# Synthesis note (arena, 2026-10-13)

- **Candidates.** Three Claude models answered one brief blind: candidate 1 (Opus), candidate 2 (Sonnet), candidate 3 (Fable). No dropout. Candidate 1's report came back as text and the parent saved it to its folder.
- **Cross-judge.** A fourth Claude (Fable), read-only, saw the three as A, B, C. Scores of 35: A 32, C 26, B 25. It recommended A as the base. The parent's own reading picked the same base before the judge returned.
- **Base.** Candidate 1. Its arithmetic reproduced in full, its first variable uses nothing from the night it weights, and it alone states the width under a drifting mean.
- **Grafted from candidate 3.** The block-bootstrap baseline. The g correction and the ceiling computed from the weight series alone. The cap on one year's share of the weight. The data gate on Yahoo opens. The risk-limit payer. The point that a future's price leaves out the cash rate.
- **Grafted from candidate 2.** The error of a fitted mean away from the average state. The stratified estimate for when the weighted mean and the plain mean differ. The calibration check of variance on squared VXN.
- **Rejected.**
  - Candidate 3's first variable, weights from the night's own realised variance. The judge's simulation reads a bias of +3.3 to +7.2 bp for a within-night correlation of -0.3 to -0.7 between return and variance, and the candidate's pass band admits about 3 bp.
  - Candidate 2's Asia and Europe indices as controls. Their own night means are premia known no better than NQ's.
  - Candidate 2's 750-night burn-in. It cuts the sample to 3,080 nights, and inference needs only that a weight not depend on the night's sign.
  - Candidate 1's fourth variable, last-hour signed volume. No free series.
  - SPY from 1993 in place of QQQ. 1,580 more nights at a guessed correlation, against a measured 0.857 on QQQ.
  - "Revised" and "late" as reasons against GDP growth. Neither holds for inference.
- **Agreement of all three.** Inverse-variance weights from VXN are the first lever. A longer sibling record reads 0.84 to 0.93 bp only if the mean was constant, and no test here sees a change under about 6 bp. No variable for the mean narrows the estimate. The dollar ranks above inflation, inflation above GDP growth, and GDP growth is dropped.
- **Corrected in the base.** Financing by the hours QQQ is held and not by calendar days: bias 0.17 bp, not 0.22. The gap error at a share of 8%: 4.17 bp, not 4.12.
- **Verified by the parent.** FRED `VXNCLS` first row 2001-02-02 and `DFF` first row 1954-07-01, both fetched without a key. The sd of log VXN on 2010-06-08 to 2026-08-13 is 0.29 (4,073 closes), so the ceiling ratio is 0.845 by the lognormal formula and 0.844 from the series. 70 / sqrt(3832) = 1.131. sqrt(70 sqrt(1/252)) = 2.10. The sibling factor at correlation 0.857 is 0.821. No return was read.
- **Not verified.** g, the two papers (search results and abstracts only), and every expected size. The test below has not been run.

---

# Tightening the estimate of the current night mean on NQ

## Verdict

- Two variables can cut the sampling error of the mean by a tenth to a quarter each: the VXN close as a night weight, and the QQQ close-to-open record as a longer stand-in. Together they take 1.13 bp to roughly 0.8 bp.
- Both cuts hold only if the mean did not change over the record. The record cannot test that. A gap between 2010-2020 and 2021-2026 smaller than 6.6 bp passes unseen, and the whole premium is 4.8 bp.
- If the mean moves by as little as 1 bp a year, the error on the current mean is 2.1 bp whatever the weighting.
- No macro state variable (inflation, growth, dollar, dealer positions) lowers the error. Each one splits the nights, and a split helps only when the true gap is larger than its own standard error.
- Of the user's three candidates, the dollar ranks first, inflation second, GDP growth last. Drop GDP growth.
- A state variable for the mean never narrows the estimate. At a current state z standard deviations from the record's average the fitted mean has error

$$SE(z) = SE_0 \sqrt{1 + z^2}$$

- That is 1.60 bp at z = 1 and 2.53 bp at z = 2.

## The arithmetic every variable is held to

Baseline error of the mean:

$$SE_0 = 70 / \sqrt{3832} = 1.13$$

Smallest effect the record resolves at 5% two-sided with 80% power (multiplier 2.80), and at the family threshold for 95 comparisons (z 3.47, multiplier 4.31):

| comparison | SE of the gap | resolves at 2.80 | resolves at 4.31 |
|---|---|---|---|
| 1 sd of a continuous variable | 1.13 bp | 3.2 bp | 4.9 bp |
| two halves of the record | 2.26 bp | 6.3 bp | 9.7 bp |
| a state on 20% of nights | 2.83 bp | 7.9 bp | 12.2 bp |
| a state on 8% of nights | 4.17 bp | 11.7 bp | 18.0 bp |

For a state with share p of the nights:

$$SE_{gap} = 70 / \sqrt{3832\,p\,(1-p)}$$

When a split helps. Let the current state hold share c of the nights and let the true gap be delta. Estimating on the current state alone raises the variance and removes a bias:

$$\text{added variance} = SE_0^2\,(1/c - 1)$$

$$\text{bias removed} = (1-c)\,\delta$$

The split lowers the error only when

$$|\delta| > SE_0 / \sqrt{c\,(1-c)} = SE_{gap}$$

A lever 1 variable therefore tightens the current mean only if its true effect exceeds one standard error of its own estimate. Below that it moves error from bias into variance and the total grows.

When the mean drifts. Let the mean follow a random walk with step variance q per night. The steady error of the filtered current mean is

$$SE_{now} \approx \sqrt{\sigma \sqrt{q}}$$

$$q = d^2 / 252$$

with d the drift in bp per year.

| drift d | effective window | SE of current mean |
|---|---|---|
| 0 | 3,832 nights | 1.13 bp |
| 1 bp a year | 1,111 nights | 2.10 bp |
| 2 bp a year | 555 nights | 2.97 bp |

Levers 2 and 3 lower sigma. Lever 1 lowers q. Under drift the error falls with the fourth root of the variance, so a 20% cut in variance buys 5%.

## Ranked variables

| rank | variable | lever | expected SE of current mean | reduces SE |
|---|---|---|---|---|
| 1 | VXN close before the night | 3, checked against 1 | 0.88 to 1.00 bp | yes, if the mean in bp does not rise with volatility |
| 2 | QQQ close-to-open in excess of fed funds, 1999 on | 2 | 0.93 bp | yes, if 1999-2009 has the same mean |
| 3 | calendar time, break at 2021-01-04 | 1 | 1.13 to 2.1 bp | no, it sets the honest width |
| 4 | dollar, same-night change | 2 | 1.11 bp | by 1.4%, inside rounding |
| 5 | dealer net position in NQ, CFTC | 1 | unchanged | no |
| 6 | inflation level | 1 | 1.22 bp | no, raises it |
| 7 | GDP growth | 1 | 1.18 bp or worse | no, dropped |

### 1. VXN close before the night (lever 3)

- **(a) Mechanism.** A night's variance is forecastable at 16:15 from index option prices. Averaging nights of unequal variance with equal weights lets the loud nights set the error. Who pays the premium is a separate question: holders with overnight margin or risk limits sell before the close and pay those who carry gap risk. If that payment scales with risk, the mean rises with VXN and the weighting changes what is estimated.
- **(b) Series.** FRED `VXNCLS`, daily close, first observation 2001-02-02, free CSV at `fred.stlouisfed.org/graph/fredgraph.csv?id=VXNCLS`.
- **(c) Sign and size.** The weight is the inverse squared VXN. Let v be the standard deviation of log VXN over the record, expected 0.25 to 0.35. The error ratio is

$$SE_w / SE_0 = e^{-2v^2}$$

  - v = 0.25 gives 0.88, so 1.00 bp. v = 0.35 gives 0.78, so 0.88 bp.
  - v is computed from VXN alone, before a return is loaded. Measured on the record's dates on 2026-10-13: v = 0.29, ratio 0.845, so 0.95 bp.
  - That is the ceiling, reached when the night's variance moves one for one with squared VXN. If the variance moves as VXN to the power 2g, the ratio is

$$SE_w / SE_0 = e^{v^2 (2 - 4g)}$$

  - g = 1 gives the ceiling and g = 0.5 gives no gain. g is the slope of log realised night variance on log VXN, and it uses no return's sign.
  - The payer behind the size of the premium: an intermediary with a fixed overnight risk limit sheds inventory at the close when implied variance is high, and the limit is set by a capital rule and not reset each night (Hendershott, Livdan and Rösch, 2020, recalled from memory).
- **(d) Test.** Ratio of the robust error of the weighted mean to the error of the plain mean. Null: ratio 1. Placebo: VXN shifted in time by a random offset. Full design in the last section.
- **(e) Spurious if.**
  - The weights use anything from the night itself. Realised night variance is higher on down nights, so weighting by it removes losses and biases the mean up. Only the prior close is allowed.
  - The gain comes from 2020 alone. Down-weighting 60 nights of March to May 2020 would give most of it. The leave-one-year-out check covers this.
  - The mean rises with volatility. Under a constant ratio of mean to volatility the weighted mean reads lower than the plain mean by the factor

$$e^{-2v^2} = 0.78 \text{ to } 0.88$$

  - That is 0.6 to 1.1 bp lower. The error of that gap is

$$\sqrt{1.13^2 - 0.88^2} = 0.71$$

  - So the gap sits at z 1.1 to 1.5. The record cannot tell a constant mean from a constant ratio.
  - Prior scoring: VXN as a hold rule read a gain of 1.27 (0.56 to 3.22), and inverse-volatility sizing deepened the worst 1%. Those scored a position. This scores an estimator and adds one comparison.
- **Standard error.** Falls to 0.88 to 1.00 bp for the precision-weighted mean. That number is the current mean only under a mean constant in bp. Under a constant ratio the current mean is the ratio times tonight's volatility, with relative error

$$1 / (0.069 \times \sqrt{3832}) = 23\%$$

  - 23% of 4.8 bp is 1.1 bp, the same as before. The weighting tightens the estimate under one model and not under the other, and the data do not choose.

### 2. QQQ close-to-open in excess of fed funds (lever 2)

- **(a) Mechanism.** The QQQ night and the NQ night share one deviation. Their correlation on the 3,832 common nights is 0.857 (mdgraph entry of 2026-10-12). QQQ has 3,068 more nights. The gap between the long QQQ mean and the overlap QQQ mean measures how lucky the overlap years were, and that luck is subtracted from NQ.
- **(b) Series.** QQQ daily open, close and dividends from Yahoo Finance or Stooq, first bar 1999-03-10. FRED `DFF`, daily, from 1954-07-01. The future earns no financing and QQQ does, so the stand-in is

$$X_t = \frac{O_{t+1} + D_{t+1}}{C_t} - 1 - \frac{DFF_t\,h_t}{8766}$$

  - h is the hours from the close to the next open, 17.5 on a weeknight and 65.5 over a weekend, and D a dividend on its ex-date.
- **(c) Sign and size.** The estimator and its error ratio:

$$\hat m = \bar r + \beta\,(\bar X_{long} - \bar X_{overlap})$$

$$f = \sqrt{1 - \rho^2\,(1 - n/N)}$$

$$f = \sqrt{1 - 0.734 \times 0.445} = 0.82$$

  - 1.13 bp becomes 0.93 bp.
  - Financing is not small. Fed funds averaged about 3.2% in 1999-2009 and 1.6% in 2010-2026, which is 1.03 against 0.52 bp a night at 28.3 hours a night on average. Left in, it biases the estimate by

$$0.75 \times 0.445 \times 0.51 = 0.17$$

  - 0.17 bp is most of the gain of 0.20 bp.
- **(d) Test.** Overlap regression of the NQ night on X. Null: the 1999-2009 mean of X equals the 2010-2026 mean. The era gap has error near 2.2 bp, so it resolves 6.2 bp.
- **(e) Spurious if.**
  - The early era differs. 1999-2002 holds one bear market, so the early mean rests on one episode.
  - The windows differ. QQQ holds 16:00-18:05 and 09:25-09:30 and the future does not. Under the 200-day average the two series read +7.95 and -0.27 bp. Beta may differ by state, so its stability across halves is tested.
  - The stand-in has no look-ahead problem because lateness is allowed.
- **Standard error.** Falls to 0.93 bp if the mean was the same in 1999-2009. If it was not, the estimate is biased by 0.33 bp for each 1 bp of era gap (0.75 times 0.445), and the test cannot see an era gap under 6.2 bp. The error moves from sampling into an untested assumption.

### 3. Calendar time, break at 2021-01-04 (lever 1)

- **(a) Mechanism.** Dealers who absorb end-of-day selling are paid when European trading starts (Boyarchenko, Larsen, Whelan, RFS 2023). The same authors report on 2026-07-01 that the 02:00-03:00 return on ES went from 3.7% a year in 1998-2020 to about zero in 2021-2025, and that the dispersion of last-hour signed volume fell from 6.5% to 2.9%. The payers stopped needing the service at the same price.
- **(b) Series.** The night date. No download.
- **(c) Sign and size.** Negative after 2021. The ES night read by eye from that post went from about 9.5% to 5% a year, near 3.8 to 2 bp a night. Expected gap on NQ: 0 to -3 bp.
- **(d) Test.** Mean after 2021-01-04 less mean before, about 1,414 against 2,418 nights:

$$70\,\sqrt{1/1414 + 1/2418} = 2.34$$

  - Resolves 6.6 bp at 2.80 and 10.1 bp at 4.31. A halving of the premium is about 2.9 bp, z 1.2.
  - Second statistic: likelihood ratio for q = 0 in the local-level model, and the 90% upper bound on q.
- **(e) Spurious if.**
  - The break date came from the ES result on the same years, so this is a replication on a correlated contract and not an independent test.
  - Two regimes give one degree of freedom. Any variable that changed around 2021 fits equally.
  - A contract-splice error once counted 1.9% as drift in 2026 (mdgraph correction entry). The record must be the corrected one.
- **Standard error.** Does not fall. Using only post-2021 nights gives

$$70 / \sqrt{1414} = 1.86$$

  - This variable decides whether 1.13 bp was ever the right width. It is ranked third because variables 1 and 2 are only as good as its answer.

### 4. The dollar (user candidate, kept, lever 2)

- **(a) Mechanism.** The same overnight news moves the dollar and the index, so the dollar's change explains part of the deviation. As a driver of the mean the case is weak: foreign buyers trade in their own morning, and no published result ties the premium to the dollar's level.
- **(b) Series.** FRED `DTWEXBGS`, daily, from 2006-01-02. FRED `DTWEXM`, daily, 1973-01-02 to 2019. Both are noon rates, so the change runs noon to noon and covers more than the night. No free series is aligned to 18:05-09:25.
- **(c) Sign and size.** Correlation with the night about -0.2, unstable in sign.

$$f = \sqrt{1 - 0.04 \times 0.714} = 0.986$$

  - 1.13 bp becomes 1.11 bp.
- **(d) Test.** Regression of the night on the same-date dollar change. Null: correlation zero. As a mean driver, the floor is 3.2 bp per standard deviation of the 200-day dollar trend.
- **(e) Spurious if.** The correlation flips sign between 2010-2019 and 2022. The dollar's own mean is not zero by theory, since interest parity sets it to the rate gap. About five dollar cycles sit in the record.
- **Standard error.** Falls by 0.016 bp, or 1.4%. It does no harm and no good.

### 5. Dealer net position in NQ (lever 1)

- **(a) Mechanism.** Dealer inventory measured directly. Larger dealer longs mean more risk carried and a higher charge.
- **(b) Series.** CFTC Traders in Financial Futures, futures only, weekly as of Tuesday, free zip files by year. The CFTC page lists a 2006-2016 archive and says the complete file starts September 2009.
- **(c) Sign and size.** Positive, under 1 bp per standard deviation.
- **(d) Test.** Night return on the last published dealer net position over open interest. Floor 3.2 bp per standard deviation nominal, with about 840 weekly values that move slowly.
- **(e) Spurious if.** Weekly positions are not the end-of-day inventory the theory names. Positions trend with the index level.
- **Standard error.** Unchanged. The expected effect is a third of the floor.

### 6. Inflation level (user candidate, kept, low)

- **(a) Mechanism.** Above about 3% inflation, bonds stop offsetting equity losses and the 08:30 releases carry more risk. Holders who cannot sit through a release pay more to pass the night.
- **(b) Series.** FRED `CPIAUCSL`, monthly, from 1947-01-01, year-on-year change, lagged one month for the release date.
- **(c) Sign and size.** Positive, 0 to 3 bp on high-inflation nights.
- **(d) Test.** Nights with inflation above 4% against the rest. About 525 nights, share 0.137:

$$70 / \sqrt{3832 \times 0.118} = 3.29$$

  - Resolves 9.2 bp at 2.80.
- **(e) Spurious if.** The high state is one episode, April 2021 to mid 2023, which also holds the 2022 bear market and the years the drift is said to have faded. One episode is one observation. CPI mornings already read t 0.68.
- **Standard error.** Rises. Estimating on low-inflation nights alone gives

$$70 / \sqrt{3307} = 1.22$$

  - The split pays only if the true gap exceeds 3.3 bp, and one episode cannot show that.

### 7. GDP growth (user candidate, dropped)

- **(a) Mechanism.** Risk premiums rise in recessions because holders can bear less risk.
- **(b) Series.** FRED `A191RL1Q225SBEA`, quarterly, from 1947-04-01. FRED `GDPNOW`, from 2011-07-01.
- **(c) Sign and size.** Negative on growth, under 2 bp.
- **(d) Test.** Nights in negative-growth quarters against the rest. About 315 nights, share 0.082, gap error 4.12 bp, resolves 11.5 bp.
- **(e) Spurious if.** The record has one recession, 2020, of about 40 nights. Two quarters of 2020 hold most of the spread of growth, so the slope is about 125 nights against the rest.
- **Standard error.** Rises to about 1.18 bp on the positive-growth nights alone, with no effect large enough to pay for it.
- **Why this one goes.** It is quarterly, so the record holds about 64 values of it, one recession carries the spread, and its information is already in the Baa-spread and yield-curve states, which read no gain. The dollar is daily and costs nothing to keep. Inflation's route through the 08:30 releases is a change in the size of the night's move, which the weights of variable 1 already carry. A future's price leaves out the cash rate, so the inflation level adds no drift to it.

## Pre-registered test for variables 1 to 3

One notebook section. Every constant below is fixed here.

### Data

- NQ night record: 3,832 nights, 2010-06-08 to 2026-08-13, the corrected file already in the plan.
- FRED `VXNCLS` and `DFF`. QQQ daily open, close, dividends from 1999-03-10. A year of QQQ bars is used only if its open equals the previous close on under 2% of days.
- No other series, weight, break date or volatility measure. Any later variation is exploratory and adds to the comparison count.

### Cells

1. **Declare.** A dictionary with every threshold below, committed before the next cell runs.
2. **Covariates only.** Load VXN, DFF, QQQ bars. No return is formed. Print v, the standard deviation of log VXN on the 3,832 dates, and the predicted ratio.
3. **Second moments, blind.** Load the nights and subtract each series' own mean without printing it. Compute the statistics of tests A and B that use no mean: the ratio, the placebo, the correlation, beta by half.
4. **Stability.** The era gap, the break gap, the likelihood ratio.
5. **Unblind.** Print the estimates the decision rules select.
6. **Verdict table.** One row per hypothesis: supported, killed, or unresolved.

### Test A: VXN weights

- Weight: one over the squared VXN close of the session before the night. The night's own realised variance is never a weight: falling nights swing more, and a simulation by the arena's judge put the bias of that weighting at +3.3 bp for a within-night correlation of -0.3 between return and variance.
- Baseline: SE_B, the standard error of the plain mean from 5,000 draws of 20-night blocks. R and every other gain is a ratio to SE_B and not to 1.13.
- Before any return is read: v, the ceiling ratio, g, the effective count of nights under the weights, and the share of the total weight in each calendar year.
- Statistic: R, the robust error of the weighted mean over the error of the plain mean.

$$V_w = \frac{\sum w_t^2 (r_t - \bar r_w)^2}{(\sum w_t)^2}$$

- Placebo: 1,000 circular shifts of VXN by at least 250 nights, seed 20261013. p is the share of shifts with R at or under the observed R.
- **Supported:** R at or under 0.90, placebo p at or under 0.01, no calendar year above 12% of the total weight, the slope of realised night variance on squared VXN between 0.6 and 1.4, and R at or under 0.95 with each calendar year removed, with February to June 2020 removed, and with 2021-2023 removed.
- **Killed:** R at or above 0.97, or placebo p above 0.05. R above 1.00 is recorded as VXN misordering the nights.
- **Unresolved:** anything between. The plain mean stays.
- Estimand guard: D is the weighted mean less the plain mean, with error from 5,000 draws of 20-night blocks. Predicted sign negative.
  - z of D at or beyond 1.96 either way: the weighted mean is not reported as the mean. The ratio model (mean equals a constant times VXN) is reported at the latest VXN.
  - z at or beyond 3.47: claim that the mean depends on volatility, in the direction of the sign.
  - z inside 1.96: report both models. The stated interval is the union of their 95% intervals.
  - When the guard fires, the estimate of the plain mean is the stratified one: the weighted mean inside each third of nights by VXN, combined by each third's share of nights.

### Test B: long stand-in

- Statistic 1: correlation of the NQ night with X on the overlap, and beta on each half of the overlap.
- Statistic 2: z of the era gap, the mean of X in 2010-06-08 to 2026-08-13 less the mean in 1999-03-10 to 2010-06-07, with unequal variances.
- **Supported:** correlation at or above 0.80, the two betas within 0.10, and z of the era gap inside 1.96. The estimate is the regression estimator with error from 5,000 draws of 20-night blocks over both series.
- **Killed:** correlation under 0.70, or betas apart by 0.10 or more, or z of the era gap at or beyond 1.96 either way. A negative z is passed to test C as support for decay. A positive z is recorded as the early era being the weak one.
- **Unresolved:** correlation between 0.70 and 0.80. Report, do not pool.
- The result line must carry the sentence: an era gap under 6.2 bp was not detectable.

### Test C: constancy

- Statistic 1: z of the mean after 2021-01-04 less the mean before, unequal variances.
- Statistic 2: likelihood ratio for q = 0 in a local-level model whose observation variance is a constant times squared VXN. Critical value 2.71. Also the 90% profile upper bound on q, expressed as drift in bp a year.
- **Decay supported:** z at or under -1.96. The current mean is then estimated on post-2021 nights alone, weighted if A passed, and test B's pooled estimate is withdrawn. A claim of decay on NQ needs z at or under -3.47.
- **Decay killed:** z at or above +1.96. The full record stands.
- **Constancy accepted as a working assumption:** z inside 1.96, likelihood ratio under 2.71, and the upper bound on drift under 0.5 bp a year.
- **Unresolved:** z inside 1.96 with the upper bound at or above 0.5 bp a year. Report the full-record estimate and the post-2021 estimate side by side. The stated interval is the union.

### Multiplicity

- Five statistics: R, z of D, z of the era gap, z of the break, the likelihood ratio. With about 90 already scored the family is 95, so a claim of a new effect on the mean needs z 3.47.

$$0.05 / 95 = 0.00053$$

- Dropping an assumption needs only 1.96. A dropped assumption widens the interval and cannot produce a finding, so the loose threshold errs toward the wider answer.
- R and the correlation use no mean and are computed blind, so choosing the estimator cannot select on the answer.

### What each outcome reports

| A | B | C | estimate of the current mean | expected error |
|---|---|---|---|---|
| supported | supported | constancy accepted | weighted regression estimator, 2001 on | about 0.8 bp |
| supported | killed or unresolved | constancy accepted | weighted NQ mean | 0.88 to 1.00 bp |
| any | any | unresolved | full-record and post-2021 estimates, union interval | 1.1 to 1.9 bp |
| any | withdrawn | decay supported | post-2021 nights only | 1.6 to 1.9 bp |

- The expected row is the third. The break test resolves 6.6 bp and the plausible gap is 0 to 3 bp.