The next useful redesign is localized thermal computation with dense output only when needed, backed by an implicit or exact linear thermal propagator. The writing equation has considerably more exploitable structure than a generic coupled simulation: temperature obeys a linear diffusion/cooling equation with a prescribed moving Gaussian source. Deposit and nutrient do not feed back into temperature in this writer-only experiment.
The same physical extent (2,2,2) and five-unit duration were simulated at 384³ and 768³. The 768³ Metal run used 1600 steps of .003125, took 651.77 seconds of evolution, and exported finite native deposit, temperature and dose fields. Its compact carrier uses five Float32 volumes,9.06 GB; the former nineteen-array layout would require 34.43 GB. Default versus compact 96³ runs matched exactly for 120 steps. This allocation improvement is implemented, not prospective.
| Native grid | Five-array base | Three-field export | Explicit runtime estimate |
|---|---|---|---|
| 768³ | 9.06 GB | 5.44 GB | 10.86 min measured |
| 1152³ | 30.58 GB | 18.35 GB | ~1.4 h |
| 1536³ | 72.48 GB | 43.49 GB | ~5.8 h |
| 2304³ | 244.61 GB | 146.77 GB | ~44 h |
Estimates scale the measured 768³ time by (n/768)^5: cubic cell count times
quadratic explicit-diffusion step count. They exclude compilation, exports and
rendering, and assume comparable effective GPU bandwidth/utilization. They are
planning estimates, not measured limits. Thermal stability bounds at those four
grids are approximately .0033405, .0014970, .0008445 and .0003761. The 768 run
used a slightly smaller step than its bound.
Another 2× or 3× per axis therefore costs roughly 32× or 243× with this explicit thermal algorithm. Acoustic CFL instead gives approximately 16× or 81× for the same physical duration. Larger memory on another accelerator, distributed execution, streaming, or a better representation can change feasibility; this 96 GB host is not a universal resolution ceiling. Base state also is not peak memory: staging, solver scratch, runtime and rendering need headroom.
1. Remove the diffusion timestep restriction while retaining the physical field. For constant diffusivity and periodic boundaries, the discrete thermal operator separates by axis. Directional implicit diffusion solves can retain real Float32 arrays and use GPU batched periodic tridiagonal solves. Alternatively, a Fourier propagator can apply the exact exponential of the discrete linear operator. This is the same diffusion/cooling model with a changed time integrator; source and threshold-dose quadrature still need accuracy checks. Fourier state and work buffers can erase memory savings, especially full complex storage. A directional implementation should benchmark scratch and bandwidth before choosing FFTs by default.
This can replace the quadratic step-count growth with an accuracy-controlled step count. At fixed acceptable temporal error, another 2×/3× per axis approaches 8×/27× work, rather than 32×/243×. It does not permit arbitrary large steps: beam motion, pulsing, threshold crossings and dose integration remain. The current path's speed is bounded by about 3.18 distance units/time, and its Gaussian width is .018. At dt=.003125 it can move .0099 units, over half a source width. Thus the existing 768 step is already a temporal-accuracy question, not free headroom for a much larger implicit step. Source quadrature along each step and adaptive threshold integration are more valuable than merely disabling CFL.
A practical first implementation is a positivity-preserving directional backward-Euler thermal step with explicit source quadrature, followed by a higher-order variant if error/time comparisons justify it. Require homogeneous cooling/decay checks, Fourier-mode diffusion, a stationary Gaussian source, moving-path timestep refinement, and nonnegative temperature/dose. Compare physical volume integrals and local profiles, not only finite output.
2. Exploit localized support and eliminate redundant evolving arrays. Measured 768³ dose exceeds .01 in only .323% of the domain; deposit exceeds .18 in .312%. Those fractions are evidence of locality, not a guarantee that only those cells need thermal computation. Allocate active physical tiles around the source and thermal wake, including diffusion halos; retain written dose tiles when the hot region moves away. Bin activation must look ahead along the swept source path. Halo exchange precedes each diffusion stage; shared tile faces must use the same stencil and boundary values. Dense output can be streamed from tiles, filling certified inactive regions with their physical defaults.
A Gaussian and diffusion have nonzero mathematical tails. Skipping tiles is therefore a controlled approximation unless a bound proves their temperature can never cross the writing threshold. Choose an absolute temperature/dose error budget; bound omitted source plus propagated heat and cooling, and expand halos until below it. Test the bound against dense 768 data. A halo chosen only from observed deposit occupancy is not sufficient. Preserve the original periodic domain; cropping the thermal domain and imposing zero edges changes its boundary conditions.
For this isolated writer, nutrient obeys dn/dt=-conversion*n*max(T-threshold,0)
and dose integrates the same nonnegative exposure. With initial nutrient 1 and
no other material process, nutrient=exp(-conversion*dose) and
deposit=1-nutrient. Thus dose can be the sole persistent material variable;
reconstruct nutrient/deposit during output. This removes two evolving arrays
and repeated material writes. It is an analytic invariant of this writer-only
model, but will differ slightly from the reference's repeated Float32 rounding.
Validate the difference and conserve the reconstructed pair by construction.
It does not apply unchanged to mixed metabolic, erosive or externally fed worlds.
3. Use the linear PDE directly where that beats stepping a full grid.
The continuous infinite-space temperature from a past Gaussian source at ageτ
is another Gaussian with variance width²+2 Dτ, amplitude factor
(width²/(width²+2 Dτ))^(3/2)*exp(-cooling*τ), centered at the past beam position.
Integrating these contributions over the prescribed path gives temperature;
adaptive integration above threshold gives dose. Periodic images restore the
periodic model when needed. This is a concrete route to evaluating only the
written support, without a global diffusion timestep or evolving dense thermal
array. It changes spatial discretization from the current finite-difference
solver, so assess continuum convergence rather than expecting bitwise parity.
Naively summing every past source sample at every voxel can be worse than the current solver. It needs temporal clustering, Gaussian splatting, tile bounds or another fast evaluation strategy. Cooling limits useful history at a chosen error tolerance; source motion and threshold exposure guide adaptive quadrature. A bounded stationary/moving-source prototype would determine the crossover before committing to a full implementation. This option is promising because of the actual equation's structure; it need not be dismissed merely because a short unoptimized prototype is slow.
First implement the tiled writer with dose-only material state and a controlled thermal propagator; keep a dense reference path for comparison. Independently prototype analytic Gaussian-history evaluation on the same physical paths. These target empty-space work and the timestep bottleneck together. Multi-GPU slabs or out-of-core GPU tiles remain useful when dense states are required: they need halo exchange, source ownership, overlap and synchronized output, and do not by themselves remove the explicit quadratic step-count cost.
Native 384/768 outputs and measured timing records remain under outputs/refinement-20261015/; the comparison includes smaller
timesteps at 768 and is not an isolated spatial convergence-order measurement.
DoseThermalWriter now implements the three-array alternative: temperature,
thermal scratch and dose. It shares the exact thermal/exposure helper with
ThermalWriter; its private fixed uniform material inventory is reconstructed
only for observations. World nutrient/deposit fields are untouched. An ordinary
material-writing process can coexist on its separate World reservoir; this does
not exchange material with the private writer. Scheduling the same embedded
thermal state twice is rejected through existing ownership preflight.
Thirty-four CPU assertions pass, including exact temperature/dose parity, pulsing, shutter stop, fractional conversion, private inventory accounting and slab readout. A 2000-step saturation test reached zero nutrient; maximum absolute nutrient difference from incremental exchange was 3.70e-12 and deposit difference 1.02e-6. These are measured rounding differences, not an all-parameter error bound. A warm 96³ CPU comparison measured 49.13 ms original versus 35.50 ms dose-only (1.38×). A bounded 12³/25-step Metal comparison also passed temperature, dose and material readout parity. Warm Metal throughput measurements appear below.
studies/refinement_dose.jl and experiments/refinement/dose-writing1152.toml drive the active run:
three-array 18.35 GB base state,22 GB recipe ceiling, dt=5/3600 below .001497,
and four-plane exports without allocating whole nutrient/deposit readout arrays.
A five-step 16³ CPU smoke run actually exported the expected native fields.
The bounded Green-function probe uses the existing dense writer at 32³/64³/96³
and a stationary Gaussian analytic reference. At dt=.001 and t=.2, temperature
relative differences at sampled points fell to about .14%, while dose remained
about .61% different. Thus finite time quadrature visibly matters even when
spatial resolution improves; this prototype supports the linear-PDE route but
is not a validated moving-source replacement. Exact data are in
outputs/refinement-20261015/thermal-green-probe.json.
The Julia dose writer now also swaps two preconstructed thermal handles instead
of copying the temperature field after every step. No extra grid arrays are
allocated. Its temperature/scratch handles are borrowed for one step: reacquire
p.writer.temperature after advancing, or copy to retain a snapshot. Dose keeps
its array identity. Both handles participate in ownership checks; readout rejects
aliases of temperature, scratch or dose. The original ThermalWriter keeps stable
buffer identities. Thirty-four CPU tests cover these contracts.
This removes eight bytes/cell/step of copy traffic: 12.23 GB per 1152³ step, or about 44 TB over 3600 steps, before cache effects. All future implementation paths described here are Julia work; diagnostic measurements from other stacks do not authorize a second algorithm implementation or production backend.
The paired warm Julia Metal benchmark now measured 96³ original 1.887 ms versus three-array ping-pong 1.423 ms (1.326×), and 384³ 36.225 versus 29.926 ms (1.210×), with exactly equal temperatures after matched steps. These are shared-host component measurements. The 1152³ four-step allocation pilot also passed: 18.346 GB state, 907 ms median, all finite. The 3600-step production completed at native 1152³ in 3414.43 seconds (56.91 minutes), using 18.35 GB for three evolving arrays. Final 360-step median: .9089 s/step. Its actual runtime is 5.24× the five-array 768³ bake; changes in representation, timestep and shared-host load make this a workload comparison, not an isolated scaling exponent.
The run and frozen source live in outputs/refinement-20261015/writing-1152/.
An independent chunked scan verified all 1,528,823,808 values in each of the three
exports: finite and nonnegative, exact expected file sizes (6,115,295,232 bytes
each), deposit maximum .99810237. Reconstructed deposit agrees with
-expm1(-18*dose) within 5.96e−8. Its volume integral is .01283649, about .20%
below the 768³ result; spatial and temporal resolution both changed. Frozen
writer and study hashes match the completed-run metadata.
The host has 96 GiB RAM. At 1536³, three evolving arrays alone require 43.49 GB; feasibility depends on scratch, output staging and competing jobs.
Original Julia kernels now also execute on hbox through AMDGPU. The dose writer measured 3.707 ms/step at 384³ and 29.145 ms/step at 768³ in bounded warmed probes. These are component timings, not complete bake or cellular-arena throughput. Runtime and pinned environment