Live: fib896.com
A WebGL visualization of the ×2 mod 9 doubling orbit — six numbers wound around a Fibonacci phyllotaxis cone, breathing.
Start at 1. Double it. Take the result mod 9. Repeat:
1 → 2 → 4 → 8 → 7 → 5 → 1 → ...
Six values. Period 6. The numbers 3, 6, and 9 are structurally excluded — they form their own closed system under ×2 mod 9 and never enter this orbit. 9 does not appear.
Each value has an echo: its complement summing to 9 (1↔8, 2↔7, 4↔5). The visualization runs two helix strands π radians apart — Strand A carries the orbit, Strand B carries the echoes. Every rung connects a node to its echo across the axis.
Labels are written in balanced ternary centered on 6: digits 5=−1, 6=0, 7=+1. The zero of the trit system is 6 — the nil element, the number that isn't there. The arc at step 21 (Fibonacci F₈) is where the golden-angle spiral nearly closes on itself: the next structure to build.
| Tab | Panel | Description |
|---|---|---|
| 01 | DIVISOR LATTICE | 896 = 2⁷×7 divisor structure |
| 02 | 1/89 CONVERGENCE | 1/89 = ΣF(n)/10ⁿ⁺¹ convergence |
| 03 | φ SPHERE | Fibonacci/Lucas nodes on a unit sphere |
| 04 | MoE ROUTING | Kimi K3 mixture-of-experts token routing |
| 05 | GREEK LETTERS | π, φ, τ, ω — connections to 896 |
| 06 | SUNFLOWER | Golden angle phyllotaxis disc |
| 07 | TRIT MATRIX | Balanced ternary digit matrix |
| 08 | HELIX | JENNIE 21 — the main event |
| 09 | TERNARY VS CLOCK | Balanced ternary vs clock arithmetic |
| 10 | ORBIT CYCLE | Full period-6 cycle animated |
| 11 | OLIVER 42 | oliver42 framework: orbit × complement |
| 12 | EXPERIMENTS | Penrose Tribar experiment results |
| 13 | GNN MIRROR | Orbit GNN skip-connection topology |
| 14 | BUCKMINSTER | C₆₀ — nil coordinate container |
| 15 | ORBIT MUSIC | Heptagon WebAudio orbit arpeggio |
| 16 | MUSIC | Extended orbit music sequencer |
| 17 | TIME-TREE | Project timeline — four streams from origin |
| 18 | JO BURROWS | Worm through filigree — hypotrochoid spirograph, lemons |
| 19 | 3I/ATLAS | Third interstellar object; ω=128=K, retrograde, e=6.14 |
| 20 | P-WAVE DETECT | Live seismic detection — StreamingNet, orbit buffer, pre-P frontier |
| Control | Action |
|---|---|
| Drag | Rotate |
| Scroll | Zoom |
| AUTO-ROTATE | Toggle rotation |
| COMPLEMENT | Toggle trit labels ↔ decimal values |
| TRIBAR | Toggle Penrose tribar inter-cycle fold overlay |
| SIDE | Camera: tornado/DNA silhouette |
| TOP | Camera: Fibonacci sunflower from above — reveals hexagram |
| HERO | Camera: low heroic angle, large nodes |
| Hover a node | Tooltip with trit, echo, group membership |
The HELIX runs 3 complete orbit cycles (18 nodes per strand + apex). Each cycle covers one full period of [1,2,4,8,7,5]. Nodes at the same orbit position across cycles form natural triangles — the same value recurring at p, p+M, and p+2M (M=6).
The edges of these triangles define three arm groups, each connecting adjacent cycles:
| Arm | Color | Pairs | Reading |
|---|---|---|---|
| A | gold | cycle 0 → cycle 1 | the first fold |
| B | cyan | cycle 1 → cycle 2 | the second fold |
| C | orange | cycle 2 → cycle 0 | the impossible return |
Arm C is impossible: the helix ascends continuously, so the return edge would have to close a triangle whose third vertex is below the starting point — a contradiction in 3D that is visible only in projection. The Penrose tribar is the classical diagram of exactly this impossibility. Arm C is drawn as lines only (no fill) to preserve that contradiction.
Arms A and B are filled with translucent face shading (centroid-split into two sub-triangles per orbit position). This produces 12 filled sub-triangles per cycle pair — 12 faces total across the two filled arms.
TOP view: From directly above, the 6 tribar triangles (one per orbit position, A+B filled) radiate from the helix center as a hexagram — a 6-pointed star. The impossible Arm C lines complete the outer silhouette of each triangle without filling it.
What it shows: The tribar makes visible the mod-3 structure inside the mod-6 orbit. Three cycles is not arbitrary — it is the natural folding depth where the orbit closes on itself modulo 3. The impossible arm encodes the fact that closure is periodic, not spatial. This is the same structure that appears in GNN skip connections: Arm A and Arm B are long-range edges that carry information across cycle boundaries; Arm C is the ghost edge that would make the graph cyclic at the wrong scale.
orbit position p: node[p] node[p+M] node[p+2M]
●────────────●────────────●
│ Arm A │ Arm B │
└────────────┴────────────┘
Arm C (impossible)
All experiments are in src/experiments/. They progress from early proofs-of-concept
through the Penrose Tribar architecture and into seismology applications.
| File | Description |
|---|---|
poc_penrose_tribar.py |
Original PoC — orbit permutation × gated skip × LayerNorm on Fashion-MNIST |
poc_tribar_fashion_k4.py |
K=4 ablation |
poc_tribar_fashion_k32.py |
K=32 benchmark (tri 80.9% vs base 74.8%, +6.1%) |
poc_tribar_seismo_ethz.py |
ETHZ seismology dataset |
poc_tribar_seismo_stead.py |
STEAD dataset — 7373 eq + 7373 noise, +2.61% at σ=0.3 |
poc_tribar_early_detection.py |
P-wave early detection — >8s warning before S-wave |
poc_tribar_seismic_system.py |
Full streaming seismic detection system |
poc_tribar_optuna.py |
Optuna HPO for Tribar hyperparameters |
Full results in ABLATION.md.
| File | Description |
|---|---|
exp_1_error_correlation.py |
Error correlation: do orbit and random models make errors on the same examples? (Jaccard, Cohen's κ) |
exp_2_ensemble_mixing.py |
Ensemble mixing: does adding orbit models improve a random-perm ensemble? |
exp_3_variance_source.py |
Variance source: why does orbit have 3.3% std vs random's 0.4%? (init, data split, sequence) |
Findings: Orbit perm hurts ensembles (−0.35pp F1 at 50/50 mix). Variance is primarily init sensitivity (H1 explains). Canonical sequence [0,1,3,7,6,4] is the most stable of all orbit-value orderings (std 2.56% vs shuffled mean 3.62%), but doesn't offset performance gap vs random. Production sensor uses random perm.
Full experiment arc documented in SEISMIC_DETECTION.md.
| File | Description |
|---|---|
poc_adaptive_buf.py |
Learnable DECAY/STRENGTH; gradient confirms attractor at 0.82–0.84 |
poc_percycle_buf.py |
Per-cycle buffer params; marginal gain, not worth complexity |
poc_streaming.py |
Streaming inference (single-0s training); precision drops OOD |
poc_streaming_trained.py |
Streaming-aware training; warm [-1s,-0.5s] → classify 0s |
poc_optuna.py |
Optuna HPO, 50 trials; best: threshold=0.48, decay=0.876 → 88.0%/95.7% |
poc_orbit_ablation.py |
Orbit vs random vs identity perm; orbit high-variance, random stable |
poc_random_perm_prod.py |
Production random perm, 5 seeds; 86.4%/97.9% mean |
poc_orbit_ceiling.py |
Orbit ceiling, 10 seeds; best seed → 92.0%/89.9% |
poc_perm_sweep.py |
Warmup-based perm selection; anti-predictive (negative result) |
poc_early_detection.py |
Early detection at -0.5s; 84.4%/98.5% vs 87.7%/96.1% at 0s |
poc_horizon_sweep.py |
Full horizon sweep; -0.5s sweet spot, -1s cliff, ~3.2pp/0.5s |
poc_dual_horizon.py |
Dual-head (early@-0.5s + late@0s); and-gate → 89.8%/97.5% |
poc_cycles_ablation.py |
CYCLES ∈ {1-6}; CYCLES=1 best mean 85.0%, CYCLES=3 was suboptimal |
poc_c1_dual.py |
Champion: CYCLES=1 + dual and-gate → 92.3%/95.0% ±1.79% (5 seeds) |
poc_early_detection_v2.py |
Pre-P frontier (3 seeds): H-0.5s AUC 0.988 beats H+0.0s baseline 0.983 |
poc_early_detection_v3.py |
Extended frontier: 7 horizons H-3.0s→H+0.0s + magnitude head (in progress) |
Champion configuration:
CYCLES = 1 # not 3; over-smoothing at higher depth
BUF_DECAY = 0.876
BUF_STRENGTH = 1.429
LR = 2.78e-3
THRESHOLD = 0.480
PERM = torch.randperm(K) # random fixed, seeded per run
TRAIN_MODE = "dual-horizon" # loss = 0.5*CE(early) + 0.5*CE(late)
EVAL_STRAT = "and-gate" # both heads must agree
# Result: 92.3% precision / 95.0% recall ±1.79% (floor: 89.6% across 5 seeds)| File | Description |
|---|---|
poc_echo_moe_final.py |
Final Echo MoE with orbit-gated expert selection |
poc_echo_moe_gaussian.py |
Gaussian noise robustness sweep |
poc_echo_moe_top2.py |
Top-2 expert routing variant |
poc_echo_moe_training.py |
Training loop diagnostics |
poc_echo_moe_warmup.py |
Warmup schedule exploration |
poc_echo_moe_temp.py |
Temperature scaling for routing |
poc_echo_init_stats.py |
Expert initialization statistics |
| File | Description |
|---|---|
poc_orbit_gnn.py |
Graph neural network with orbit skip topology |
poc_sgd_orbit.py |
SGD learning rate mapped to orbit sequence |
poc_ternary_sweep.py |
Ternary activation sweep |
poc_ternary_orbit.py |
Orbit-structured ternary quantization |
poc_ternary_hard.py |
Hard ternary (no straight-through) |
poc_trib_3layer.py |
3-layer Tribonacci balance (33/33/33 result at fc3) |
poc_trib_improved.py |
Tribonacci attractor improved |
poc_trib_hard.py |
Hard Tribonacci constraints |
poc_tribonacci.py |
Tribonacci sequence baseline |
poc_jennie22.py |
jennie22 hypothesis test |
| Path | Contents |
|---|---|
data/MNIST/ |
Fashion-MNIST raw files (auto-downloaded by torchvision) |
data/silk_research.md |
Silk / compass thread research — orbit permutation as signal conduit |
The orbit (×2 mod 9, period 6):
n, seen, orbit = 1, set(), []
while n not in seen:
seen.add(n); orbit.append(n); n = (n * 2) % 9
# [1, 2, 4, 8, 7, 5]Balanced ternary (centered on 6):
def to_bt(n):
if n == 0: return '6'
digits = []
while n != 0:
r = n % 3
if r == 0: digits.append('6'); n //= 3
elif r == 1: digits.append('7'); n = (n - 1) // 3
else: digits.append('5'); n = (n + 1) // 3
return ''.join(reversed(digits))
# 1→'7' 2→'75' 4→'77' 5→'755' 7→'757' 8→'765'Node positions (golden angle phyllotaxis cone):
const GA = 2 * Math.PI * (2 - (1 + Math.sqrt(5)) / 2); // ≈ 137.508°
const nodePos = (step, phiOffset = 0) => ({
x: (0.28 + step * 0.13) * Math.cos(step * GA + phiOffset),
y: step * 0.68,
z: (0.28 + step * 0.13) * Math.sin(step * GA + phiOffset),
});
// Strand A: phiOffset = 0
// Strand B: phiOffset = Math.PI (echo strand, opposite side)Accordion breath (all geometry shares one scalar):
const breath = 1 + 0.10 * Math.sin(t * 0.50); // ±10%, ~12.6 s cycle
node.position.y = baseY(step) * breath;make # show all available targets
make sync # install python deps (uv) + npm deps
make dev # vite dev server at localhost:5173
make build # production build → src/ui/dist/
make deploy # build + sync to s3://hak4 + CloudFront invalidation
make test # run all tests (python + js)Stack: Svelte · Vite · Three.js (WebGL + CSS2DRenderer)
The three Lean 4 formal proofs are permanently inscribed on Bitcoin (2026-08-14):
| Proof | Inscription |
|---|---|
| orbit_proof.lean — mod-9 doubling orbit, period 6, complement {0,3,6} | 62a33e0…i0 |
| primitive_root_proof.lean — 2 is a primitive root mod 9 | 2143c07…i0 |
| ideal_proof.lean — ideal structure of the complement {0,3,6} in ℤ/9ℤ | 5089bf2…i0 |
Inscription IDs: tags/inscriptions.json
CC BY-NC-SA 4.0 © 2026 Ryan Scarbery and Traci Johan
src/
ui/ # WebGL visualization (Svelte + Vite)
index.html
main.js
App.svelte # shell: canvas, tabs, nav, controls
app.css # global styles + responsive breakpoints
lib/
state.js # active scene store
scenes/
index.js # scene registry (17 panels)
shared.js # Three.js setup, R singleton, tooltip helpers
s0.js – s16.js # individual scene modules
tests/
unit/ # vitest unit tests
e2e/ # playwright e2e tests
experiments/
poc_penrose_tribar.py # Penrose Tribar architecture (main)
poc_tribar_*.py # Tribar variants: Fashion-MNIST, seismology, HPO
poc_echo_moe_*.py # Echo MoE with orbit-gated routing
poc_orbit_gnn.py # Orbit GNN topology
poc_sgd_orbit.py # SGD with orbit learning rate schedule
poc_ternary_*.py # Ternary activation experiments
poc_trib_*.py # Tribonacci attractor experiments
poc_jennie22.py # jennie22 hypothesis
poc_adaptive_buf.py # Streaming buffer series (see SEISMIC_DETECTION.md)
poc_streaming_trained.py # Streaming-aware training
poc_optuna.py # Optuna HPO (threshold=0.48, decay=0.876)
poc_early_detection.py # Early detection at -0.5s before P-wave
poc_c1_dual.py # Champion: CYCLES=1 + dual and-gate → 92.3%/95.0%
TRIBONACCI_ATTRACTOR.md # Tribonacci attractor writeup
SEISMIC_DETECTION.md # Streaming buffer / seismic detection series writeup
data/
MNIST/ # Fashion-MNIST raw files
silk_research.md # Silk / compass thread — orbit as signal conduit
896 = 2⁷ × 7. Seven doublings of 1, multiplied by 7. τ(896) = 16. dr(896) = 5. dr(897) = 6 — the nil neighbor.
The ×2 mod 9 orbit of 1 has period 6: {1,2,4,8,7,5}. At step 21 (F₈), the golden-angle spiral has accumulated 21 × 137.508° ≈ 7.7° past 8 full rotations — the Fibonacci near-return where the sunflower arm pattern folds back.
The Penrose Tribar overlay makes visible the mod-3 folding structure inside the orbit. Three cycles of period-6 produce 6 triangles whose edges are the tribar arms. From TOP: a hexagram. Arm C is the impossible return — periodic closure drawn as a ghost edge. This connects to the oliver42 topology: arm A is complement, arm B is bridge, arm C is collapse.
The Penrose Tribar architecture (gated skip + orbit permutation + LayerNorm) outperforms baseline MLP at every tested noise level on Fashion-MNIST and STEAD seismology datasets. Peak advantage at ambient noise (σ=0.3) — the operating regime of real seismic detection. Wall time: 61 seconds for the full seismology benchmark.
Ablation (2026-08-16): Three experiments (error correlation, ensemble mixing, variance source)
found that the orbit permutation is not load-bearing for the seismic sensor. Random fixed
permutations outperform orbit on mean precision (86.1% vs 84.2%) and are more stable. Mixing
orbit models into an ensemble hurts F1 by 0.35pp. Orbit variance is primarily init sensitivity:
the periodic tiling creates a rugged loss landscape with multiple basins. The canonical sequence
[0,1,3,7,6,4] is the most stable ordering of the orbit values, but cannot close the gap to random.
Production sensor uses random perm. Math (Lean proofs, Bitcoin inscriptions) stands independently.
See ABLATION.md for full results.
The streaming buffer series extended this to real-time P-wave detection. Champion result (2026-08-05): CYCLES=1 + dual-horizon and-gate → 92.3% precision / 95.0% recall ±1.79% across 5 seeds on STEAD chunk2, streaming-aware training, threshold=0.48.
Pre-P detection (2026-08-06): A model trained at H-0.5s (0.5s before P-wave arrival) achieves AUC=0.988 — beating the post-arrival baseline (AUC=0.983) by +0.5pp AUC / +3.3pp precision. Cross-trained to H+0.0s loses only −1.1pp. This result has no known prior in the seismic ML literature: all 28 papers in a 2025 systematic review use post-arrival P-wave data. Pre-P detection at AUC=0.988 is outside the current state of the art.
Extended frontier (v3, in progress): 7 horizons from H-3.0s to H+0.0s with a magnitude regression head. Preliminary results (seed 0): AUC rises monotonically from 0.811 at H-3.0s to 0.893 at H-1.0s; magnitude MAE breaks below 1.0 magnitude unit at H-1.0s (MAE=0.971). A transfer wall at H-1.5s prevents earlier models from generalizing to H-0.5s/H+0.0s. Best deployment horizon: H-1.0s (AUC 0.893, transfers to H-0.5s at −2.2pp, magnitude useful).
See src/experiments/SEISMIC_DETECTION.md for full results.