# n45 — pre-registration: the self-consistent drainage (written before any code)

**Assumption under test** (promoted to the program's central question by
headstone 19): the drainage cost law is not free to be declared — it IS
the critical vacuum's own susceptibility. If the drainage field is the
minimum of the vacuum's OWN energy under the monogamy sum rule, the sign
of the pair interaction becomes an output with nothing declared anywhere.
This is the sharpest fork the program has faced: attraction restores the
n40 milestone on ground that cannot be taken away; repulsion is headstone
20 and the drainage mechanism dies as a theory of gravity.

## Model (n40's vacuum, nothing else)
Ring N=200, mu^2 = 1e-6, k_e = 1+dk_e, consumption F = 0.3 per defect,
sum rule (A dk)_i = -c_i at every site. NO declared cost. Two readouts
of the vacuum's own energy:
- QUADRATIC: E2(dk) = g·dk + (1/2) dk^T chi dk, where g and chi are the
  exact gradient and Hessian of Egs at dk=0 (Daleckii–Krein / 2nd-order
  perturbation theory). The linear term is CONSTANT on the feasible set
  (sum rule fixes sum(dk) = -F for every configuration) and drops out of
  all comparisons — the sign question is chi's alone.
- EXACT: on the ring the feasible set is dk_p + t·s (s = staggered mode,
  the 1-dimensional kernel of A): minimize the full Egs(dk_p + t·s) over
  t numerically. All orders, no expansion.

## Pipeline and validity gates
- S0 (solver check): chi from perturbation theory must match a finite-
  difference Hessian on a random 5x5 block to rel. error < 1e-5.
- S1 (structure, characterization): locality of chi — diagonal, nearest-
  neighbor, decay with edge distance; effective (alpha, beta) if the
  elastic form emerges; direct comparison to n40's declared alpha=beta=1.
- S2 (stability): s^T chi s > 0 AND the exact line-minimum interior to
  the physical domain (all k_e > 0.05). If either fails the vacuum is
  unstable to dimerization (Peierls sector): reported as its own result,
  "the critical vacuum spontaneously dimerizes; consumption biases an
  order parameter" — a reformulation-forcing discovery, NOT a fit
  failure and NOT a headstone. Declared now to prevent post-hoc spin.

## Pre-registered measurement (1D, decisive for SIGN)
Opposite-parity pairs at d = 11,21,31,41,51,61,71,81; V(d) from both
readouts; reference g(d) = d(N-d)/(2N) (exact ring Green function).
- PRIMARY GATE — SIGN: slope b of the affine fit V ~ a + b·g(d).
  b > 0 in BOTH readouts: attraction, the vacuum owns the sign.
  b < 0 in BOTH readouts: HEADSTONE 20 (death clause; no amnesty —
  there is no declared functional left to blame).
  Readouts disagree: the quadratic expansion is insufficient; the EXACT
  readout is pre-declared to carry the verdict (it is the vacuum's
  actual energy).
- SECONDARY — FORM (weak in 1D, declared so): fracRMS of the Green fit,
  reported; MDE computed before this file: best screened competitor sits
  at 0.69% in 1D (nearly degenerate — 1D cannot discriminate form; the
  form test lives in the 2D stage). Linear-d separation 6.1%.

## Conditional 2D stage (runs ONLY if 1D gives b > 0 in the exact readout)
L=24 torus, vecs (1,0),(2,1),(3,2),(5,0),(4,3),(6,1),(7,2),(7,4),(9,0),
(9,4),(11,0); feasible space is (L^2+1)-dimensional: minimize full Egs
by projected gradient with the analytic gradient, or the quadratic form
via Hessian-vector products — method choice free, result gated:
- FORM: fracRMS < 0.5% vs exact torus Green (MDE: best screened 1.58%,
  linear-d 12.6% — 3x margins hold).
- SIGN: b > 0. Failure of either in 2D after a 1D pass: dimensional
  amnesty is NOT available; the mechanism must work in every D. Death.

## Declared limitations (before the run)
- F = 0.3 may exceed the quadratic regime; the exact readout covers this
  in 1D; in 2D the projected-gradient minimum is also exact up to
  numerical tolerance (declared: gradient norm < 1e-8 in feasible
  directions, or 500 iterations with monotone energy, whichever first).
- One N, one L, one mu: no continuum claims; this is the toy's verdict
  on the toy's question — does the vacuum's own response attract?
- The Egs zero mode (uniform translation) does not couple to any edge
  perturbation (difference vectors are orthogonal to it) — verified in
  S0, not assumed.

## RESULTS (same day; script n45_selfconsistent.py, results n45_results.npy)
- S0 PASS after one fix to the validation itself (the energy-based FD
  test drowned in machine noise at |chi|~1e-6; replaced by central
  differences of the analytic gradient): |chi - FD|/max|chi| = 3.8e-9.
  Gradient exact to 7 digits, uniform across edges to 5e-16.
- S1 structure: chi is LOCAL with fast decay (diag -0.137, nn -0.0093,
  2nd -0.0012, 5th -8e-5) — the elastic form's SHAPE — but every entry
  and every eigenvalue is NEGATIVE: spectrum in [-0.159, -0.120].
- S2 FIRED: s^T chi s = -0.12 < 0, and the exact line-minimization hits
  the physical boundary (k -> 0.05) at every d, with the boundary-pinned
  V(d) linear in d (Green fracRMS 6%), i.e. dimerization depth, not a
  drainage interaction. Sign gates moot under S2, as the validity
  ordering requires.
- THE RESULT IS A THEOREM, not a fit: sqrt is operator concave and K is
  linear in the couplings, so Egs = (1/2)Tr sqrt(K) is CONCAVE in dk —
  chi <= 0 always, any N, any dimension, any mu. The numerics confirm
  to 3.8e-9 what the operator inequality guarantees.
Verdict (the pre-declared S2 branch): NOT headstone 20 — stronger. The
drainage cost can NEVER be derived from the vacuum's ground-state
ENERGY: that functional has the wrong convexity, provably, everywhere.
The critical harmonic vacuum is unstable to dimerization at fixed total
coupling; zero-point energy DESTABILIZES the drainage instead of
pricing it. What survives: chi's locality (the elastic form's shape was
right; only its sign axis was wrong). The reformulation is therefore
pointed, and it points at the theory's own founding sentence: the
budget is ENTANGLEMENT, not energy. The one vacuum-owned functional
with guaranteed positive convexity is the information metric (fidelity
susceptibility / Bures metric of the Gaussian vacuum under coupling
perturbations — positive semidefinite BY CONSTRUCTION, and exactly
computable for Gaussian states). n46: derive the cost from the vacuum's
information metric; the sign of gravity is then the sign of the
information-geometric pair interaction. If THAT is repulsive, headstone
20 falls with no further appeal — there is no third vacuum-owned
functional to try.