# PRE-REGISTRATION — the veil, attempt 3: in log space, with the precision limit designed in

Written 29 September 2026, after two failures, before this run.

## Two failures, both mine, both published

**Attempt 1** (`PREREG_veil_at_the_bottom`): the leading-order law within 5% across p. **Worst 16.07%.
FAILS.** Cause: I tested an asymptotic law at p = 0.05, where the gap is 0.664 — far outside the "small g"
the expansion itself assumes.

**Attempt 2** (`PREREG_veil_exact_attempt2`): the exact fixed point within 10⁻⁹ at every p. **No
convergence at p ≥ 0.8825. FAILS.** Cause: my bisection for the top bracketed u from 10⁻¹⁴, and the top
veil at p = 0.95 is `(C/k)^(1/(1−p)) = (0.018975)^20 ≈ 10⁻³⁴`.

**Both failures were in my test design, not in the mathematics.** That is worth saying twice because it is
the same error twice: I chose a numerical range without first asking what range the quantity actually
lives in. Where attempt 2 converged it was exact to 2.55×10⁻¹⁵, machine precision.

## The precision limit, stated in advance this time

The top attractor sits at `x = 1 − u`. In IEEE double precision, `1 − u` is **indistinguishable from 1**
once u < ~10⁻¹⁶. So beyond that there is no "true root" to compare against — not because the flow stops,
but because the representation does. Any test of the top must either work in u (or ln u) directly, or stop
where doubles stop. **This is a property of arithmetic, fixed here before the run, not a threshold chosen
after seeing an error.**

So this attempt works in **log space** and tests the top only where a true root is representable.

## Criterion, fixed before the run

**LEG 1 — the bottom, everywhere.** The exact fixed point matches the true bottom attractor to within
**10⁻¹²** at all 41 values of p ∈ [0.05, 0.95]. No exclusions: the bottom gap never goes below 0.039, so
it is representable throughout.

**LEG 2 — the top, where doubles reach.** Solving `2C(1−u)/((2−u)u) = k[u^(−p) − 1]` for **ln u** by
bisection on ln u ∈ [−700, −10⁻⁹], the recovered u matches the true top gap to a **relative** error below
**10⁻⁹**, at every p for which the true gap exceeds **10⁻¹²**.

**LEG 3 — the house.** At p = 0.35 both reproduce the published bottom −0.910797 and heaven 0.997239 to
within **10⁻⁵**.

**REPORTED, NOT SCORED:** how the top gap behaves as p → 1. The leading order says `u* = (C/k)^(1/(1−p))`,
which falls super-exponentially. The measurement is recorded across p, and the p beyond which the gap
leaves double precision is reported as a fact about the arithmetic.

**WHAT KILLS IT:** a failure of leg 1, which has no precision excuse at all — the bottom gap is never
smaller than 0.039.

**Prediction, recorded now:** leg 1 and leg 3 hold. Leg 2 holds where it is tested. I make **no**
prediction about the exact p at which the top gap leaves double precision.

## Discipline

Both earlier verdicts are published beside this one. Three attempts, two failures, and the failures were
mine.