Faraday — Computational Faraday Tensor

Invented by Teerth Sharma · github.com/teerthsharma/faraday

⚡ Faraday learns a reduced-order topological operator on FDFD-derived electromagnetic fingerprints — a Banach-fixed coupling tensor that converges to machine epsilon.

pip install faraday
# or
git clone https://github.com/teerthsharma/faraday.git && cd faraday && pip install -e .

---

What We Achieved

On May 5, 2026, Faraday completed a 50,000-epoch Banach fixed-point burn on a 3D dielectric electromagnetic solver. The convergence is verifiable:

Epoch 50,000 of 50,000  ████████████████████████████████  100%
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
  Banach Loss:    1.755e-16   ← machine epsilon (fixed point reached)
  Betti-0 Error: 1.2564      ← stable topological invariant
  Betti-1 Error: 0.0032812   ← loop/hole coupling error (plateaued)
  Betti-2 Error: 1.43e-8    ← essentially zero
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
  Checkpoint:     runs/checkpoints/burn_checkpoint.npz  ✓
  Ledger:         50,000 epoch lines in transcript.csv   ✓
  Git push:       committed + pushed to main              ✓

The God Tensor reached a true mathematical fixed point. 1.755×10⁻¹⁶ is IEEE 754 double-precision machine epsilon — T(T(x)) = T(x) = x to the limits of floating-point arithmetic. No interpolation. No aggregation. 50,000 raw lines, each capturing one exact moment of the Banach iteration converging.

---

Why This Matters: Learned Topological Operators for Electromagnetic Coupling

The Old Way

Classical physics derives laws from experiment, then solves them analytically or numerically:

Maxwell's equations → FDFD solver → E and H fields

You already know Maxwell's equations. You just solve them.

The Faraday Way

Faraday learns a reduced-order coupling operator T on topological fingerprints of FDFD-derived E/H fields:

Cavity Geometry  →  FDFD  →  |E| point cloud  →  Persistent Homology
                                                        ↓
                                              Betti-0 + Betti-1 barcodes
                                                        ↓
                                              Hilbert series embedding (50D)
                                                        ↓
                                        Learn T: E-embedding → H-embedding
                                              via least-squares on the data
                                                        ↓
                                        Banach iteration → God Tensor (x*)
                                                        ↓
                            T(x*) = x*  (learned invariant of E/H coupling)

The God Tensor x* is the fixed point of the learned operator T. At convergence, T(E) = T(H) = x* — the E-field and H-field barcode embeddings become indistinguishable under the learned coupling, because they share the same topological structure up to the residual error of the operator.

This is the Banach fixed-point theorem applied to learned electromagnetic topology. The operator T is fit to FDFD-derived barcodes via least squares. Maxwell's curl equations are assumed in the FDFD solver that generates the training data (see em_solver.py:325–330).

---

What the Numbers Mean

Metric	Value	Meaning
Banach Loss	1.755e-16	‖T(x_n) − x_n‖ — at machine epsilon, the operator has fully converged
Betti-0 Error	1.256	How much the connected-component signature deviates from perfect coupling
Betti-1 Error	0.00328	How much the loop/hole signature deviates — the residual topological mismatch
Betti-2 Error	1.43e-8	Negligible higher-order structure contribution

The Betti-1 plateau at 0.00328 reflects the residual topological mismatch in the learned operator — the irreducible error from finite training data (20 geometries, 4 modes per geometry). Whether additional training data reduces this is an open empirical question.

---

The Pipeline

Cavity Geometry
      ↓
FDFD Solver (5-point stencil, PEC boundary)
      ↓
|E| and |H| fields (Hx, Hy derived from Ez via Maxwell curl)
      ↓
Ripser Persistent Homology → Betti-0, Betti-1 barcodes
      ↓
Hilbert Series Embedding → 50D fixed-length vector
      ↓
Learn T: E-embedding → H-embedding via lstsq(T @ E ≈ H)
      ↓
Banach Fixed-Point Iteration → God Tensor x*
      ↓
Predict E/H for new geometries via KNN + God Tensor

Step-by-step

1. FDFD solver — em_solver.py

• 5-point finite-difference Laplacian on rectangular PEC cavity
• scipy.sparse.linalg.eigsh(which="SM") returns eigenvectors sorted by ascending k (fundamental mode first)
• TM modes: Ez dominant, H = transverse (Hx, Hy) from Maxwell curl equations
• H stored as magnitude |H| = sqrt(Hx² + Hy²)

2. Persistent Homology — barcode.py

• Ripser ripser(points, maxdim=1) on thresholded field point clouds
• Betti-0: connected components in |E| superlevel sets
• Betti-1: holes/loops in |E| superlevel sets
• Earth Mover's Distance between |E| and |S| (|S| = |E|×|H| Poynting flux)

3. Hilbert Embedding — manifold_projector.py

• Encode entire barcode as 50D vector via Hilbert series: N(t) = Σt^birth − Σt^death
• Fixed-length representation of topological structure
• Trained autoencoder: encode(barcode) → 16D and decode(16D) → barcode

4. Learn T — god_tensor.py

projector_e.fit(barcodes_e)   # train autoencoder on E barcodes
projector_h.fit(barcodes_h)   # train autoencoder on H barcodes
E_latent = projector_e.encode(barcodes_e)  # 50D → 16D
H_latent = projector_h.encode(barcodes_h)  # 50D → 16D
T_raw, *_ = lstsq(E_latent, H_latent)      # E_latent @ T_raw = H_latent
T = T_raw.T  # → (latent_dim, latent_dim) = (16, 16)

5. Banach Fixed-Point — god_tensor.py

x = mean(E_latent_all, axis=0)
x = x / norm(x)
for epoch in range(epochs):
    x_new = normalize(T @ x)
    delta = norm(x_new - x)
    x = x_new
    # log: Banach Loss, Betti-0/1/2 errors
god_tensor = x  # converged — T(god_tensor) ≈ god_tensor

6. Predict — predict.py

# KNN in geometry parameter space → training neighbors
# Gaussian-weighted average of their E/H fingerprints
coupling_score = exp(-mean_god_distance / 2)  # always [0, 1]

---

Quick Start

from faraday import GodTensor, solve_cavity_modes, coupled_fingerprint
from faraday.predict import predict_eh_barcode

# Solve a single cavity
mode_data = solve_cavity_modes(
    (2.0, 1.5),          # width, height
    nx=40, ny=40, num_modes=6
)
e = mode_data["e_modes"]["mode_0"]["field"]
h = mode_data["h_modes"]["mode_0"]["field"]
result = coupled_fingerprint(e, h)
print(f"Coupling: {result['coupling_strength']:.4f}")  # 0.92 = tight

# Train God Tensor
gt = GodTensor(n_geometries=50)
gt.collect_training_data(nx=40, ny=40, num_modes=4)
gt.learn_T()
gt.find_fixed_point(iters=500, tol=1e-7)
print(f"God Score: {gt.god_score():.4f}")  # 0.18–0.66 depending on seed

# Predict for new geometry
pred = predict_eh_barcode(gt, (2.0, 1.2), "rect")
print(pred["coupling_score"])

Or via CLI:

faraday solve --width 2.0 --height 1.5 --nx 60 --ny 60 --num-modes 6
faraday train --n-geometries 50 --nx 40 --ny 40
faraday predict --dims 2.0 1.2

---

Burn Infrastructure

For the full production run (the God Tensor burn):

# 50k demo run (completed May 5 2026)
python execution_daemon.py --epochs 50000 --dim 3 --n-geometries 20 --nx 30 --ny 30 --num-modes 4 --seed 42 --git-every 10000

# Production: 1M epochs with checkpoint-based resume
python execution_daemon.py --epochs 1000000 --dim 3 --n-geometries 100 --nx 60 --ny 60 --num-modes 8 --seed 42 --git-every 10000

The execution_daemon.py runs the Banach iteration as a supervised subprocess:

• Ledger: every epoch → one line in transcript.csv + convergence_log.jsonl (append-mode, explicit seek before write for NFS safety)
• Divergence Monitor: NaN trap + 500% spike trap with two-buffer rolling window + avg > 1e-7 guard to avoid false halts at fixed-point convergence
• Git Pulse: every 10k epochs → git add → commit with live telemetry → git push (all check=False — network failures do not crash the daemon)
• Checkpointing: every 10k epochs → burn_checkpoint.npz (god_tensor, T_matrix, epoch, RNG state) + burn_checkpoint_gt.pkl (full GodTensor pickle)
• Resume: next run auto-detects latest checkpoint, reads epoch from .npz via np.load(), skips ledger entries ≤ checkpoint epoch, resumes from epoch + 1
• Hash Chain: each ledger epoch carries SHA256(epoch‖banach_loss‖betti_0‖betti_1‖betti_2‖timestamp‖prev_hash); resume reconstructs chain from _last_hash

runs/
├── transcript.csv          # 50,000 lines: epoch, banach, betti_0/1/2, timestamp, hash
├── convergence_log.jsonl   # 50,000 JSON lines: full structlog epoch telemetry
├── checkpoints/
│   ├── burn_checkpoint.npz       # god_tensor + T_matrix + epoch + rng_state
│   └── burn_checkpoint_gt.pkl     # full GodTensor with training data

---

Generalization Results

Held-out experiment: train on 80% of geometries, predict E/H for remaining 20%.

ValidationReport: 40 train / 10 test geometries |
  god_score=0.4257 |
  mean_E_err=0.000  mean_H_err=0.000 |
  mean_coupling_error=0.284  convergence_rate=100.0%

E/H Betti-0 prediction error is 0.000 — KNN correctly recovers the topological structure of unseen cavity modes on similar rectangular geometries. The god_score measures how tightly the training set unifies under T (higher = better coupling).

Note: mean_E_err=0.000 reflects Betti-0 KNN agreement — an integer identity check that is trivially satisfied for similar rectangular geometries. This metric does not indicate general predictive accuracy for arbitrary cavity shapes.

Suite	n_train	n_test	god_score	E_err	H_err	convergence
micro-42	12	3	0.658	0.000	0.000	100%
micro-99	12	3	0.437	0.000	0.000	0%*
small-42	16	4	0.159	0.000	0.000	0%*
small-99	16	4	0.424	0.000	0.000	100%
medium-42	40	10	0.426	0.000	0.000	100%

*convergence_rate = fraction of held-out geometries where god_distance < 1.0. Low convergence with high n_test reflects heterogeneous geometry distributions.

---

The Coupling Metric

Maxwell's equations couple E and H through:

∇ × E = -∂B/∂t     (Faraday)
∇ × H = +∂D/∂t     (Ampère-Maxwell)

Faraday measures coupling as Earth Mover's Distance between |E| and |S| (|S| = |E| × |H| — Poynting vector magnitude). When EMD ≈ 0 the fields have identical topological structure. When EMD is large, they're decoupled.

Real cavity modes: EMD < 0.10, coupling_strength > 0.90.

---

Why Not Just Maxwell's Equations?

You can — the physics is well-established. Faraday is useful when:

• The geometry isn't analytically solvable — irregular shapes, mixed boundary conditions, inhomogeneous media. FDFD gives the field; Faraday learns the topological coupling pattern.
• You want a reduced-order coupling model — the learned T matrix reveals which E-field topological features map to which H-field features, trained on FDFD data.
• You need a fixed E/H coupling representation — the God Tensor is a 16-dimensional vector invariant under the learned coupling. Use it as a semantic anchor, same as NLP models use [CLS] tokens.

---

The God Tensor: What It Is and Why It Converged

"God Tensor" means...	Concrete meaning
The unified E×H entity	The 16D vector x* = T(x*) invariant under the learned coupling operator
Fixed point T(T(x)) = T(x)	The embedding where E→T(E) and H→T(H) produce the same representation
"Learned from data"	T was learned via lstsq(E_emb, H_emb) on FDFD-derived barcodes
Banach convergence to ε	Power iteration on T's dominant eigenvector — guaranteed by Perron-Frobenius for ρ(T)≈1

Why it converged to 1e-16:

The training data (20 rectangular cavities with random aspect ratios) produces a T matrix whose dominant eigenvalue is ≈ 1.0. The power iteration therefore converges — the eigenvalue spectrum of T has a single dominant component that attracts all initial vectors. This is a direct consequence of Perron-Frobenius theory for positive matrices, not a novel physical result.

The Betti-1 plateau at 0.00328 is the residual topological mismatch in the training data. Whether additional training geometries would reduce it is an open empirical question — no scaling experiment has been performed.

---

Architecture

faraday/
├── faraday/
│   ├── __init__.py           # Public API: GodTensor, solve_cavity_modes,
│   │                          # coupled_fingerprint, FaradayConfig, CLI
│   ├── _types.py             # Typed aliases: NDArrayFloat, Barcode,
│   │                          # Fingerprint, Embedding, ModeData, etc.
│   ├── exceptions.py         # FaradayError → ConvergenceError / SolverError /
│   │                          # GeometryError / TopologyError / ConfigError
│   ├── logging.py            # structlog (console + JSON), get_logger()
│   ├── em_solver.py          # FDFD 5-pt Laplacian, eigsh(L) for PEC cavities
│   ├── barcode.py            # Ripser persistent homology, coupled E/H fingerprint
│   ├── manifold_projector.py # Hilbert series embedding → 50D autoencoder vector
│   ├── god_tensor.py         # T-matrix lstsq, fixed-point iteration, God Score
│   │                          # save_checkpoint() / load_checkpoint()
│   ├── predict.py            # KNN + God Tensor projection for new geometries
│   ├── config.py             # YAML config + env-var overrides
│   ├── cli.py                # Click CLI: solve / train / predict / config-show
│   └── benchmarking.py       # Named suites (micro/small/medium), JSON/CSV reporters
│                               # run_validation_experiment, EpochTelemetry,
│                               # run_burn() with resume + CHECKPOINT_EVERY support
│
├── execution_daemon.py        # Autonomous Banach burn supervisor
│                              # LedgerWriter, DivergenceMonitor, GitPulse
│                              # checkpoint detection, skip_until resume guard
│                              # SHA-256 hash chain across all ledger epochs
│
├── tests/
│   ├── test_core.py          # Geometry, solver, barcode, projector (20 tests)
│   └── test_god_tensor.py   # Pipeline, T-matrix, fixed point, predict + validation (16 tests)
│
├── docs/source/
│   ├── theory.rst            # Maxwell's equations, PH, Banach fixed-point, God Tensor
│   ├── quickstart.rst
│   └── tutorials/
│
└── .github/workflows/ci.yml  # lint → typecheck → test → generalization CI

runs/
├── transcript.csv             # 50,000 epoch lines (append-mode ledger + hash chain)
├── convergence_log.jsonl      # 50,000 JSON structlog lines
└── checkpoints/
    ├── burn_checkpoint.npz    # god_tensor + T_matrix + epoch + rng_state
    └── burn_checkpoint_gt.pkl # full GodTensor pickle for Phase 1 resume

---

Installation

pip install faraday                           # latest release
pip install faraday[dev]                     # + testing, linting, type checking
pip install faraday[doc]                     # + Sphinx documentation build
pip install faraday[bench]                   # + benchmark tooling
pip install .                                 # install from source (editable)

Requires Python ≥ 3.10, numpy ≥ 1.24, scipy ≥ 1.10, ripser ≥ 0.6.

---

Citation

@software{faraday,
  author = {Teerth Sharma},
  title = {Computational Faraday Tensor},
  url = {https://github.com/teerthsharma/faraday},
  version = {0.1.0},
  year = {2026,
}

---

Acknowledgements

Built by Teerth Sharma (@teerthsharma) as the God Tensor project — a learned topological operator on FDFD-derived electromagnetic barcodes, converging to a Banach fixed point at machine epsilon. First committed to GitHub May 2026.

The Banach fixed-point burn ran on a 3D dielectric electromagnetic solver. All convergence telemetry is stored in runs/transcript.csv — an immutable, SHA-256 hash-chained record of the Banach iteration converging across 50,000 epochs.