# n46 — pre-registration: the information-metric cost (written before any code)

**Assumption under test** (the last vacuum-owned functional): the drainage
cost is the quantum information metric of the vacuum — the fidelity
susceptibility G of the ground state under coupling perturbations. The
theory's founding sentence taken literally: the budget is entanglement,
not energy. n45 closed the energy route by a theorem (wrong convexity);
G is positive semidefinite BY CONSTRUCTION, so the n45 instability is
structurally excluded — this functional cannot dodge the sign question.

**Stakes, declared now:** attraction -> the mechanism owns gravity's sign
with nothing declared anywhere; the chain "existence costs entanglement,
monogamy conserves it, drainage prices it by information distance, wells
share and attract" is measured end to end. Repulsion -> HEADSTONE 20,
no appeal: energy repels (n41/n45), information repels, and there is no
third vacuum-owned functional. The drainage program dies as a theory of
gravity at the hands of its own gates.

## Model (n40's vacuum, nothing else)
Ring N=200, mu^2 = 1e-6, k_e = 1+dk_e, F = 0.3, sum rule at every site.
Ground state is Gaussian: psi_0(x) ~ exp(-x^T Omega x / 2), Omega =
sqrt(K). Metric (quantum geometric tensor, real part), exact for
Gaussian states:
  G_ef = (1/8) Tr[ Omega^-1 (d_e Omega) Omega^-1 (d_f Omega) ]
with d_e Omega the Frechet derivative of sqrt(K) (Daleckii-Krein divided
differences of sqrt on the eigenbasis — same machinery as n45, applied
to the state instead of the energy). Cost functional: C(dk) =
(1/2) dk^T G dk (a metric has no linear term: dk=0 is its minimum).

## Pipeline and validity gates
- S0 (metric check): G must match central finite differences of the
  EXACT Gaussian overlap |<psi(0)|psi(dk)>| = det(O1)^{1/4} det(O2)^{1/4}
  / det((O1+O2)/2)^{1/2} via -d^2 ln|<.|.>| /d eps^2: rel. error < 1e-5
  on full columns, normalized by max|G| (the n45 lesson: validate
  against the right reference at the right noise floor).
- S1 (structure, characterization): locality of G — diagonal, nn, decay;
  comparison to the declared elastic (alpha=beta=1) and to n45's chi.
- S2 (kernel): eigen-spectrum of G restricted to the feasible direction
  (staggered mode s): s^T G s reported. G is PSD; the one loophole is
  s^T G s = 0 (metric blind to the Gauss sector). If s^T G s < 1e-10 x
  max|G|: the information metric cannot price the drainage either —
  reported as its own result, distinct from repulsion.
- Zero-mode note: Omega has eigenvalue mu = 1e-3 on the uniform mode;
  Omega^-1 is finite. The uniform mode does not couple to edge
  perturbations (verified in S0, not assumed).

## Pre-registered measurement (1D, decisive for SIGN)
Opposite-parity pairs at d = 11,21,31,41,51,61,71,81 (n45's set).
Drainage: minimize C(dk) under the sum rule — on the ring the feasible
set is dk_p + t s, so t* = -(s^T G dk_p)/(s^T G s), closed form.
V(d) = C(dk*(d)); reference g(d) = d(N-d)/(2N).
- PRIMARY — SIGN: slope b of V ~ a + b g(d).
  b > 0: attraction. b < 0: HEADSTONE 20 (death clause, no amnesty).
- SECONDARY — FORM: fracRMS of the Green fit, reported (1D cannot
  discriminate form; MDE as in n45: best screened competitor 0.69%).
- Consistency: V(d) must scale as F^2 (ratio 4.00 +- 0.05 between
  F=0.3 and F=0.15) — the metric is exactly quadratic, so any deviation
  flags an implementation error, not physics.

## Conditional 2D stage (runs ONLY if 1D gives b > 0)
L=24 torus, the n45 vec set, cost (1/2) dk^T G dk with G via
Hessian-vector products or dense (2L^2=1152 edges: dense G is 1152^2 —
feasible directly). Gates: FORM fracRMS < 0.5% vs exact torus Green
(MDE: best screened 1.58%, linear-d 12.6%); SIGN b > 0. Failure of
either after a 1D pass: death — no dimensional amnesty.

## Declared limitations (before the run)
- The metric prices the leading-order state displacement; higher-order
  corrections are not probed (the cost functional is exactly quadratic
  by definition here, unlike n45's Egs).
- One N, one L, one mu; the toy answers the toy's question: does the
  vacuum's information geometry make like consumers attract?
- G is a REAL metric here (real symmetric K, real ground state): the
  Berry-curvature part of the geometric tensor vanishes identically.