Noether Capacity — How Many, Counted Exactly

Cislunar space is not a smooth field you sprinkle craft into — it is a crystal of stable orbit families separated by bands of chaos, and two craft whose resonances overlap tear each other's orbits apart. So a region has a carrying capacity: a hard maximum number of collision-free families. Everyone estimates it by sampling; sampling is blind to the exact edge, and the honest count — enumerate every arrangement — is 2^N and dies at a few dozen shells.

Noether Capacity counts it exactly. The overlap rule snaps to an integer-exact threshold, so the capacity at every perturbation level follows from a finite computation done once — and the answer ships with a two-sided proof: a set of craft that achieves the count, and a set of witnesses that forbids one more. No enumeration, at any scale.

A sampled capacity is a guess with error bars. This one is a number with a proof that it cannot be beaten.

THE PROBLEM How many craft can a cislunar shell lattice hold at a given perturbation level — exactly, and certifiably — without enumerating arrangements no machine can enumerate?

THE RESULT The exact count, both directions, at a scale the referee cannot reach:

The count, exactly. A 128-shell lattice holds exactly 34 craft at the reference perturbation — with the full census (8.7×10¹⁶ allowed configurations) counted, not sampled, in milliseconds.
Proven both ways. The certificate carries 34 pairwise-disjoint shells (the count is reached) and 34 piercing points (no 35th arrangement exists) — a third party checks both with no enumeration, in exact arithmetic.
Where the referee is impossible. Brute enumeration of that lattice is 2¹²⁸ = 3.4×10³⁸ configurations — 10¹⁹ years at a trillion checks a second. The exact census is polynomial; the answer is milliseconds.
Every perturbation level at once. The whole capacity curve — how the count falls as the region is stressed — is a single exact staircase, computed once, verified from the file.

The census — and the proof that stands in for it

Step	What was established	The number
The snap	the overlap rule between two orbit shells snaps from a floating-point test to an integer-exact threshold — the whole exclusion structure over all perturbation levels from one finite computation	exact thresholds · computed once
Three-way check	at the reference perturbation, the exact census agrees with the research crystal's own float model and a direct enumeration — and reproduces the recorded Fermi level	agree 3-way · capacity 5 at small scale
The dual witness	at 128 shells: a set that reaches the count, and a set that forbids one more — both verified in exact arithmetic, no enumeration	34 disjoint + 34 piercing
Beyond the referee	the brute census is 2¹²⁸ configurations; the exact census and the full capacity staircase are computed and verified in milliseconds-to-seconds	2¹²⁸ → ms

Capacity steel thread — measured audits; the exclusion model (Chirikov overlap) is the cislunar research arc's own, with the physics validated there; this instrument transports the model's exact consequences and states its model tier plainly.

The census — and the proof that stands in for it

How — the snap, the intervals, the witness