Thirty-One Signals, Six Bets — §3 · Thirty-one signals are six bets

Thirty-One Signals, Six Bets · part 6 of 10

§ 3

The central result, continued

3.2Twenty-five of thirty-one are one cluster

Figure 2 — Eigenvalue spectrum of the IC correlation matrix If the 31 signals were independent every eigenvalue would equal 1.0 (dashed line) and the participation ratio would be 31. Instead the leading eigenvalue is 10.70 — a third of the total on its own — and the first three carry 65.5%. Twenty-four of the thirty-one fall below 1.0, meaning they contribute less than one signal's worth of independent variation each. Effective breadth, the participation ratio (Σλ)²/Σλ², is the number that belongs under Grinold's square root: 5.57.

Table 7 — Breadth, both universes, 447 sampled days
Measure	Large cap	Small cap
Signals in catalogue	31	31
Effective breadth (IC correlation)	5.57	6.19
Effective breadth (score correlation)	8.40	8.85
The multiplier √breadth	2.36×	2.49×
√31 if independent	5.57×	5.57×
Top-3 eigenvalue share	65.5%	62.6%
Largest cluster at \|r\| ≥ 0.5	25 of 31	24 of 31

Momentum, reversal, moving averages, RSI, Bollinger position, drawdown depth, close location, volatility and skew are all chained together at |r| ≥ 0.5. Only negative_skew and rev_1d stand alone in both universes.

Some pairs are near-identities that the family labels concealed. mom_1m against rev_21d is −0.97 — they are the same number with opposite signs, filed under different families. lowvol_60 against downside_dev is +0.97. The same pairs top the matrix in both universes, which is what a structural result looks like rather than a fitted one.

3.3The framework is confirmed; the inputs are empty

Equal-weighting N independent signals should multiply IC by √N_eff. Using the a-priori signs — not the signs that worked, which would be fitting 31 binary parameters to the same decade the answer is read from — it does.

Table 8 — Combination lift against prediction
Quantity	Large cap	Small cap
Mean individual \|IC\|	0.0140	0.0159
Mean individual IC (signed)	−0.0058	−0.0039
Combined IC	−0.0130	−0.0145
Achieved lift	2.22×	3.71×
Lift predicted from breadth	2.36×	2.49×

Combining lifts the IC by roughly what the eigenvalues predict — not by √31, and not by nothing. The breadth machinery works as advertised. It amplified a negative number.

A note on precision: the large-cap agreement (2.22 against 2.36) is closer than this statistic deserves. Its denominator is a mean IC near −0.005, close enough to zero that the ratio is unstable, and the small-cap replication gives 3.71 against 2.49. The directional claim survives; the apparent 6% accuracy was luck.