One progression: wall traction, exact trace and external force

Read finite wall friction, then compatible no-slip, then uniform forcing and Poiseuille startup. These three native operations share one fixed flat-column model and one endpoint basis. They do not compose a general liquid solver. The original six 3D reference laboratories remain separate analytical/dense-solve teaching mechanisms.

What stays fixed, and what each step adds

The slab has fixed height H and area A, periodic lateral directions, zero normal motion and layer-uniform tangential velocity. Density rho is constant and positive; dynamic viscosity mu is in Pa s. Liquid dual lengths omega_j give V_j=A omega_j and m_j=rho V_j. Each viscous edge has stiffness mu A/h. The retained first/last wet-center value extends constantly to the physical wall, so a constrained trace can be exact while the continuum field retains endpoint spatial error. This is not an evolving free-surface/pressure update.

The no-slip API requires the incoming trace to match its prescribed fixed speed. It is stateless and cannot infer an intercall wall-speed change when the caller supplies a matching state. Two no-slip walls on one wet node are refused because their separate reactions are nonunique. A mixed one-node law can have a unique reaction. No large artificial beta replaces a constraint.

A common momentum and work language

Let delta be the combined explicit increment before storage rounding, and let J contain no-slip reaction impulses. The most general step in this progression is

$$m_j\delta_j=\Delta t(f_{bulk,j}+f_{Navier,j}+f_{body,j})+J_j.$$

Free nodes have J=0. Compatible constrained nodes have delta=0, so reaction cancels every unconstrained contribution, including body force and any Navier traction sharing that node. Body force is applied to its actual liquid dual volume even at constrained endpoints; wall reaction is not obtained by dropping those nodes from the external impulse.

With bulk and relative Navier dissipation measured at the incoming velocity,

$$\Delta T+\Delta t(P_{bulk}+P_{Navier})-W_{wall}-W_{body,old} -\tfrac12\delta^TM\delta-W_{round}=0, \qquad W_{body,old}=\Delta t\sum_j f_{body,j}\cdot u_j.$$

Momentum changes by body impulse plus total wall impulse plus stored-f32 rounding momentum. Prescribed wall work is wall velocity dotted with delivered wall impulse and can be negative. Compatible no-slip relative reaction work J dot (u-w) vanishes without removing bulk dissipation. A stationary wall does no work while still exchanging momentum. From rest, body old-speed work is zero on the first step, but the explicit squared-increment energy is positive. This combined-step convention differs from midpoint applied work in the separate box force stage.

The free-row explicit admission is unchanged by a constant body source. No-slip rows have zero increment; arithmetic checks still apply. All gates and cancellation callbacks precede output publication, with the same five fixed scratch vectors and no per-call allocation. These are native runtime contracts, not an implicit integrator, general geometry theorem or whole-program memory certificate.

Keep the reference problems separate

For symmetric finite slip, u(y)=U(y+ell)/(H+2ell). The native discrete equilibrium is U(jh+ell)/((L-1)h+2ell). For lower slip and an upper no-slip wall, the respective denominators are H+ell and (L-1)h+ell. Both no-slip Couette references are Uy/H and Uj/(L-1).

For stationary walls driven by uniform q=G, the continuum steady reference is Gy(H-y)/(2mu), with flow per width GH³/(12mu). The retained discrete equilibrium is Gh²j(L-1-j)/(2mu). At fixed G, density changes relaxation but not steady amplitude; at fixed acceleration, G=rho a changes too. The Poiseuille lab uses a body source and does not solve a pressure gradient.

Only the forcing lab supplies a closed-form exact-real explicit discrete startup comparison. Its binary64 evaluation has trigonometric, power and sum rounding, while the native assembly also rounds before f32 storage. There is no certified reference-evaluation bound. A steady reference at an early saved time is not a transient oracle. Distance to equilibrium includes unfinished relaxation; decreasing endpoint deviation does not establish temporal order.

Proofs and recorded demos

The integrated native proof source has 57 public theorems and 77 audited declarations at accepted PR24 source. The older expansion receipt’s 42/60 counts remain historical, with their own source identity. Definitions and generated helpers are not added to the public-theorem count.

The native lab hub connects all three recorded packets when supplied to the book builder. Every case/time control is explicitly snapshot playback. The PR22 plot is its final saved profile; its CSV contains four saved times. PR23/24 sliders choose five actual saved profiles. They do not recompute arbitrary parameters or run a live fluid solver. The builder verifies frozen input bytes and performs zero numerical calls. A source-only CI build without recorded packets states that they are absent, rather than silently substituting new runs or fabricated results.

These steps qualify local wall/force mechanisms. Continue to the requirements map and next geometry contract before extending beyond the fixed slab.