Chamber Controls
3D Rational Chamber
Live Verification
Selected State
Six-Cardinal Projection
Firewall meaning: this projection is lossy; the diagonal is not recoverable from the six channels. The full vector r is shown in the selected-state box.
The six-cardinal projection is a directional summary, not the whole object.
Null Baseline
Null baseline = random walks of equal length from the same start. Compare your path's shell statistics against this before reading any pattern as structure.
Run null baseline to compare shell transitions.
Shell Counts
| q | actual | expected J₃(q) | status |
|---|
Computable Closures
Jordan totient: Jd(q)=qd∏p|q(1−p−d).
Conservation: ∑q|nJ3(q)=n3 for residue classes. The 1≤k≤n chamber is shell-count equivalent when k=n acts as residue 0 for GCD shell counts.
Origin parity: with x=2k/n−1, the origin is a chamber point only when k=n/2 exists under the selected convention.
Claim posture: relations are flagged as observed, verified, null-consistent, convention artifacts, proved, or speculative bridges.