Three core batteries plus six combo batteries. Together they cover: seed exactness,
asymptotic correction, theory verification, scale comparison, ACF/memory depth,
Gram geometry, model comparison, Li criterion, and rolling transitions.
The minimum to understand the project is B2+B3+D. The full suite runs in ~5 minutes.
THE KEY NUMBER
AR-2: 80.6%
vs M1: 43.9% on held-out 50k
LR statistic
19,635
df=2 · p≈0 · AR-2 vs Markov(1)
Alternation flip
N≈60,000
>50% flips to <50% (Battery F)
γ₁ rank at N=100k
#139
was #1 at N=500 (Batteries A, F)
lag-2 ACF(r)
-0.407
dominant over lag-1=-0.029 (Battery B)
ACF(|r|) lag-2
-0.724
NEW FINDING — magnitude has memory too
Alt-ind lag-1 ACF
-0.606
crossings anti-cluster (Battery B)
KS p(uniform)
0.9839
magnitude null confirmed (B2, B3)
Run order: B1 → B2 → B3 → A → B → C → D → E → F.
Skip B1 and E to skip mpmath (~1 min total). Full suite ~5 min.
BATTERY 1
B1 · Seed
N=200 · dps=50 · 57 checks · ~60s
High-precision primer. Verifies the shell decomposition coordinate system, catches the z=3.441 alternation crossing at N=200, kills the sum=π hypothesis (t=−23.26, p≈10⁻²⁶). The myth-generation battery — where γ₁ rank-1 looks plausible.
✓49 ✗2 ⚠6 — 2 fails are findings, not errors
The shell is the center; the law is in the crossing.
BATTERY 2
B2 · Asymptotic
N=100k · Odlyzko · 12 blocks · ~5s
The decisive empirical battery. Loads Odlyzko's 100k zeros. Kills the 68.5% alternation story (reverses to 47.26%, z=−17.35). Finds lag-2 ACF=−0.407 dominant. Kills γ₁ rank-1.
✓~35 ✗~5 — fails are findings (reversal, Markov, rank)
Distance is null. Memory is law.
BATTERY 3
B3 · Theory Verify
N=100k · ~40 checks · ~5s
Post-N=100k theory verification. 6 CONFIRMED + 4 REVISED + 5 CANON TESTS. The manifesto-to-metrics battery. Tests each canon sentence quantitatively. THE KEY NUMBER: AR-2 vs Markov(1) improvement on held-out 50k zeros.
✓40+ ✗0 — 5 canon sentences all pass
Every canon sentence must survive quantification.
BATTERY A
Combo A · Scale
N=200→500→1k→5k→100k · ~10s
Direct scale comparison. Tracks alternation, lag-2 ACF, KS p, γ₁ rank, sign balance, run-2 fraction across all N. Shows exactly which findings flip and which are scale-invariant.
✓9+ — flip detection, scale invariants confirmed
Scale corrects myth.
BATTERY B
Combo B · ACF
ACF/spectral/PACF/AR order · ~5s
ACF to lag-50. Optimal AR order by AIC (finds order 8, confirms AR-2 dominant). PACF via Yule-Walker. Power spectral density top-10 peaks. NEW FINDING: ACF(|r|) lag-2 = −0.724 — magnitude has 2-step memory too.
✓8 ✗1 NEW FINDING — magnitude ACF(lag-2) is non-zero
The process remembers two steps.
BATTERY C
Combo C · Gram
Gram's law · deserted shells · ~5s
Shell gap distribution (shared/solo/deserted). Gram's law failures (76%). ANOVA across gap types (p≈0). Gap-conditional sign persistence. Weyl law accuracy. Deserted positions n=1,3,6,8,18 confirmed.
✓10 ✗0 — all Gram geometry confirmed
Local geometry writes the sign.
BATTERY D
Combo D · THE KEY NUMBER
M0/M1/AR-2/AR-3 · held-out 50k · ~30s
THE KEY NUMBER. Trains 4 models on first 50k zeros, tests on second 50k. AR-2 accuracy = 80.6%. M1 = 43.9%. Improvement = +36.7 pp. LR = 19635, p ≈ 0. AR-3 marginal (+0pp). AR-2 is the dominant memory step. This is the paper number.
✓6 ✗0 — AR-2 wins cleanly, LR p≈0
The process remembers two steps. AR-2 is the dominant step.
BATTERY E
Combo E · Li / λₙ
Li criterion · mpmath · ~120s
Tests Li's criterion: λₙ > 0 for all n is equivalent to RH. λ₁ closed form = 0.023095708966 > 0. Partial sum convergence (from above). λ₁ through λ₅ all > 0 at N=200 partial sum.
✓6 ✗0 — λ₁ > 0 confirmed, floor holds
λ₁ > 0. The floor holds.
BATTERY F
Combo F · Rolling
Scale transitions · flip point · ~15s
Rolling window finds exact transition points. Alternation flip: N≈60,000 (not 2,000 as naively expected). γ₁ rank at N=100k = #139. Sliding window confirms stability. Sign balance stable throughout.
✓7 ✗0 — flip at N≈60k confirmed
Scale reveals; myth dissolves.
CONFIRMED — ALL BATTERIES
Shell decomposition is exact. Magnitude is null-like at all N.
γₙ = Sₙ − rₙ is exact arithmetic. KS p > 0.50 at every checkpoint
from N=100 to N=100k. All taxonomy classes within 2% of uniform null. The shell is not
an attractor at any scale.
THE KEY NUMBER — BATTERY D
AR-2 accuracy = 80.6%. Markov(1) = 43.9%. Improvement = +36.7 pp.
Trained on n=1..50,000. Tested on n=50,001..100,000. LR=19,635, p≈0.
AR-3 improvement over AR-2: only −0.0004 pp. AR-2 is the dominant memory step.
This is the number for any paper.
NEW FINDING — BATTERY B
ACF(|rₙ|) at lag-2 = −0.724. Magnitude has 2-step memory.
The published story said magnitude is structureless (KS confirms uniformity).
But |rₙ| itself has strong lag-2 autocorrelation. The sign process is not the only
carrier of structure. The magnitude process has its own 2-step rhythm.
This was not in the theory before today.
TRANSITION — BATTERY F
Alternation flip: N≈60,000. Not 2,000.
The flip from above-50% to below-50% alternation happens at
N≈60,000, not the small-N regime. This refines the earlier estimate. The waltz
structure takes 60,000 zeros to fully assert itself.
# Install once:
pip install mpmath scipy numpy statsmodels requests
# Fast path (skip mpmath batteries, ~1 min):
bash run_all_batteries.sh --skip-b1 --no-li
# Full suite (~5 min):
bash run_all_batteries.sh
# Single battery:
python3 combo_battery_d_models.py # THE KEY NUMBER standalone
# Log directory: battery_logs/battery_D_TIMESTAMP.log
# Look for: AR-2 vs M1 improvement line
# Pack combos:
make seed → Battery 1
make truth → Batteries 2 + 3
make memory → Batteries B + D
make scale → Batteries A + F
make gram → Battery C
make li → Battery E
make all → all 9