Periodic fitted seam correction and scope erratum

This successor corrects the declared periodic Powell–Sabin geometry. It preserves all bytes and commit identities of research evidence 2db90e8c and native assembly evidence c995671c. Those checkpoints are genuine algebra/numerical evidence for the actual mesh they constructed, but their declaration of periodic Powell–Sabin geometry and its rank-33 pressure image was incorrect.

Actual mismatch and corrected construction

The frozen reference used the physical-boundary rule at both periodic seam edges: split at the midpoint, then identify the nodes. On the reference strip, its left seam point was (0,1/2) with mesh derivative (1/8,0). The neighboring macrotriangle centers on the periodic covering plane are P=(1/12,3/4) and Q=(-1/12,1/3). Their line intersects the seam at (0,13/24), not (0,1/2). At cap speed 1/4 the intersection derivative is (13/96,5/192).

The intended Powell–Sabin construction, section 1 joins neighboring macrotriangle interior points through the shared macro edge. A periodic seam has that neighbor across the period; it is not a physical boundary edge. The native repair records the actual neighboring macrotriangle on both seam edges, translates its centroid by the fixed period when computing the left intersection, and represents the right intersection as its exact periodic translate. Dual-number differentiation follows this construction, including motion of both neighboring centroids. The fixed-period translation has zero derivative. Physical cap and bottom edges retain midpoint splits and the existing affine velocity trace constraints. The other geometry and operator contracts remain unchanged.

The repaired rational reference independently constructs both intersections and checks their exact translated equality. Native regression checks the seam line, (0,13/24), its (13/96,5/192) derivative and matching right-seam motion. The previous midpoint construction fails this direct declared-model contract. Merely changing a rank constant would fail the geometry test.

Changed rank and conditioning limits

With periodic neighbor intersections, the exact constrained divergence image and integrated pressure matrix have rank 32 for the 4-column fixture, compared with 33 for the frozen midpoint seam. The alternating divergence relation at the singular seam vertex is now exact. Native tests check the corresponding 8C mode count on C=2,3,4,6,8,12 snapshots, positive pressure Gram pivots, the selected velocity-column/basis correspondence and constant pressure representation. The previous 8C+1 claim is superseded for the intended periodic construction.

This correction does not establish uniform pressure stability. Exact rational perturbations of the two seam heights by 1/64, 1/4096 and 1/2^30 each restore rank 33. These are pressure-algebra perturbation experiments only; no trajectory or GCL claim is made for those deliberately altered snapshots. The additional mode approaches dependence as the geometry approaches the singular seam. This sensitivity is material to conditioning and to geometric roundoff, rather than a reason to retain the incorrect midpoint rule.

Native geometry/gradients are floating point. The constructor checks the numerical rank using its explicit tolerance and expected topology count; a roundoff-sized extra mode is not an exact-real rank proof. Pivot ratios and positive finite Gram tests are not inf-sup bounds. Actual element divergence remains a separately measured diagnostic; a future pressure solver cannot infer adequate physical continuity from an arbitrary pressure residual alone. No published stability theorem is inherited for the added boundary trace restriction or nodal mass lumping.

Identities retained, notation clarified

The corrected full reference witness still has exact zero moving-dual GCL, third-component momentum rate, finite transport energy residual and affine traction residual. Liquid area remains 19/16, mass per velocity component 57/16, and unrestricted affine strain power 475/512. There are still 22 nonzero nodal mass rates and 76 nonzero shared flux pairs, with corrected individual values. For the same nonconstant third-component fixture,

D_adv = 23503996984881317 / 52895810764800000,
Uᵀ K U = 2595531373 / 1259712000,
T' = −132491397549700517 / 52895810764800000.

These are the constrained semidiscrete third-component witness, not a genuinely coupled moving free-surface numerical solution. The affine stress patch uses prescribed traction and is not a globally periodic homogeneous free-cap flow. The smooth flat-cap predictor is an instantaneous analytic benchmark. Those scope distinctions continue to apply.

Equation (3) in the frozen formulation should explicitly use unordered face pairs. The correct finite donor identity is

T(U_adv,m_new)−T(U_old,m_old)
  +½ Σ_i m_old,i |U_adv,i−U_old,i|²
  +½ Σ_{unordered {i,j}} |F_ij| |U_adv,i−U_adv,j|² = 0.

Each shared physical face is counted once, as in the implementation and rational witness. An ordered sum over both (i,j) and (j,i) would double this dissipation. No implementation energy factor is changed by this notation clarification.

Qualification and continuation

The repair adds a direct seam regression to the native suite, a corrected bounded rational reference, exact singular-vertex and rank-sensitivity tests, and actual rejection of the frozen geometry by the intended seam contract. The complete Rust feature matrix, focused release tests, independent normal and optimized reference/native validations, actual binary hashes and corrected native figure are frozen in evidence/fitted-height-periodic-repair/ separately from the source commit. All old evidence and claimed limitations are retained; the predecessor is also verified in a detached checkout.

No new Lean theorem is claimed. The preserved 15 conditional finite exact-real identities are still applicable to their explicit algebraic hypotheses; they never proved the geometric seam or a physical solver. Native managed payload is unchanged: 55,408 bytes for the 4-column workspace, including rank and operation scratch. The pressure coefficient array now has 32 modes for that fixture.

The next work remains physical space-time geometry/full-momentum transport and compatible pressure/strain solve qualification, including the differentiated constraint B’RZ, actual integrated face provenance/GCL, numerical residual/work gates, and a single bounded accepted/candidate facade. Uniform pressure stability, pointwise traction, temporal accuracy and native advancing/rendered liquid dynamics remain unqualified. Existing advancing-viscosity refusal, accepted flow owners, phase geometry authority and all simulation clocks are unchanged by this repair.