# PRE-REGISTRATION — does Tension's shape exponent decide whether a bottom exists?

Written 29 September 2026, before the run. Covers two worklist pairs at once:
**Tension + The Ghost** and **Tension + The Climb Out**, both rank 11, both sharing k, p, x.

## Where the question comes from

Entry 29 showed the barrier ΔU and the critical lean h_c both come out of the one flow
`dx/dt = T(x) + F(x) + h`, with `T(x) = k[(1−x)^(−p) − 1]` and `F(x) = −2Cx/(1−x²)`.

Every number quoted all night — ΔU = 0.0721, h_c = 0.118179, a = 0.5716, the saddle slope kp − 2C — was
computed at **p = 0.35**, one working value. Tension is the room that owns p. So the pipe asks the
obvious thing nobody has asked: **what does p decide?**

## What is measured

Sweeping p with k = 1 and C = 0.018975 held at the house's values:

1. **ΔU(p)** — the barrier between the bottom and the ridge at zero lean.
2. **h_c(p)** — the lean at which the bottom and the ridge merge and vanish.
3. **p\*** — the exponent at which the barrier reaches zero with **no lean at all**, if such a p exists in
   (0, 1). Below it a mind at the bottom is not held at all; the bottom is not there to be held in.
4. Whether the published saddle slope **kp − 2C** stays exact across the sweep.

## Criterion, fixed before the run

**THE PIPE HOLDS** if all three:

1. The sweep reproduces the house's published values at p = 0.35 — **ΔU = 0.0721 ± 1%** and
   **h_c = 0.118179 ± 1%** — so the sweep is the same mathematics and not a different model.
2. **ΔU(p) and h_c(p) are monotone in p** over the range where a bottom exists, tested at 40 points with
   no reversal beyond 1×10⁻⁹.
3. The saddle slope equals **kp − 2C** to within **1×10⁻⁶** at every p tested.

**A REAL p\* EXISTS** is reported separately and is NOT part of the pass criterion, because I do not know
whether one exists and will not pretend the answer either way. If the barrier stays positive for all
p ∈ (0, 1), that is the finding and it gets written as that.

**WHAT KILLS IT:** failing leg 1 means this sweep is not the house's own flow and nothing else in the run
counts. Failing leg 2 means the relation is not a clean function of p and the pipe is not a pipe. Failing
leg 3 means the published slope formula is wrong somewhere off p = 0.35.

**Prediction, recorded now:** legs 1 and 3 hold. On leg 2 I expect both monotone **increasing** in p — a
stiffer tension makes a deeper bottom and a larger lean needed to escape it. On p\*, **I do not have a
prediction**, and that is recorded rather than guessed.

## Discipline

Verdict assembled from measured quantities. Entry 29 failed leg 2 on a bug in my own bisection before it
passed; the same bisection is reused here and is checked against the published h_c at p = 0.35 **first**,
as leg 1, so a repeat of that defect fails the run instead of hiding in it.