Conservative represented liquid-volume transport

This slice implements Chapter 21’s shared-face volume ledger as a separate, transactional LiquidVolumeState on a fixed Cartesian grid. It stores dimensionless cell-average liquid fractions, a constant positive represented-liquid density, version and time. Dry/mixed/full occupancy is derived from accepted fractions. Those labels are not a pressure-unknown or mesh-solid mask: supplied carrier velocity is held over an explicit interval, with no free-surface feedback. The existing smoke tracer, sampler, force and pressure implementations are unchanged.

Face transfers, boundaries and sources

For each Cartesian face compute one signed, positive-axis interval transfer T=dtAu*f_upwind in m³. Interior upwind fractions come from old cells. Inflow uses explicit constant outside fractions on the complete box side; outflow uses the old adjacent cell. Zero velocity transfers zero. Adjacent cells reuse the same stored transfer with opposite signs; all axes update together from old fractions. This is first-order unsplit donor-cell transport, not PLIC or another geometric interface reconstruction.

Each cell starts with amount m=fV, subtracts outward transfers, adds inward transfers and adds dts, with signed borrowed source rates s in m³/s per cell. Source saturation or post-update fraction clamping would change the volume ledger and is forbidden. Raw candidate fractions must be finite and within [0,1]. The independently checked local summed outward Courant number is at most the declared limit in (0,1]. Direct carrier divergence is independently measured and gated. For exact divergence-free flow, zero sources and compatible inflow donors, the update is a convex combination when outward Courant is at most one. Courant alone does not ensure bounds for nonzero divergence or arbitrary sources. Even a small accepted carrier residual can trigger raw fraction rejection near full cells.

The accepted report verifies

V_liquid_after - V_liquid_before + V_outward - V_inward - V_source = error.

Compensated reductions and a budget of 64epsilon times the sum of absolute old, new, boundary and source amounts make the binary64 acceptance convention explicit. This is not a certified floating-error bound. Checked products (including face areas, intermediate transfers, sources and represented amounts) and quotients (Courant, divergence and candidate fractions) require normal finite nonzero results or exact zero. Subnormal results, nonzero values rounded to zero, nonfinite arithmetic and unresolved time are rejected with ArithmeticFailure or the time-specific error. This intentionally rejects some usable extreme scales, including exact subnormal intermediates, rather than rescaling them. The Courant numerator dtoutward_speed is checked before division by spacing. The constant density scales represented mass as rho*V_liquid; it does not change a carrier pressure coefficient or introduce two-phase inertia.

For example, one cell with h=[2^538, 1.52^-538, 1.52^-538], dt=2^538, unit positive X velocity on both faces, an initially dry cell and left inlet fraction 1/2 has exact updated fraction 1/2. The X-face area is 9/16 of the smallest binary64 subnormal and rounds up to that subnormal. Without the scale check the evaluated fraction becomes 8/9 while Courant is 1, divergence is 0 and the rounded volume ledger balances exactly. The scale check rejects this case without publishing fractions, time or version. These checks do not certify all IEEE rounding errors or extend the exact finite-algebraic proof scope.

State and ownership

LiquidFlowInterval borrows the exact immutable grid and all three finite full face fields, together with an explicit start and dt. Its end is start+dt; start must match the accepted volume clock exactly. Velocity, inlet and source stamps are caller-owned identities copied into successful reports. The accepted volume stamp version increments only on publication. Geometry, timing, input shape and settings failures precede callbacks. The inlet snapshot specifies outside liquid fractions rather than silently extending an appearance tracer at inflow.

The state owns two f64 cell arrays and three f64 face-transfer arrays. Its limit covers actual retained Vec capacities, including the transferred initial Vec: nominal payload 16Ncell+8Nface bytes. Flow/source arrays, allocator overhead and RSS are excluded. Advances allocate no heap buffers. Cancellation or failure may leave scratch partial but preserves accepted fraction bits, volume stamp and time. Retry overwrites scratch and agrees with fresh execution.

Publication is transactional for this volume-only state. It does not roll back a separate accepted carrier simulation or constitute a complete liquid step that publishes pressure, geometry and phase together. Tests explicitly demonstrate that separation. Future coupling must establish the appropriate liquid/air pressure mask, velocity-validity mask and unified accepted-state transaction.

Volume and spatial evidence

Signed transfers in all axes, an anisotropic closed circulation, oblique constant transport, signed sources and thin merging supports exercise conservation and bounds. Both accepted PCG implementations supply a readonly 1D projected carrier in a separate fixture; liquid fractions do not feed back into that pressure solve. A 3D translated unit cube remains bounded and retains volume one, yet its cell-average L1 volume error is 0.65625 after one unsplit step. Conservation does not establish geometric accuracy.

An independent binomial oracle checks axis-aligned slab transport. The slab starts at x=[0.25,0.5], moves at 0.25 m/s for 0.5 s, and is compared with exact translated cell overlaps at x=[0.375,0.625]. With outward Courant 0.5, 16/32/64 cells give cell-average L1 volume errors 0.09375, 0.068359375 and 0.04909515380859375. Total volume remains exactly 0.25 m³ and centroid is correct. Halving dt at fixed 64 cells (Courant 0.25) increases the error to 0.06039030345194791: this first-order scheme becomes more diffusive as dt decreases at fixed spatial stencil. Record both refinements instead of claiming that smaller dt alone improves interface shape. These mechanisms are not calibrated pouring, thin-film or droplet physics.

VolumeLedger.lean supplies eight exact finite theorems and three definitions for shared-face/source/boundary conservation, constant-density mass scaling, conditional nonnegative/unit-interval donor updates and constant preservation. Explicit Courant and post-clamp counterexamples name missing assumptions. Its incidence/weights are supplied; it does not certify Rust assembly, IEEE execution, geometry or continuum accuracy. The historical proof inventory and accepted book/PDF inputs are preserved. See the executed evidence.

Remaining work includes geometric reconstruction and interface normals, active free-surface pressure/air extension, density-dependent correction, unified liquid step publication, mesh-solid classification/cut cells, moving swept volume, viscosity/traction, wetting and capillarity. This milestone earns a represented volume-transport claim, not a general liquid solver or moving mesh wall claim.