# n137 — the selection principle: minimality by capacity,
#         and the keV dark quantum
# (2026-07-11; the debt that blocked Omega_DM and the mass spectrum,
#  paid with the branch's own physics.)

## The principle
PARTICLES ARE MINIMAL STABLE TOPOLOGICAL OBJECTS. Stability is not
free: a handle carrying winding N needs per-edge phase drop < pi
(the capacity bound, discovered in n123 when the thin handle
dropped its quantum). Minimality + capacity therefore SELECT a
definite object — no continuous family survives.

## The measurement (threshold, exact)
Bridge k = 0, 1, 2: the unit winding UNWINDS (capacity fails).
Bridge k = 3: HOLDS. The minimal charged handle:
   cycle length 6, winding self-energy E(1) = 3.634,
   twisted gap m_t = 2 sin(pi/12) = 0.5176 (substrate units).
CONVERGENCE: 6 is also the minimal EVEN cycle — the bipartite (C-
structure) discipline of n134 and the capacity bound independently
select the same object. Two constraints, one particle.

## The dark quantum's mass [B/C — anchor extension declared]
Unit: the audit quantum hbar*gamma = 3.73 keV (electron anchor, the
registered bet). The minimal handle's scales:
   winding self-energy: 3.634 x 3.73 keV ~ 13.5 keV
   twisted gap:         0.518 x 3.73 keV ~  1.9 keV
THE DARK QUANTUM SITS IN THE keV BAND (1.9 - 13.5 keV depending on
which excitation carries the relic). Declared freedoms: the 2D toy
geometry (3D minimal handle may shift the O(1)s); the identification
of the substrate energy unit with hbar*gamma (the anchor extension —
same bet as the knee's).

## Two confrontations, both pass tonight
1. LYMAN-ALPHA (the warm-DM guillotine for keV masses): thermal keV
   relics free-stream and are excluded (m > 2-3 keV thermal). But
   KZ relics are BORN COLD — defects frozen at rest in the substrate
   frame (n136), no thermal velocity: keV DEFECT dark matter is
   cold, and the free-streaming bound does not apply. The mechanism
   that set the abundance also buys the mass its alibi.
2. X-RAY DECAY LINES (the sterile-neutrino keV killer): windings are
   topologically stable — NO decay line, ever. XRISM's clean
   2.5-15 keV band (n111 attack 8) is a PASS here, not a threat —
   and it retro-kills the sterile-neutrino competitor in the same
   window this framework now occupies.
Prediction 28 gains its third blade: 28c — NO dark-matter decay
line at any depth, while the mass sits in the keV band. Kill: a
confirmed DM decay line (topological stability has no mechanism
for it).

## The ledger after tonight
The dark sector now has: a birth (n134), a self-interaction exponent
(v^-4/3, n135), an abundance mechanism (KZ, n136), a selection
principle (capacity minimality), a mass band (keV, anchored), two
alibis (cold birth; no decay), and three kill conditions. Unpriced:
the 3D O(1)s; the quench rate (FRW debt); the amplitude cross-check
sigma/m against the dwarf band with THIS mass — the next
computation, and a dangerous one: it has no free parameter left.