# Test 1: change the medium, not the state

**Exploratory plan. Draft for Colson's review, October 14, 2026. Nothing has been run.**

Colson chose this as the first of three proposed directions. Test 2 (a medium that remembers) and Test 3 (spread by
descent) stay separate and unapproved. Choosing this one doesn't commit us to the others: we review this result before
deciding whether either deserves to be built.

## The question

In the oscillator model, the pull between oscillators is changed for 200 time units and then set back exactly. Does
the system behave differently afterwards? Two things are measured:

- how its coherence recovers;
- how fast a tiny gap grows 200 to 500 time units after the pull is back.

Each is compared with leaving the run alone, and with pushing the phases directly to the same coherence.

Why this question: the Cause Test (October 12) only ever pushed the oscillators' state, and its FAIL applies to that
kind of change. The follow-up found that the natural link between coherence and later growth is carried by coherence
over the past 100 to 500 time units, while a push mostly moves the fast part. Changing the medium for a while lets the
system's own dynamics do the reshaping, so it might reach what a push didn't. It might also fade just like a push.
Either answer is worth having.

## What changes, and what doesn't

- **Strength, not topology.** In this model every oscillator feels its whole group's average rhythm. There's no
  network of who is connected to whom. Changing topology would mean building a network version of the model, which is
  a new assumption like the ones Tests 2 and 3 need, so it stays out. Test 1 changes coupling strength only.
- **Which strength.** The pull within each group, K = 1.9, is raised or lowered by the same fraction in both groups
  for 200 time units, then set back to exactly 1.9.
  - The faint link between the groups (ε = 0.1) stays as it is.
  - K is chosen because it's the coupling that most directly sets coherence, the quantity the Cause Test was about.
  - Changing ε instead would be a separate, later option.
- **Nothing new is added.** There is no memory variable and no new scalar. The model's equations and settings don't
  change, apart from K during the window. The model code is used unchanged, as in the Cause Test.
- **The phases are untouched when the change starts.** The medium arm begins from exactly the saved state; only the
  pull changes.
- **The model sits right at its threshold** (K + ε = 2Δ).
  - Raising the pull moves the model above its tipping point, where the groups start drawing together.
  - Lowering it moves the model below, where coherence fades.

  So a small change matters a lot, and raising and lowering may not move coherence by equal amounts. That's why the
  size of the change comes from a calibration step rather than a guess (below).

## The conditions

All of them start from the same saved state at each moment T.

| Condition | What happens |
| --- | --- |
| Untouched | The run continues unchanged. |
| Medium, raised and lowered | K is raised (or lowered) for 200 time units from T, then set back exactly at R = T + 200 and followed for 500 more. |
| Push, matched to each medium arm | At R, the untouched run's phases are pushed so that each group's coherence equals the medium arm's at R. Then it's followed for 500 units. The push uses the Cause Test's dense method: every oscillator is moved a little toward or away from its group's rhythm. |

**Why the push is matched at R.** From R on, the medium arm and its matched push run on identical settings with
identical coherence. Any difference after R then comes from something other than the coherence level at R. It could
be the coherence history built up over the 200 units, or how the phases are arranged. This is how the plan handles
the point that changing the coupling itself changes coherence.

**The tiny gap.** As in the Cause Test, each arm carries two copies a hair apart (10⁻⁷), measured and put back every 5
time units. Any change is applied to both copies alike.

- During the window the gap grows under the changed pull. That growth is recorded, but it isn't an outcome.
- The 200-unit wait before the growth window lets the gap settle under the restored pull, as in the Cause Test.

**Not included:**

- **A sham push.** The main growth comparison is raised against lowered, and any common effect of disturbing the
  system cancels in it.
- **A kept-changed arm**, where the pull is never restored. It isn't needed for the question, because restoration is
  exact. It appears once in the calibration, coherence only, to show what permanently changed settings look like.

## What's recorded

- Each group's coherence every time unit, in every arm: from T to R + 500, or from R for the pushes.
- The timing between groups A and B.
- The gap's growth every 5 time units.
- At T and R: each group's coherence, its timing, and the higher-order measures |z₂| to |z₄|, as in the Cause Test.
- The pull in use in every stretch of every arm, so restoration can be checked.

**Main measures**

1. **Persistence.**
   - The coherence difference from the untouched run, averaged over moments, from T to R + 500.
   - The share of the difference at R that is still there over (R + 200, R + 500], medium against matched push. In
     the Cause Test, about 30% of a push's change was still there over the same stretch.
2. **Later growth.** Gap growth over (R + 200, R + 500], the same measure as the Cause Test.
   - The direction-dependent part, S = (lowered − raised) / 2, for the medium arms and for the pushes.
   - The common part: the average of raised and lowered, minus untouched.

Reported beside these but not used for the readings: growth during the window and over (R, R + 200], the A–B timing,
and the higher-order measures.

## Competing explanations and what each predicts

| Explanation | Coherence after the pull is back | Later growth, raised vs lowered | Medium vs matched push |
| --- | --- | --- | --- |
| **Fades like a push.** Once the pull is back, the phases carry the whole effect. | Returns to normal on the usual time scale. | Small, as in the Cause Test (where S was 0.00031 against a line of 0.00104). | About the same. |
| **The medium reaches the slow part.** 200 units of changed pull reshape the slow part of coherence, where the natural link lives. | The change lasts longer than after a matched push. | Raising slows later growth and lowering speeds it up, by at least half the benchmark. | The medium's effect is clearly larger. |
| **Any disturbance.** Changing the pull either way just knocks the system onto a different path. | Fades. | Raising and lowering shift growth the same way, with no direction-dependent part. | A similar common shift. |
| **A lasting switch.** The temporary change tips the system into a different long-lived state. | The difference in coherence or A–B timing stops fading. | Depends on the new state. | They differ and stay different. |

**The benchmark** is what the natural link predicts for the coherence difference at R: 0.1602 × half the difference
between the raised and lowered arms' coherence at R. The slope 0.1602 is the Cause Test's, from the untouched runs.
Because the pushes are matched at R, the same benchmark applies to them.

If coherence at a moment were itself the lever, both the medium and the push would reach the benchmark. The Cause Test
already found that pushes don't.

## The size of the change: calibration first

Before the pilot, the calibration runs on two old starts (16 and 17), ten moments each. Both starts were used for
earlier checks, so they stay out of the results.

- Raise and lower K by 0.5%, 1%, 2%, 5% and 10% for 200 units. Look at coherence up to R only; no gap growth and
  nothing after R.
- Choose the smallest size that meets both conditions:
  - raising lifts average coherence at R to at least 1.5 times the untouched run's (the size of the Cause Test's
    change);
  - the matched pushes can be built at 9 or more moments in 10.

  Lowering uses the same size, whatever it does.
- Run the chosen size once with the pull kept changed for 700 units, coherence only. This is the reference for
  permanently changed settings.
- If no size qualifies, stop and bring it back to Colson. Nothing is forced.

Also checked before anything else:

- a branch with the pull changed for zero units matches the untouched run bit for bit;
- the restored arms run on the unchanged model.

## Sample size

- **12 fresh starts, 103 to 114 (seeds 102 to 113), with ten moments each: 120 moments.**
  - These starts have never been run.
  - Starts 61 to 102 stay untouched for the drafted hold-vs-pulse test, which is paused.
- **Moments.** Ten per start, 330 time units apart after one random offset per start (a multiple of 5 from 0 to 325),
  from t = 2,000. Every branch ends before t = 6,000.
- **The starts are the units.** Each start's moments are averaged first, then the 12 starts. Error bars are the larger
  of a resampled value and SD/√12, as in the Cause Test.
- **What it can see.**
  - From the Cause Test's spread, the error bar on S should be about 0.0004.
  - If the medium changes coherence at R by about as much as the Cause Test's pushes did, the benchmark is about
    0.0021, about 5 error bars, and half of it is about 2.5 error bars.
  - So a benchmark-sized effect would show clearly. If nothing is there, the pilot rules out half the benchmark about
    3 times in 4.
  - Bigger changes in coherence make it more sensitive.
  - 8 starts would save about a third of the time, but would rule out half the benchmark only about half the time.
- **Size of the job.** Under a third of the Cause Test's computing:
  - roughly 15 minutes for the checks and calibration;
  - 45 to 60 minutes for the pilot.

  Both run in foreground pieces of under 5 minutes each.

## What counts as informative

These readings are fixed before the run. "Clear" means more than 3 error bars from zero. "Ruled out" means the
estimate plus 2 error bars is below the line.

- **The medium reaches the slow part:** the medium's S is clear, at or above the half-benchmark line, and clearly larger
  than the push's. Inside the model, a temporary change to the medium did something a push of the same size didn't.
  The next step would be finding out what it changed, by reading the saved runs. A confirmation test would be a
  separate decision.
- **Fades like a push**, a useful null: the medium's S is ruled out at the half line, and its persistence is about the
  same as the push's. In a model whose links don't remember, changing the medium briefly is no better than pushing the
  state. That sharpens Test 2's question (does memory in the medium make the difference?) without committing to it.
- **Any disturbance:** the common part is clear while S isn't. Intervening itself moved later growth, whichever way.
  For harm reduction, that means judging any intervention against an untouched comparison.
- **A lasting switch:** the medium arms' difference in coherence or A–B timing shows no clear decline between
  (R + 200, R + 300] and (R + 400, R + 500], while the push's clearly declines. A temporary change would have had a
  lasting effect. First confirm the pull really was restored, then look at what holds the change.
- **Inconclusive, never negative:** the calibration finds no usable size, more than 1 moment in 10 can't get a matched
  push, or the error bars can't tell half the benchmark from zero.
- **Anything in between** is reported as it stands, without forcing it into one pattern.

## What it can't tell

- It's a toy model. It shows structure inside the model, not evidence about cells.
- Matching at R makes coherence equal at one moment, not over its history. If the medium differs from the push, the
  pilot can't say whether the history or the arrangement of the phases did it.
- Even ordinary dynamics may make coherence that was built up gradually fade at a different pace from pushed
  coherence. So persistence alone isn't enough to call something new, and the growth contrast carries more weight.
- It tests one pull (K), one size, one window length (200) and one push method (dense).
- It's exploratory: it can suggest, not confirm.

## How it would run

1. Colson reviews this plan, and it's changed as needed.
2. With his go: write the code, run the code checks and the calibration (old starts, coherence only), and report
   back. Nothing runs on the fresh starts at this stage.
3. With a second go: the pilot on starts 103 to 114. It runs in foreground pieces of under 5 minutes each, and each
   task is saved as it finishes. Nothing is left running in the background.
4. Report every reading exactly as fixed here. Anything added after seeing the results is labelled as after the fact.

Everything stays in this folder. The model's files are only read, never changed. Nothing else is touched: not the
Cause Test folders, the Spiral Staircase pages, the website, cancer-research.md or map.md.