Affine forms map: Pass 5 probe results, authority map, and the tape target's pointer
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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@ -5,7 +5,8 @@ the Self opened the `cultmath-tapes` campaign from pass 2; pass 3 revises the
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two CultLib cuts after their first Hands reports and the operator's challenge
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to the march, and changes the site field to envelope-then-noise; pass 4
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(section "Pass 4") revises `interval-ops` to r3 after its second Hands report
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and Soul's verdict, and names the one shipped void (Q9). Where a later pass and
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and Soul's verdict, and names the one shipped void (Q9). Pass 5 (section "Pass 5") maps
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affine forms against CultLib `main` at `a7966142`. Where a later pass and
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an earlier section disagree, the later pass wins. Where this map and the Body
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disagree, the Body wins and this map is stale.
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@ -1007,3 +1008,279 @@ beside the two CultLib r2 specs.
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- `:root:has(body[data-slug])` for the ritual seeds: supported in every
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current browser; a browser without `:has` gets the default seeds, which is
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acceptable for a fallback of a fallback.
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## Pass 5: affine forms (ruling `affine-forms-unparked`)
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Imagination pass 5, 2026-10-03. It maps the affine-arithmetic piece of the
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tape target, unparked on its own by ruling
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`cultmath-tapes:ruling:affine-forms-unparked`. It answers follow-up
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`cultmath-tapes:follow_up:wells-oracle-shortfall-affine-forms`. No tape,
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pruning or interpreter is built; tape steps 2-5 stay parked under
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`tape-target-unparks-when-asura-stable`. Pinned head: CultLib `main` at
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`a7966142`. Every `file:line` in this section is against that commit.
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### What was measured (body facts)
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The probe is a renamed copy of `NoiseBoundTests.cs` with five more bounds for
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`FogField`. It lives on the local branch `scratch/imagination-affine-probe`
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(`5fa8e9ce`). That branch was never pushed to origin and is not for merging.
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It was run on Yggdrasil through the ygg-verify stopgap (`dotnet/sdk:10.0`,
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about 3 minutes). It re-runs `Run(FogField.Wells(w), 8, 64, 0x5A7E + 8)`, so
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the tiles, rays and seeds are those of the landed (c) row at `N = 8`. V0
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reproduces the landed figures exactly (0.378 and 0.194).
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Constants the probe needed, each sampled over 2e5 seeded points with a 1.25
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margin (probe grade, not provenance):
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- the Frobenius norm of the Hessian of `snoise(float3)`, max 55.15, so
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`M = 68.94`;
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- the gradient length of `snoise(float2)`, max 7.35, so `L2 = 9.19`;
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- for comparison, the mean `|snoise_grad|` is 2.59, against `L = 10.10`.
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The bounds:
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- **V0**: the landed bound.
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- **V1**: V0 with the detail octave's ball no longer enlarged by the warp. The
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detail reads the unwarped point (`NoiseBoundTests.cs:860`, `Noise.Detail(p, ...)` of the unwarped `p`), yet `Bound`
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hands it `ball.w`, which carries `D` (`:875`, `:890`).
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- **V2**: V1 with the warp centred. Each phase's coverage ball is centred at
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`c + flow(c) shift_k`, with radius `r + |shift_k| rho`, where
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`rho = FlowAmplitude L2 FlowFrequency r sqrt 2` bounds how far the flow moves
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over the ball. This is the mean-value form of the warp: interval arithmetic
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only, no affine noise.
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- **V3**: V2 with the centred form of the noise,
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`n(c) +- min(L R, |g(c)| R + M R^2 / 2)` from one `snoise_grad`.
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- **V4**: V3 with a reduced affine form. Three shared symbols (depth `t`,
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slopes `u` and `v`) carry the envelope's plane term and every noise term's
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linear part, and one error symbol takes the rest. It applies where the fade
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is the point 1, and is intersected with V3.
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- **V5**: V4 with one shared symbol, depth `t`, plus one error symbol:
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Gamito's `t0 + t1 e1 + t2 e2`.
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A gradient evaluation is counted as 2 snoise, an estimate. V3 to V5 gate a
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probe on the best result any centre value and gradient could give.
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Range ceiling. A cell is range-nonempty if the density is positive at any of
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5 x 5 slopes over the tile's slope square (corners included) times 9 depths
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over the cell, or if it is in the oracle mask. Its efficiency is the oracle
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march's cost over the cost of marching the range mask, with probes free. The
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sampling can miss nonzero points, so this figure is an upper estimate. No
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sound enclosure can beat it.
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| (c), `N = 8` | efficiency (oracle) | cost ratio | dense cells/ray | probe overhead | cull |
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| --- | --- | --- | --- | --- | --- |
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| warp 0, V0 landed interval | 0.378 | 2.74x | 12.40 | 0.961 | 0.952 |
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| warp 0, V2 centred interval | 0.368 | 2.67x | 12.40 | 0.937 | 0.952 |
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| warp 0, V3 centred noise | 0.466 | 3.38x | 8.67 | 0.868 | 0.974 |
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| warp 0, V5 affine, one symbol | **0.511** | **3.71x** | 7.81 | 0.859 | 0.979 |
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| warp 0, V4 affine, three symbols | 0.534 | 3.88x | 7.37 | 0.853 | 0.981 |
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| warp 0, range ceiling | 0.859 | | | | 0.995 |
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| warp 60, V0 landed interval | 0.194 | 1.39x | 31.55 | 0.993 | 0.842 |
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| warp 60, V1 detail unwarped | 0.205 | 1.47x | 27.74 | 0.963 | 0.864 |
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| warp 60, V2 centred interval | 0.347 | 2.49x | 13.71 | 0.938 | 0.945 |
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| warp 60, V3 centred noise | 0.441 | 3.16x | 9.52 | 0.870 | 0.969 |
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| warp 60, V5 affine, one symbol | **0.477** | **3.42x** | 8.71 | 0.864 | 0.974 |
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| warp 60, V4 affine, three symbols | 0.502 | 3.60x | 8.16 | 0.856 | 0.977 |
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| warp 60, range ceiling | 0.854 | | | | 0.995 |
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Every variant's transmittance equals the dense march's (max |dT| 0). No
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variant's pre-pass proved a cell empty where any of the 225 samples per cell
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was nonzero (0 enclosure violations).
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What this establishes:
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1. **The oracle cannot be reached by any sound bound.** The oracle mask
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samples one depth per ray. A bound must cover the whole slice of the tile,
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so the best any exact enclosure could do is about 0.86 of the oracle. That
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matches the 0.57-0.85 that conservative visibility methods reach against
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exact sets (`docs/research/culling-efficiency-oracle-prior-art.md`). An
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oracle target of 0.90 is unreachable by construction.
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2. **The warp diagnosis was half right.** At warp `D` most of the gap is the
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warp enlarging the radius by its maximum (V0 to V2: 0.194 to 0.347). That
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is fixed by evaluating the flow at the ball's centre and enlarging the
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radius only by how far the flow can move across the ball. It takes no
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affine noise, and it costs one more snoise per probe, for the second phase
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ball. The detail octave's spurious warp (V1) is worth little on its own.
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3. **Affine noise buys the rest:** the centred form, then the shared depth
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symbol. Against the centred interval, affine with one symbol closes about a fifth
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of the oracle gap at warp 0 and at warp `D` (efficiency 0.368 to 0.511 and
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0.347 to 0.477). Its combined cost ratio is 1.39x and 1.37x the centred
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interval's, and 1.35x and 2.46x the landed interval's. It reaches 0.59 and
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0.56 of the range ceiling.
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4. **Three symbols buy about 5% over one** (0.534 against 0.511, 0.502
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against 0.477). One symbol fits a `float3`. Three need a struct or a
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`float4` plus a `float`, and the HLSL mirror's harness compares struct
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returns but has no struct parameters (`packages/cultmath/docs/design.md:126-135`).
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5. **The probe overhead falls** from 0.96 to 0.86 under every centred or
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affine bound. Each probe costs a gradient and proves more, so the total
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cost still drops by 1.35x to 2.46x. Today's (c) assertion, overhead at
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least 0.90 (`NoiseBoundTests.cs:1558`, `:1588`), would reject the cheaper
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march. That is question `affine-wells-contract`.
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### The model
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An affine form is a transient value, not a persistent kind. The persistent
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things this pass adds or changes:
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| Thing | Identity | Lifecycle | Authority |
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| --- | --- | --- | --- |
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| `SNOISE_HESSIAN` (C#), `CULTMATH_SNOISE_HESSIAN` (HLSL, lowered to GLSL) | The constant's name; one value in `math.Affine.cs`, mirrored | Measured once by `NoiseBoundTests.MeasureHessian` (slow, explicit), as `SNOISE_LIPSCHITZ` is. It is re-measured and re-pinned whenever the `snoise` kernel changes. | `MeasureHessian` sets it; `HessianConstantPinsSampledCurvature` pins it; no consumer carries its own |
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| `af_*` families in `fixtures/glsl-parity.json` | Family signature string, appended after the last existing family | Regenerated only by `CULTMATH_WRITE_GLSL=1`; earlier cases stay byte-identical | `GlslMirrorTests` generator |
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| The (c) contracts | `IntervalSkipHalvesEvaluations` assertions | Changed only by a ruling | Operator (question `affine-wells-contract`) |
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| `SNOISE2_LIPSCHITZ` (C#), mirrored in HLSL | The constant's name in `math.Interval.cs`, beside `SNOISE_LIPSCHITZ`, with an `iv_snoise_ball(float2, float)` overload | Measured by `MeasureLipschitz2` (slow, explicit) and pinned, as `L` is | CultMath. The (c) field's flow is `snoise(float2)`, and a bound on a CultMath function belongs to CultMath, not to a test-local helper. A consumer's own flow, such as the site's texture, keeps its own variation bound |
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### Representation
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**A reduced affine form with one shared symbol: `float3(x0, a, e)`, meaning
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`x0 + a·ε + e·δ`.** Here `ε ∈ [-1, 1]` is shared by every form over the same
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region, and `δ ∈ [-1, 1]` is a private error, with `e ≥ 0`.
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- This is Messine's AF1 as Gamito and Maddock use it for gradient noise,
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`t0 + t1 e1 + t2 e2`.
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- Sharp and Jacobson's measurements show that this fixed-symbol form is the
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fast one on GPUs: 1.0x against 8.4x for full affine.
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- It fits one `float3` register, as an interval fits a `float2`. It needs no
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struct parameter, so the mirror, lowering and fixture machinery take it
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unchanged.
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- V4 against V5 prices the two extra symbols at about 5% efficiency.
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For a tile probe `ε` is the depth parameter: `z = z_m + h ε` with
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`h = (z1 - z0) / 2`. The footprint goes into the error. For a single ray it is
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Gamito's form exactly.
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A point in space is not a new type. It is the three arguments
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`(centre: float3, axis: float3, radius: float)`, the set
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`{centre + axis ε + w : |w| ≤ radius}`. `af_snoise` and `af_fbm` take a point
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in that form, and `af_frustum_axis` and `af_frustum_ball` produce one for a
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tile slice.
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Ops, named by the (c) consumer, plus `af_mul` because affine arithmetic is
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defined by its product, as `iv_mul` is for intervals:
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- `af_point`, `af_symbol(x0, a)`, `af_from_iv`, `af_range` (the bridge to
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`iv_*`);
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- `af_add`, `af_sub`, `af_neg`, `af_scale`, `af_add_iv`, `af_mul`;
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- `af_snoise`, `af_fbm`;
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- `af_frustum_axis`, `af_frustum_ball`.
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No others until a consumer names one, which is the rule `design.md` already
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states for intervals.
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The interval side gains `SNOISE2_LIPSCHITZ` and the overload
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`iv_snoise_ball(float2, float)`. The (c) field's flow is `snoise(float2)`, and
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centring the warp needs a bound on it, which belongs to its owner.
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`af_snoise(centre, axis, radius)` makes one `snoise_grad` call. With
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`R = |axis| + radius`, it returns the narrower of two sound forms:
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- the Lipschitz form `(n, 0, L R)`;
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- the centred form `(n, g · axis, |g| radius + M R² / 2)`.
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`e` absorbs the op's own float rounding and an allowance for a GPU's gradient
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error. The allowance is derived in the doc comment and measured on WebGL2 by
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the fixture's enclosure check, as `iv_frustum_ball`'s is.
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The warp stays the consumer's, so CultMath owns no flow. The rule, stated in
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`design.md` and tested for the (c) field: a warp whose value at the centre is
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known, and whose variation over the ball is at most `ρ`, is enclosed by moving
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the centre by `w(c)` and adding `ρ` to the radius. Interval balls benefit from
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it too (V2).
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### Authority map
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- **Owner.** `src/CultMath/math.Affine.cs` owns affine-form semantics on the
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CPU. `shaders/CultMath.Affine.hlsl` mirrors it bit for bit, and is the shader
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semantic authority. `shaders/CultMath.glsl` is derived from the HLSL.
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- **Inputs.**
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- `snoise_grad`, whose `.w` is bit-equal to `snoise`;
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- `SNOISE_LIPSCHITZ`;
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- `SNOISE_HESSIAN`;
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- the caller's forms and points.
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- **Outputs.** `float3` forms, and `float2` intervals through `af_range`.
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- **Derived state.**
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- The GLSL text is generated by `GlslLowering`, unchanged.
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- The fixture families are generated.
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- The (c) efficiency rows are printed measurements, not state.
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- **Forbidden writers.**
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- A consumer linearising noise itself: the probe's V4 and V5 did this by
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hand, and Hands must call `af_snoise`.
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- A second curvature constant anywhere.
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- Hand-edited `CultMath.glsl`.
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- An affine or interval evaluator over `CultMath.Expr` (the parked tape's,
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see collisions below).
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- A flow or envelope function in CultMath. Consumers compose those, as for
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intervals.
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- **Shared paths.** The test field's `FogField`, the site ground and
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Aetheria's nebula all reach affine noise through `af_snoise` or `af_fbm`,
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and a tile slice through `af_frustum_*`.
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- **Deletion line.**
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- In `FogField.Bound` (`NoiseBoundTests.cs:875`), the warp-enlarged ball is
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replaced by the centred warp, and the detail ball stops carrying the warp
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(`:890`).
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- The (c) probe-overhead assertion (`:1558`, `:1588`) is replaced as the
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ruling on `affine-wells-contract` says.
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- `design.md`'s "What they do not: ... affine forms" (`:584-587`) and "That
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gap belongs to the tape target's affine forms" (`:623-624`) are rewritten.
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- The tape target's open item on representation is closed.
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### Cut order
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1. **`affine-forms`, the one she can see.**
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- It adds C# `math.Affine.cs` and `AffineTests`, plus `SNOISE_HESSIAN`'s
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provenance.
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- The (c) field gains the centred warp and the affine bound.
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- (c) is re-measured with three bound rows on the same tiles: landed
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interval, centred interval, affine. The range ceiling is printed beside
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the oracle.
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- An explicit, env-gated test writes a cull-map PNG of one (c) frame: dense
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cells per ray for each bound, and the transmittance. Self publishes it.
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- `design.md` is updated.
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- It is blocked on `affine-wells-contract` only for the assertion's shape.
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2. **`affine-shaders`.**
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- It adds `CultMath.Affine.hlsl` and its Unity copy and `.meta`, the
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include, and the regenerated `CultMath.glsl`.
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- Fixture families are appended, with an enclosure check for `af_snoise`
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and `af_frustum_ball`.
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- The dxc and glslang compiles and the WebGL2 golden-fixture run happen on
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Starfire.
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- Both cuts share the branch `hands/cultmath-affine-forms` and merge
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together after cut 2's Soul pass. Main never carries a C#-only `af_*`, so
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three-runtimes-agree holds at every merged commit.
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Writing the C# before the HLSL inverts the usual order, because HLSL is the
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semantic authority. The cost is small: the ops are plain arithmetic in the
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common subset. Cut 2 changes the C# if the HLSL dialect forces something,
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never the reverse.
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### Verification venue
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Heavy work runs on Yggdrasil through the stopgap: `dotnet test`, Stryker scoped
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to `math.Affine.cs`, the (c) re-measure and the cull-map render. Starfire runs
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only the Windows and GPU legs, one job at a time:
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- `dxc -T lib_6_3 -HV 2021` on `CultMath.hlsl`;
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- `glslangValidator` 16.6.0 on the lowered GLSL (ruling
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`operator-compiler-downloads`);
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- the WebGL2 golden-fixture run on the GTX 1070, under ANGLE D3D11 in a
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browser. It is seeded from the glsl-lowering Soul pass's `gpu.js`
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(follow-up `glsl-browser-parity-run`).
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### Collisions
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- **`cycles-converters` r1** edits the same seams:
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- the include list (`CultMath.hlsl:509-510`);
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- `HlslMirrorTests.cs:64-67`;
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- `build-unity-package.ps1:82-87`;
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- the fixture's family list.
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Whichever merges second rebases, regenerates `CultMath.glsl` and the fixture
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(`CULTMATH_WRITE_GLSL=1`), and appends its families after the first's.
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Nothing semantic collides.
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- **`expr-ir` r1 and question `material-ir-in-cultmath`.** No file collides,
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and neither cut waits on the other.
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- Under `cultmath-ir-now`, the parked tape's step 2 will add interval and
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affine evaluators over `ExprGraph`, and those will call `iv_*` and `af_*`
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as op semantics. That is one more reason the affine ops are plain value
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functions over `float3`, with no struct state.
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- Under `aetheria-local`, nothing here changes.
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- The `expr-ir` spec's negative check forbids `Interval(Evaluator)?` under
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`Expr`. Self may add `Affine` to it at its next revision.
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- **`site-ground` r3** (not dispatched): the void (e) and the site's flow
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texture are not measured here. Follow-up `site-ground-affine-and-centred-warp`
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carries both. Centring the warp in the site needs a bound on the flow
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texture's variation, and the texture stays the one motion owner (it is
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sampled at the centre, never re-derived).
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@ -108,6 +108,9 @@ source.
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- Run the (a) spike first, or go straight to (c)?
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- Pick the interval representation (plain intervals or affine arithmetic) and
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the noise bound strategy.
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Affine forms for noise and warp were unparked on 2026-10-03 (ruling
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`cultmath-tapes:ruling:affine-forms-unparked`): one shared symbol, mapped in
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`docs/cultmath-interval-ground-cut.md`, "Pass 5".
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- Choose the bytecode register model for the HLSL interpreter, given GPU
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register pressure.
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- Decide whether the brush-grammar station project lives in Aetheria or in
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