Fixed-flat column momentum in the accepted MAC owners

This prerequisite connects the conservative column parcel representation to Rheon’s existing velocity, pressure, liquid-volume and reconstructed-column owners. It materializes a prescribed tangential field, projects it with matching liquid masses and flux measures, and publishes both owner revisions together. It does not advance physical time or move the interface. Ordinary and viscous steps refuse the resulting mode until a compatible coupled advance exists.

The dependency is: conservative parcels → mass-consistent MAC transfer → compatible pressure → shared publication → moving-geometry transport and strain. The preceding remap already supplies within-column overlap and explicit cap exchange. Establishing its connection to the accepted MAC owners comes before varying-height strain: different collapsed cap slabs produce nonconforming lateral interfaces, which need a separate flux/dual-mass derivation. Fixed flat columns have conforming lateral slabs and permit an independently assembled pressure graph. The existing periodic flat-shear prototype has different wall conditions and a P1 trial basis; it is not silently attached to this bridge.

One geometry and three mass partitions

Take constant density ρ, normal spacing h, common height H=(k+f)h, 0≤f<1, and tangential column footprint A. The existing column contract requires h/2 < H ≤ (N−1/2)h. There are L=k+1[f>1/2] wet pressure nodes. Parcel dual slabs are [jh,(j+1)h] for j<L−1 and [(L−1)h,H] for the final node. Write their lengths ω_j and define V_j=Aω_j, m_j=ρV_j. The cell-lattice parcel components are means on these slabs, not P1 nodal samples. The phase fraction array retains its original grid-cell representation; an air-centered partial cap is folded into the last wet pressure slab. Compensated phase mass and parcel mass are compared explicitly with the unchanged 64ε extensive budget.

For each tangential component, interior MAC face mass is m_j and each fixed outer wall face owns m_j/2. Its interior flux area is h_other ω_j, and wall/dry flux is zero. Equal adjacent slab masses justify averaging their two parcel values. Wall velocity is prescribed zero; wall impulse and removed kinetic energy are explicit boundary terms, even though wall dual mass is retained.

For the normal component, the bottom wall mass is ρAh/2; interior face masses are ρAh. The top pressure face has mass ρA[H−(L−1/2)h]. Interior and top flux areas are A. Above the top face, mass and flux vanish. Each component’s complete face mass partition, including the zero-velocity wall masses, equals Σ_j m_j per column. The top slot is a virtual free-surface pressure degree of freedom in existing MAC storage. It does not make ordinary sampling or phase transport compatible with this control-volume geometry. That is why dynamics are refused.

FlatColumnMacGeometry borrows the accepted ColumnSurfaceView; it owns no second geometry or mass cache. All columns must have exactly the same stored height decomposition. Geometry, density, coefficient scale and active support are checked before transfer. No small-cell floor, fraction clipping or damping coefficient is introduced. Crossing a cell-center threshold changes L and the collapsed slab layout; just above that threshold the top normal mass can become small. This static qualification supplies neither a continuity argument nor a moving-domain treatment for that transition.

Transfer, adjoint restriction and explicit losses

For an interior tangential face with parcel velocities a,b, set v=(a+b)/2. Equal values copy the existing f32 value exactly, retaining the constant-field correction in the preceding remap successor. For a two-cell row with cell masses m, the output face mass is m and the two wall masses are m/2:

P_MAC − P_parcel = −m(a+b)/2 + P_round
E_parcel − E_MAC = m(a²+b²)/4 + m(a−b)²/8 − W_round.

The first energy term is prescribed wall removal; the second is interior averaging loss. These are not viscous dissipation. Restriction computes each cell’s average of its two bounding tangential faces. It is the mass adjoint of lifting with wall velocities fixed, and preserves weighted tangential momentum up to measured f32 rounding. It is a lossy averaging operation, not an inverse. Its report explicitly records the normal momentum and energy omitted from the two-component output. Re-prescribing an exported field also records the change from the full previous accepted three-component state, including omitted normal energy; the repeated demo cannot be interpreted as unforced liquid evolution.

All transfers use a declared 3F f64 workspace, nominally 8F bytes where F is the sum of the three MAC face counts. Actual vector capacities are charged. Outputs are copied only after every arithmetic, ledger and cancellation gate. Restriction reuses the first N entries of two workspace vectors; it allocates no per-call snapshot. Inputs and stored candidates require normal-or-zero finite values. Nonzero subnormal results, nonzero product/division underflow and invalid f32 stores are rejected in the bridge’s checked arithmetic. This does not establish such a guarantee for every internal PCG intermediate.

Matching pressure graph and energy accounting

Let B take signed integrated face flux, lower minus upper, on each wet slab. Let M contain the face masses above. The operator and correction are

A_pressure = B M⁻¹ Bᵀ,     b = B u / Δt,
v_raw = u − Δt M⁻¹ Bᵀ p,  p_air = 0,
κ_face = q_face² / m_face.

Here Δt scales the pressure; it does not advance a clock. Tangential cap coefficients scale with ω_j. The top normal coefficient corresponds to the actual pressure-node-to-interface distance. RHS, matrix application, Jacobi and SGS neighbor coefficients, correction and true residual divergence all use the same mass/flux geometry. Divergence is integrated flux divided by the wet slab volume, not by the original full cell volume. Legacy operator modes retain their original coefficients and scaling.

With δ=v_raw−u, D=½δᵀMδ, residual work W_res=−Δt pᵀBv_raw and f32 store work W_round, the measured identity is

E_stored − E_before + D − W_res − W_round = 0.

The allowance for energy increase contains positive measured residual work, absolute rounding work and the existing 64ε extensive budget. Momentum changes are checked against pressure impulse plus rounding; sealed walls and atmospheric pressure can exchange momentum. This is not a closed-system momentum claim. True linear residual, predicted volume divergence and actual stored-f32 divergence retain their existing settings; none is relaxed for this evidence. The independent oracle assembles B separately and solves small cases with a dense SPD matrix. It also reconstructs the native paired/compensated flux accumulation when comparing a tiny reported residual-work budget; its separate six-term incidence calculation remains checked against a flux-magnitude roundoff bound. This changes no native acceptance tolerance.

The MAC placement, discrete gradient/divergence relationship and atmospheric pressure boundary direction follow the primary Bridson and Müller SIGGRAPH course notes, sections 4.1–4.5. The collapsed-slab mass and flux construction above is the specific restricted model qualified here; those notes do not validate it.

Publication through the existing facade

LiquidTransportSimulation::materialize_flat_column_profiles takes an exact ColumnMacStateStamp pair, borrowed two-component profiles, a pressure scaling Δt and a cancellation callback. The first materialization requires the fresh rest carrier; subsequent materializations require the same static mode. An already advanced legacy carrier is rejected rather than reinterpreting its full-cell inertia. Both density owners must agree and the accepted fractions must match the current reconstructed-column stamp.

Preparation reuses Simulation candidate velocity and PressureWorkspace. Previous accepted momentum/energy and prescribed changes are reported. After all transfer, pressure, actual divergence, energy, phase-mass and cancellation gates, one final publication barrier copies held pressure, swaps only the accepted velocity vectors, increments carrier generation and liquid revision, and updates both end and held-pressure geometry stamps. Fractions, heights, tracers and physical time retain their accepted values. No accepted snapshot is allocated for rollback. Export takes the actual revision pair, borrows those accepted MAC fields, and transactionally restricts into caller-owned output.

On a 64-bit target, the existing column facade has nominal array storage 16F+80N+48C bytes, where C is the number of columns. The bridge workspace adds 8F, for total 24F+80N+48C. Reports charge actual capacities. This accounting is for these arrays, not a whole-process memory cap; native evidence generation also owns output snapshots outside the production API.

Eight native contract tests cover an independently assembled partial-mass matrix and discrete adjoint, hand-computed transfer losses, all three normals and both PCG methods, exact paired revisions, every callback occurrence from a nonzero accepted state, output rollback, export normal omission, stale stamps, nonflat/paused/overflow/scale failures, late actual-divergence failure and exact retry, single-node and decimal-spacing phase-mass identity, and legacy advance refusal. Accepted velocity, pressure, phase, tracer, clocks, geometry and stamps are compared bit for bit on cancellation/failure.

Measured behavior and proof scope

The native packet has 22 cases and 76 static publications. Its measured figure shows prescription/transfer/pressure energy and both refinement measures. Eighteen pulse cases cover X/Y/Z normals, cap fractions 0.25/0.5/0.75 and Jacobi/SGS. Each subsequent prescription is exactly the preceding accepted export. JSON holds every source, lifted, accepted and exported f32 field, accepted pressure and identity/ledger; initial/final grayscale PNGs are generated from actual MAC velocity. Independent verification checks every stored candidate, phase mass, epoch and pixel.

Four instantaneous circulation cases use u_x=sin(πx)cos(πz), u_z=−cos(πx)sin(πz), u_n=0 over a fixed 1 m square. This is a kinematic manufactured transfer/projection reference with no stress boundary or material calibration. Its mass-weighted velocity error uses the sum of all three component masses (3 times liquid mass):

Instantaneous transfer error decreases approximately fourfold per refinement; stored-f32 derivative roundoff grows with resolution. All cases satisfy the unchanged 1e−5 stored divergence limit. This is not temporal convergence, moving-interface accuracy, a viscosity benchmark or general fluid validation.

The 13 Rheon.ColumnMac Lean theorems establish conditional finite exact-real mass partitions, constant transfer, wall momentum, a two-cell mass adjoint, energy partitions, positive pressure weights, pressure power, divergence/ gradient adjointness and an energy implication given residual work. Pinned Lean/mathlib builds and transitive axiom audits pass; injected sorry and extra axioms are rejected, and restored source passes. They do not prove arbitrary geometry or basis integration, Rust/IEEE execution, PCG convergence, transaction atomicity or moving-surface dynamics. These depend on native tests and measured evidence only within their stated scope.

Still missing are compatible three-component moving momentum transport, varying-height lateral flux/mass geometry, moving-interface pressure publication, full strain on that geometry, adhesion and density constitutive laws, capillarity and general reconstruction. The preceding frozen fixtures, including the known f32 viscosity refinement accuracy failure, are retained without altered bounds.