# n50 — pre-registration: the second-order metric (written before any code)

**The resurrection condition of headstone 20, made testable.** At first
order the drainage theory is quadratic: superposition is exact, a test
dipole's self-binding is position-independent, clocks see only tides —
no redshift channel, structurally. But the information metric is STATE-
DEPENDENT: G = G[vacuum], and a mass deforms the vacuum. At second
order the local coupling a test system feels — its local 1/G_N — can
vary with position in the mass's field. That is exactly the structure
of gravitational redshift: local physics rescaled by the potential.
This experiment either finds the metric channel at second order or
closes the geometry question for this mechanism (headstone 21).

## Model
Ring N=200, mu^2 = 1e-6. Background mass: neutral pair {40, 139},
F_M in (0.15, 0.3). Background drainage dk_M via the bare metric G0
(n46). DEFORMED metric G' = information metric recomputed at K(dk_M)
(same Daleckii-Krein machinery, evaluated at the deformed vacuum).
Probe: test dipole (opposite-parity consumer pair, f_t = 0.01, linear-
response regime) at cell position s.

## Registered observable
B(s) = self-binding energy scale of the test dipole computed in the
deformed background minus in the bare vacuum:
  B(s) = [V'_pair(delta; s) - V0_pair(delta)] / V0_pair(delta),
where V_pair is the info-metric interaction of the test pair (the n46
machinery with metric G' and the sum rule), delta = dipole separation.
Cell-averaged over the two parity placements at s (declared: the same
averaging that made n47's clocks parity-blind — if the channel survives
cell averaging it is a genuine smooth channel, the thing headstone 20
said the theory lacks).

## Gates
- M1 (existence of the smooth channel): B(s), cell-averaged, is nonzero
  beyond numerical floor (|B| > 1e-8) at cells within 20 of the mass and
  varies with s (range > 3x floor). Fails -> HEADSTONE 21: the theory
  has no second-order metric channel either; "Newton yes, Einstein no,
  structurally" goes on the page at full prominence.
- M2 (linearity in the source): B(s) at F_M = 0.3 vs 0.15: ratio
  2.0 +- 0.1 at the 5 cells nearest the mass (this is what GR demands
  and what first-order clocks failed).
- M3 (potential tracking): affine fit of B(s) against the cell-averaged
  ENVELOPE of dk_M over cells 3..40 from the mass: fracRMS < 10%.
- M4 (universality — EP of clocks at second order): dipole sizes
  delta = 1 and delta = 3: fractional shifts agree within 10% ->
  the channel is a property of PLACE, not of the probe: a metric.
  Disagreement > 50% -> probe-dependent: a force, not geometry —
  recorded as such.
- Sign: reported, not gated (which sign corresponds to GR redshift is
  an interpretation question — deeper wells weakening local binding
  mimics slower clocks; the sign is recorded and discussed after).

## Declared limitations
- Second order only: G' is recomputed exactly at the deformed point,
  but the background dk_M is still computed with G0 (the first
  iteration of the self-consistency loop; full fixed-point iteration is
  declared future work if M1 passes).
- 1D, static, envelope-level; units of the shift are lattice-internal.
- The numerical floor (1e-8) is calibrated by the F_M = 0 control,
  which must return |B| < 1e-10 identically (declared validity check).

## RESULTS (same day; n50_second_order_metric.py)
- Control: 2e-14. M1 PASS: the second-order channel EXISTS — B = +6.7e-2
  at the nearest cell (binding strengthened in the well), four orders
  above floor, smooth after cell averaging.
- M2 FAIL: B(0.3)/B(0.15) = 4.11 at every near cell — QUADRATIC in the
  source, not linear. M3 FAIL (17% vs the linear envelope; consistent
  with tracking envelope^2). M4 PARTIAL (11%).
- THE STRUCTURAL CONCLUSION, stated plainly: the parity obstruction
  survives to second order. The deformed metric's shift that is linear
  in dk_M is staggered; cell-averaged (parity-blind) observables cancel
  it and see only the even part — response goes as M^2, never M. First
  order (n47b) and second order (n50) fail the same way because they
  ARE the same way: every local, parity-neutral observable in this
  architecture is even in the staggered amplitude. CONJECTURED THEOREM
  (n51a, analytic): on a bipartite substrate with the Gauss sector at
  q = pi, no cell-averaged observable responds linearly to consumption.
  If proven, geometry is closed for this architecture — not by a failed
  fit but by symmetry.
- THE ONE ARCHITECTURAL ESCAPE, named: move the Gauss sector to q = 0 —
  a SIGNED sum rule (divergence-form incidence), where the conserved
  flux is smooth and clocks could couple linearly. n37 killed the
  signed (vector) channel for ATTRACTION under the elastic cost — but
  the elastic cost is dead anyway (n45), and nobody has run the signed
  sum rule with the INFORMATION METRIC. The info metric already
  reversed one verdict the energy functional got wrong (n41 -> n46).
  n51: signed drainage under the information metric — sign of the
  interaction AND linearity of the clock channel, one experiment. If
  like charges attract there too, the theory keeps Newton AND gains
  Einstein's clocks; if they repel, this architecture's choice is
  final: attraction OR geometry, not both.