Prescribed normal box flux

This slice supplies a pressure/velocity boundary interaction for one connected, all-fluid, fixed rectangular control volume. Each complete side has a constant outward normal speed (metres/second): negative is inflow, positive outflow. Lower-side stored Cartesian speed is minus that value; upper-side speed equals it. No pressure is simultaneously prescribed on those faces. It follows the book’s compatible incidence and boundary-work accounting, without changing the accepted book sources or PDF. It is a standalone library projection primitive; the existing closed-box Simulation and passive tracer stages are unchanged.

Affine constraint and pressure

Let C be integrated outward interior-face incidence (entries ±face area), Q the known outward boundary flux per cell, M diagonal interior-face masses rho*cell volume, and u the provisional interior speeds. The constraint is C v + Q = 0. Pressure solves K p = -(C u + Q)/dt with K = C M⁻¹ Cᵀ, then v = u + dt M⁻¹ Cᵀ p. This uses the preserved PressureOperator coefficients and PressureWorkspace::solve_rhs, including both original solver identities, gauge cell zero and the all-cell true-residual acceptance gate. External input samples are validated but replaced by prescribed values in RHS and output.

Known total flux is checked independently of the provisional field: a compensated sum of six side-area/speed products must lie within 64epsilonsum(abs(products)). There is no mean subtraction, pressure opening, or provisional-RHS-based relaxation. Nonzero products that underflow to zero, nonfinite arithmetic and invalid masses are rejected. This rounding budget is a numerical acceptance convention, not a certified exact compatibility predicate. After correction, direct divergence uses every stored f32 face value, including outer values and the gauge cell; a separate physical 1/second limit must pass.

Work and ownership

Only interior unknown faces have kinetic degrees of freedom. Prescribed samples are excluded from that kinetic sum. With R = C v + Q and eta = M(v-u)-dt Cᵀp, the reported ledger is

E(v)-E(u)+E(v-u) = -dt pᵀQ + dt pᵀR + etaᵀv.

Boundary pressure work can raise energy. The 3×1×1 through-flow example, rho=2, dt=0.5, speed=±0.25, starts with zero interior energy and ends at 0.125 joules; correction energy is 0.125 and boundary work is 0.25. Declaring this projection unconditionally dissipative would be incorrect. Stored f32 correction and floating coefficient error are included in eta; actual divergence supplies R. The ledger error is diagnostic and is not an additional acceptance tolerance.

The workspace owns its immutable grid/density, six original f64 pressure arrays and one extra f64 cell array. Actual Vec capacities are capped (56 bytes/cell when exact-sized); caller input/output arrays, allocator overhead and RSS are excluded. It copies one immutable boundary value with caller-owned unique identity/version per request. No allocation occurs in the solve. Inputs are borrowed read-only; output is caller-owned disposable scratch. Cancellation or failure after correction starts may leave output partial and pressure scratch unaccepted. Publish output and consume pressure only with a successful report. Retry overwrites scratch and is checked against a fresh workspace.

Verification and limits

Eleven contract methods test signed flow on all axes and both methods, unequal side areas against an independently assembled dense system, the affine adjoint and gauge invariance, work accounting, zero-flux legacy bits, net-flux rejection under enormous provisional values, actual stored divergence rejection, cancellation/retry, iteration failure, singleton/capacity and invalid/extreme arithmetic. A real red fixture exposed a nonzero boundary product disappearing through underflow; rejecting that product fixes the demonstrated violation.

AffineProjection.lean contains seven exact finite algebraic theorems and five definitions for the supplied C, masses and residuals. It proves the work identity, compatibility, gauge-invariant boundary work and conditional zero-flux energy nonincrease. It does not certify Rust assembly, IEEE arithmetic, continuum convergence or geometric predicates. See the executed evidence.

This is uniform normal inlet/outlet data on a fixed box, with no tangential no-slip rule. Mesh solid classification, cut cells/connectivity, translating wall swept volume, moving-domain work, free surfaces, pressure openings, conservative tracer transport and full time-step boundary integration remain outside this slice. No native graphical run or isolated performance benchmark is claimed.