# From a coordinate to an executable edit

The core object is a field, not a text instruction. For source pixel p, store its OKLab lightness L_p, chroma C_p, hue direction, and normalized image position (x_p,y_p). Keep the source immutable.

## Operator

Let B be a positive, normalized 6×6 Gaussian spatial basis. Seventy-two controls z in [0,1] choose 36 expansion coefficients and 36 contraction coefficients. Their per-pixel influence matrix E incorporates the fixed protected region, a smooth transition around it, and each pixel's sRGB gamut headroom. Chroma gain is g_p = 1 + (Ez)_p. The implementation limits gains to [0.35,1.65].

Render L'_p=L_p, a'_p=g_p a_p, b'_p=g_p b_p. Positive gains preserve the source hue direction in OKLab; source pixel coordinates are unchanged. This moves the distribution of chromatic emphasis, not objects. It is a color-adjustment family, not intrinsic-image decomposition.

## Why the target is solvable as a constrained problem

The source hue assigns each pixel to one of six fixed 60-degree bins. For every bin h, require sum(C_p (Ez)_p) over p in h = 0. These six linear equalities preserve the bin's chroma mass and, together, preserve total chroma M.

With M fixed, the edited centroid is affine in the controls:

    center(z) = center(0) + [mx; my] z
    mx_j = sum_p x_p C_p E_pj / M
    my_j = sum_p y_p C_p E_pj / M

A target centroid adds two equality constraints. The set of feasible controls is a box intersected with affine constraints: a convex polytope. Its image in centroid coordinates is also convex. SciPy HiGHS first tests feasibility; SLSQP then minimizes a quadratic color-change/smoothness objective over the feasible set. No learned interpretation or language-model response is involved.

The ablation uses the same controls and objective without the six hue-budget equalities. Its total chroma can change, so the centroid target is expressed as sum C_p g_p (x_p-target_x)=0 and likewise for y. Those constraints are still linear in z for a fixed target.

## Refusal witness

For an infeasible target t, a quadratic projection proposes a nearby feasible point. Its residual supplies a direction n. A separate support LP computes the maximum reachable n·center. If n·t exceeds that bound, t lies outside a reachable half-space. Store the separation margin and the LP's dual-gap/stationarity residuals. A refusal is marked verified only when the margin is positive and those residuals pass tolerance.

This is a numerical certificate for this bounded operator family, under this implementation's floating-point tolerances. It is not proof that no other image-editing method can fulfill the request. The displayed outline samples 32 support directions; it is a visualization, not the feasibility oracle.

## Export is part of the contract

An exact mathematical coordinate need not survive RGB conversion and quantization. All acceptance checks reload the written PNG and recompute coordinates from its pixels. Protected source pixels are stored exactly. The initial protocol used 8-bit files. A recorded post-hoc precision check uses identical solved parameters at 16-bit and the same gates; no target or constraint was relaxed.

## What remains unrepresented

Semantic object identity, material/illumination separation, geometric motion, texture identity, perceptual intent and artistic merit are not certified here. Fixed OKLab L does not imply fixed ISR display-luma or fixed physical lighting. Six hue sums do not specify the entire color distribution. Global metric agreement alone cannot establish image equivalence.

## Prior art and inherited components

- Color transforms: the user's Color Kernel v0.4.8 functions, extracted verbatim from cell 11. Underlying color space: [Björn Ottosson's OKLab](https://bottosson.github.io/posts/oklab/).
- Structural readout: the user's ISR `evidence_fields` and `field_summary`; measured after export, not used to claim invariant ISR mass.
- Optimization: SciPy HiGHS linear programming and SLSQP constrained minimization. No new optimizer is claimed.
- Relevant existing image-control work: [DiffVG](https://people.csail.mit.edu/tzumao/diffvg/), [DragDiffusion](https://yujun-shi.github.io/projects/dragdiffusion.html), [Saliency Driven Image Manipulation](https://arxiv.org/abs/1612.02184).

The experiment-specific work is the field-to-control mapping, source-conditioned gamut/protection bounds, feasibility witness, reversible coordinate state, export verification and comparative harness. Scientific novelty remains an open question.