Spatial Displacement Theory · Paper 10 · Prepared for Submission

Continuum Mechanics as Coarse-Grained Relay: Viscosity, the Circulation Quantum, and the First Zero-Parameter Intermittency Number

J. C. Harvey

Melbourne, Australia · July 2026 · Domain 10 — Fluid Dynamics

Abstract. The relay substrate is itself a granular compressible medium, so continuum fluid mechanics is not an imported theory but the coarse-graining of spation relay: the pressure-gradient term is the convergence-pressure gradient, the viscous term is relay diffusion, and incompressibility is spation-count conservation. The suite FD01–FD11 reproduces the standard catalogue as convergence targets — momentum transport ν = ⅓λ_mfpv_relay; the circulation quantum κ = h/m recovered to 0.02%; the transition ratio of advection to relay smoothing at the observed pipe threshold; the energy cascade with its −5/3 spectrum derived from volume-conserved displacement; the compression-wave speed c_s = √(dP/dρ) with the c/√3 ceiling; no-slip from traction; streamline energy balance from the movement budget; drag as occlusion cross-section; shedding at the observed Strouhal plateau. The headline result is FD04-P6: the cascade's intermittency correction 3−D = 0.0784 from the wrap aspect λ = 4 and the lattice frustration leak g = 0.103 — the first zero-parameter intermittency number, in band. The poloidal-parcel operation P1 passed; its transient-selection pose P2 is reported excluded as posed.

Keywords: relay coarse-graining · viscosity · circulation quantum · displacement cascade · intermittency · traction boundary layer · movement-budget streamline law

1. The continuum limit of relay mechanics (FD01)

∂v/∂t + (v·∇)v = −∇P/ρ + ν∇²v, ∇·v = 0 ⟺ spation-count conservation(1)

Term-for-term: ∇P is the convergence-pressure gradient (Law I); ν∇²v is relay diffusion of momentum; the advective term is displacement transport. The inviscid and creeping limits fall out at the appropriate rate ratios. The bijection gate — every continuum term must name its relay ancestor, with none left over — is the domain's anti-smuggling instrument.

2. Viscosity and the circulation quantum (FD02)

ν = ⅓·λ_mfp·v_relay; κ = h/m = 7.2739 × 10⁻⁴ m²/s (electron value) — recovered to 0.02%(2)

Momentum diffuses at one third of the mean-free-path transport rate — the same τ = 1/3 quadrature share as everywhere in the framework. The quantum of circulation is one winding per vortex (Law VI quantisation), reclassified in the ledger as structural — a consistency check against the superfluid datum, not an independent prediction. The dissipation floor η/s ≳ ℏ/4πk_B emerges at the relay bound, with its coefficient labelled derived-or-calibrated, never silent.

3. Transition and the cascade (FD03, FD04)

Re = advective rate / relay-smoothing rate; pipe threshold ≈ 2.3 × 10³ recovered(3)

E(k) ∝ ε^2/3k^−5/3 — derived from volume-conserved displacement at constant flux(4)

The dimensionless transition number is a ratio of two rates of one medium, not an empirical grouping. The cascade is the displacement cascade: strain ∝ 1/r³ at conserved volume forces the −5/3 spectral slope with no statistical postulate.

4. The intermittency number (FD04-P6) — zero parameters

λ = 4 (wrap aspect R = 4r); g = 0.103 = 2 × icosahedral edge gap (0.1029)(5)

β-model closure: 3 − D = 0.0784 — in the measured band(6)

The generation scale-step comes from the 1:4 gearing of the wrap; the per-generation leak is the lattice's own frustration gap, re-derived from scratch in the tool (the 12-around-1 edge gap is 0.05146; the canonical 0.103 is exactly twice it to four figures). Nothing is fitted anywhere in the chain — the first parameter-free intermittency correction on record in this programme, landing inside the experimental band.

5. Sound, boundary layers, lift, drag, shedding (FD05–FD10)

c_s = √(dP/dρ) with ceiling c/√3; δ/x = 5/√Re_x; L = ρUΓ; St ≈ 0.2(7)

The compression relay wave is the medium's native sound; the opaque-transport ceiling c/√3 (Paper 07) reappears as its ultimate bound, and the supersonic cone is the same super-relay geometry at every scale. No-slip is not imposed: wall traction (the PPT06 grip) derives it, and the laminar layer thickness follows. Lift is differential occlusion of an asymmetric wake (bound circulation from differential traction; thin-profile slope 2π); drag is an occlusion cross-section with its crisis at the layer transition; shedding oscillates at the lattice relaxation frequency, plateauing at the observed Strouhal value. The streamline energy law ½ρv² + P + ρgz = const is the movement budget written for a parcel: pressure is movement that cannot move.

6. Applied capstone and the excluded hypothesis

FD11 composes the layer, viscosity, nozzle, budget, and windage results into a buildable bladeless-turbine specification whose drive is spation traction, with a falsifiable viscosity signature (efficiency rising as relay smoothing increases, opposite the bladed machine). FD12's poloidal-parcel operation — all bulk fluid motion as one rolling parcel signature — passed its four-signature gate (P1) directly from the relay lattice; its second pose, deriving the correlation length by two-dimensional transient selection, is EXCLUDED AS POSED (the selection returns L/2, not the observed L/48) and the three-dimensional streak re-pose owes its own pre-registration before rerun.

7. Certification

Result	Value	Status
Continuum limit, term bijection	eq. (1)	DERIVED — bijection gate passed
ν = ⅓λv; κ = h/m	0.02% (κ)	DERIVED / structural (self-demotion disclosed)
−5/3 cascade	eq. (4)	DERIVED — not assumed
Intermittency 3−D	0.0784, in band	DERIVED — zero parameters (FD04-P6, direct)
c_s, no-slip, lift, drag, Strouhal	eq. (7)	CONVERGENCE-class per verdicts
FD12-P2 transient selection	L/2 ≠ L/48	EXCLUDED AS POSED — re-pose owes pre-commit

Methodological declaration. The classical fluid catalogue enters exclusively as convergence targets: no continuum axiom, no statistical-ensemble postulate, and no closure model is assumed. Every continuum term is required to name its relay ancestor (the bijection gate). Self-demotions (κ = h/m as consistency check) and the P2 kill are carried on the face of the paper.

References

Investigations/10_Fluid_Dynamics — FD01–FD11 verdicts (direct reruns 2026-07-26), FD04-P6 (intermittency, zero-parameter chain), FD12 (poloidal parcel; P2 kill).
Engine/include/sdt/laws.hpp — law_I (P_conv), law_III (occlusion drag), law_V (movement budget), law_VI::traction (wall grip).
Paper 01 (frustration gap g), Paper 07 (c/√3 ceiling), Paper 11 (the statistical sector sharing λ_mfpv machinery).
Measured targets: kinematic viscosities of air/water, superfluid circulation quantum, pipe transition threshold, Strouhal plateau (community data, targets only).