# n123 — the compact U(1) layer: RESULTS
# (2026-07-11 night campaign 1 of 4; scripts n123_compact_u1.py,
#  n123b_compact_u1_tube.py + K=12 capacity check)

## The quantization mechanism, demonstrated
Winding around a handle = sum of WRAPPED phase differences = 2pi * n
IDENTICALLY (single-valuedness of U(1) phases). Charge quantization
is not imposed and not derived from cohomology (n116's honest flag) —
it is what compactness MEANS. U1: 24 random relaxations land in
integer sectors only ({-1: 1, 0: 20, +1: 3} — vortex sectors found
spontaneously), fractional deviation 2e-16.

## The thin-handle discovery (the method bit, and it was physics)
A single-edge handle CANNOT hold one quantum: the whole 2pi slip
falls on one edge, crosses pi, unwinds (n123 first run: every N
relaxed to zero). A tube of length k spreads the drop: capacity
N_max grows with tube length (K=4 holds N=1; K=12 holds N=1,2,3).
CHARGE CAPACITY IS A GEOMETRIC PROPERTY OF THE HANDLE: high winding
demands long/thick topology. (Echo, flagged not wired: why nature's
elementary charges are small integers.)

## The ladder (K=12 tube)
E(N)/N^2 = 1.413, 1.343, 1.230 for N = 1, 2, 3: quadratic ladder
with compact SOFTENING (the cos saturates) — U2 PASS. Linear anchor
U3: E(1)/E_lin = 0.91 (compactness only softens) — PASS. And E(1)
drops with tube length (3.14 at K=4 -> 1.41 at K=12): the charge's
self-energy is geometric, as the branch demands.

## U4 — the alpha bracket [recorded, NOT claimed]
Natural charge unit e = 2pi (one turn). Composing with the audit's
alpha = 1/S_inf reading brackets alpha^-1 in [S_inf/2pi, S_inf/pi] =
[78.6, 157.2]. Nature's 137.036 sits INSIDE. The point is not picked
tonight (F10: picking among normalizations post-hoc is numerology);
picking it honestly = the full compact-audit computation — the named
follow-up of this campaign, and n122 item #2a's doorway.