# n51 — pre-registration: signed drainage under the information metric
# (written before any code)

**The architectural fork.** The bipartite (unsigned) sum rule gives
attraction but its Gauss sector lives at q = pi: every parity-neutral
observable is even in the field, no linear clock response at any order
(n47b, n50). Moving the Gauss sector to q = 0 — a SIGNED sum rule,
divergence form — makes the drainage field smooth: clocks can couple
linearly. n37 killed the signed channel for attraction under the
ELASTIC cost; the elastic cost is dead (n45) and the information metric
has already reversed one energy-functional verdict (n41 -> n46). This
experiment runs the signed rule under the information metric and asks
BOTH questions at once.

**Declared expectation (before the run):** the Poisson analogy predicts
like charges REPEL under any positive-definite quadratic cost with a
signed constraint (E_int = 2 f^2 ker(d), decreasing). If that holds
while the clock channel turns linear, the recorded result is the
DRAINAGE DICHOTOMY: within quadratic drainage, attraction (bipartite,
no geometry) and linear clocks (signed, repulsion) exclude each other.
Measured from both sides, that is an impossibility result worth its own
page. If instead like charges ATTRACT, the mechanism gains Newton and
Einstein's clocks simultaneously and everything before n51 was prologue.

## Model
Ring N=200, mu^2 = 1e-6. Signed incidence: (A_s dk)_i = dk_i - dk_{i-1}
(divergence). Compact neutrality: consumption is jellium-compensated,
c = f(delta_a + delta_b) - 2f/N. Cost: the information metric G (n46).
Drainage: min (1/2) dk^T G dk s.t. A_s dk = -c.

## Gates
- Z1 (zero mode): ker A_s = uniform edge mode; W_s = A_s G^-1 A_s^T has
  its zero at q = 0 exactly; dispersion W_s(q)/q^2 finite as q -> 0.
- Z2 (SIGN): V(d) for like pairs, d = 11..81 (parity now irrelevant,
  all d): slope vs ring Green. b < 0 = repulsion (expected); b > 0 =
  attraction (the resurrection). Either way: recorded, no amnesty.
- Z3 (FORM): Green-fit fracRMS reported (Coulomb expected either sign).
- Z4 (CLOCKS): background mass pair {40, 139} (jellium), F_M in
  (0.15, 0.3); the three n47 clock constructions, cell-averaged (cells
  now trivial — no staggering — averaging kept for comparability):
  shifts vs the SMOOTH potential lambda(r); gates: linearity exponent
  in F_M = 1.00 +- 0.05; proportionality fracRMS < 5%; universality
  spread of kappa across C1/C3 < 10% (C2 reported; it is the local
  probe). PASS = the signed architecture has GR-shaped clocks.
- Z5 (the dichotomy verdict): Z2 repulsion + Z4 pass -> the dichotomy
  is measured from both sides and goes on the site as a theorem-shaped
  result (analytic proof owed as n51a). Z2 attraction + Z4 pass ->
  the unification; every downstream page changes. Z2 repulsion + Z4
  fail -> the signed architecture is sterile; bipartite Newton stands
  alone.

## Declared limitations
- 1D, static; the signed rule's relativistic story (n37's causality
  wall was about the VECTOR gauge cone) is untouched here — this is
  statics of sign and clocks only.
- Jellium compensation is the declared treatment of compact
  non-neutrality (the same treatment n40 used).

## RESULTS (same day; n51_signed_drainage.py)
- Z1 PASS: zero mode exactly at q=0 (2e-14), quadratic dispersion,
  S_signed = 32.4.
- Z2: REPULSIVE (b = -2.78e-3), machine-Coulomb (Z3: 0.019%) — the
  Poisson expectation confirmed. Like consumers repel under the signed
  rule, information metric or not.
- Z4: the clocks turn EXACTLY LINEAR — F-exponent 1.000, 1.000, 0.998
  across all three constructions (the smooth field delivers what the
  staggered one could not) — but the registered proportionality and
  universality gates FAIL (29-34% vs lambda; C1=C2 to 0.5%, C3 at
  half), and the diagnosis is structural: the shifts track the local
  FLUX dk (a step between sources), not the POTENTIAL lambda (a tent).
- THE DRAINAGE DICHOTOMY, measured from both sides and SHARPENED:
  (i) bipartite architecture: attraction, but every parity-neutral
  observable is even in the field — no linear clock response at any
  order (n47b, n50);
  (ii) signed architecture: linear clock response, but like masses
  repel — and even the linear response reads the field, not the
  potential.
  Root cause, one sentence: in GR the potential is itself a local field
  (g00), while in ANY flux-conservation drainage the potential is a
  nonlocal integral of the flux — local matter can only couple to the
  flux, so potential-coupled clocks (gravitational redshift) cannot
  arise in this class of theories, in either architecture. Geometry is
  closed for quadratic drainage. What a resurrection would require,
  named precisely: the coupling field itself must BE the potential,
  with consumption sourcing its Laplacian — which is no longer emergent
  drainage but a metric put in by hand. n51a (analytic proof of the
  dichotomy) remains owed; the static program's structural map is
  otherwise complete.