Spatial Displacement Theory · Paper 06 · Prepared for Submission
Koppa Kinematics: Gravitation and Orbital Mechanics from the Depth Field z = ϟ/r, with Neither G nor Weighed Mass
Melbourne, Australia · July 2026 · Domain 06 — Gravitation & Orbital Mechanics
Abstract. The entire gravitational sector is carried by one velocity-defined length, the koppa ϟ = R/k² = v²R/c² (k = c/v), and one depth field z = ϟ/r with one bound-motion law v = c√z. Neither the gravitational constant nor a weighed mass appears anywhere in the formalism. The Depth–Closure Theorem identifies local closure, local relay speed, local clock rate, and the gravitational spectral shift as one quantity. Results: solar spectral shift to 0.03%; accumulated-depth delay to 0.0000% in the standard log form; limb deflection 1.750″ vs 1.751″; the anomalous perihelion closure returns k☉ = 686.4 from orbital observables alone, whence c = k·vsurf = 299,795,136 m/s (+0.0009%) — c as an output; the inspiral chirp reads as a length, fisco = c/(π·63/2·ϟtot) = 67.65 Hz vs 68.0 measured for the first observed merger event; and one two-body exchange ledger spans the Sun–Jupiter pair and the hydrogen seat — twenty orders of magnitude — at 0.4%, with kseat(H) = 137.0726 recovered at 1.2 ppm. Orbits are closed least-resistance paths in the depth gradient; nothing pulls.
Keywords: koppa · depth closure · bound-motion law · perihelion closure · accumulated-depth delay · koppa-metry of inspirals · baryon census
1. The koppa and the depth field
One number in metres per body: ϟEarth = 4.43 mm, ϟ☉ = 1477 m, ϟH = α²a₀ = re. The numerical coincidence with the rival's composite GM/c² is a consequence of the closure, never an input. The convergent flux pushes all bodies; a closed least-resistance path through the radial depth gradient is an orbit — no attraction is posited at any point.
2. The Depth–Closure Theorem
Local closure, local relay speed, local clock rate, and the spectral shift are one quantity. Solar surface: z = ϟ☉/R☉ = 2.123 × 10⁻⁶, matching the measured shift to 0.03%. The velocity law v = c√z spans Earth's surface to the proton interior (1.830c at the sub-boundary seat) on one line. Proven content: C1 (shift = depth) and C2 (one law across ~15 orders); the absolute closure ℓP,∞ and the internal radii carry the Seed-Theorem caveats of Paper 01, stated, not hidden.
3. Accumulated depth: delay and bending
The delay is accumulated depth along the ray — the local light speed never drops below its local c; nothing slows "in a well." Bending is the transverse gradient of the same field: optics and gravitation are one gradient-index calculation (Paper 15).
4. c as an output: the perihelion closure (GOM04)
The anomalous precession 42.98″/cy inverts to the speed ratio k with no c and no mass anywhere in the inversion; multiplying by the surface velocity reconstructs c from geometry alone. The precession benchmark itself (E24) returns 43.0″/cy vs 43.1 ± 0.1 observed. Disclosure: vsurf is itself an orbital inference; the caveat is carried in the ledger.
5. Koppa-metry of inspirals (GOM06)
For the first observed merger (ϟtot ≈ 96 km) the chirp scale is a length, with no G and no mass in any dynamical relation; read inversely, the interferometer is a koppa-meter. Limitation: applied to the rival's ISCO expression this is an algebraic identity of the bridge (disclosed); the SDT content is conceptual — the chirp scale is a length — plus a divergent prediction: no breathing mode, and post-merger echoes at Δt ≈ 58 ms where the rival predicts none. The 96/5 radiation coefficient remains CONVERGENCE-PENDING.
6. One ledger, twenty orders: the exchange test (GOM17)
The two-body exchange ledger — the same kinematic bookkeeping with no gravitational constant — closes on the Sun–Jupiter pair and on the hydrogen seat simultaneously: the planetary ledger matches the atomic Rydberg ledger to 0.4%, and the hydrogen seat ratio is recovered as kseat(H) = 137.0726, i.e. 1/α to 1.2 ppm, from orbital-form input. All gates passed on direct rerun. The same formulas govern the planets and the particles: v²r = c²ϟ holds from the proton to the black hole at the Galactic centre, and no rival framework owns that single line. The baryon census follows without kilograms:
Disclosure: the conventional tabulation of ℓP encodes the rival's constant by construction; this is stated in the ledger rather than laundered.
7. Certification
| Result | Value | Status |
|---|---|---|
| Solar spectral shift = depth | 0.03% | DERIVED — class C (C1 proven) |
| Accumulated-depth delay | 0.0000% | DERIVED — log form recovered |
| Limb deflection | 1.750″ / 1.751″ | DERIVED — gradient of z |
| Perihelion closure → c | +0.0009% | DERIVED — class C; c an output |
| fisco from ϟtot | 67.65 / 68.0 Hz | class C — identity disclosed; echoes Δt ≈ 58 ms = divergent prediction |
| Exchange ledger, 20 orders | 0.4%; kseat(H) 1.2 ppm | ALL GATES PASS (GOM17, direct rerun) |
| Baryon census (9) | no G, no kg | C-flagged — ℓP convention disclosed |
Methodological declaration. No gravitational constant, no weighed mass, no curved metric, and no geodesic postulate enters any derivation. Every observable is computed from {v, R, r} and the whitelist. Where an expression coincides algebraically with the rival's (eqs. 4, 8), the identity is disclosed and the divergent predictions (echo timing, no breathing mode, mass-independent drift of E21) are named as the discriminants.
References
- Engine/include/sdt/laws.hpp — namespaces bridge (koppa, k-hierarchy, GOM04 closure, GOM06 koppa-metry), depth_closure (Depth–Closure Theorem, Shapiro form, a₀ floor), law_V (T16–T17).
- Investigations/06_Gravitation_and_Orbital_Mechanics — GOM01/02/04/05/06, GOM15–17 (exchange ledger).
- Papers/Depth_Closure_Theorem — keystone derivation and numbers.
- IAU 2015 nominal solar values; planetary ephemerides (measured observables).
- Published strain data of the first binary merger event (measured anchor for eq. 8).