Spatial Displacement Theory · Paper 03 · Prepared for Submission
Handed Occlusion Without a Charge Substance: The Force Law F = (π/4)PeffR₁²R₂²/r² and the Provenance of Its Coefficient
Melbourne, Australia · July 2026 · Domain 03 — Electromagnetism & Charge
Abstract. Electrostatics is exhibited as occlusion of convergent relay throughput with handedness. There is no charge substance: "+e" and "−e" name the two handednesses of the redirection a vortex imposes on the convergent phase, and e itself is a unit bridge. Attraction is a shadow deficit; repulsion is the exact inverse — a focused surplus. The universal force law F = (π/4)·Peff·R₁²R₂²/r² carries the Coulomb interaction with Rcharge = √(Rpre) = 1.5396 × 10⁻¹⁵ m, reproducing the hydrogen force datum to 0.12%. We give the coefficient's provenance in full: Peff = 4kee²/(πRp²re²) = 5.225 × 10³¹ Pa is CALIBRATED at the hydrogen anchor (class E) in the engine ledger, while the electropause closure Peff = mp²me²c⁵/(4παℏ³) supplies a closed whitelist form with α the single remaining input. The identity kee² = αℏc is a definitional tautology (class F) and is claimed as nothing more.
Keywords: occlusion · handedness · unit bridge e · effective pressure · transfer function · electropause
1. Force as occluded convergence (Theorems T3–T4)
The 1/r² structure and the R₁²R₂² product are derived from shadow geometry (class C; they pass the delete-test). The occluded solid angle of a boundary R at range r is Ω = πR²/r². Nothing pulls: the force is the unbalanced antipodal remainder of an isotropic flux.
2. Handedness replaces charge
A circulating vortex does not merely block the convergent phase — it redirects it with a handedness fixed by its circulation sense. Two structures of opposite handedness deepen each other's deficit (net push together); identical handedness focuses a surplus between them (net push apart). Repulsion is thus the exact inverse of convergence-deficit attraction, not a separate mechanism (EMC03). Gravity is the scalar, unhanded limit of the same occlusion — one sign only — which is why it is universally attractive while the handed channel carries two signs.
The elementary quantity e is a unit bridge, not a substance count: EMC02 (INCONCLUSIVE, disclosed) established that the whitelist {ℓP, c, ℏ, kB, TCMB, d = 3} carries no current dimension, so no algebraic combination yields coulombs; e stays an irreducible measured input.
3. The interaction radius and the hydrogen check
Evaluated at the hydrogen seat: 8.23 × 10⁻⁸ N computed against 8.24 × 10⁻⁸ N measured (0.12%). This match is the calibration anchor of the coefficient, not an independent prediction, and is labelled so. The geometric mean in (3) resolves why e–e, p–p, and e–p couplings share one strength: the handed force signature is a count product times one universal unit area, (Z₁Z₂)·(Rp²re²). For a Z-nucleus acting on an electron the occluded area scales with Z, so the effective radius is √Z·Rp, giving F = Z·kee²/r² identically — the measured scaling; the superseded R = Z·Rp form, which gave F ∝ Z², is retracted in the canon and recorded here.
4. Provenance of Peff — stated in full
Equation (5) fails the delete-test — the hydrogen calibration fixes the scale — and the engine keeps the label. Equation (6), obtained from the electropause force balance (Paper 16), expresses the same magnitude from {mp, me, c, α, ℏ} alone, with α the one remaining empirical anchor. The two values' coexistence is recorded as an open provenance question, not silently resolved. The transfer function
maps cosmological convergence pressure to interaction pressure and inherits the class-E status; its upgrade to class C is contingent on the CR05 closure of RCMB from the pressure tensor, itself pending the GOM05 structure derivation.
5. The coupling identity, disclosed as identity
Since α ≡ kee²/ℏc by definition, (8) is a tautology and is claimed as nothing more. The physically load-bearing object is the product kee²; neither factor is separately fundamental. A non-definitional SDT path to (8) would require deriving α from W = 1 mode structure — an open problem (the wake ratio g(1) ≡ α is likewise definitional; PPT02 located α as the koppa of the hydrogen ground state, which still consumes e).
6. Certification
| Result | Value | Status |
|---|---|---|
| Force-law structure (1/r², R₁²R₂²) | eq. (1) | DERIVED — class C |
| Peff (engine) | 5.225 × 10³¹ Pa | CALIBRATED — class E, fails delete-test (disclosed) |
| Peff (electropause closure) | mp²me²c⁵/(4παℏ³) | closed whitelist form; provenance external to engine — coexistence OPEN |
| Rcharge = √(Rpre) | 1.5396 fm | class E — composed of measured radii |
| Hydrogen force datum | 0.12% | CALIBRATION ANCHOR — not a prediction |
| kee² = αℏc | identity | class F — tautology, disclosed |
| √Z nuclear–electron scaling | F = Z·kee²/r² | DERIVED (corrected canon; Z² form retracted) |
| e derivable from whitelist? | — | EMC02 INCONCLUSIVE — e stays an irreducible input |
Methodological declaration. No field is taken as primitive, no charge substance is posited, no gauge structure and no photon-particle enters. The measured Coulomb datum appears once, as the disclosed calibration anchor of Peff. All identities that merely restate a definition (eq. 8; g(1) ≡ α) are labelled class F and carry no evidential weight in this framework's favour.
References
- Engine/include/sdt/laws.hpp — namespaces law_III (P_eff, R_charge, F_occlusion, F_nuclear_electron), coulomb_identity (class-F disclosure in situ).
- Investigations/03_Electromagnetism_and_Charge — EMC02 (charge quantisation, INCONCLUSIVE), EMC03 (repulsion as inverse of convergence), EMC04 (electropause derivation).
- Theory/00_Ruleset.md — prohibition register (no charge substance; no fields as primitives).
- CODATA 2018 (ke, e, α, re; measured anchors).
- Paper 14 of this series (collective circulation: the ℓ = 2 sector and induction programme).