ElBalor/AlienX-2D-Isotropic-Stencil

The AlienX-Operator, with Isomorphic Spatial Net (ISN) — 2D Release v3: Isotropic Stencil + Gross-Pitaevskii

Isomorphic Spatial Net : A neural operator that operates on continuous geometric manifolds instead of fixed grids, with native rotation and scale invariance.

Strict operator-level verification: pred(R·k) = R·pred(k), bitwise exact under D4 (90/180/270°) with live message passing — fallback branch closed at the roundoff floor by the sign_lock. Rerun it yourself: python test_equivariance.py --device cuda

The grid is dead. The manifold is awake.

Created by Eric Heylel Danjuma Yaka, Capital Software / Next Gen Tech.

What's in this release

File	Purpose
model.py	AlienXOperator + ISNBlock + isotropic 24-neighbor stencil (importable module)
data.py	GPU Darcy flow generator with pull-back rotation and analytical gradients
train.py	Training + full evaluation suite (modular, imports model/data)
colab.py	Single-file self-contained version — paste into Colab, identical to the as-run script
test_equivariance.py	Strict operator-level equivariance test: pred(R·k) = R·pred(k)
RESULTS.md	Full run logs for two independent v2 runs + v1→v2 comparison + strict equivariance results + GP cross-PDE runs
PAPER.md	The complete v3 paper (now with §9: Cross-PDE Demonstration — Gross-Pitaevskii)
AlienX 2D × GP/	Cross-PDE demonstration: both GP training runs, both 5-test eval suites, raw logs, Dark Necromancer graph, dark-field rollout

The architecture in one paragraph

Each node lives on a spatial graph with an isotropic 24-neighbor circular stencil (radius √8 — uniform angular coverage, no cardinal spikes). A local SO(2) frame (e1 from ∇k, inertia-tensor fallback where ∇k is weak) makes displacements rotation-equivariant. Edge features encode the local angle as a complex harmonic embedding with even harmonics (cos 2θ, sin 2θ, cos 4θ, sin 4θ) — message weights are invariant under θ → θ + π. A constant physical scale depth makes the operator scale-blind by construction, which is exactly what produces zero-shot transfer across resolutions. No data augmentation. No learned geometry.

Results (v2, Interior L1)

Test	Result
Scale invariance (16→256, 256 zero-shot)	0.0036–0.0087, no drift
Rotation equivariance (0/45/90/180/270°)	essentially identical — 45° residual eliminated
Arbitrary angles (13/27/77/123/199°)	< 0.003 deviation at ≥ 32×32; < 0.002 at 64×64

Full logs in RESULTS.md.

Figures

Dark Necromancer Graph — Run B training diagnostics (loss, val RelRMSE, LR schedule, K-unroll curriculum)	Rollout comparison — split-step Fourier solver vs AlienX, |ψ|² at four time steps

Continuous rotation sweep — error vs angle across the full circle

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Cross-PDE: Gross-Pitaevskii (v3)

The architecture is PDE-agnostic — only the outer shell changes (k = |ψ|², regime parameter g, invariant ∫|ψ|² = 1). ψ is natively complex and the ISN harmonic machinery operates on complex features natively; Gross-Pitaevskii is the test where that pays off. Two runs, 416,150 params each, single-step interior RelRMSE:

Test	Run A (precision, fixed ICs)	Run B (generalization, K-curriculum)
Eval RelRMSE	0.3439%	1.0686% ± 0.2052% (16 unseen ICs)
Phase ablation	108.58× MSE degradation	61.4×
Rotation sweep (24 angles)	0.4080% ± 0.0315%	1.0347% ± 0.0806%
Scale, zero-shot 16→256	0.324–0.367%	1.05–1.10%

Run A's IC family is deterministic — a mechanism-precision claim, not an IC-generalization claim; Run B carries generalization. The claims are kept separate on purpose. Fix arc: plateau 0.056 → 0.000004 (14×) after even-harmonic edge features + k = |ψ|² frame source. Full detail: AlienX 2D × GP/README.md, paper §9, raw receipts in AlienX 2D × GP/*.txt.

The evaluation suite measures accuracy of the rotated problem. The strict statement — rotating the input rotates the output identically — is tested directly in test_equivariance.py:

python test_equivariance.py --checkpoint alienx_best.pt --device cuda

Three tests:

1. Exact D4 commutation (90/180/270°): R·k and R·pred(k) are exact index permutations of the flat fields — no interpolation, no generator rounding. Result on the v2 architecture with message passing live: max |pred(R·k) − R·pred(k)| = 0.000e+00 — bitwise exact. With the gradient branch active there are no cross-node reductions, and every within-node reduction runs in bitwise-identical order in the rotated run (aligned slot permutation; 2-element sums are IEEE-commutative). Control probe: mis-aligning only the slot order moves the output by ~1.4e-06 — summation order is the sole roundoff carrier, and it is aligned. (Earlier draft's "1–2e-08" was an artifact of zero-init res_scale — messages were a no-op; retracted.)
2. Continuous angles (13/27/77/123/199°): evaluated against bilinearly interpolated R·pred(k) — max ≈ 1e-01, mean ≈ 5e-03–8e-03, uniformly flat across angles (interpolation + discretization floor, not operator error). Informational regression tracker.
3. sign_lock verification: the eigh sign boundary is CLOSED. eigh's eigenvector sign is unspecified under rotation (dot(R·e1, e1_rot) = −1.0000 at 90°/180°), which previously broke the odd cos/sin edge input at 1.5e-03 when the fallback was forced. Fix (ported from the 3D QSA_ISNBlock3D): anchor the eigenvector sign to the k-weighted centroid — s = ⟨v, m⟩ is odd in v (cancels the arbitrary flip) and D4-invariant. With the fallback branch forced active, all D4 angles read CLEAN at the roundoff floor (1.4e-06 – 1.7e-06). The gradient branch — 100% of real workloads on this field — is exact.

[TEST 1] Exact D4 commutation:  pred(R.k) = R.pred(k)   (grid 32x32, interior)
     angle |     max|d| |    mean|d| |       rel
        90° |  0.000e+00 |  0.000e+00 |  0.00e+00  PASS
       180° |  0.000e+00 |  0.000e+00 |  0.00e+00  PASS
       270° |  0.000e+00 |  0.000e+00 |  0.00e+00  PASS

[TEST 3] sign_lock verification: fallback branch forced (grid 32x32)
    Case A     90°: max|d| = 1.431e-06   CLEAN (sign_lock holds)
    Case A    180°: max|d| = 1.669e-06   CLEAN (sign_lock holds)
    Case A    270°: max|d| = 1.669e-06   CLEAN (sign_lock holds)
    Case B   90°: max|d| = 1.315e-03   (zero-field degenerate; S = 0 makes
           the eigenvector arbitrary — ill-posed by construction, excluded)
    verdict: eigh sign boundary CLOSED — fallback branch is
    equivariant at all D4 angles with the sign_lock active.

Note: with no checkpoint present, the harness injects res_scale = 1.0 into all blocks — zero-init would make message passing a no-op and the D4 test would pass trivially (exactly 0). The injection makes the test exercise the real gather → frame → harmonic gate → aggregation path.

Reproduce

# Single T4 GPU, ~15 minutes for 500 epochs
python colab.py

# or modular:
python train.py   # trains, evaluates, saves alienx_rotation_continuous.png

Config: AdamW lr 2e-3, weight decay 1e-4, cosine annealing over 500 epochs, resolutions {16, 32, 64, 128} with 16 samples each, random rotations in [0, 360), interior MSE with dilation-scaled boundary crop. Best checkpoint: alienx_best.pt.

The debugging log (no failures hidden)

• Energy collapse in early runs
• Random gates producing noise
• Numerical gradient artifacts at 45° → analytical gradients + pull-back rotation
• FP16/AMP NaNs → geometry ops forced to FP32
• OOM with 24 neighbors → chunked gather + dynamic batch sampler
• Gradient instability → accumulation + gradient clipping
• Domain stretching at 45° → pull-back rotation
• Physical radius collapse at high resolutions → dynamic dilation + constant physical depth

Every bug was owned, diagnosed, and killed.

Design boundaries

1. Scale-blindness by construction — correct for Darcy flow (scale-free PDE), needs extension for scale-carrying physics (turbulence spectra, multifractal permeability). See PAPER.md §7.1.
2. Resolution-dependent frame fallback — the grad_k_mag < 0.1 threshold is a raw gradient value; both branches are equivariant, but the branch mixture differs by resolution. Cleaner formulation: threshold on ||∇k||·σ. The former eigh sign-flip boundary inside the fallback branch is CLOSED by the sign_lock (TEST 3: clean at the roundoff floor at all D4 angles). See PAPER.md §7.2.

Next: the n-Dimensional Organism

Clifford algebra Cl(4,0) substrate, per-node blade masks, NecroGraft expansion, fiber-bundle message passing. The 3D frontier (SO(3)) lives in ../AlienX-S03-Invariance/.

License

From The Grimoire of Elbàlor The Digital Necromancer.

AGPL-3.0 — see LICENSE.