# n164 — the four-budget kernel: the breather watched, the TT pair
#         dark, two residuals named
# (2026-07-11, mandate tick. The organ forced by headstone 34, built
#  in first draft — and it works where it had to.)

## The construction
9 edge directions (3 axes + 6 face diagonals: all h_ab present);
4 constraint rows: energy (unsigned incidence, the founding budget)
+ 3 momentum rows (ansatz: direction-weighted signed incidence,
d^i(1 - e^{-iq.d}) - declared as ansatz, derivation owed).

## What works (the two things that HAD to)
1. THE OLD BREATHER IS WATCHED: |C.v| = 0.85-2.2 - the momentum
   rows catch exactly the mode that killed the single-constraint
   class. Cooled, stiffened, dead as a radiator: PSR B1913+16's
   demand is MET by the four-budget structure.
2. THE FULL TT PAIR IS DARK: '+' (trace 0.000, transverse) and 'x'
   (pure h_xy) both in the kernel - the second tensor polarization
   arrives with the diagonal edges, as predicted in n157.

## The residuals (named, next)
- A longitudinal-VECTOR dark pair (h_xz + h_yz, q-long ~ 1.0):
  unsourced by planar binaries (I_xz = I_yz = 0) - GR's gauge-mode
  analogs; their thermostat status and generic-source coupling to be
  settled (conservation should render them non-radiative; check).
- MODE 3: a zz-longitudinal hybrid carrying partial trace
  (0.24-0.56) with xx+yy content ~ 0.33-0.39: THE RESIDUAL RISK -
  if its transverse-trace content is same-sign, eccentric binaries
  source it at ~10% of the old breather's overlap, and the
  Hulse-Taylor bound re-engages at reduced pressure. Its fate hangs
  on (i) the sign structure (extract next) and (ii) the momentum-row
  DERIVATION (the ansatz freedom must be closed from the Action -
  the honest route; design-spec tuning is forbidden).

## The scoreboard
dark physical modes: 4 (target 2) - down from "always d-1, always
one fatal scalar" to "TT pair + two constrained-looking residuals".
The constraint-counting road to GR's radiation content is one
derivation (the momentum incidence) and one sign-extraction from
done. The mandate continues.