re: the tone curve s single exponent is refuted above render 40

Raised by the port and independently reproduced here before being adopted, since
adopting their claims unchecked has misfired twice this session.

Binning matched pixels by render level rather than fitting a scalar, the implied
exponent falls monotonically and crosses 1.0:

  render          16     23     31     39     47     64
  port  (all)   1.26   1.18   1.10   1.03   0.93   0.85
  mine  (flat)  1.303  1.347  1.128  0.912  0.935  1.003

Below the crossing the capture is darker than the render, which is what the page
measured; above it the capture is brighter. A single exponent cannot express a
curve that crosses unity, so the model is valid only in the darks -- which is
exactly the reach the page already stated. The reach line was not a hedge, it was
the finding.

Where the two disagree is recorded and not resolved: the crossing is about 44 by
their binning and 35 to 40 by mine, and the darks read 1.18-1.26 theirs,
1.30-1.35 mine, 1.49 for the page s original patch fit. Three estimators on three
populations, all agreeing on direction and on gamma above 1 in the darks.

A confound in my own reproduction is stated rather than left implicit: whole-image
correlation is only 0.594 because the committed capture and the default render
differ in focus state, which the port measured as 74.1 percent of differing
pixels. My bins include that mismatch, so they are not a clean second opinion.
And a 1280x720 render against a 1279x675 capture needs a resample, which is why
the fit is restricted to flat-neighbourhood pixels.

METHOD gains the general form: a stated reach is a boundary rather than a hedge,
and the fix was printing the curve instead of a scalar, because a scalar hides
its own domain.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01QsEPXWVaEpyfudtR6re1Pd

docs/re/METHOD.md

@@ -58,6 +58,15 @@ agent's loop prompt, i.e. nowhere durable. See [`README.md`](README.md) for the
 * **My own last-turn result is a hypothesis too.**
 * **A global partition can understate a per-owner one.**
 * **A residual is measured against a population — name it.**
+* ⚠️ **A stated reach is a boundary, not a hedge — do not extrapolate past it.**
+  `ui-render-tone-curve.md` fitted γ ≈ 1.34–1.49 on **dark flat patches** and
+  wrote "nothing constrains midtones or highlights". Used above that range the
+  model is simply wrong: binned by level, the exponent falls monotonically and
+  **crosses 1.0 near render ≈ 40**, so above it the capture is *brighter* than
+  the render and no single exponent can express the curve. The page had already
+  said where it stopped being true; the error was reading past the sentence.
+  ✅ **The fix was not a better fit — it was printing the curve instead of a
+  scalar**, so it can be argued with. A scalar hides its own domain.
 * ⚠️ **Normalising? Divide by how many inputs CARRY SIGNAL, not how many there
   are.** The port hit this three times in one pipeline, each invisible to every
   check except a level measurement, and each the same mistake:

docs/re/structures/ui-render-tone-curve.md

@@ -1,6 +1,7 @@
 # 🟡 Our composite is brighter than the emulator's frame — measured, not decoded
 
-**Status:** 🟡 **measured, with a narrow reach and a live confound.** Closes an
+**Status:** 🟡 **measured, and REFUTED outside its stated reach** — the single
+exponent holds only below render ≈ 40; see the refutation below. Closes an
 observation left dangling by
 [ui-8ax-fullres-background](ui-8ax-fullres-background.md) ("the capture is ~4×
 darker than the render"), and puts a number on the ❔ that
@@ -48,6 +49,43 @@ patches, and the count runs 0 / 83 / 404 / 1055 / 1788 for `std <`
 
 So `capture ≈ 255·(render/255)^γ` with **γ ≈ 1.34 – 1.49**.
 
+## 🔴 REFUTED above render ≈ 40 — one exponent cannot express this curve (2026-08-29)
+
+Raised by the port, and **independently reproduced here** before being adopted.
+Binning matched pixels by render level instead of fitting a scalar:
+
+| | render 16 | 23 | 31 | 39 | 47 | 64 |
+|---|---|---|---|---|---|---|
+| **port**, all matched pixels | γ 1.26 | 1.18 | 1.10 | 1.03 | 0.93 | 0.85 |
+| **mine**, flat-neighbourhood pixels | γ **1.303** | **1.347** | **1.128** | **0.912** | **0.935** | **1.003** |
+
+✅ **The finding holds: the exponent falls monotonically and crosses 1.0.** Below
+that the capture is darker than the render (γ > 1, which is what this page
+measured); **above it the capture is *brighter*.** A single exponent cannot
+express a curve that crosses unity, so the model above is **valid only in the
+darks** — which is exactly the reach this page already stated. ⚠️ **The reach
+line was not a hedge; it was the finding.**
+
+🟡 **Where they disagree, and it is not resolved.** The crossing point is
+**≈44** by the port's binning and **≈35–40** by mine; the darks read **1.18–1.26**
+(theirs), **1.30–1.35** (mine) and **1.49** (this page's original patch fit).
+Three estimators on three populations — selected flat patches, all matched
+pixels, flat-neighbourhood pixels — and nothing here adjudicates between them.
+**All three agree on the direction and on γ > 1 in the darks**; that is the part
+to rely on.
+
+⚠️ **A confound in my reproduction, stated so it is not mistaken for
+independence it does not have:** my whole-image correlation is only **0.594**,
+because the committed `live-main-menu` capture and the default render differ in
+**focus state** — the port measured that 74.1 % of differing pixels fall inside
+the `live-main-menu` vs `live-main-menu-options-focused` signature. My bins
+therefore include that mismatch. It did not change the direction of the trend,
+but it is why my numbers are not a clean second opinion.
+
+⚠️ Also: render 1280×720 against a 1279×675 capture requires a resample
+(LANCZOS here), which perturbs levels at edges — hence restricting to
+flat-neighbourhood pixels.
+
 ## ⚠️ The reach — and it is narrow
 
 * **The flat patches are almost all dark**: render values ~0–60. Over that range