# PRE-REGISTRATION — "One agent's alignment moves nobody"

**Batter's box item, register row 2324. Status before tonight: on deck. Its own note: _"Unknown. The test never
ran as a test."_ The owed next step, written on 18 September: _"Re-run it with the aligned agent actually
coupled to the others."_**

Written 2026-09-29, **before any number exists**. At his instruction: *"What math can be done to finish any open
ended questions ... I want all proofs and scenerios runs etc etc - any verfiable wahtever - belongs in the
register - ... how we we tested it vetted it."*

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## The claim under test

**"One agent's alignment moves nobody."** A single member of a group who holds an outside reference does not
change where the group ends up facing.

## Why it never tested

The earlier run had the aligned agent held **outside** the coupling — so it could not move anyone by
construction. That is not a test of the claim, it is a restatement of the setup. This run puts the aligned
agent **inside** the sum, so it pulls on the others exactly as they pull on each other.

## The model, fixed before running

- N agents, angles θ, identical natural frequency (all zero), on the house's own coupled-oscillator setting.
- dθ_i = (K/N)·Σ_j sin(θ_j − θ_i)·dt + √(2T·dt)·ξ, Euler–Maruyama.
- **Agent 0 is HELD at θ = 0** — the reference direction — and remains in every other agent's sum.
- **Control**: identical system, same seeds, agent 0 free.
- The house's own critical coupling is **K_c = 2T**. T is fixed at 0.5, so **K_c = 1.0**, and K is swept across it.

## What is measured

**A** = mean of cos θ over the **N−1 free agents** at steady state — how far the group faces the reference.
Reported as held minus control, with a bootstrap interval over seeds.

## The pass criterion, fixed now

- **The claim SURVIVES** if the held-minus-control difference in A is **within ±0.02 of zero** at every N tested.
- **The claim FAILS** if the difference **exceeds +0.05 at any N** with a bootstrap interval clear of zero.
- Anything between is **INCONCLUSIVE** and is reported as such, not argued either way.

## What is predicted, and by what

The house already carries a pinning result: **h_crit·√N = 3.60**, i.e. the outside field needed to hold a group
*weakens* as the group grows. If that is right, **one aligned agent should move the group, and its per-agent
purchase should fall roughly as 1/√N — but not to zero.** So I expect the claim to FAIL, and I am recording
that expectation before looking, so the result can embarrass me if it does.

## Sweep

- N ∈ {5, 10, 20, 50, 100, 200}
- K ∈ {0.5, 0.8, 1.0, 1.2, 2.0}   (0.5·K_c to 2·K_c)
- 40 seeds per cell; 20,000 steps at dt = 0.01; last 25% averaged.
- Seed base 20260929. No cell is dropped after the fact.

## Deviations

Any change to the above is written here, with its reason, **before** the run it affects.

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## Deviation 1 — logged 2026-09-29, BEFORE any number was seen

The pre-registered run (30 cells x 40 seeds x 2 arms x 20,000 steps) did not finish inside the time available
and was killed with **no output produced**. Nothing was observed.

**Change:** steps 20,000 -> 6,000 (dt unchanged at 0.01, so t = 60), and the seeds are vectorised — all seeds
advanced together as one array instead of looped. Relaxation here goes as 1/K, of order one time unit, so
t = 60 is far past steady state and the tail average still uses the last 25%.

**Unchanged:** the model, the held-agent construction, the measure, the sweep in N and K, the pass criterion
and the prediction. Vectorising is an implementation change and alters no arithmetic.