease to superpose as isolated shadows: the medium carries one collective pressure gradient. The koppa law of Paper 06 is recovered exactly in the dilute limit; the collective regime is governed by the root relation (2) with no fitted constant (GD06). Eclipse saturation is the geometric statement: when integrated disk occlusion reaches its critical fraction, the shadow stops deepening with radius and v(r) = c√(ϟ/r) flattens onto the floor (1).
3. The audited rotation-curve ledger — passes and failures together
| Construction | Dataset | Result | Status |
|---|---|---|---|
| Canonical 3-D eclipse | real SPARC photometry | RMS 66% | FAILED — prior "success" was a circular mock, VOIDED |
| Twin-regime (bulge ⊕ disk), M4 convergence floor | SPARC | RMS 23.8% | PASS-with-disclosure — crossover shape borrowed from the rival's simple form |
| Drafting crossover (pilot) | real SPARC pilot | 8.8% (direct rerun); pilot 23.7%; BTFR slope 3.58; control 6× | RE-EARNED 2026-07-23 — shape still borrowed, labelled |
| Mass-free koppa-density regression | 693 systems | 5.9% | PASS — no mass, no G (GOM14) |
| Streak de-biasing of halo inference | SPARC | +35% → +0.2% | PASS — bias identified and removed |
The fit problem for full rotation curves is, at present, ill-posed in this framework's own terms (the GD05 finding), and that is stated rather than papered over. What survives audit: the floor (1), the collective-displacement root (2), the mass-free regression, and the de-biasing result. What does not survive is listed above with its kill.
4. The winding arithmetic and spiral structure
Differential rotation would wind material arms ~62 times in 10 Gyr; observed grand-design structure therefore cannot be material. In the collective-displacement medium the arms are standing pressure waves — interference spokes of the disk's own displacement field — with a natural clock of 162 Myr per spoke cycle. No density-wave amplitude is fitted; the clock follows from the same kinematics as (2).
5. Certification
| Result | Value | Status |
|---|---|---|
| a₀ = cH₀/2π | 1.04 × 10⁻¹⁰ m/s² | DERIVED, C-flagged; wrong-floor control ×6 |
| zd = h(𝔇) | 0 parameters | DERIVED root (GD06) |
| Rotation-curve constructions | see ledger §3 | MIXED — kills and disclosures carried in full |
| Koppa-density regression | 693 systems, 5.9% | PASS — mass-free |
| Spiral spoke clock | 162 Myr | DERIVED from winding arithmetic |
Methodological declaration. No dark-matter halo, no halo profile, no mass-to-light fit, and no modified-inertia axiom enters any derivation. Where a construction currently borrows the rival's interpolation shape, the borrowing is labelled in place and treated as debt, not achievement. The photometric database enters as the measured target; a prior mock-circular validation is reported as voided. Alignment of the floor's magnitude with the rival's phenomenological constant is treated as convergence of independent origins, not derivation from the rival.
References
- Engine/include/sdt/laws.hpp — depth_closure (a0_floor with provenance block; z_galactic_floor; c_infinity).
- Investigations/08_Galactic_Dynamics — GD05 (audit, streak, twin-regime), GD06 (collective displacement root), APS03 (drafting crossover, direct rerun 2026-07-23), GOM14 (koppa-density regression).
- SPARC photometry and rotation-curve database (measured target).
- Paper 06 (koppa kinematics — the dilute limit), Paper 07 (H₀ provenance).