# n121 — fermions, interrogated
# (2026-07-11; the branch had spin-half and the mass gap; FERMION means
#  exclusion. Toy: exact sectoral demo below; the exchange step is
#  Finkelstein-Rubinstein, cited not rerun.)

## The demo (exact, residual 0.0)
Ring of l = 9 with k antiperiodic bonds (cuts), T = translation by one
site, [H,T] = 0 identically:
  0 cuts: T^l = +I, ground mode 0.0000 (gapless)
  1 cut : T^l = -I, ground mode 0.1206 (gapped)
  2 cuts: T^l = +I, ground mode 0.0000 (two cuts gauge away)
  3 cuts: T^l = -I, ground mode 0.1206
A 2pi rotation is T^l. On every odd-twist handle it is -1 ON EVERY
STATE OF THE SECTOR — not a property of a special mode; anything made
of that handle carries the minus sign. Even twists are untwisted in
disguise. TWIST PARITY IS A Z2 CHARGE, and it controls BOTH:
  statistics parity (2pi = -1)  AND  the mass gap (no zero mode).

## From -1 to Pauli: the Finkelstein-Rubinstein step [T, cited]
FR 1968: in 3D, exchanging two identical solitons is homotopic to a
2pi rotation of one of them. With the sectoral -1 above: odd-twist
handles are FERMIONS (exchange antisymmetry, exclusion), even-twist
are bosons. Spin-statistics is orientability arithmetic.
NOT DERIVED (flagged): that our lattice complex supports the FR
homotopy — needs the 3D embedding of handle configurations; plausible
(3D complexes embed), unproven here. The branch's honest gap.

## What one bit buys (the fermion, assembled)
One bit — is the handle orientable? — delivers, with receipts:
  spin-half        (half-integer mode ladder, n115/n118)
  rest mass        (no zero mode: the Möbius gap, n118 — m = twisted hum)
  2pi = -1         (sectoral, exact, today)
  exclusion        (via FR)
  indestructibility (class property: audit-dark, n120 barrel A)
A fermion is a non-orientable handle; everything else follows.
And "identical particles" stops being a mystery: two electrons are
indistinguishable because they are the same topology — identity of
particles is identity of classes, not similarity of objects.

## Confrontations with nature (the reflex)
1. NO MASSLESS FERMION, STRUCTURALLY: every odd-twist sector is
   gapped — a massless fermion is impossible in this branch. The
   Standard Model's massless neutrino was therefore impossible;
   oscillation data (Delta-m^2 > 0) retro-agrees. LIVE EDGE: the
   LIGHTEST neutrino could still be exactly massless in current
   data — the branch says it is not. Registered as prediction 27.
2. PAULI IS EXACT, FOREVER: statistics here is a homotopy class —
   topology does not do "slightly". Every parametrized small
   violation (quon models, beta^2 deformations) is structurally
   zero. VIP-2/Gran Sasso keep returning null at every future
   precision. Registered as prediction 26. Kill: any confirmed
   Pauli-principle violation.
3. Massive bosons (W, Z, Higgs) are NOT excluded: even-twist
   sectors have massive excited modes; only masslessness is
   fermion-forbidden. The photon (massless) sits in the even
   sector: consistent. Which even structure is W/Z: unpriced.
4. Composites: twist parity is additive mod 2 — three odd-twist
   constituents make an odd (fermionic) baryon, two make a boson:
   the quark arithmetic of hadron statistics comes out of Z2
   bookkeeping. (Color itself: absent, honest — headstone 25's
   sector stays silent.)

## Predictions registered from this interrogation
26. Pauli exclusion is exact at every precision, forever.
    Kill: one confirmed violation (VIP-class anomalous X-rays,
    forbidden atomic states, forbidden nuclear transitions).
27. There is no massless fermion: the lightest neutrino has m > 0.
    Kill: a proof that the lightest neutrino is exactly massless
    (or any confirmed massless fermion).