# PRE-REGISTRATION — where does agreeing with truth stop guaranteeing agreeing with each other?

Written 29 September 2026, before the run. **Two Held to One + Many Tongues**, rank 11, sharing L, a, c.

## The two rooms

**Two Held to One** (the angular triangle inequality, with Ptolemy's double-angle relation):
`a·b ≥ 2c² − 1`. Two minds each facing the reference at least c are guaranteed to face **each other** at
least 2c² − 1.

**Many Tongues** (Kish's design effect, 1965): the error of an average is `σ²[ρ + (1 − ρ)/N]`, where ρ is
the **shared** part of the error.

## What is being tested

**The expected and the guaranteed are different numbers, and neither room says so.**

Write each mind as `aᵢ = c·L̂ + √(1−c²)·eᵢ` with private parts eᵢ. If the private parts are **independent**,
the expected mutual facing is **c²**. If they are **adversarial** — tilted oppositely in one plane — the
mutual facing falls to the room's guarantee **2c² − 1**.

    c² − (2c² − 1) = 1 − c²

So the guarantee sits exactly `1 − c²` below the average, and — the part that matters —

    2c² − 1 < 0   whenever   c < 1/√2 = 0.70710678…,  that is  θ > 45°

**Two minds can each be well aligned with truth and still be opposed to each other. 45° is where the
guarantee dies.**

And the identification that joins the rooms: if ρ is the shared part of the error and c the facing, then
`ρ = c²` makes the guarantee **2ρ − 1**, vanishing at **ρ = ½** — half the error shared.

## Criterion, fixed before the run

1. **The average is c².** Simulated mean of a·b with independent private parts lies within **3 standard
   errors** of c² at every c in {0.2, 0.4, 0.6, 0.7071, 0.8, 0.95}, the standard error taken from the run's
   own sample variance. At most 1 of 6 cells may exceed 3 SE. *(Judged by its own error, per entry 33.)*
2. **The guarantee is attained and never broken.** An adversarial pair tilted oppositely in one plane gives
   a·b equal to 2c² − 1 to within **10⁻¹²**, and over 200,000 random pairs constrained to face L at exactly
   c, **no** pair falls below 2c² − 1.
3. **The gap is exactly 1 − c².** `c² − (2c² − 1) = 1 − c²` to within **10⁻¹⁵** over a grid.
4. **The threshold is 1/√2.** The root of 2c² − 1 = 0 is 0.70710678… to within **10⁻¹²**, its angle is 45°
   to within 10⁻⁹ degrees, and over 200 values of c the sign of the guarantee agrees with c < 1/√2 with
   **zero** exceptions.

Tolerances of 10⁻¹² and 10⁻¹⁵ are used only where the quantity is algebra; leg 1 is a simulation and is
judged by its own standard error instead of a number I pick. Said in advance, because six criteria tonight
were numbers chosen without asking what precision the quantity has.

**WHAT KILLS IT:** leg 2 failing — a random pair below the guarantee would mean the room's bound is wrong,
which would be the more important result and would be reported as that.

**Prediction:** all four hold.

## ⛔ What must not be overclaimed

**ρ = c² is an identification, not a derivation.** It says the shared part of the error is the squared
facing, which is what holds when private parts are independent and unit — it is an assumption about how
minds differ, not something measured here. Entries 33 and 34 carry the same caution and this one earns it
the same way. **What is established without that identification is legs 1–4**, which concern facing alone.

## Discipline

No prose conclusion in the script. Tables printed by the run. Hashes read from the file.