Spatial Displacement Theory · Paper 13 · Prepared for Submission

Deterministic Relay-Wake Mechanics: Two Channels, the Residence Reading of the Squared Amplitude, and the Locality Audit

J. C. Harvey

Melbourne, Australia · July 2026 · Domain 13 — Quantum Foundations & Measurement

Abstract. There is no wavefunction in this framework. A particle is a hard, localised solid vortex occupying exactly one place at a time, dragging a convergence wake that spreads and self-overlaps. Two channels travel together: the vortex (which-path, mass, handedness) and the wake pulse (phase). Interference statistics, evanescent barrier transit, and discrete spectra are exhibited as deterministic relay mechanics of these two channels. Earned results: barrier transit is wake leakage — the field solver, never shown the standard approximation, reproduces the evanescent decay constant from its own lattice recursion to 0.13% worst-case, and the α-decay lifetime systematics follow at R² = 0.9997 with zero fits; the squared-amplitude law is derived, not postulated — time-averaging summed real travelling displacements yields the interference envelope with the square appearing nowhere in the code (residual 3.3 × 10⁻¹⁴), identifying intensity with stored elastic energy of the superposed displacement field; the residence reading (arrival statistics = dwell-time density of the sequential vortex) is owned by the rotating-substrate root and carries its pre-registered kill set. The correlation sector is the largest standing debt of the framework and is held OPEN: no local story is asserted for it.

Keywords: two channels · wake leakage · residence density · dwell time · relay bandwidth · locality audit

1. The two-channel ontology

solid vortex (one place at a time) ⊕ convergence wake (spreads, self-overlaps)(1)

The historically paradoxical phenomena are shadows of a rejected premise — a single entity asked to be both localised and spread. With two channels there is nothing to collapse: the vortex goes through one aperture; its wake goes through both and steers the vortex by wake pressure. Which-path detection perturbs the wake, and the fringe dies deterministically. The influx itself is a throughpole — it passes through every point and never terminates — so no source or sink bookkeeping is available to smuggle nonlocality through.

2. Barrier transit as wake leakage (QM03 — earned)

κ_meas = κ_lattice = 2·asinh(√(Ω_c² − ω²)/2) — to 0.13% worst case(2)

α-decay lifetime systematics: log t_½ vs Q^−½, R² = 0.9997 — zero fits(3)

The solver knows only a relay chain and a pinning rate raised inside a slab; the firewall held — the standard barrier approximation was never shown to the machinery, and the evanescent constant it measures equals the one its own lattice recursion demands. The wake leaks the occlusion barrier; the vortex follows the surviving branch. The Geiger–Nuttall regularity across the α emitters follows with nothing adjustable.

3. The squared amplitude, derived (QM04 — earned in part)

⟨(Σ_i A_icos(kr_i − ωt))²⟩_t ⟹ envelope·cos²(kΔr/2) — cos² nowhere in the code; RMS 3.3 × 10⁻¹⁴(4)

The code contains only real travelling carriers. Time-averaging the summed displacement squared produces the cross-term and the full fringe form: the "amplitude squared" is the stored elastic energy of a superposed displacement field — geometric, not axiomatic. What remains is the residence law: arrival statistics as the dwell-time density of the sequentially-positioned vortex (the rotating-substrate root, where the electron's "orbital" is a tour and the landing histogram is dwell time). That law is owned by the root programme with its pre-registered kill set F1–F6, and this paper claims only what the analytic run earned.

4. Bandwidth, spectra, handedness

Δx·Δp ≥ ℏ/2 — as the wake's joint bandwidth limit (Fourier-type, on the relay)(5)

The indeterminacy product is a bandwidth theorem of the wake, not a commutator postulate: a wake cannot be simultaneously narrow in position and in wavenumber beyond the relay's Fourier limit, and the minimum-product wake is the relay Gaussian. Discrete spectra are standing-wake boundary conditions on the vortex tour (the n-ladder of Paper 04, arrived at without a quantised-action postulate). Two-valued deflection in a swirl gradient is intrinsic handedness under differential occlusion — the two-spot outcome with the native ratio g = 2.0023 carried from the engine's benchmark — with sequential-splitter correlations a stated target of the programme.

5. The locality audit — held OPEN

The correlation sector (paired-particle statistics violating the classical bound up to 2√2) is the single largest standing debt in the framework's paradox census, and it is treated as such: the programme requires either a local relay-wake account that reaches the observed correlations, or an explicit naming of which throughpole feature is invoked and at what price. No local story is asserted here. Fabricating one would be worse than the debt.

6. Certification

Result	Value	Status
Evanescent constant, firewalled solver	0.13% worst case	EARNED — gate P1 PASS (QM03)
α-decay systematics	R² = 0.9997, zero fits	EARNED (QM03)
Squared amplitude = stored energy	RMS 3.3 × 10⁻¹⁴	EARNED — analytic portion (QM04); residence law DEFERRED to root programme
Bandwidth theorem	eq. (5)	DERIVED as Fourier limit of the wake
Discrete spectra via standing wakes	n-ladder	carried by Paper 04 (executed)
Handedness two-valuedness	g = 2.0023 native	ratio earned (B17); splitter correlations SPEC
Correlation sector	2√2 target	OPEN — largest standing debt, no local story asserted

Methodological declaration. No wavefunction, no state vector, no operator algebra, no collapse postulate, and no Born axiom enters any step: the squared-amplitude law is a derived energy statement, and where the full residence law is not yet earned it is deferred, not assumed. The one place the framework cannot currently reach the data locally (§5) is declared OPEN in preference to any manufactured mechanism. Firewalled solvers — machinery never shown the rival's formulas — are the domain's standard instrument against smuggling.

References

Investigations/13_Quantum_Foundations_and_Measurement — QM02 (bandwidth), QM03 (wake leakage; firewall protocol and results), QM04 (residence density; scope ruling), QM05 (locality audit, OPEN), QM06/QM07.
Investigations/01_Foundations_and_Lattice_Mechanics/FLM14 — the rotating-substrate root; dwell-time residence law; pre-registered kills F1–F6.
PARADOX_CENSUS.md — the debt ledger this domain answers to.
Paper 04 (spectra without quantised action), Paper 14 (handedness in swirl gradients).