The 68.5% at N=500 was driven by shared-shell opposite-sign pairs (38.3% of early pairs, 78.9% opposite). At N=100k that effect is diluted. The dominant pattern: runs of length 2 in same sign (58.8% of all runs). P(+->-)=0.4726 < 0.5 at scale. Same-sign persistence is the global truth.
All Findings
What 100,000 zeros actually say
F-FLIP - FINDING
The alternation rate INVERTED at N=100k: 68.5% to 47.26%
At N=500, alternation was 68.5% (z=+8). At N=100k: 47.26% (z=-17.35). REVERSED. The 68.5% was inflated by shared-shell opposite-sign pairs in early zeros (38.3% of pairs, 78.9% opposite-sign). At scale: P(+->-)=0.4726, P(->=+)=0.4725 — both below 50%. Same-sign persistence is the global truth.
F-RANK - FINDING
gamma1 is NOT rank-1 at N=100k — n=21962 has |r|=10^-8
n=21962 (gamma~19586.26) is essentially on its shell to float64 precision. u=10^-8. gamma1 (u=0.00155) is ~154,000x farther. u1/E[u_min at N=100k]=98.96. Correct claim: gamma1 is rank-1 among the FIRST 500 zeros only.
F-PERIOD2 - FINDING
The sign process is period-2: lag-2 ACF=-0.407, alt-indicator ACF=-0.606
Lag-1: -0.029 (weak). Lag-2: -0.407 (STRONG). Lag-4: +0.370 (STRONG). Crossings anti-cluster (alt-indicator ACF=-0.606). 58.8% of runs have length 2 vs 24.9% for Markov. The zeros travel in (-,+) or (+,-) PAIRS with 2nd-order memory, not simple Markov alternation.
F-GAPFLIP - FINDING
Alternating pairs have LARGER gap than same-sign (mean 0.872 vs 0.639, t=123)
Same-sign consecutive zeros are CLOSER TOGETHER. Shared-shell pairs (close in gamma, opposite sign 78.9%) create local alternation but at global scale: small gap -> same-sign persistence. Gap and sign are structurally linked.
F-UNIFORM - CONFIRMED
Magnitude uniformity confirmed to 1.32% max deviation at N=100k
Perfect phase divider. P(r>0)=P(r<0)=0.5 to 4 decimal places at scale.
F-TAXONOMY - CONFIRMED
All 9 taxonomy classes match uniform null within 2%
FLAGSHIP 0.979, DEEP 0.987, STRONG 1.001, LOCKED 0.987, ANTI-SHELL 0.988. Magnitude is flat across all tiers. Shell-attraction hypothesis has zero support.
Negative at lag-2 (-0.407): two steps back anticorrelated. Positive at lag-4 (+0.370): correlated. Classic period-2 pattern. Alt-indicator ACF=-0.606: crossings anti-cluster (a crossing makes next crossing less likely). n=21962 is rank-1 at N=100k with |r|=10^-8 - gamma1 is NOT globally special.
Taxonomy - N=100k - All Classes Match Null
Every class within 2% of uniform — magnitude is genuinely flat
class
observed
expected (null)
ratio
FLAGSHIP
623
636.6
0.9786
DEEP
3140
3183.1
0.9865
STRONG
12743
12732.4
1.0008
LOCKED
987
1000.0
0.987
ANTI-SHELL
4940
5000.0
0.988
MID-PHASE
50030
50000.0
1.0006
FLOOR
49997
50000.0
0.9999
CEILING
50003
50000.0
1.0001
Theory After N=100k
Magnitude is generic. The structure is in the 2-step memory.
OLD: zeros seek pi-shells / alternation is the primary law
NEW: The nontrivial zeros define a shell-centered phase process. Magnitudes u_n are genuinely Uniform(0,1). Signs are balanced (50/50). The signed residue r_n exhibits a dominant period-2 oscillatory structure: zeros travel in matched pairs of opposite sign with 2nd-order memory. The structure is not in the magnitude; it is in the 2-step rhythm of the sign.
CONFIRMED AT N=100k
Magnitude ~ Uniform(0,1) exactly
Sign balance 50/50 to 4 decimal places
All taxonomy classes match null <2%
No shell attraction in magnitude
Dominant period-2 structure (lag-2 ACF=-0.407)
Crossings anti-cluster (alt-indicator ACF=-0.606)
Same-sign persistence at scale (P(+->-)=0.473 < 0.5)
REVISED / CORRECTED
gamma1 rank-1: first 500 only, not global
68.5% alternation: N=500 artefact
Simple Markov model: WRONG (period-2 required)
Alternating pairs have LARGER gap (reversed)
Same-sign pairs cluster at small gap
Five canon sentences after N=100k:
Magnitude is flat; the 2-step rhythm is the law.
The shell is a reference plane for a period-2 oscillation, not a target.
gamma1 is the champion of the first 500 zeros, not the universe.
Sign is balanced; pairing is the structure. The signed residue process has 2-step memory. That is the deepest pattern in 100k zeros.