Imagination pass 3d: footprint LOD. The cavity cost as a function of hollow radius, camera offset and ramp width; Q8 retired for a steps-per-pixel contract

The march is footprint-aware: step = max(k z theta_px, min(S/4, lambda_fine/2)),
one smooth weight per octave, the bound enclosing the truncated field. The
cavity cost model is a grid, not a number; the shipped point needs about 6
steps and 12 snoise per pixel against about 35 fixed steps.

Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>

docs/cultmath-interval-ground-cut-interval-ops.r2.spec.json

@@ -9,10 +9,8 @@
   "depends_on": [],
   "first": [
     "Follow-up on cut report cultmath-tapes:cut_report:cut-interval-ops.h1 (head c9bd003b, worktree F:\\Projects\\CultLib-iv). Rulings in force: operator-tile-amortized-march (the bound stays; one probe per screen tile), operator-compiler-downloads. The operator's challenge, verbatim in the map's Pass 3 section: r1 measured a uniform noise slab with no empty space to skip, the worst case for intervals, and the density that matters is an analytic envelope (a pre-distortion SDF) modulated by noise, where the envelope's interval is tight and proves whole ray lengths empty in one probe. Map: docs/cultmath-interval-ground-cut.md, 'Pass 3'. r1's ops and bounds stay; this revision touches math.Interval.cs, CultMath.Interval.hlsl (both tracked copies), NoiseBoundTests.cs, design.md and README only.",
-    "The field, from Aetheria (Assets/Shaders/Volumetric.cginc:165-221, :91-102; Scripts/ServerShared/Zone.cs:369-397 GetHeight, :426 PowerPulse; Resources/Settings.asset DefaultEnvironment and PlanetSettings). Pre-distortion SDF s(p) = p.y + h(p.xz), h >= 0 the well map: h = PowerPulse(|p.xz| / (2R), 2) * 64 (the zone bowl) + sum over wells of PowerPulse(|p.xz - c_b| / r_b, 16) * d_b, with PowerPulse(x, e) = (1 - 4x^2)^e for x in [0, 1/2] and 0 beyond, r_b = 500 M^0.25, d_b = 30 M^0.175 (Aetheria's GravityRadius/GravityDepth curves).",
-    "(Field, continued.) Noise only where s < S (SafetyDistance 30): fade(s) = 1 - smoothstep(0.75 S, S, s); displaced height s' = s + A fade(s) n(p) with n the warped two-phase coverage noise plus half the detail octave (r1's field shape); density = max(0, (F - s') / B) (Aetheria's floor: FloorOffset F = -20, FloorBlend B = 10; the patch band and the 1e-9 fill are left out). Above the safety distance density is exactly 0, and so is the noise cost: Aetheria's dense march gates the noise pointwise (`if (dist < _SafetyDistance)`); the test's dense march does the same.",
-    "The composed bound over a tile's frustum slice [z0, z1], every step an existing iv_* op. Envelope over the slice's exact box, not the ball: with slopes m in [m_c - s, m_c + s] per axis and z in [z0, z1], y = iv_mul([z0, z1], m_y) and x = iv_mul([z0, z1], m_x) (plus the tile's translation o); h's interval over the xz box is the sum over bowls of [pulse(far), pulse(near)], near and far the least and greatest distance from the box to the bowl's centre (pulse is monotone decreasing in distance, so each term is exact; the only slack is the sum).",
-    "(Bound, continued.) iv_s = iv_add(y, iv_h); iv_fade = the ordered endpoints of fade over iv_s (monotone); iv_n = iv_snoise_ball(ball.xyz * F0, ball.w * F0) with ball = iv_frustum_ball(..., warp = D); iv_s' = iv_add(iv_s, iv_scale(iv_mul(iv_fade, iv_n), A)); the slice is provably empty iff iv_s'.lo >= F. When iv_fade.hi == 0 the noise term is identically zero and iv_snoise_ball is not evaluated: the envelope alone proves the slice empty at no snoise cost. That short-circuit is the compounding the ruling expects from tiles.",
+    "The field, from Aetheria (Volumetric.cginc:165-221 and :91-102; Zone.cs:369-397 GetHeight, :426 PowerPulse; Settings.asset). SDF s(p) = p.y + h(p.xz), h >= 0 the well map: h = PowerPulse(|p.xz| / (2R), 2) * 64 (the zone bowl) + sum over wells of PowerPulse(|p.xz - c_b| / r_b, 16) * d_b, with PowerPulse(x, e) = (1 - 4x^2)^e for x in [0, 1/2] and 0 beyond, r_b = 500 M^0.25, d_b = 30 M^0.175 (Aetheria's GravityRadius/GravityDepth curves). Noise only where s < S: fade(s) = 1 - smoothstep(0.75 S, S, s); displaced height s' = s + A fade(s) n(p) with n the warped two-phase coverage noise plus half the detail octave (r1's field shape); density = max(0, (F - s') / B) (Aetheria's floor: FloorOffset F = -20, FloorBlend B = 10; the patch band and the 1e-9 fill are left out). Above the safety distance density is exactly 0, and so is the noise cost: Aetheria's dense march gates the noise pointwise (`if (dist < _SafetyDistance)`); the test's dense march does the same.",
+    "The composed bound over a tile's frustum slice [z0, z1], Envelope over the slice's exact box, not the ball: with slopes m in [m_c - s, m_c + s] per axis and z in [z0, z1], y = iv_mul([z0, z1], m_y) and x = iv_mul([z0, z1], m_x) (plus the tile's translation o); h's interval over the xz box is the sum over bowls of [pulse(far), pulse(near)], near and far the least and greatest distance from the box to the bowl's centre (pulse is monotone decreasing in distance, so each term is exact; the only slack is the sum). iv_s = iv_add(y, iv_h); iv_fade = the ordered endpoints of fade over iv_s (monotone); iv_n = iv_snoise_ball(ball.xyz * F0, ball.w * F0) with ball = iv_frustum_ball(..., warp = D); iv_s' = iv_add(iv_s, iv_scale(iv_mul(iv_fade, iv_n), A)); the slice is provably empty iff iv_s'.lo >= F. When iv_fade.hi == 0 the noise term is identically zero and iv_snoise_ball is not evaluated: the envelope alone proves the slice empty at no snoise cost. That short-circuit is what tiles compound.",
     "Where the warp lives in the bound. Aetheria warps only the noise argument (triNoise3d(pos + fl * ...)); the surface height is read at the unwarped xz and the displacement is applied to y afterwards. So the envelope is evaluated on the pre-distortion SDF over the unwarped slice (no D), and only the noise ball is enlarged by D (iv_frustum_ball's warp argument). A warp applied to the envelope's argument would be unsound without enlarging the box by D; the field is defined so that it is not. The test EnvelopeBoundEnclosesDensity pins this: 2,000 seeded slices x 64 warped points, the composed bound encloses density(p) with no tolerance (every op is monotone or already widened in r1; if a seeded case fails at an endpoint, widen in the implementation, never the test).",
     "iv_frustum_ball(float2 centreSlope, float z0, float z1, float footprintPerDepth, float warp) -> float4(centre, radius), in C# and both HLSL copies, is the one new CultMath function: the noise ball of a tile over a depth segment. Camera: pinhole at the origin looking +z, focal length f pixels; a pixel's ray is p(z') = (m z', z'); a tile's slopes lie in m_c + [-N/(2f), N/(2f)]^2 (full footprint, so sub-pixel jitter is covered), s = N / (sqrt(2) f). Centre c = (m_c z_m, z_m), z_m = (z0 + z1) / 2; radius r = ((z1 - z0) / 2) |(m_c, 1)| + z1 s + warp. Proof: a point of slope m_c + e (|e| <= s) at depth z' is c + (z' - z_m)(m_c, 1) + z'(e, 0), the terms bounded by the two radius terms, and the warp moves it by at most D (WarpedPointsStayEnclosed). The slice is a convex polytope, so every point between rays is enclosed too. The two vectors are never parallel, so the inequality is strict and float32 cannot cross it.",
     "(Tile ball, continued.) Test TileBallEnclosesEveryRaySegment: 2,000 seeded tiles (N in {1, 4, 8, 16}, f in [500, 3000], |m_c| <= 1.1, segments across each scenario's z range), 64 points each with sub-pixel offsets, both depth ends and a flow of length <= warp; no tolerance. Hand mutations that must die, listed in the file header: z0 for z1 in the lateral term; |m_c| for |(m_c, 1)|; the warp dropped. A moving camera needs nothing more: every frame's pre-pass evaluates iv_frustum_ball from that frame's camera and never carries a ball, a mask or a probe result across frames; the only cross-frame state is the TAA history, which is colour, not a soundness input. The test says so by drawing each tile's camera independently.",
@@ -21,8 +19,10 @@
     "The march, on cell ranges. The dense grid is a scenario-owned list of cell boundaries z_0 < ... < z_J (uniform for the slab scenarios; Aetheria's quadratic spacing z_j = near + (j/J)^2 (far - near), J = 256, for the Aetheria-like ones, from CloudShader.shader:116). MarchTile takes a range [j0, j1): the envelope probe over the slice (z_j0, z_j1); if empty, skip and double the next range; else if the noise probe is affordable (item 2) and proves empty, skip; else if the range is more than one cell, halve; else the cell is dense. It runs to the grid's end once per tile (a pre-pass does not know any ray's transmittance) and yields the dense cells; each of the N^2 rays integrates exactly those cells in order with the dense march's own Integrate and early-out at 0.02. Counted separately: snoise evaluations and envelope evaluations (pointwise in the dense march; interval probes in the tile march), per ray with probes divided by N^2.",
     "The cost model, printed and reasoned about in snoise units: a pointwise envelope evaluation or an envelope probe costs e = 0.2 (a handful of pulses or a height-map fetch against ~150 flops of snoise; stated as an estimate, both counts printed so any weight can be applied). Per scenario the test prints: dense snoise and envelope per ray, tile snoise and envelope probes per ray, the combined cost ratio, and the probe-free ceiling (dense cost over the cost of the dense cells alone). The 2x contract is on the combined cost at N = 8 for scenarios (a) and (c); (b) and (d) are printed as the worst cases and not asserted.",
     "Scenarios (IV-REPORT lines, each with 64 tiles at N in {1, 4, 8, 16}, warp in {0, D}): (a) height fog: h = 0, camera at y = S + 30, rays with slope m_y in [0, 0.5] (level and up), Aetheria's grid, expected near-total skip (0 snoise either way; the envelope count carries the ratio). (b) inside the fog: h = 0, camera at y = F - 10, rays level, expected near 1x: every ray saturates within a few cells and the pre-pass still probes the whole grid. (c) Aetheria-like: the zone bowl R = 2000 and four wells of mass {100, 1000, 10000, 1000} at seeded positions inside R, camera at a seeded xz in the bowl at y = max(0, S + 10 - h(camera)), rays gazing across with m_y in [-0.25, 0.1], Aetheria's units (noise scale 414 so F0 = 1/414, D = 15 * 8 / 2 = 60, A = 20 with snoise in [-1, 1]), Aetheria's grid. (d) the uniform slab: r1's field and parameters, kept as the reported worst case. The headline is (e) at N = 8, 200 tiles, printed with both its numbers; (a) and (c) are asserted.",
-    "(e) the void, the shipped scene (rulings operator-ground-void-brush and operator-void-true-cavity; site-ground r3): a cloud volume everywhere, carved by one negative pseudo-Gaussian sphere. d = |p - c_v|; g(d) = (1 - (d / Rc)^2)^4 for d < Rc, else 0; Rc = 640, CARVE = 1.5, so density = K max(0, 1 - CARVE g(d')) is exactly 0 inside the hollow radius Rh = Rc sqrt(1 - CARVE^(-1/4)) = 198 and ramps to full by Rc. Noise only near the wall: fade(d) = smoothstep(Rh - S, Rh, d), S = 50; d' = d + A fade(d) n(warp(p)), A = 20, F0 = 0.01, D = 10; K = 1/30 (optical depth 1 per 30 units of full density). The sun sits at the bottom of the hollow (y = c_v.y - Rh + 12.5); the camera is 120 behind it along the orbit tangent and 45 above it, 186 from c_v: inside the hollow, low. Rays over the whole frustum (vertical FOV 60, f = 935): every ray crosses empty interior (14 to 380 units) and ends in a wall. Uniform grid, StepDense 10, 128 cells to 1280 = 2 Rc; termination by transmittance < 0.02.",
-    "Predictions to confirm or refute, N = 8, combined cost: (a) above 50x. (b) 0.8-1.0x. (c) 2-4x: clear cells collapse to a few probes per tile, fog cells cost the same on both sides; the noise ball proves nothing at Aetheria's scale (L r F0 >= 1.6), so item 2 keeps it unevaluated and the saving is the envelope's. (d) 1.3-1.5x: the ceiling at L = 10.099261. (e) 1.0-1.2x, under 2x, said plainly: with no sky every ray ends in a wall, the hollow costs the dense march only envelope evaluations (about 20 x 0.2 per ray against about 15 wall cells x 3.2), and the pre-pass can only remove those. What it buys there is sample placement: at the site's S the samples land in the 150-unit wall ramp, not the whole ray; the test prints marched length over wall length per ray (predicted 2.0-2.6x). (e) asserts per the fork cavity-contract and skips naming it until ruled. If (a) or (c) misses 2x, skip naming saving-2x-scenarios. Never tune the fields, the grids or e.",
+    "(e) the void, the shipped scene (rulings operator-ground-void-brush, operator-void-true-cavity; site-ground r3): a cloud volume everywhere, carved by one negative pseudo-Gaussian sphere, parametrised by the hollow radius Rh (density exactly 0 inside) and the ramp width S (density rises to full over [Rh, Rh + S]); the brush (1 - (d / Rc)^2)^e with CARVE gives Rh = Rc sqrt(1 - CARVE^(-1/e)) and S from the exponent, so (Rh, S) are the authored knobs and (Rc, e) derived. Noise displaces the wall only: fade(d) = smoothstep(Rh - S, Rh, d); d' = d + A fade(d) n(warp(p)), A = 20, F0 = 0.01 (wavelengths 100 and 25), D = 10; K = 1/30 (optical depth 1 per 30 units of full density), termination at T < 0.02. Shipped point: Rh = 198, S = 50, the sun on the hollow's bottom, the camera 120 behind it along the orbit tangent and 45 above it (186 from the centre, 'low'). Rays over the whole frustum (vertical FOV 60, f = 935, so theta_px = 1/f). Every ray crosses empty interior and ends in a wall.",
+    "Footprint LOD (the operator's challenge, map 'Footprint LOD'): theta_px = 2 tan(fov/2) / H; the footprint at distance z is z theta_px. Step: Delta(z) = max(k z theta_px, min(S / 4, lambda_fine(z) / 2)), k = 4: never finer than four footprints, otherwise the finer of the two content limits (four samples across the ramp, two per wavelength of the finest live octave). Octaves: octave i of wavelength lambda_i carries weight w_i(z) = 1 - smoothstep(lambda_i / (2m), lambda_i / m, z theta_px), m = 4: one smooth blend per octave, no popping; TAA hides per-frame sparsity. The weights are part of the field, so the bound encloses the truncated field exactly: the noise interval over a slice is the sum over octaves of iv_mul([w_i(z1), w_i(z0)], iv_snoise_ball(c F_i, r F_i)), w_i monotone in z. At the shipped parameters (wavelengths 100 and 25, z <= 2 Rc) every weight is 1 in the frustum and the content limits set the step; the footprint bites when S < 4 k z theta_px.",
+    "(e) runs three marches per tile and prints them: the fixed-step reference (Delta_fixed = min(10, S / 4) everywhere, the pointwise envelope gate, the baseline a march that does not know where the wall is must pay); the masked fixed-step march (the same cells, so transmittance agrees exactly: the soundness pin, as (a)-(d)); and the masked LOD march, the shipped one (steps from Delta(z) inside the unmasked range only, octave weights applied), whose transmittance must be within 0.02 of the reference (printed; an approximation by design, not a soundness claim). Printed per configuration: steps per pixel and snoise per pixel for all three, the combined cost ratio reference / LOD, probes per tile. The grid for (e): Rh in {100, 200, 400, 1000} x camera offset in {centred, mid (0.5 Rh), low (0.94 Rh, the shipped)} x S in {10, 50, 150}, 32 tiles each at N = 8; the shipped point with 200 tiles.",
+    "Predictions to confirm or refute, N = 8, combined cost: (a) above 50x. (b) 0.8-1.0x. (c) 2-4x: clear cells collapse to a few probes per tile, fog cells cost the same on both sides; the noise ball proves nothing at Aetheria's scale (L r F0 >= 1.6), so item 2 keeps it unevaluated and the saving is the envelope's. (d) 1.3-1.5x: the ceiling at L = 10.099261. (e), from Imagination's model (scratchpad cavity_model.py): the LOD march needs about 6 steps and 12 snoise per pixel at every grid point with S <= 50 (the content limits S / 4 and the detail half-wavelength 12.5 set the step, not the footprint); S = 150 needs 14 steps and 36 snoise. Against the fixed-step reference at the shipped point: steps 35 to 6 (5-8x fewer), combined cost 1.7x, snoise alone 1.25x; cost 3.2x at S = 10, 1.4x at S = 150, 2.5x at Rh = 1000; camera offset barely matters. (e)'s contract is steps per pixel (Q8 retired). If (a) or (c) misses 2x, skip naming saving-2x-scenarios. Never tune the fields, the grids or e.",
     "Stryker on Yggdrasil from a plain clone: --since:c9bd003b -m math.Interval.cs; report timeouts separately from kills (a timeout is not a demonstrated kill); if the saving test is what times mutants out, exclude it from Stryker's test selection (a measurement, not a pin) rather than raising the timeout. Compile checks on Starfire, one job, detached with a log, polled in the foreground: FXC cs_5_0 over every cultmath_iv_* including the new function, both tracked copies; then tools/get-dxc.ps1 (pinned, ruling operator-compiler-downloads) and dxc -T lib_6_3 -HV 2021 on both copies. Carried from r1 and still out of scope, name them in the report: the Unity CultMath.dll is not rebuilt; the Unity HLSL copies stay one blob with packages/cultmath/shaders; public asfloat(uint) stays. Branch from c9bd003b; if main moved by docs only, say so as r1 did."
   ],
   "deletes": [
@@ -57,7 +57,7 @@
   "adds": [
     "packages/cultmath/src/CultMath/math.Interval.cs: iv_frustum_ball after iv_fbm_ball, doc comment carrying the derivation and that the caller scales by F0 and adds its translation.",
     "packages/cultmath/shaders/CultMath.Interval.hlsl: cultmath_iv_frustum_ball, same text in both tracked copies.",
-    "tests/CultMath.Tests/NoiseBoundTests.cs: Scenario (grid, wells, camera, ray slopes, units), SavingField with the envelope (PowerPulse wells, fade, floor), EnvelopeBound (the composed interval, test-side; iv_* ops only), MarchTile, the per-tile Run; TileBallEnclosesEveryRaySegment; EnvelopeBoundEnclosesDensity; IntervalSkipHalvesEvaluations prints the scenario x N x warp grid and asserts (a) and (c) at N = 8.",
+    "tests/CultMath.Tests/NoiseBoundTests.cs: Scenario (grid, wells, camera, ray slopes, units), SavingField with the envelope (PowerPulse wells, fade, floor), EnvelopeBound (the composed interval, test-side; iv_* ops only), MarchTile, MarchLod (footprint step and octave weights, the shipped march), the per-tile Run; TileBallEnclosesEveryRaySegment; EnvelopeBoundEnclosesDensity; IntervalSkipHalvesEvaluations prints the scenario x N x warp grid and asserts (a) and (c) at N = 8.",
     "packages/cultmath/docs/design.md, 'Intervals': the frustum-ball paragraph after :401; the envelope-then-noise paragraph (how a consumer composes the bound, the short-circuit, where the warp lives); the rewritten saving paragraph with the four scenarios and the headline.",
     "packages/cultmath/README.md :38-43: iv_frustum_ball named; one sentence that the bounds compose with an analytic envelope."
   ],
@@ -88,7 +88,7 @@
         "path": "packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs",
         "line": 409
       },
-      "change": "IntervalSkipHalvesEvaluations: scenarios (a)-(e) x N in {1, 4, 8, 16} x warp in {0, D}, 64 tiles each; headline (e) at N 8, 200 tiles; assert >= 2x combined cost for (a) and (c) at N = 8; (e) per the cavity-contract ruling, else Skip naming it."
+      "change": "IntervalSkipHalvesEvaluations: scenarios (a)-(e), N in {1, 4, 8, 16}, warp in {0, D}; the (e) grid Rh x offset x S with three marches; assert >= 2x combined cost for (a) and (c) and >= 2x fewer steps per pixel for (e) at the shipped point, else Skip naming saving-2x-scenarios."
     },
     {
       "location": {
@@ -159,7 +159,7 @@
       },
       {
         "name": "NoiseBoundTests.IntervalSkipHalvesEvaluations",
-        "pins": "scenarios (a) and (c) at N = 8 cost at most half the dense combined cost with transmittance within 1e-3; (e) prints its cost ratio and its placement ratio and asserts per the cavity-contract ruling; every scenario, N and warp printed with both counts and the ceiling."
+        "pins": "scenarios (a) and (c) at N = 8 cost at most half the dense combined cost with transmittance within 1e-3; (e) at the shipped point: the masked LOD march takes at most half the fixed-step reference's steps per pixel, the masked fixed-step march agrees exactly, the LOD march within 0.02; the (e) grid and every scenario, N and warp printed with both counts and the ceiling."
       },
       {
         "name": "IntervalTests.EveryOpEnclosesItsPoints (existing)",
@@ -185,7 +185,7 @@
       },
       {
         "pattern": "Assert\\.Skip",
-        "scope": "NoiseBoundTests.cs: at most two, in IntervalSkipHalvesEvaluations, naming saving-2x-scenarios and cavity-contract; 0 once both are met or ruled."
+        "scope": "NoiseBoundTests.cs: at most one, in IntervalSkipHalvesEvaluations, naming saving-2x-scenarios; 0 if (a), (c) and (e) meet their contracts."
       },
       {
         "pattern": "10\\.099261|0\\.35f",
@@ -215,7 +215,5 @@
     "cultmath-tapes:ruling:operator-compiler-downloads",
     "cultmath-tapes:ruling:operator-campaign-home"
   ],
-  "questions": [
-    "cavity-contract (map, Q8): which number the shipped cavity scene promises"
-  ]
+  "questions": []
 }

docs/cultmath-interval-ground-cut-site-ground.r3.spec.json

@@ -15,11 +15,13 @@
     "Map: F:/Projects/CultLib/docs/cultmath-interval-ground-cut.md, 'Pass 3' (the void scene, the composed bound, the pre-pass, the budget), then 'The shader design' for what pass 3 does not restate (sampled wash, palette input, pausing, contrast). Rulings in force: cultmath-tapes:ruling:operator-ground-void-brush (the scene), operator-tile-amortized-march (the march), site-masthead:ruling:operator-chromeless-column; wash-home-only's authority map. Re-pin base to main after wash-home-only lands. Stardust and DoF stay the next cut; leave the named composite slot. r2's token, readback, controller, vendoring, parity and fallback items stand unless this revision names them.",
     "The original, read once (Aetheria at e6181c11, 2021-03-13, Assets/Scenes/Main Menu.unity; Prefabs/RPG/Planets/Sun.prefab; Materials/Brushes/Gravity Well.mat, Main Menu Sun Fog Tint.mat, Main Menu Ambient Fog Tint.mat, Sector Boundary Fog.mat; Materials/Fog Marching Main Menu.mat; Shaders/Raymarching/VolumeRender.shader at that commit; Shaders/Brushes/PowerBrush.shader). What survives, in the original's units: the brush is PowerBrush (1 - x^2)^16 with depth 40 over radius 256 (half depth at r = 52, pseudo-Gaussian sigma about 45), the sun's own gravity-well child; a wide boundary bowl (depth 75, power 0.25, radius 1500) is flat across the scene and acts as a constant 75; floor offset -16.46, floor blend 39.6, so the fog top is at y = -91 outside the void and dips to -131 at the sun; the sun sphere (radius 12.5) sits at y = -130, the bottom of the void.",
     "What survives (continued): the sun is a child of 'Center', which rotates 5 degrees per second about y (72 s period), at orbit radius 175; the camera is a Cinemachine vcam following the sun (framing transposer, distance 150, damping 10, handheld noise amplitude 1 at frequency 0.25) and looking at 'Look Target' (0, -130, 0), the orbit centre at the sun's depth; the saved camera sits at y = -27, 64 above the fog top. The lighting was authored, not marched: the sun's 'Fog Tint' brush (1 - x^2)^3, depth 2, radius 128, colour (1, 0.4, 0.1) orange, painted into the tint map, times a scene-wide ambient tint (0.18, 0.59, 1) blue, depth 0.01. Stardust: 256^2 particles, spacing 2, floor -60, ceiling 0, height exponent 800 / (ceiling + wellDepth) so particles climb higher where the well is deeper (the void fills with stardust). Noise: strength 150, frequency 0.01, safety 50. Not surviving: the 2021 d() details, the tint map's runtime contents, the video.",
-    "The field (ruling operator-void-true-cavity: a spherical hollow inside a cloud volume that surrounds it on every side, not the 2021 bowl). Units are the original's. Outer volume: an unbounded uniform fill, chosen over a shell because every ray then ends in a wall and the march terminates by opacity, never by a far plane; the far plane Z1 = 2 Rc bounds the pre-pass range only. The carve: d = |p - c_v|, g(d) = (1 - (d / Rc)^2)^4 for d < Rc, else 0 (the pseudo-Gaussian, exponent 4 so the wall is a ramp and not a skin), Rc = 640, CARVE = 1.5. density = K max(0, 1 - CARVE g(d')): exactly 0 inside the hollow radius Rh = Rc sqrt(1 - CARVE^(-1/4)) = 198, half at d = 0.49 Rc, full at Rc. Noise displaces the wall only: fade(d) = smoothstep(Rh - S, Rh, d), S = 50; d' = d + A fade(d) n(warp(p)), A = 20, F0 = 0.01. K = 1/30 with the extinction, so a wall is opaque within about 150 units of entry. Rc grew from the brush's 256 because a 3D hollow must hold the camera (next item).",
+    "The field (ruling operator-void-true-cavity: a spherical hollow inside a cloud volume that surrounds it on every side, not the 2021 bowl). Units are the original's. Outer volume: an unbounded uniform fill, chosen over a shell because every ray then ends in a wall and the march terminates by opacity, never by a far plane; Z1 = 2 Rc bounds the pre-pass range only. The carve is the pseudo-Gaussian brush g(d) = (1 - (d / Rc)^2)^e, d = |p - c_v|, density = K max(0, 1 - CARVE g(d')), authored by two knobs: the hollow radius Rh (density exactly 0 inside; Rh = Rc sqrt(1 - CARVE^(-1/e))) and the ramp width S (density full at Rh + S; set by the exponent: e = 4 gives S = 1.3 Rh, e = 0.25 gives S = 0.12 Rh). Shipped: Rh = 198, S = 50 (so e about 0.6, Rc about 220), CARVE = 1.5, K = 1/30: a wall is opaque within about 150 units of entry. Noise displaces the wall only: fade(d) = smoothstep(Rh - S, Rh, d), d' = d + A fade(d) n(warp(p)), A = 20, F0 = 0.01.",
     "Scene motion. The sun orbits the origin at radius 175 in xz with period 72 s (5 degrees per second, the original's Center pivot). The cavity centre is c_v = sun + (0, Rh - 12.5, 0): the sun sphere (radius 12.5) rests on the hollow's bottom and the hollow moves with it. The camera follows the sun 120 behind it along the orbit's tangent and 45 above it, so it sits 186 from c_v, 12 inside the hollow (the handheld drift is 2 units, two low-frequency snoise on time), low in the cavity, looking at the sun's position plus (0, 60, 0) so the lit lower wall and the dark upper wall share the frame. The 2021 follow distance 150 and height 25 do not fit a hollow a camera can stand in; 120 and 45 are the re-derivation, and the report says so. worldAt(t) in ground.js is the one owner of sun, cavity and camera, uploaded as uniforms; the shaders never integrate motion.",
     "The flow field, authored around the void: flow(xz) = SWIRL * tangent(xz - sun.xz) * (1 - (r / Rw)^2)^2 + GLOBAL * (snoise(xz Ff + o1), snoise(xz Ff + o2)) (the swirl is largest mid-slope and zero at the rim and the centre; the global term is r2's). The flow texture is a world-space window of side 4 Rw centred on the sun, quarter of the canvas resolution, re-rendered every frame; warp, hue, motion vectors and (next cut) the stardust read it, so D4 holds with a moving scene. D = FLOW_MAX * PERIOD / 2 as before, with FLOW_MAX the authored maximum of the swirl plus the global amplitude.",
-    "The march (tile pre-pass, interval-ops r2's design). Pre-pass at W/8 x H/8: per tile, the loop over the slab's cell ranges (uniform cells, StepDense = 10, 128 cells to Z1 = 1280 = 2 Rc along +view depth): the envelope probe first (exact box: y and xz ranges by cultmath_iv_mul of the depth and slope intervals, d's interval is the box's least and greatest distance to c_v, exact; g, fade and the carve are monotone in it), then the noise ball (cultmath_iv_frustum_ball with D, cultmath_iv_snoise_ball) only when fade.hi > 0 and L r F0 leaves room to prove anything; writes the 96-bit mask of cells not proven empty to an RGBA32UI texel.",
-    "(March, continued.) Main pass: texelFetch the mask, place S samples stratified over the unmasked cells only (exact: masked cells are zero), integrate transmittance and the depth-weighted mean distance, and apply the light below. Every tile's interior cells retire in a few envelope probes and its wall cells are marked; there are no sky tiles. Behind a fully transmissive pixel the resolve shows --gamecult-nebula-base; by construction only an unfinished ray (budget exhausted before the wall, measured, must be under 1%) reaches it.",
+    "The march (tile pre-pass, interval-ops r2's design). Pre-pass at W/8 x H/8: per tile, the loop over the slab's cell ranges (uniform cells, StepDense = min(10, S / 4), 128 cells to Z1 = 2 Rc along +view depth): the envelope probe first (exact box: y and xz ranges by cultmath_iv_mul of the depth and slope intervals, d's interval is the box's least and greatest distance to c_v, exact; g, fade and the carve are monotone in it), then the noise ball (cultmath_iv_frustum_ball with D, cultmath_iv_snoise_ball) only when fade.hi > 0 and L r F0 leaves room to prove anything; writes the 96-bit mask of cells not proven empty to an RGBA32UI texel.",
+    "(March, continued.) Main pass: texelFetch the mask, then march the unmasked range with the footprint step Delta(z) and the octave weights (next item), the per-frame sample count capped by the budget S_budget and the start jittered so TAA completes what one frame does not; masked cells are skipped exactly; integrate transmittance and the depth-weighted mean distance, apply the light. Every tile's interior cells retire in a few envelope probes and its wall cells are marked; there are no sky tiles. Behind a fully transmissive pixel the resolve shows --gamecult-nebula-base; by construction only an unfinished ray (budget exhausted before the wall, measured, must be under 1%) reaches it.",
+    "Footprint LOD (the operator's challenge, map 'Footprint LOD'): theta_px = 2 tan(fov/2) / H; the footprint at distance z is z theta_px. Step: Delta(z) = max(k z theta_px, min(S / 4, lambda_fine(z) / 2)), k = 4: never finer than four footprints, otherwise the finer of the two content limits (four samples across the ramp, two per wavelength of the finest live octave). Octaves: octave i of wavelength lambda_i carries weight w_i(z) = 1 - smoothstep(lambda_i / (2m), lambda_i / m, z theta_px), m = 4: one smooth blend per octave, no popping; TAA hides per-frame sparsity. The weights are part of the field, so the bound encloses the truncated field exactly: the noise interval over a slice is the sum over octaves of iv_mul([w_i(z1), w_i(z0)], iv_snoise_ball(c F_i, r F_i)), w_i monotone in z. At the shipped parameters (wavelengths 100 and 25, z <= 2 Rc) every weight is 1 in the frustum and the content limits set the step; the footprint bites when S < 4 k z theta_px.",
+    "Where the scene sits and what to turn (Imagination's model, scratchpad cavity_model.py; the test and the browser confirm): at the shipped point the LOD march needs about 6 steps and 12 snoise per pixel for a converged frame, against about 35 steps for a fixed-step march; the step is set by S / 4 and the detail octave's half-wavelength, not by distance, because footprints are under 4 units within the frustum. What makes it cheaper without losing the mood, in order: (1) a lower per-frame S_budget, which LOD and the mask make nearly lossless (S_budget = 4 converges in about three frames at the walls); (2) a thinner ramp S (S = 10 is 3.2x on cost but reads as a surface; 25-50 keeps the fog); (3) a longer detail wavelength or no detail octave near the far wall (the 12.5-unit half-wavelength is the floor for S >= 50); (4) a larger hollow (Rh = 1000: 2.5x) costs nothing per pixel and changes the composition. Camera offset barely matters at these scales.",
     "Lighting, the cheapest version that keeps it moody (the original's own mechanism, made 3D): at each sample, light = SUN_COLOUR * (1 - (d / Rl)^2)^3 for d = |p - sun| < Rl, Rl = Rh = 198 (the original's 128 scaled with the hollow), plus AMBIENT * ambientTint; colour = density * (light + ambient) * hue(flow). No light march: zero extra snoise, one length and one pow per sample. The lower wall within Rl of the sun glows and the upper wall stays dark because the pulse is zero beyond Rl, which is the moody split; the sun inside the hollow is occluded by nothing, so the sun itself is a small additive disc drawn in the resolve at its projected position, occluded by the accumulated transmittance along its ray. Not in this cut, measured before it is ever added: one envelope-only transmittance sample toward the sun per primary sample (self-shadowing by the bowl's geometry), about +25% march cost; it is desktop-only if ever.",
     "TAA with a moving camera. ground.js uploads this frame's and the previous frame's view-projection. For each pixel the resolve reconstructs the world point at the march's depth-weighted mean distance (written by the main pass into a second history channel, R16F), moves it back by the field motion (p + flow(p) * dt: the pattern moves at -flow), projects it with the previous VP and fetches history at that uv. The same 3x3 min/max clamp, alpha = max(1/frames, 1/32), out-of-bounds alpha 1; history reset on resize, first frame and budget step. Transmittance, colour and the depth channel are all reprojected by the one motion vector. ?ground=debug-mv shows |mv| with the camera moving and the flow frozen, and with the flow moving and the camera frozen, so both terms are seen separately.",
     "Stardust distribution and the sun disc are the next cut (site-stardust), authored to the void: particles spawned in the bowl's volume with the original's height rule (more headroom where the well is deeper), lit by the same pulse, moving with the swirl. This cut leaves the composite slot and the depth history they need.",
@@ -143,8 +145,8 @@
         "pins": "text contrast with the lit cavity."
       },
       {
-        "name": "browser pane, /?ground=debug vs /?ground=dense: mean luminance within 1/255; mean snoise per pixel printed for both; masked-cell fraction and probes per tile printed",
-        "pins": "the mask culls only provably empty cells; the saving on the shipped scene is a number."
+        "name": "browser pane, /?ground=debug vs /?ground=dense: converged mean luminance within 1/255; steps and snoise per pixel printed for both, debug at most half of dense's steps; mask fraction printed",
+        "pins": "the mask culls only provably empty cells; the shipped march takes at most half the fixed-step steps; the saving on the shipped scene is a number."
       },
       {
         "name": "browser pane, /?ground=debug-mv: |mv| non-zero with the camera moving and flow frozen, and with flow moving and camera frozen; zero under reduced motion",
@@ -236,7 +238,6 @@
   ],
   "questions": [
     "Q4 in the map: palette input tokens (a, recommended)",
-    "Q5 in the map: carry the last sampled wash across page loads (a, recommended)",
-    "Q8 in the map (cavity-contract): which number the shipped cavity scene promises"
+    "Q5 in the map: carry the last sampled wash across page loads (a, recommended)"
   ]
 }

docs/cultmath-interval-ground-cut.md

@@ -332,7 +332,7 @@ TFLOP available; estimates, replaced by measurement):
 | --- | --- | --- |
 | Flow texture, quarter res | 160 k texels × 2 snoise | 0.05 |
 | Tile pre-pass (pass 3), 1/64 res, `N = 8` | 40 k tiles × ≈ 30 probes in the cavity (interior retired in a few, the wall's ≈ 15 cells at 1.5 probes each), envelope-only because the noise ball cannot prove anything at this scale; ≈ 0.1 snoise-equivalents per pixel; worst case ≤ 192 probes per tile | 0.1 (≤ 0.8) |
-| March at `S = 8` | 2.6 M × 8 snoise, every pixel marches a wall (no sky pixel ends free), samples placed in unmasked cells only, plus the analytic sun light (one length, one pow per sample, ≈ 5%) | 3.3 |
+| March at `S = 8` | 2.6 M × 8 snoise per frame, every pixel marches a wall; LOD and the mask decide where the frame's samples land, not how many (the controller does) | 3.3 |
 | TAA resolve + DoF gather | ~19 texel reads per pixel, plus the depth-history fetch and one matrix multiply for the camera reprojection | 1.1 |
 | Sun self-shadowing sample (named, not taken) | one envelope-only transmittance sample per primary sample, ≈ +25% of the march | +0.8, not on the phone |
 | Stardust layer, half res, 4,096 particles | fill-bound, small | 0.2 |
@@ -340,12 +340,13 @@ TFLOP available; estimates, replaced by measurement):
 | **Total at `S = 8`, the cavity (pass 3)** | | **≈ 4.8** |
 | Total at `S = 6` / `S = 4` | | ≈ 3.9 / 3.1 |
 
-Said plainly: at `S = 8` the cavity with the pre-pass, the analytic light and
-the camera reprojection does not fit the 4 ms phone budget on the estimate;
-`S = 6` sits at the line and `S = 4` is under it. The controller will settle
-a mid-range phone at `S = 4–6`, which the TAA covers (convergence in ~32
-frames, about a second), and the pre-pass is what makes `S = 4` look like
-more: every ray's samples land in its wall, none in the hollow. What gives, in order, is the
+Said plainly: LOD does not change the per-frame cost, which the controller
+fixes by `S`; it changes how much a frame's samples are worth. At `S = 8` the
+cavity does not fit the 4 ms phone budget on the estimate (≈ 4.8); `S = 6`
+sits at the line and `S = 4` is under it. What gives is `S`, and with the
+mask and LOD that is nearly free: a converged wall needs about 12 snoise per
+pixel in total, so `S = 4` converges in about three frames through the TAA
+instead of thirty, and `S = 8` in two. Then `dpr`, as before. What gives, in order, is the
 sample count, then the resolution; the stardust and the DoF are never dropped
 before the clouds' samples; the self-shadowing sample is not taken on any
 device until measured. On desktop at `S = 16` the stack is ≈ 1.9 ms.
@@ -522,12 +523,76 @@ warped points of 2,000 slices, no tolerance).
 | (b) inside the fog | `h = 0`, camera at `y = F − 10`, rays level | 0.8–1.0x: every ray saturates in a few cells; the pre-pass still probes the whole grid |
 | (c) Aetheria-like | zone bowl `R = 2000` + four wells (masses 100, 1000, 10000, 1000), camera above the fog in the bowl, rays gazing across (`m_y ∈ [−0.25, 0.1]`), Aetheria's units and 256-sample quadratic grid | 2–4x: the clear cells before the fog surface collapse to a few probes per tile; the fog cells cost the same on both sides. The noise ball proves nothing at Aetheria's scale (`L r F0 ≥ 1.6` at the finest cell with `D = 60`), so it is not evaluated; the saving is the envelope's |
 | (d) uniform slab | r1's field, kept as the worst case | 1.3–1.5x: the ceiling at `L = 10.099261` with the cheaper baseline |
-| (e) the void, the shipped scene | the true cavity in the original's units (`Rc = 640`, `Rh = 198`), camera 186 from the centre, low, rays over the whole frustum; every ray crosses 14–380 units of interior and ends in a wall | **1.0–1.2x on combined cost, under 2x.** With no sky the interior costs the dense march only envelope evaluations (≈ 20 × 0.2 per ray against ≈ 15 wall cells × 3.2) and the pre-pass can only remove those; the noise ball proves nothing (`L r F0 ≥ 1.5`). What the pre-pass buys here is sample placement: at the site's `S` the samples land in the 150-unit wall ramp instead of the whole ray, a 2.0–2.6x effective sample density (printed as marched length over wall length) |
+| (e) the void, the shipped scene | the true cavity, hollow radius `Rh = 198`, ramp `S = 50`, camera 186 from the centre, low; every ray crosses interior and ends in a wall; three marches (fixed-step reference, masked fixed-step, masked LOD) | **steps per pixel ≈ 35 → 6 (5–8x fewer), combined cost 1.7x, snoise alone 1.25x**, from the model below. The noise ball proves nothing (`L r F0 ≥ 1.5`); the interior is retired by the envelope, the ramp is sampled at the content limit |
 
 The 2x contract on combined cost is asserted on (a) and (c) at `N = 8`; (b)
-and (d) are printed. (e) is the headline and prints both its numbers; which
-one it promises is Q8 (`cavity-contract`), and until that is ruled its
-assertion skips naming the question. A shortfall skips the assertion naming `saving-2x-scenarios`; the
+and (d) are printed. (e) is the headline: its contract is steps per pixel
+(below), and it prints cost and snoise beside it.
+
+### Footprint LOD, and the cavity cost as a function
+
+The operator's words: "Surely this depends on the size of the cavity and the
+camera placement, no? The slope of the wall's density gradient offset
+determines the region we have to sample in. How much we actually sample that
+region depends on how close it is to the camera. A 10-wide gradient 1,000
+units away only needs one raymarching step worth of detail, right?"
+
+Right. Pass 3b costed every wall at a fixed step. The march is now
+footprint-aware, in the site and in the saving test:
+
+- `θ_px = 2 tan(fov/2) / H`; the footprint at distance `z` is `z θ_px`.
+- Step `Δ(z) = max(k z θ_px, min(S/4, λ_fine(z)/2))`, `k = 4`: never finer
+  than four footprints, otherwise the finer of the two content limits (four
+  samples across the ramp, two per wavelength of the finest live octave).
+- Octave `i` of wavelength `λ_i` carries weight `w_i(z) = 1 −
+  smoothstep(λ_i/2m, λ_i/m, z θ_px)`, `m = 4`: one smooth blend per octave,
+  no popping; TAA hides the per-frame sparsity.
+- The weights are part of the field, so the bound encloses the truncated
+  field exactly: the noise interval over a slice is `Σ_i iv_mul([w_i(z1),
+  w_i(z0)], iv_snoise_ball(c F_i, r F_i))`, `w_i` monotone in `z`.
+  `intervals-enclose` holds for what the shader evaluates.
+- The pre-pass says where each ray's ramp begins; LOD says how finely to
+  sample it once there. They compound. The reference a march that does not
+  know where the wall is must pay is the fixed step `min(10, S/4)` everywhere.
+
+The cavity is parametrised by the hollow radius `Rh` and the ramp width `S`;
+the brush `(1 − (d/Rc)²)^e` with `CARVE` gives `Rh = Rc √(1 − CARVE^(−1/e))`
+and `S` from the exponent (`e = 4` gives `S ≈ 1.3 Rh`, `e = 0.25` gives
+`S ≈ 0.12 Rh`), so `(Rh, S)` are the knobs and `(Rc, e)` derived. Shipped:
+`Rh = 198`, `S = 50` (`e ≈ 0.6`, `Rc ≈ 220`), `K = 1/30`.
+
+Imagination's model (scratchpad `cavity_model.py`; the test confirms or
+refutes), 1080p, `N = 8`, per pixel for a converged frame; "ref" is the
+fixed-step reference's steps:
+
+| `Rh` | camera | `S = 10` steps / snoise / cost ratio | `S = 50` | `S = 150` |
+| --- | --- | --- | --- | --- |
+| 100 | low | 6 / 12 / 2.6x | 6 / 12 / 1.6x | 14 / 36 / 1.3x |
+| 200 | low (**shipped**, ref ≈ 35 steps) | 6 / 12 / 3.2x | **6 / 12 / 1.7x** | 14 / 36 / 1.4x |
+| 400 | low | 6 / 12 / 4.0x | 6 / 12 / 1.9x | 14 / 36 / 1.5x |
+| 1000 | low | 5 / 9 / 8.2x | 6 / 12 / 2.5x | 14 / 36 / 1.7x |
+| 1000 | centred | 4.3 / 7 / 12.9x | 6 / 12 / 2.9x | 14 / 36 / 1.8x |
+
+Centred and mid offsets differ from low by under 10% at every point. What
+the table says: within the frustum the footprint is under 4 units, so the
+step is set by the content limits (`S/4`, and the detail octave's 12.5-unit
+half-wavelength), not by distance; the "10-wide gradient 1,000 units away"
+row is where the footprint finally bites (4.3 steps, 7 snoise). The sample
+count is nearly flat in `Rh` and camera offset and rises with `S` only once
+the detail wavelength, not the ramp, sets the step. What makes the shipped
+scene cheaper without losing the mood, in order: a lower per-frame sample
+budget, which LOD and the mask make nearly lossless (`S_budget = 4` converges
+in about three frames at the walls); a thinner ramp (`S = 10` is 3.2x on cost
+but reads as a surface; 25–50 keeps the fog); a longer detail wavelength or
+no detail octave at the far wall; a larger hollow, which changes the
+composition and costs nothing per pixel.
+
+**Q8 retired.** At the shipped point the shipped march takes 5–8x fewer steps
+than the fixed-step reference and 1.7x less combined cost; the contract for
+(e) is the number that scales with GPU time, steps per pixel: ≥ 2x fewer than
+the fixed-step reference at the shipped parameters, with snoise per pixel and
+combined cost printed beside it and promised nothing. The test asserts that;
+the browser measures it (`?ground=debug` against `?ground=dense`). A shortfall skips the assertion naming `saving-2x-scenarios`; the
 fields, grids and `e` are not tuned. The honest summary: at a Lipschitz bound
 near 10 the noise ball is weak everywhere; what intervals buy on these fields
 is the envelope's empty space, found exactly, and the tiles make finding it
@@ -679,19 +744,10 @@ timeouts are reported as timeouts, not kills.
 cloud volume that surrounds it on every side, including above; not the 2021
 bowl.
 
-**Q8. `cavity-contract`: which number the shipped cavity scene promises.**
-The proving-ground contract's "2x fewer evaluations at equal transmittance"
-cannot hold in a cavity with no sky (prediction 1.0–1.2x, above), because a
-pointwise-gated dense march already pays almost nothing in the hollow. The
-pre-pass's real gain there is where the samples land.
-- (a) (e) is asserted on the placement ratio (marched length over wall
-  length per ray, ≥ 2x, predicted 2.0–2.6x); the cost-ratio 2x stays on (a)
-  and (c). **Recommended**: it is the number that describes the shipped
-  gain, and it is still a measured property of the same march.
-- (b) (e) is printed only; the contract stays on (a) and (c); the browser
-  measurement (variance at the walls, snoise per pixel) is the shipped
-  scene's evidence.
-- (c) Relax the contract for (e) to the measured cost ratio.
+**Q8. `cavity-contract`.** Retired in pass 3d (see "Footprint LOD"): the
+contract for (e) is steps per pixel against the fixed-step reference at the
+shipped parameters, ≥ 2x fewer, predicted 5–8x; cost and snoise are printed
+beside it and promised nothing.
 
 Specs for pass 3: `docs/cultmath-interval-ground-cut-site-ground.r3.spec.json`
 beside the two CultLib r2 specs.