# n41 — pre-registration: the dimension gate (written before any lattice code)

**Assumption under test** (the mechanism page's next executioner): drainage
dilution is DIMENSION-BLIND. Gravity's 1/r² is claimed to be nothing but
flux conservation over spheres; therefore the exact n40 mechanism, moved to
2D and 3D bipartite tori with NO new ingredient, must deliver the lattice
Coulomb interaction of the ambient dimension (potential ~ ln r in 2D,
~ -1/r in 3D) with the same attractive sign between like consumers.
Secondary suspicion under test: the 1D attraction rides the staggered
carrier (ker A); if the higher-D attraction vanishes or changes form, the
n40 milestone was a one-dimensional artifact.

## Model (inherited verbatim from n40, nothing new allowed)
- Vacuum: critical harmonic lattice, mu^2 = 1e-6, link couplings k_e = 1+dk_e.
- Sum rule at every site: sum of dk over incident edges = -c_i.
- Drainage: minimize E = (AL/2)|dk|^2 + (BE/2)|D dk|^2 under the sum rule,
  AL = BE = 1 (D = oriented edge-difference operator of the ambient lattice).
- Consumption F = 0.3 per defect (n40's value); linearity check at F = 0.15.
- Observable: V(vec) = ground-state energy Egs of the harmonic lattice with
  couplings 1+dk(pair at displacement vec). Constant offset absorbed by the
  affine fit; no single-defect (non-neutral) solves needed.
- Lattices: 2D L=48 torus; 3D L=12 torus (even L: bipartiteness required).

## Pre-registered displacement sets (opposite parity: odd coordinate sum)
- 2D (25): (1,0),(2,1),(3,0),(3,2),(5,0),(4,3),(6,1),(5,4),(7,0),(7,2),
  (9,0),(9,4),(11,0),(11,4),(13,0),(13,2),(15,0),(15,4),(17,0),(17,2),
  (19,0),(19,4),(21,0),(21,2),(23,0)
- 3D (18): (1,0,0),(1,1,1),(2,1,0),(3,0,0),(2,2,1),(3,1,1),(3,2,0),
  (4,1,0),(3,3,1),(4,2,1),(5,0,0),(4,3,0),(5,2,0),(4,4,1),(5,3,1),
  (6,1,0),(5,4,2),(6,3,0)

## Reference curves (exact, no asymptotic approximations)
g_dim(vec) = G(0) - G(vec), G = zero-mode-removed Green function of the
discrete Laplacian on the SAME torus (FFT-exact). Torus wrap is thereby
handled exactly, not treated as an error.

## Minimum-detectable-effect analysis (program rule from headstone 18,
computed BEFORE this file was committed, Green functions only)
Fractional RMS residual of the best AFFINE fit of each competitor to the
true lattice Coulomb over the sets above:
- 2D: best screened exp(-d/lam), lam scanned 0.5-30: 2.2% (lam=5);
  linear-d (1D law persisting): 11.5%.
- 3D: best screened: 2.4% (lam=1); linear-d: 14.1%.
The measurement is deterministic; thresholds below are systematic-error
budgets with >= 3x margin to the nearest competitor.

## Gates
- G1 (FORM, each dimension): affine fit V = a + b*g_dim over the full set:
  fractional RMS residual < 0.7%; AND the best screened fit (lam scanned
  0.5-30) has residual > 3x the Coulomb fit's; AND linear-d likewise.
- G2 (SIGN, each dimension): b > 0 (energy grows with separation =
  attraction), conditional on G1 passing in that dimension.
- DEATH: G1 or G2 fails in EITHER dimension at F=0.3 AND at F=0.15.
  (The F=0.15 clause is the pre-declared amplitude-distortion amnesty: if
  F=0.3 fails form but F=0.15 passes form AND sign, record as nonlinearity
  of the Egs readout, mechanism survives with a flag. Sign failure with a
  clean fit has NO amnesty: b < 0 anywhere = headstone.)
- Secondary (characterization, not gates): b(F=0.3)/b(F=0.15) ~ 4
  (charge-product scaling); staggered-carrier check below.

## Controls
- C1 (conservation is the cause of the long range): non-conserving control
  — dk = -c_i/(2*dim) on each edge incident to a consumer, zero elsewhere
  (sum rule satisfied AT the consumer, violated at its neighbors: local
  dressing without propagated drainage). Run on the first 8 displacement
  vectors of each set. Expected: fails G1 (short-range defect-Casimir).
  If the control PASSES G1, the gate is not probing conservation and the
  whole experiment is inconclusive — declared now.
- C2 (carrier): Fourier spectrum of the drainage field dk (per edge
  orientation, as a field on sites): report the fraction of |dk|^2 within
  |q - (pi,...,pi)| < pi/4. Characterization only: if attraction + Coulomb
  form hold but the carrier is NOT staggered, the 1D-artifact worry
  dissolves (the staggered mode was a 1D representation, not the cause).
- C3 (emergent neutrality in every D): a same-parity pair must leave a
  nonzero staggered component in the constraint (infeasible sector).
  Report the residual; expected nonzero in both dimensions.

## Declared limitations (before the run)
- One lattice size per dimension; finite size is handled exactly by the
  torus Green function, but no continuum extrapolation is attempted.
- mu = 1e-3 correlation length (~1/mu = 1000) >> L: effectively critical;
  the massless limit is assumed, not scanned.
- Egs is a nonlinear functional of dk; the F=0.15 clause bounds this.
- The drainage COST law remains declared (elastic), as in n40: this gate
  tests dimension-blindness GIVEN the cost law, not the cost law itself.

## RESULTS (same day; script n41_dimension_gate.py, results n41_results.npy)
- 2D: G1 FORM PASS — fracRMS 0.44% (F=0.3) and 0.40% (F=0.15), beats
  best-screened (1.93%) and linear-d (11.5%) beyond the 3x margin.
  G2 SIGN FAIL — b = -3.35e-3: REPULSIVE. F-scaling exact (ratio 3.99 ~ 4).
- 3D: G1 marginal FAIL (1.08% vs 0.7% gate; still beats competitors ~2x),
  G2 SIGN FAIL — b = -1.60e-3: REPULSIVE. F-scaling exact (3.99).
- Controls: C1 non-conserving control fails Coulomb (18.2% / 11.2%) —
  conservation IS the cause of the long range, in every D. C3 neutrality
  confirmed in every D. C2: carrier NOT dominantly staggered in higher D
  (43% in 2D, 7.5% in 3D).
- DEATH CLAUSE MET on the registered observable (sign, both dimensions,
  both F values, no amnesty available).

## POST-HOC DIAGNOSTIC (flagged as such; run after the gate fired)
1. Re-running the COMMITTED n40 script today gives slope -3.455e-3 =
   REPULSIVE in 1D too, and nonlinearity 11.1% (fails its own 10% gate).
   The recorded n40 milestone ("like charges attract") is NOT REPRODUCED
   by the committed code; the run that produced it exists only in a
   session transcript. Same archival hole as n43's runner.
2. The ELASTIC-COST readout E_drain = (AL/2)|dk|^2+(BE/2)|D dk|^2 of the
   same configurations is ATTRACTIVE in ALL dimensions and is the lattice
   Coulomb EXACTLY (fracRMS 0.000% in 2D and 3D — as the quadratic theory
   demands): b = +0.81 (2D), +1.17 (3D), +0.144/step (1D).
3. Reading: the two energy functionals disagree in every dimension, and
   the cost term is ~200x the vacuum-energy term. Dimension-blindness of
   the conserved-flux FORM is CONFIRMED (the strongest version of it:
   exact Coulomb). The SIGN is not an emergent result: it is set by which
   functional is declared to be the physical energy. If the declared cost
   is an energy, attraction dominates trivially — but then attraction was
   an INPUT, not a discovery, and the n40 milestone claim is demoted.
Verdict: headstone 19 — "the attractive sign was measured" dies as a
claim; what is actually measured and survives is: conserved Gauss-
protected Coulomb response in every dimension + emergent neutrality +
conservation-caused long range. The sign now stands or falls with the
derivation of the cost law from the vacuum's own susceptibility (the
program's next declared step) — until that derivation exists, the
mechanism has gravity's FORM and borrows gravity's SIGN.
Numerical note: benign Accelerate-BLAS RuntimeWarnings (overflow in
matmul on the singular KKT) appear in all runs incl. n40; all outputs
verified finite; the exact-Coulomb cost curves double as a solver check.