Algorithm boxes (complete forms)

Notation: F is the discrete Fourier transform; · a Fourier coefficient; the 2/3 rule zeroes the top third of wavenumbers (dealiasing). These are the complete pseudocode forms of the in-body algorithm boxes. Each is written so that a reader, or an AI, can re-implement the result directly from this appendix—without opening the bundle—and reach the stated tolerance. [LOCK]

Notation: F is the discrete Fourier transform; · a Fourier coefficient; the 2/3 rule zeroes the top third of wavenumbers (dealiasing).

These are the complete pseudocode forms of the in-body algorithm boxes. Each is written so that a reader, or an AI, can re-implement the result directly from this appendix—without opening the bundle—and reach the stated tolerance. Notation: $\mathcal F$ is the discrete Fourier transform; $\hat\cdot$ a Fourier coefficient; the 2/3 rule zeroes the top third of wavenumbers (dealiasing).

Pillar I — 2D Euler/Navier–Stokes invariants

Input: grid , viscosity $\nu$ , band-limited initial vorticity $\omega_0$ , time , domain $L=2\pi$ .\\ Steps:

Build wavenumbers ; ; inverse $k^{-2}$ (zero at $\mathbf k{=}0$ ); dealias mask $|k_{x,y}|\le\frac23\max|k|$ .
Stream function $\hat\psi=\hat\omega/k^2$ ; velocity $u=\mathcal F^{-1}(ik_y\hat\psi)$ , $v=\mathcal F^{-1}(-ik_x\hat\psi)$ .
Nonlinear term $N(\hat\omega)=-\mathcal F\!\big(u\,\omega_x+v\,\omega_y\big)\cdot\text{mask}$ .
Time step by integrating-factor RK4 using the exact viscous factor $e^{-\nu k^2\,dt}$ (so the linear part is integrated exactly).
Each step record $E=\tfrac12\!\int|u|^2$ and $Z=\tfrac12\!\int\omega^2$ .

Output: series E(t),Z(t)

; for $\nu{=}0$ the drifts $dE/E_0,\,dZ/Z_0$ ; for $\nu{>}0$ the ratio $|dE/dt+2\nu Z|/(2\nu Z)$ .\\ pass: $\nu{=}0\Rightarrow|dE/E_0|,|dZ/Z_0|\le10^{-5}$ ; $\nu{>}0\Rightarrow$ budget ratio $\le10^{-4}$ and Z(t)

monotone non-increasing.

Pillar II — the RCCI capacity identity

Input: published audit samples, each a tuple $(N,r,\varphi,s,L)$ .\\ Steps:

For each sample compute the predicted count from the identity $N_\star\,r^2=\varphi\,s\,L^2$ , i.e. $N_\star=\varphi s L^2/r^2$ .
Residual $\Delta=|N-N_\star|$ (and the analogous residuals of any derived metric reported in the audit).

Output: $\max_{\text{samples}}\Delta$ and the metric residuals.\\ pass: identity reproduced to $\le10^{-15}$ (machine precision); derived-metric residuals $\le2.83\times10^{-12}$ .

Pillar III — length selection and the one-half law

Input: dispersion offsets $\mu$ , curvature $\varepsilon$ , hyper-diffusion $\sigma$ over a range (linear part), and a Swift–Hohenberg field (nonlinear part).\\ Steps:

Linear growth rate $\lambda(k)=\mu+\varepsilon k^2-\sigma k^4$ .
Maximize: $\lambda'(k)=0\Rightarrow k_\star=\sqrt{\varepsilon/2\sigma}$ , independent of $\mu$ ; selected length $L_\star=2\pi/k_\star$ .
Sweep $\sigma/\varepsilon$ ; fit slope of $\log L_\star$ vs $\log(\sigma/\varepsilon)$ .
(Nonlinear) integrate Swift–Hohenberg to a steady pattern; measure the dominant wavenumber.

Output: $k_\star(\sigma/\varepsilon)$ and the fitted exponent $\hat\beta$ .\\ pass: $k_\star$ independent of $\mu$ ; $\hat\beta\to\frac12$ ( $0.46\!\to\!0.50$ refining the grid); the nonlinear pattern locks at $k_\star$ (matching DNS $\hat\beta\approx0.501$ ).

Pillar IV — budget closure and Onsager saturation

Input: injection , viscous coefficient $a\nu$ , binding coefficient , range of $\nu$ .\\ Steps:

Evolve $dE/dt=I-a\nu E-bE$ to steady state $E_\star=I/(a\nu+b)$ .
Split dissipation: viscous $\eps_\nu=a\nu E_\star$ , binding $\eps_{\rm bind}=bE_\star$ .
Check closure $\eps_\nu+\eps_{\rm bind}=I$ ; track $\eps_{\rm bind}/I$ as $\nu\to0$ .

Output: budget residual $|{\eps_\nu+\eps_{\rm bind}-I}|$ ; ratio $\eps_{\rm bind}/I$ vs $\nu$ .\\ pass: residual $\approx0$ for all $\nu$ ; $\eps_{\rm bind}/I\to0.998$ as $\nu\to0$ (anomaly). A naive contact-merger closure leaves a residual $\approx0.49$ (the gate).

Pillar IV — events carry the flux (Burgers Duchon–Robert)

Input: viscosity $\nu$ , grid , time ; decaying Burgers from $u=\sin x$ .\\ Steps:

Integrate spectral Burgers (IF-RK4, dealiasing) until a shock forms.
Locate the shock (steepest descent, $\arg\max(-u_x)$ ); measure the jump $\Delta u$ across it.
Viscous dissipation $\int\eps_\nu=\int\nu u_x^2$ .
Duchon–Robert flux density $\Pi_\ell(x)=\tfrac14\!\int \phi_\ell'(r)\,[u(x{+}r){-}u(x)]^3\,dr$ with a Gaussian filter $\phi_\ell$ , at an inertial scale $\ell\!\sim\!8\,dx$ .
Concentration: fraction of $\int|\Pi_\ell|$ within $\pm0.15$ of the shock.

Output: $\int\eps_\nu$ , the event flux $(\Delta u)^3/12$ , the concentration fraction.\\ pass: $\int\eps_\nu\approx(\Delta u)^3/12$ (to $\sim10\%$ ), $\nu$ -independent, with $\sim100\%$ of the flux at the shock.

Pillar IV — 2D NS flux concentrates on events (multid_flux.py)

Input: grid , viscosity $\nu$ , time , filter scale $\ell$ ; band-limited random vorticity (energy near $k\!\sim\!8$ ).\\ Steps:

Evolve 2D NS (vorticity, IF-RK4, dealiasing) to a filamented snapshot.
Gaussian-filter at $\ell$ : $\hat{\bar\omega}=\hat\omega\,e^{-k^2\ell^2/2}$ ; recover $\bar u,\bar v,\bar\omega$ via the stream function.
Subfilter vorticity flux $\sigma_j=\overline{u_j\omega}-\bar u_j\,\bar\omega$ (filter the product, subtract the product of filtered fields).
Local enstrophy flux $\Pi=-(\sigma_x\partial_x\bar\omega+\sigma_y \partial_y\bar\omega)$ ; palinstrophy density $g^2=|\nabla\bar\omega|^2$ .
Sort cells by (events first); cumulate the positive flux $\Pi_+$ ; read the top- $20\%$ fraction, the area carrying half, and $\mathrm{corr}(\Pi,g^2)$ . Repeat for at fixed physical $\ell$ .

Output: top- $20\%$ forward-flux fraction; half-flux area fraction; $\mathrm{corr}(\Pi,g^2)$ ; their stability in

.\\ pass: top- $20\%\approx60\%$ , half-flux area $\approx12$ – $16\%$ , $\mathrm{corr}\approx0.28$ , stable across

— the events carry the inter-scale flux to the small scale.

Pillar IV — 3D NS energy flux concentrates on strain events (ns3d.py)

Input: grid , viscosity $\nu$ , step , forcing energy E_f , filter $\ell$ ; random divergence-free IC.\\ Steps:

Evolve 3D NS (velocity/rotational form, IF-RK4, dealias), holding the energy in $|k|\!\in\![1,2]$ at (low- forcing), until developed: the velocity-derivative skewness $\langle(\partial_x u)^3\rangle/\langle(\partial_x u)^2\rangle^{3/2}\to\approx-0.5$ .
Gaussian-filter at $\ell$ ; form the subfilter stress $\tau_{ij}=\overline{u_iu_j}-\bar u_i\bar u_j$ and the filtered strain $\bar S_{ij}=\tfrac12(\partial_i\bar u_j+\partial_j\bar u_i)$ .
Local energy flux $\Pi=-\sum_{ij}\tau_{ij}\bar S_{ij}$ ; strain density $|\bar S|^2=\sum_{ij}\bar S_{ij}^2$ .
Sort cells by $|\bar S|^2$ (events first); cumulate the positive flux; read the forward fraction, the top- $20\%$ share, the half-flux volume, and $\mathrm{corr}(\Pi,|\bar S|^2)$ . Repeat for at fixed physical $\ell$ .

Output: forward fraction; $\langle\Pi\rangle/\eps$ ; top- $20\%$ share; half-flux volume; correlation; convergence in

.\\ pass: inviscid energy conserved to $\sim10^{-16}$ (solver check); $\langle\Pi\rangle>0$ ,

– $95\%$ forward; top- $20\%\approx60$ – $66\%$ , half-flux volume $\approx10$ – $14\%$ , $\mathrm{corr}\approx0.80$ , converged across

Event RG — the $\Pi_L$ plateau (event_rg.py)

Input: exponents $\alpha,\psi,\chi$ ; coarse-graining factor $\lambda$ ; steps; initial $(L,\eps_{\rm bind},\sigma_{\rm eff})=(1,1,1)$ .\\ Steps:

Each step record $\Pi_L=L^{1+\alpha}\eps_{\rm bind}/\sigma_{\rm eff}$ .
Coarse-grain: $L\leftarrow\lambda L$ , $\eps_{\rm bind}\leftarrow\lambda^{-\chi}\eps_{\rm bind}$ , $\sigma_{\rm eff}\leftarrow\lambda^{\psi}\sigma_{\rm eff}$ .
Gap $g=(1+\alpha)-\psi-\chi$ .

Output: the $\Pi_L$ sequence and

.\\ pass: $g=0\Rightarrow\Pi_L$ constant (plateau, fixed point); $g>0\Rightarrow\Pi_L$ grows; $g<0\Rightarrow\Pi_L$ decays. Measured: $1.0\to1.0$ , $1.0\to7.5$ , $1.0\to0.18$ .

Mechanism M — forced co-rotation, merger, pumping (corotation.py)

(M1) Frustration $\to$ co-rotation. Build the contact graph of a square and a triangular lattice; greedy two-colour it (counter-rotation attempt); count frustrated edges.\\ pass: square (bipartite $\Rightarrow$ counter-rotate, net ); triangular =48 (not bipartite $\Rightarrow$ co-rotate forced).\\ (M2) Co-rotation $\to$ larger rotation. 2D Navier–Stokes; two strongly overlapping Gaussian vortices, same or opposite sign; integrate; count connected strong-vorticity blobs ( $|\omega|>0.4\max$ ).\\ pass: same-sign $\to1$ (merged); opposite $\to2$ .\\ (M3) Rotation $\to$ through-flow. Ekman spiral $W(z)=W_\infty(1-e^{-(1+i)z/\delta})$ , $\delta=\sqrt{\nu/\Omega}$ ; radial inflow transport $M=\int(W_\infty-\mathrm{Re}\,W)\,dz$ ; fit $\log M$ vs $\log\Omega$ .\\ pass: slope -1/2 ( $M\propto\sqrt{\nu/\Omega}$ ), measured -0.500 .

G-Q — the 81{+}1 lattice-with-inflow (lattice_inflow.py)

Input: 3D lattice $\#\{R^2\le6\}=81$ co-rotating cells nozzle; planar disk slice $\{x^2+y^2\le6\}$ for the Biot–Savart test; rotation $\Omega$ , $\nu$ .\\ Steps:

Net circulation around a loop enclosing the disk cells, for co-rotating signs and for an alternating control.
: net radial flux of a regular divergence-free field through an enclosing sphere (Gauss regularity, no external source).
: axial through-flux $Q_z=2\pi R\,M$ fed by the Ekman radial inflow (open rotation axis).

Output: circulations; $Q_{3D}$ ; $Q_{2D}$ .\\ pass: co-rotating circulation adds ( =21

) vs control ( $\approx0$ ); $Q_{3D}=0$ (nozzle closes); $Q_{2D}>0$ (sustained). The dimensional split matches E3, so the falsification test passes.

Universality U1–U3 (universality.py)

(U1) Burgers anomalous dissipation. Integrate Burgers at several $\nu$ ; record $\int\eps_\nu$ . pass: $\int\eps_\nu$ saturates ( $\nu$ -indep, $\sim6\%$ spread).\\ (U2) Capacity identity in dimension . For d=2,3,4,5 form $N v_{\rm eff}$ and $\phi V_{\rm core}$ from a packed core. pass: $|Nv_{\rm eff}-\phi V_{\rm core}|\le10^{-16}$ (identity, dimension-general).\\ (U3) Reaction–diffusion selection. Two-species dispersion; locate the fastest-growing mode $k_\star$ . pass: $k_\star$ matches the extremum-selection prediction to – $8\%$ .

Extensibility E1–E4 (extensibility.py, rigid_shell.py)

(E1) Genome fluid. Condensate identity $N v_{\rm eff}=\phi V_{\rm core}$ in d{=}3 ; droplet preferred size at jamming $\phi^\ast\approx0.64$ . pass: residual .\\ (E2) Droplet length selection. Rayleigh–Plateau dispersion of a liquid column; maximize growth. pass: $k_\star a=0.697$ , $\lambda_\star=9.01\,a$ .\\ (E3) Plane breathes, sphere closes. Planar continuity $u_r=-Ar\Rightarrow w=2Az$ (axial outflow); spherical $u_r\propto1/R^2$ (no axis). pass: 2D sustains, 3D closes.\\ (E4) Rigid shell. $\#\{R^2\le6\}=81=3^4$ , R^2=7 empty (Legendre), so 82=81{+}1 ; -fold rate $\nu_n=n\,\pi^{2(n-1)}$ ; quantum length triangulated three ways. pass: spread $0.039\%$ ; relative spread of $L_\star$ narrows $3.4\%{\to}1.8\%$ as coordination grows.

Marginal substrate — relaxed moduli and the fluidity intercept (marginal_fluidity.py)

setup: D bidisperse ( 50{:}50 , ratio 1.4 ) harmonic spheres, $V(r)=\tfrac12(1-r/\sigma_{ij})^2$ for $r<\sigma_{ij}$ , periodic box V{=}1 ; $N\in\{128,256\}$ , $\phi\in[0.650,0.720]$ ( values) $\times$ seeds.\\ minimize: L-BFGS $\to$ FIRE $\to$ L-BFGS to residual force $<10^{-7}$ . backbone: iteratively remove rattlers (contacts <d{+}1 ); $z=2N_c/N_{\rm bb}$ .\\ moduli (analytic, FD-verified): affine (Born) shear $G_{\rm Born}=\sum_b\,[\,V''\,(n_xn_y r)^2+V'\,r\,n_y^2(1-n_x^2)\,]/V$ ; bulk $B=\sum_b V'' r^2/(d^2V)+\tfrac{d-1}{d}P$ with virial pressure ; mismatch $\Xi=-\partial^2E/\partial\gamma\,\partial\xv$ assembled per bond.\\ non-affine (the trap and its avoidance): never form an eigendecomposition pseudoinverse—near the margin the soft modes make $\Xi^{\mathsf T}H^{+}\Xi$ over-subtract through near-zero $1/\omega^2$ weights. Instead pin DOF of the most-coordinated backbone particle, Cholesky-factor the reduced positive-definite , solve, apply iterative-refinement steps; then $G_{\rm rel}=G_{\rm Born}-\Xi^{\mathsf T}H^{+}\Xi/V$ with $0\le G_{\rm rel}\le G_{\rm Born}$ by construction. Keep negative accepted values if any occur (no clamping).\\ acceptance (physical, untuned): jammed ( $P>10^{-12}$ ), positive-semidefinite (Cholesky succeeds), residual force $<10^{-7}$ .\\ fit: pooled $G_{\rm rel}=a\,(z-z_0)$ ; bootstrap ( $3000\times$ ) CI for z_0 . pass: CI $\ni 2d$ ; $\langle G_{\rm rel}\rangle$ monotone in ; finite at the lowest- quartile; $c^2=B_{\rm rel}/\rho$ size-independent.\\ result: z_0=5.99 , CI $[5.49,6.29]\ni6$ ( n{=}56 ); quartile means $4.04\to12.39$ ; c^2 spread $0.4\%$ ; softening $\langle B_{\rm rel}/B_{\rm Born}\rangle=0.711$ ; 56/56 accepted, negative.

Rotation unjams; forced-radius attractor (unjam_inflow.py)

packings: as above, N{=}128 , $\phi\in\{0.652,\dots,0.720\}$ ( values) $\times$ seeds.\\ rigid rotation: extension per bond under $\mathbf u=\theta\,\hat{\mathbf z}\times\rv$ is $\hat{\mathbf n}\cdot(\hat{\mathbf z}\times\rv)\equiv0$ ; measured $\le2.8\times10^{-17}$ . PASS\\ shear reserve: per extending bond the breaking strain is $\gamma_b=\mathrm{ov}_b/(n_xn_y r)_b$ ; track the distribution ( $q_{05}$ , median) across the $\phi$ ensemble against pressure. pass: $q_{05}$ ratio $5.9\times$ , $\mathrm{corr}(\log q_{05},\log P)=0.96$ , $\mathrm{corr}(\log\mathrm{med},\log P)=0.99$ (the single first break is extreme-value noise: reported, not gated).\\ attractor: integrate $\dot x=\alpha x^{-5}-x^{-4}$ (RK4) from $x_0\in\{0.2,0.5,1,2,5\}$ with inherited $\alpha=2/\pi$ . pass: all $\to x^\star=\alpha$ to $10^{-15}$ ; $F'(x^\star)=-(\pi/2)^5=-9.5631$ .

Transition class — the G-SOC rules land on DP (transition_dp.py, 3 stages)

model: 1D ring; reserve from the pseudogap $(1{+}\theta)x^{\theta}$ ( $\theta{=}0.45$ , flagged); inactive site unjams if $\lambda\,n_{\rm act}\,U>x$ ; active re-jams w.p. q{=}0.35 ; toggled sites redraw . All-inactive is absorbing.\\ stage 1: slope-matched bisection ( L{=}30 k, T{=}4 k), then curvature-minimised refine ( L{=}80 k, T{=}20 k): $\lambda_c{=}1.2544$ , $\delta{=}0.158$ .\\ stages 2–3: spreading ( R{=}512 , T{=}6 k): $\theta_s$ windows $0.382{\to}0.356{\to}0.335$ , $\delta'{=}0.105$ , hyperscaling sum 0.662 vs d/z{=}0.6326 ; lifetimes $\nu_\parallel{=}1.637$ ( $0\%$ censored); activity correlation (fit $M_2{-}1$ ): $a{=}0.482{\pm}0.011$ vs $2\beta/\nu_\perp{=}0.504$ .\\ pass: five for five on DP ( 1{+}1 d); mean-field ( $\delta{=}1$ ) and compact-DP ( $\delta{=}0.5$ ) excluded. SOC avalanche $\tau$ : cutoff >10^6 at $0.35\%$ below threshold, thin statistics — reported, not gated.