The ramp was a0..a4 with the pixel's base interpolated straight between them.
A segment meets its neighbour at an ANGLE, and the second derivative of a tone
curve is what a gradient reads as a band — so a knob left a line across the
mid-tones, worst exactly where it was reported: BLACK +100 put its whole lift
inside 0.25 and the stretch from 0.25 up came back identical to the untouched
frame (the "transition stays grey" it was reported for), and HIGHLIGHT -100
folded a seam into 0.747, between the highlights it pulled and the shadow it
left under them.
Measured on a 1024-step luma wedge through the exported pass, second
difference through a ±1% box: BLACK +100 read 105 at 0.030 against 0.000 from
0.25 up, HIGHLIGHT -100 read 72 at 0.747. Step of the first derivative across
the knots: 0.955 at 0.25 and 1.146 at 0.75 — the curve arrived folded, and
1.146 is a sign flip, not a bend.
So each knob is now a BUMP on the identity, peaking on its own knot — BLACK on
0.00, SHADOW on 0.25, HIGHLIGHT on 0.75, WHITE on 1.00 — with the kernel
(1-u^2)^2 over a half-width (a half of the ramp for the two ends, whose knots
ARE the ends, a quarter for the two heads). Level at u = 0, so a knot moves
without a fold at its own top; level at u = 1, so a move lands on the identity
and on its neighbour without an angle; C1 everywhere between. The two bumps of
a half meet on 0.50 both on zero, which is the same fixed midpoint as before,
and DR still moves the same knots (0.12 on the toe, 0.18 on the head, half of
each on the heads beside them).
A sum of bumps can overshoot where two steep sides land on one stretch — past a
slope of 1 the curve runs BACKWARDS, a worse band than the seams this replaces,
and it is reachable: DR alone was under it, BLACK and SHADOW +100 together were
not (unguarded min slope -0.0141). The guard reads each pair at its own
steepest points, 8/(3*sqrt(3))/w per unit amplitude (TONE_BUMP_SLOPE_HALF 3.0792,
TONE_BUMP_SLOPE_QUARTER 6.1584), holds the two under one and gives them up
together past it. A single knob never reaches it (a full BLACK is 0.77, a full
SHADOW 0.77), so every slider keeps its whole travel; the worst case is DR at
full, which gives up a tenth of its head roll (0.18 -> 0.8376 on the head), and
BLACK with SHADOW both at +100, which arrive at 0.65 of their own lift instead
of folding. Guarded, the sweep over five levels of all four knobs and DR reads
a min slope of +0.0183 and a max of 1.9937, with the largest slope jump 0.00005.
After: the same wedge, the same pass. The knot steps are 0.096 at 0.25 and
0.478 at 0.75, with no sign flip — C1 across the knot instead of a fold. BLACK
+100 now carries the rework out of its own quarter: +0.139 at 0.25, +0.101 at
0.30, +0.033 at 0.40, 0.000 at 0.50, where it used to read 0.000 from 0.25 all
the way up. HIGHLIGHT -100 keeps its lift (-0.126 peak against -0.121 before) and
spends it over the quarter instead of into a line. Every knob on zero is the
identity to the last bit — the pass also runs for the stock split tones and for
DR alone — and 0.50 is still the one value no knob moves.
Checks: tone-base-check.mjs now runs the bump and the guard as arithmetic
beside the shader (with the negative control: the unguarded pair still folds,
the guard is what stops it). highlight-knee-check.mjs reads the kernel and the
two slope constants off the source, pins the four amplitudes and the guard, and
sweeps the travel of every knob as before. All ten checks that run without a
browser pass, build clean.
Skipped: the guard's ceiling is a constant, not a search for the widest travel
that still clears a band we cannot see. Add when a frame shows a band the
deflections in hand cannot explain.
3dtours
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