Most forecast bands you see on a chart are one line of code: take a per-bar sigma, multiply by the square root of the horizon, draw. It is fast, it is defensible in an interview, and on the instruments below it is wrong twice - too wide at short horizons and too narrow at long ones, which means no single correction factor fixes it.

Here are the measurements, taken on 31 July 2026 against full Binance history, so anyone can reproduce them.

Method, deliberately dull: 4h candles, full Binance history paged back to listing. For a horizon of H bars, take every rolling window of that length, compute the realised log return, and read the empirical percentiles straight off the distribution. Compare the resulting interval width against the sigma*sqrt(H) width computed on the same bars. No distributional assumption, no fitting, no parameters.

FINDING 1: THE PARAMETRIC BAND IS TOO WIDE AT 24 BARS, ON ALL FOUR

Ratio of empirical 80% width to the sigma*sqrt(H) 80% width, at a 24-bar horizon:

- BTC 0.86x

- ETH 0.89x

- SOL 0.83x

- BNB 0.75x

Every one below 1. The textbook band overstates the 10th-to-90th percentile range by 11% to 25% depending on instrument. This is the opposite of the usual intuition, which is that fat tails make the Gaussian band too narrow.

Both things are true at once, and the resolution is worth stating because it trips people up. Excess kurtosis lives in the extreme tails, not in the shoulders. An instrument can have genuinely fat tails and still have a narrower-than-Gaussian 80% interval, because the 10th and 90th percentiles are nowhere near the tails. If you size against an 80% band you are working in the shoulders, and the shoulders are thinner than the parametric formula thinks.

FINDING 2: THE SCALING EXPONENT IS NOT 0.5, AND THE ERROR REVERSES SIGN WITH HORIZON

Root-t predicts that doubling the horizon widens the interval by 1.41x and tripling it by 1.73x. Measured core-50 widths, as a fraction of spot:

- BTC: 6.9% at 24 bars, 9.9% at 48 (1.43x), 12.6% at 72 (1.83x)

- ETH: 9.0% at 24 bars, 13.6% at 48 (1.51x), 18.0% at 72 (2.00x)

- SOL: 12.3% at 24 bars, 18.5% at 48 (1.50x), 23.3% at 72 (1.89x)

- BNB: 8.1% at 24 bars, 11.5% at 48 (1.42x), 14.8% at 72 (1.83x)

Fitting width proportional to H^a gives exponents of 0.548 for BTC, 0.631 for ETH, 0.582 for SOL and 0.549 for BNB. Not one of them is 0.5.

Every instrument widens faster than diffusion, and the gap opens with the horizon: at 48 bars the measured ratios sit near the 1.41x prediction, and by 72 bars all four are well past 1.73x. That is what persistent, trending behaviour looks like in the numbers, as opposed to independent increments.

The practical consequence is the reversal. On a 24-bar horizon a root-t band is too wide, so it understates your risk of a stop that never gets hit and overstates the range. Push the same band to 72 bars and it becomes too narrow, so it understates the range instead. Applying one correction factor across horizons makes the error worse at one end.

FINDING 3: THE SAMPLE SIZE YOU REPORT IS USUALLY NOT THE SAMPLE SIZE YOU HAVE

BTC at 24 bars gives 19,582 rolling windows and 815 genuinely independent ones, because consecutive windows share 23 of their 24 bars. SOL gives 13,058 and 544.

Quantile standard errors scale with the second number. Reporting the first is the easiest way to manufacture false confidence in a backtest, and it is nearly always left unstated. Both counts belong on the output.

FINDING 4: COVERAGE FAILS IN BOTH DIRECTIONS, AND ONE DIRECTION IS INVISIBLE

Width comparisons are cheap. The only score that matters is whether the interval contains the outcome as often as it claims, tested on data the band never saw.

Our own live record: 56 resolved forecasts, each hashed and Bitcoin-timestamped before publication. The stated 50% interval contained the outcome 47 times - 83.9%. The 80% interval came out at 98.2%.

That is a failure, and it is the failure mode nobody catches. An interval that is too narrow gets caught in days: outcomes land outside, somebody complains. An interval that is too wide generates an unbroken run of successes. Every result lands inside, the scoreboard looks healthy, and there is no symptom to investigate. A band that contains almost everything can never be caught being wrong, and being uncatchable is not the same as being right.

The diagnosis, for anyone with the same problem: our first hypothesis was insufficient history, and it was wrong. A walk-forward over 20,058 observations, rebuilding every interval from prior data only, still ran 55.6% against a 50% target. Splitting by condition found it - on the calmest fifth of days the same intervals covered 67.1%. The bands carried a permanent allowance for turbulence that a quiet market had not earned, and the average across all regimes concealed it.

Conditioning the sample on the volatility regime observed at the moment the forecast opens - not at the close, which leaks the outcome you are trying to predict - brings coverage to 50.8% and the wide band to 79.6%.

WHAT THIS ADDS UP TO

Read the interval off the empirical distribution at the matching horizon. Condition the sample on the regime observable at decision time. Print both window counts. Publish out-of-sample coverage next to the band, before the outcome exists, and treat over-coverage as a defect rather than as prudence.

None of this produces an edge. It produces an interval whose width means what it says, which is a different and more defensible thing to build. No method reliably beats a liquid market.

The indicator is free and open source on TradingView if anyone wants to run these numbers on their own instruments, and the sealed forecasts with their block heights are published as a dataset.

Educational content - not financial advice.