# PRE-REGISTRATION - slow turns, attempt 3: the coefficient, not just the scaling

Written 29 September 2026. Third attempt, and the first with a prediction that has a number in front of it.

## Why attempt 2 failed, correctly diagnosed at last - and not by me

Attempt 2 fitted the deficit from a half and got exponent 0.4769, rejected at 7.08 SE. **I published the
reading that this was systematic curvature in the deficit. That reading was wrong.**

The website session found the cause: **I measured every deficit from 0.5, but the start's own dark share was
0.499223** - an ordinary 0.98-SE fluctuation, 0.000777 short of a half. So **every deficit carried a constant
+0.000777**, and a constant flattens a log-log slope most where the values are smallest. Measured from the
start's own dark share the exponent is 0.4968. **And the five windows walked the same 400,000 paths, so their
errors were shared, which is why the residual-based fit error was tiny** - not because the deviation was
systematic.

⛔ **Attempt 2's leg 3 also failed and my report of it omitted that** (ratio 0.5206 against a criterion of
"below 0.5"). It was a knife-edge criterion: if the exponent is exactly a half the ratio is exactly a half,
so "below" fails about half the time with everything correct. Published.

## The prediction, with its coefficient

Near the equator the facing diffuses as `dc = sqrt(2D) dW`, and for d = 3 the facing's density there is 1/2.
So with displacement Delta ~ N(0, 2 D tau),

    P(start dark, end lit) = (1/2) E[max(Delta, 0)] = (1/2) sqrt(D tau / pi)

**That is the deficit from the start's dark share - a square root in tau WITH a coefficient.** Checked by hand
at tau.lambda = 0.01: it gives 0.01995 against attempt 2's offset-corrected 0.019750.

## Two defects fixed by design

1. **The deficit is measured from the start's own dark share**, not from 0.5.
2. **Each window gets a FRESH independent sample**, so the five points no longer share paths and their errors
   are independent.

## Criterion, fixed before the run

At d = 3, D = 0.05, with the entry-41 geodesic step:

1. **The coefficient holds.** At each of tau.lambda = 0.0025, 0.005, 0.01, 0.02, 0.04, the measured deficit
   agrees with `(1/2) sqrt(D tau / pi)` to within **3 standard errors of that measurement**. **At most 1 of 5
   may exceed** - the allowance from the binomial, since 5 cells at 3 SE expect 0.0135 exceedances and
   P(at least two) is 9e-5.
2. **The exponent, now that the offset is gone.** Fitting log(deficit) against log(tau) gives an exponent
   within **3 fit standard errors** of 0.5. Independent samples per window, so the fit error is honest.
3. **No knife edges.** Nothing is scored as "below one half" or any other boundary a correct result sits
   exactly on.

**WHAT KILLS IT:** two or more cells in leg 1. Then the coefficient is wrong and the approach is not the
equator-crossing calculation.

**Prediction:** both hold. ⛔ **I have been wrong twice about this item's cause** - once calling it curvature,
once omitting a failed leg from my own report. The diagnosis being tested here is not mine, and the criteria
are built so a third wrong reading fails rather than passes.

## Discipline

Both earlier verdicts stay published, including the reading of mine that was wrong. No prose conclusion in
the script.