Illustrative prototype. Explore toy models and save demo settings. Browser results cannot authorize controlled execution or support research claims. View repository gates · Open notebook

Educational demonstrations

Explore structure and nulls

Two browser demonstrations use 12 handwritten illustrative entries and 148 deterministic synthetic fixtures. Generated features depend on rank, so omitting its direct axis is not independence. Runs are unregistered, exploratory, and provide no controlled or physical support.

EXP-MAP-A01 separately selects 38,042 analysis rows (one curve-number-1 representative per isogeny class) from 64,687 source rows, with k = 10, 999 SplitMix64 rank permutations, and α = 0.01. These browser protocols differ. Inspect the recorded gates.

H0 persistence and graph cycles describe a chosen finite fixture cloud. Rank is omitted as a direct metric column, but generated regulator and height values depend on it. N1 independently permutes feature columns while preserving marginal distributions. With 18 nulls, the smallest possible p-value is 1/19 ≈ 0.053; this lightweight demonstration cannot reach a 0.05 threshold.

catalog18 nulls · k = 8

These diagnostics summarize the fixture cloud against 18 column-permuted nulls. They are educational outputs, not controlled scientific evidence.

160

5 features

H0 max pers.

1.351

p = 0.684 vs N1

kNN β₁

689

p = 1.000

⟨tortuosity⟩

1.732

p = 0.053

H0 persistence

Scalar embedding

Demonstration protocol

N1 — independently permute each intrinsic column.
Features: log N, sgn log |Δ|, log |j|, Faltings h, log R.
Graph geodesics exclude direct kNN neighbors, so reported tortuosity is a multi-edge path.
Positive control uses a sampled torus in R³, not Fibonacci or golden-ratio maps.

Geodesic sample

i	j	graph	chord	τ
109	25	1.392	1.063	1.310
12	79	3.040	2.136	1.423
154	54	2.893	1.306	2.215
67	127	0.829	0.772	1.074
137	147	1.308	0.860	1.522
107	49	3.129	1.153	2.714

Caveats

Persistent H_k of a finite point cloud is not a proof that the arithmetic state space is a smooth manifold.
Rank is excluded from the primary feature metric (RTCH-E1 protocol).
Graph Betti-1 is computed on a kNN graph, not a complete Vietoris–Rips complex.
N1 independently permutes each feature column, preserving every marginal.