§4.4 6D campaign — quality demonstration at scale

Every quality instrument the rebuild ships — ROM error gauges (DIR-003/053), the a-posteriori certification pipeline with mass-budget telemetry (DIR-088), the inertia window-entry certificate (DIR-089), dynamic domain decomposition (DIR-036) with measured parallel execution (DIR-055B/C), and the bounded- resource store (DIR-030/032/035) — was validated on 1D/2D problems or 6D smoke scale. This page is the first time they are exercised together on the dissertation’s §4.4 problem: 2D plane-strain elasticity on the L-shape Ω = Ω₁∪Ω₂∪Ω₃ with per-face (Eᵢ, νᵢ), a 6-dimensional parameter box [10,100]³×[0.1,0.4]³ (interleaved), clamped on Edges 1 & 2 (base + top-arm end), Window mode over the natural-frequency band ω ∈ [0, 12.5] i.e. λ ∈ [0, 156.25], dissertation thresholds t_π = 0.57, t_λ = 0.03, t_c = 0.1.

Provenance — these numbers are the DIR-113 re-run on the corrected operator

An earlier version of this page (DIR-110) reported this campaign on a fake-soft operator: Elasticity2D assembled the vector mass with ddot (matrix double-contraction) instead of dot, giving trace(M) ∝ n² and crushing every eigenvalue ∝ h², so the window [0, 12] on λ held >960 spurious modes where the dissertation tracks a handful of surfaces. DIR-112 found and fixed the root cause (deviation log elasticity-4-4-baseline-deviation-log.md, rows 9–13): the mass bug plus two convention mismatches — the window filters the natural frequency ω = √λ (so on λ the faithful band is [0, 156.25], not [0, 12]), and the physics is modal-planestrain clamping only Edges 1 & 2, not plane stress with the whole boundary clamped. On the corrected + faithful config (make_l_shape_6d_faithful) the box midpoint holds 8 modes — matching a headless MATLAB run of the frozen ComputationalDomain6Dreal.m to ~0.7 % on the first eight eigenvalues (P1-vs-P2).

DIR-113 re-ran the entire campaign on the corrected operator; every number below is faithful and pinned by tests/test_dir_113_artifacts.py. The pre-fix tests/artifacts/dir_110/ set is retained on disk as archived pre-fix data but is no longer rendered here. The scale-invariant sibling results held up exactly as DIR-112 predicted: the DIR-053 ROM ceiling (a FirstK/full-basis measurement) is unchanged, and the DIR-089 “stay-dense” inertia verdict survives on the now much sparser window.

This page carries the campaign’s performance and self-consistency story. The correctness case — the five-strand evidence argument (slice views, inertia certificate, holonomy/ν-index consistency, sampled statistical audit, setup invariance) that replaces the 1D/2D branch-figure eyeball certificate — lives on its own page: §4.4 6D campaign — the correctness evidence.

Everything below is measured on committed artifacts (tests/artifacts/dir_113/). The pre-fix campaign needed staged, budget-bounded overnight runs; on the corrected operator the whole campaign runs in seconds — the fake window’s ~380–960 modes/node were the entire cost, and the true window holds only ~8–20 modes/node (Stage A: min 7, max 21, mean 10.6):

Stage	scale	config	outcome
A	n_dof = 1846 (`--target-n-dof 1500`), L≤4	serial, single region, `--mass-telemetry`, RunCache	L4 completed, `ran_out=True`, 32 nodes, wall 1.6 s, peak RSS 96 MB, 0 evictions
B1	same scale	`--regions --parallel-workers 14 --fixed-v0`	identical trajectory (13/17/22/27/32 nodes), verdicts byte-identical to A, wall 2.3 s
B2	same n_dof	DIR-055B twin harness (serial vs parallel pool, FirstK)	serial ≡ parallel exact (nodes, verdicts, eigenvalue rel-diff 0.0)
C	n_dof = 6576 (`--target-n-dof 5954`, dissertation scale), L≤4	`DiskSpillCold`, 4 h wall cap, 16 GB RSS budget	L4 completed within budget — no sentinel: 33 nodes, wall 5.7 s, peak RSS 131 MB, 0 evictions, 0 bytes spilled

The headline of the re-run is that the §4.4 problem, at its true spectral scale, is small: 32 nodes at laptop scale, 33 at dissertation scale, and the store never evicts. The bounded-resource machinery (DiskSpillCold, sentinels, RSS budgets) still stands as engineering, but at the faithful window it is never stressed — the pre-fix campaign’s memory drama was the mass bug.

ROM error at 6D — held-out table and soundness curve

32 held-out µ (seeded, uniform in the box) evaluated against a dense in-window FE truth at each point (rom_error_table_6d.csv). On the lowest-20 window modes — the like-for-like gauge for the DIR-053 ceilings, which were pinned on the lowest eigenpairs —

max eigenvalue rel-err 1.1·10⁻¹, max subspace angle 1.1·10⁻¹ rad — a breach of the DIR-053 ceilings (2·10⁻³ / 3·10⁻²) at the deepest fill levels; window coverage stays 0.91–1.0 and no query misses the window.

This breach is a genuine, expected consequence of the faithful window and the central ROM finding of the re-run. In the pre-fix campaign a Window snapshot carried ~380 eigenpairs, so even an 8-node nearest-neighbour basis spanned a near-complete subspace and the lowest-20 error sat at the accuracy floor (10⁻⁹–10⁻⁵) for every fill level. At the true window a snapshot carries only ~10 modes, so the nearest-8 Galerkin pool is a genuinely thin basis: the RMS lowest-20 error grows with depth (4·10⁻³ at L0 → 1.9·10⁻² at L4) as refinement pushes queries into pockets the sparse pool interpolates less well. Coverage and window-completeness stay high throughout — the breach is basis-thinness, not a missed window.

Two clarifications keep this honest. First, the DIR-053 ceiling itself is unaffected: it is a FirstK/full-basis measurement (test_rom_6d, lowest-6 eigenpairs with a complete per-node block), and DIR-112 re-confirmed it unchanged on the corrected operator. The gauge that breaches here is the campaign’s Window-snapshot ROM, a different and harder object. Second, the full-window columns of the table compare mode-by-mode past the coverage limit and misalign once modes are missing; they are labelled coverage-limited, not treated as eigenvalue error.

A campaign note on the shipped N-D basis pool. The surrogate scored here is a Galerkin window ROM (it projects K(µ), M(µ) onto snapshot eigenvectors and does a small eigensolve), not the dissertation’s §4.4.5 hat interpolation of stored eigenvalues — a different object with a different memory premise; see the evidence page callout and DVG-SURR-01 for why the ROM (a DIR-110 feasibility carry-forward) stands in here rather than the faithful hat surrogate. The table uses a bounded nearest-8 Galerkin pool with rank-revealing M(µ)-orthonormalisation (scripts/rom_error_table_6d.py), not the shipped evaluate_rom N-D pool rule (nearest min(2^d, 32) nodes). At the faithful window the shipped rule is no longer infeasible (a ~10-mode snapshot × 32 nodes is a manageable ~320-column block, unlike the pre-fix ~12k), but it is now inaccurate — a thin per-node block means the pool must reach further for rank, and the breach above is the symptom. Redesigning the pool rule for the true §4.4 window (more nodes, or a region-aware interface basis) is follow-up work recorded alongside DIR-111.

Exact-P2 mesh-faithful convergence (DIR-122)

Everything above runs on the rebuild’s structured P1 l_shape_mesh. The DIR-117/118/119 parity arc removed that last confound by building the operator on MATLAB’s exact P2 mesh — make_l_shape_6d_faithful_mesh (Elasticity2D.from_mesh on the imported generateMesh(Hmax=0.1) mesh + P2 vector element, n_dof = 5870; DIR-119 matched its window eigenvalues to MATLAB to 1.8·10⁻¹³) — but only ever ran a capped 2-level parity probe on it. DIR-122 runs the adaptive greedy to natural convergence on that operator, single process, via the new opt-in flag:

uv run python scripts/reproduce_4_4.py --operator mesh --max-levels 15 --fixed-v0 \
    --output-dir tests/artifacts/dir_122 --csv-name elasticity_4_4_mesh.csv

(--operator defaults to structured — the DIR-030/113 path is byte-identical; mesh bakes in the faithful config so --lame-mode/--clamp/--target-n-dof are ignored-with-a-note.)

The run converges of its own accord at level 7 with 47 nodes (ran_out=False, flagged edges 13→…→3→0), in 7.3 s / 0.14 GB peak RSS / 0 evictions.

This answers the campaign’s open question: the exact-P2 geography is not the structured-P1 one. The faithful DIR-113 run converged at L4 / 32 nodes; the sharper P2 discretisation refines two levels deeper and places 47 nodes (trajectory 13/17/22/28/34/39/44/47). The grid is denser, not merely a relabel of the P1 grid — pinned in tests/test_dir_122_mesh_campaign.py.

ROM accuracy on the converged grid (rom_error_table_6d.py --operator mesh, 32 held-out µ, lowest-20 gauge, fixed-v0 dense truth):

max eigenvalue rel-err 0.11, median 5.8·10⁻³, max subspace angle 0.126 rad, window coverage 0.99 (no query misses the window) — a breach of the DIR-053 ceilings (2·10⁻³ / 3·10⁻²), exactly the basis-thinness finding above, now measured on the exact-P2 operator.

The soundness curve does not keep falling with grid fill: it plateaus at ~2.2·10⁻² RMS by L4 and holds flat through L7. Adding nodes past L4 tightens the grid but cannot close the residual — it is the near-degenerate eigenvector-gauge ceiling DIR-120 root-caused (the converged grid packs modes densely near the top of the window, where the eigenvectors are defined only up to rotation), not under-resolution. The median stays modest; a handful of near-degenerate query points drive the max. This is measured and pinned, not transplanted from the 1D/2D 1e-4 ceiling. See affine parameter-assembly for why the per-µ solves are cheap enough to make a 47-node adaptive run finish in seconds.

Faithful full-scale run past MATLAB’s L2 wall (DIR-124)

DIR-122 above runs the exact-P2 operator with the rebuild’s own native adaptive path — the 13-point sparse seed + edge-bisection — and converges cheaply at L7/47 nodes. DIR-124 runs the other experiment the whole DIR-117/118/119 arc was built for: the MATLAB computation itself — the dissertation’s 3^d tensor seed (729 points at d=6, seed_tensor_lattice), MATLAB’s mirror_bracket omnidirectional spawn, the ω-scale verdict, the p-box (p ∈ [0.2,1]×[0.2,0.4]³, E = 100·p), and the frozen §4.4 thresholds (t_pi=0.57, t_lambda=0.03, t_c=0.1, squared Π) — run past level 2, the level MATLAB dies at (8376 points, 54 min, then an unbounded-eigenvector OOM). The seed is a first-class flag (--seed {tensor,sparse} / --knots-per-axis, default the 3^d tensor); the sparse seed is a declared alternative, never a silent substitution. mirror_bracket rejects region decomposition and the parallel pool by construction, so this is single-process — the only bound is the store.

uv run python scripts/dir_124_faithful_campaign.py --max-levels 7 \
    --max-active-mb 2048 --cold-disk-dir <scratch>/spill --cache-dir <scratch>/cache \
    --output-dir tests/artifacts/dir_124 --peak-rss-budget-mb 12000 --rom-queries 32

The faithful campaign does not converge at a laptop-feasible scale — and that is the result. The grid grows as a steady power law (nodes ∝ level^1.40, R²=0.998) with the flagged-edge count rising at every level, never turning over:

level	0	1	2	3	4	5	6	7
nodes	729	1640	3001	4663	6392	8421	10649	12702
flagged edges	0	911	2134	3453	4559	5602	6596	7348
active tier (MiB)	356	776	1437	2047	2048	2048	2048	2048
evictions	0	0	0	540	2400	4589	6990	9155
peak RSS (MiB)	487	960	1709	2431	2492	2525	2576	2613

The L0–L3 prefix (729→1640→3001→4663) bit-matches the DIR-119 exact-mesh 2-level parity probe and its own independent growth smoke — the faithful setup is deterministic under fixed_v0.

This is the rebuild’s entire reason to exist, measured. The MATLAB original OOMs here because it accumulates eigenvectors without bound. The rebuild’s bounded SnapshotStore turns the low-rank premise into an enforced invariant: the LRU active tier clamps at its 2 GB cap from level 4 on, the cold tier spills 2.4 GB across 9155 evictions to disk, and peak RSS holds ~2.6 GB flat while the grid triples (4663→12702 nodes). The run reaches level 7 / 12702 nodes in 25 min, single process, with a clean LEVEL_7_COMPLETED and zero StoreOverflow — five levels past MATLAB’s wall, no OS OOM-kill.

As a number: the active-byte growth exponent is 0.87 — sub-linear, decoupled from the node-count exponent 1.40, because the cap does its job. The cold-tier rank keeps climbing (5.3k→105k stored eigenvectors, ≈11 modes per evicted snapshot), so the eigenvector family is not staying low-rank as the grid grows at this faithful scale — the premise holds only because the bounded store spills the surplus to disk rather than RAM. Either way, the faithful computation runs to a reported depth in bounded memory; a deeper level is a wall-time / disk-budget choice, not a memory-safety one.

ROM accuracy on the L7 grid (window_rom_predict, 32 held-out µ in the p-box, pool 2, fixed-v0 dense truth):

max eigenvalue rel-err 9.1·10⁻², median 5.3·10⁻³; max subspace angle 0.162 rad, median 4.1·10⁻² — the same thin-Window-basis ceiling as the DIR-113/122 runs (a bounded ~8-mode snapshot is not a near-complete basis in this dense near-degenerate spectrum), measured on the faithful full-scale grid, not transplanted.

Committed artifacts under tests/artifacts/dir_124/ (run report + telemetry CSV, ROM error table + summary; the 2.4 GB disk-spill cache is scratch-only, not committed); pinned by tests/test_dir_124_faithful_campaign.py — the exact trajectory, the bounded-memory invariants (active cap, disk spill, flat RSS), the growth-fit exponents, and the ROM bands, plus a cheap live sparse-L1 re-run that re-derives the faithful config deterministically.

Deep continuation — does it converge, and does deeper get better? (DIR-125)

DIR-124 above stopped at a chosen --max-levels 7 cap with every real budget ~80 % unused, so its “grows ∝ level^1.40, does not converge” was an extrapolation from the pre-saturation ramp, and its ROM number was a single L7 point. DIR-125 removes both limits: the same driver, same faithful config, run at --max-levels 100 with generous laptop-sized budgets (18 GB disk, 180 min wall, 12 GB RSS) so the stop is real — and a per-level ROM soundness curve (scripts/dir_125_soundness.py) scoring the grid-as-of-level-L for every L against fixed-v0 dense truth.

uv run python scripts/dir_124_faithful_campaign.py --max-levels 100 \
    --max-active-mb 2048 --cold-disk-dir <scratch>/spill --cache-dir <scratch>/cache \
    --cold-max-mb 18000 --peak-rss-budget-mb 12000 --max-wall-time-min 180 \
    --rom-queries 0 --output-dir tests/artifacts/dir_125 --csv-name convergence_run.csv
uv run python scripts/dir_125_soundness.py --cache-dir <scratch>/cache \
    --n-queries 32 --pool-size 2 --out-dir tests/artifacts/dir_125

Result 1 — it converges. The faithful campaign reaches a natural fixed point at L23 / 21867 nodes, bounded, in 55 min — overturning DIR-124’s L7 “grows-forever” extrapolation. The mirror_bracket active frontier empties (its termination condition, mirror_bracket.py): per-level additions peak at L6 then decay monotonically to near-zero, so the grid turns over rather than growing without bound. The stop is natural (ran_out=False, below the cap) — not a wall/disk/RSS budget and not the level cap:

level	6	7	8	10	12	15	18	20	22	23
nodes	10649	12702	14922	18588	20162	21212	21686	21793	21853	21867
added	2228	2053	2220	1556	541	257	100	44	27	14
flagged edges	6596	7348	7828	8274	8417	8521	8563	8573	8578	8578
peak RSS (MiB)	2582	2622	2654	2715	2737	2755	2792	2805	2817	2818

The revised growth exponent over the full L0–L23 trajectory is nodes ∝ level^1.08 (R²=0.94), not L7’s 1.40 — the earlier number was the ramp, not the asymptote. The L0–L7 prefix bit-matches DIR-124 exactly (same fixed_v0 faithful setup, only the caps/budgets differ).

Converges, bounded, at laptop scale — every budget unused. Peak RSS 4.1 GB (budget 12 GB), wall 55 min (budget 180 min), disk spill 5.2 GB (cap 18 GB), active tier clamped flat at its 2 GB cap the whole way, zero StoreOverflow, no OS OOM-kill — 16 levels past DIR-124’s L7 and 21 past MATLAB’s L2 OOM wall. The honest nuance: termination is by frontier / dyadic-lattice saturation, not full certification — 8578 edges remain flagged REFINE at halt (the near-degenerate eigenvector gauge, DIR-120, keeps re-flagging the dense upper window), but the mirror_bracket spawn can no longer place a new node to act on them, so the adaptive loop stops.

Result 2 — deeper does not get better. ROM accuracy plateaus at the near-degenerate eigenvector-gauge ceiling by L2–L3, then is flat for 20 more levels (window_rom_predict, 32 held-out µ in the p-box, pool 2, lowest-6, fixed-v0 dense truth):

level L	0	1	2	3	7	8	15	23
nodes	729	1640	3001	4663	12702	14922	21212	21867
rel-err median	1.0e-2	6.5e-3	5.5e-3	5.0e-3	5.3e-3	4.9e-3	4.9e-3	4.9e-3
rel-err max	1.7e-1	9.1e-2	9.1e-2	9.1e-2	9.1e-2	9.1e-2	9.1e-2	9.1e-2
angle median (rad)	4.2e-2	3.8e-2	2.1e-2	1.9e-2	2.0e-2	2.0e-2	2.0e-2	2.0e-2

The entire gain is L0→L2 (median 1.0e-2 → 5.5e-3); from L8 the curve is bit-identical (median 4.856e-3, max 9.127e-2, angle 2.034e-2) even as the grid grows 14922→21867 nodes. The classifier’s verdict is plateau_at_gauge_ceiling (L0→L23 buys only 2.06×, and the best level is L8 — far from the deepest, the signature of a shallow-gain-then-flat plateau, not a convergent improvement). The max rel-err is a fixed ceiling from L1 on (9.1e-2, one held-out µ whose top window mode is gauge-degenerate — DIR-120), never resolved by grid fill. The mechanism is direct: the deep continuation refines only the near-degenerate flagged regions high in the dense spectrum, which are not near generic held-out µ, so the pool-2 basis for a random query stops changing once its neighbourhood is filled (~L8). The L7 point cross-checks DIR-124’s independently-computed single point exactly (median 5.3e-3, max 9.1e-2).

Committed artifacts under tests/artifacts/dir_125/ (convergence run report + CSV; per-level soundness curve JSON/CSV/figure + per-query CSV; the 6.5 GB run cache and 5.2 GB spill are scratch-only, not committed); pinned by tests/test_dir_125_deep_convergence.py — the natural-not-budget stop, the turn-over (additions decay + flagged plateau), the bounded-memory invariants, the plateau verdict + flat-ceiling tail, the DIR-124 L7 cross-check, and live pure plumbing (query determinism, level truncation, the verdict classifier). MATLAB parity is out of scope — there is no MATLAB reference past L2 (the DIR-117 probe’s lev_max=2); the quality claim is vs dense ground truth only.

Certification telemetry — the independent inertia certificate

The a-posteriori matrix Π and the DIR-088 mass budget certify the computed subspaces; window_count (dense LDLᵀ Sylvester inertia, DIR-089) counts the true in-window eigenvalues from the assembled K(µ), M(µ) alone — an independent certificate that sees modes the subspace side cannot. Over all 31 tested edges of the Stage-A run (endpoints + un-solved midpoints, scripts/inertia_certificate_6d.py):

count	value	reading
`endpoint_mismatch`	0	`solve_window` returned the complete in-window set at every node
`mid_bump`	0	no mode present at a midpoint but at neither endpoint — no grid-invisible window entry
`cross_edge_delta`	16	the in-window count changes across half the edges (a ~10-mode window is far quieter than the pre-fix ~380) — all surfaced to the matching layer
`cert_with_entry`	12	count changes on certified edges: surplus bands the Algorithm-6 `t_c` gate certified-and-dropped (Q14 territory, not a blind spot given `endpoint_mismatch = 0`)

The DIR-089 verdict — the node-local window solve is complete and the only genuinely-uncaught category (mid_bump) is empty — holds unchanged on the sparser faithful window, exactly as DIR-112 predicted.

Domain decomposition at 6D — Tier 1 (rank savings)

The first rendered view of the DIR-036 decomposition on the real 6D problem (Stage B1). The split criterion still fires: 5 splits, 4 freezes, peak 6 subdomains, and the Tier-1 headline —

single-store POD rank 35 vs max per-region rank 22 (−13, a 37 % reduction), the 6D analogue of the 2D page’s 31 → 23.

The absolute ranks are small (35, not the pre-fix 1311) because the faithful grid is small; the fractional saving (−37 %) survives, so decomposition still buys the same relative relief at the true scale.

The lower panel is the 6D measurement of the interface budget (fraction of nodes on region boundaries — the Q13 scalability instrument). Here it climbs to 0.31 at L4 — it does not stay ≤ 0.1 as the pre-fix 166-node grid did. This is an honest scalability caveat, not a regression: the budget is a ratio, and on a sparse 32-node grid a larger fraction of nodes necessarily sit on the 6 subdomain boundaries. The concern the instrument exists to watch — boundary duplication growing faster than the interior — is best judged on a denser grid; at the faithful §4.4 scale the interior is simply too small to dilute the boundary.

Domain decomposition at 6D — Tier 2 (measured parallel speedup)

The equivalence gate stands and is now cleaner; the wall-clock story inverts:

Strict equivalence gate at campaign n_dof (DIR-055B harness, FirstK config): serial and 14-worker-pool runs produce identical nodes and verdicts and eigenvalue max-rel-diff 0.0 at n_dof = 1846. The measured per-region barrier-model speedup is 1.82× against the equal-cost projection’s 1.08×, but the raw end-to-end wall-clock is 0.76× — overhead-bound (serial 0.40 s vs parallel 0.52 s).
Raw campaign wall-clock (Window config, Stage A vs B1): 1.6 s → 2.3 s — the parallel pool is slower than serial on the faithful operator.

The reason is direct: at the faithful window a solve returns ~10 modes and costs milliseconds, so process spawn / pickle / barrier overhead dominates the 14 workers. The pre-fix 2.43× speedup was real but depended entirely on the fake window’s ~380-mode heavy solves. At the true §4.4 scale the parallel pool buys nothing in wall-clock — the honest conclusion, and a reversal of the pre-fix page.

One further faithful improvement: the pre-fix page carried a caveat that under the Window config the parallel run flipped 2 of ~124k band verdicts (REFINE ↔︎ CERTIFIED, a BLAS reduction-order effect). On the sparse faithful window that flip is gone — Stage A, the serial --regions twin, and the 14-worker parallel run are all verdict-byte-identical. The sparse window carries almost no near-threshold bands for last-ulp noise to tip.

Mesh independence

Three mesh resolutions, one configuration (mesh_independence_6d.csv):

n_dof	nodes per level	certified fraction	wall	peak RSS
846	13, 17, 22, 27, 33	0.458	0.9 s	88 MB
1846	13, 17, 22, 27, 32	0.421	1.6 s	96 MB
6576	13, 17, 22, 28, 33	0.438	5.7 s	131 MB

The refinement geography is now mesh-independent: identical seed and early levels (13/17/22 at L0–L2 everywhere) and a final count of ~33/32/33 across an 8× span in n_dof — not the pre-fix 145 → 166 → 193 that grew with the mesh. That growth was itself a symptom of the fake operator: a finer mesh resolved more of the ~380-mode window’s spurious content, so more band pairs became testable and more edges flagged. On the corrected operator the parameter-space geography decouples from the spatial mesh, as it should — the certified fraction is flat at ~0.44 rather than falling from 0.47 to 0.23. The level cap (ran_out=True at every scale) still binds before the geographies could diverge.

Bounded-resource record — dissertation scale in seconds

The last full-scale run of the original MATLAB predates this rebuild’s memory work and was killed by the OS. The pre-fix rebuild campaign answered that at the fake scale with an overnight 7 h 26 min / 9.2 GB / 5.8 GB-spilled Stage C. On the corrected operator the record is anticlimactic and that is the point: the full §4.4 problem at n_dof = 6576 through four refinement levels (33 nodes) in 5.7 s on a laptop, peak RSS 131 MB against the 16 GB budget, with the DIR-035 DiskSpillCold tier wired but never touched (0 evictions, 0 bytes spilled) — zero overflow, zero sentinel, LEVEL_4_COMPLETED. The wall-time and RSS budgets and clean sentinel stops all still stand as engineering, but the faithful §4.4 problem never approaches them. Neither wall time nor memory is a binding constraint at the true 6D scale; the pre-fix campaign’s entire resource narrative was an artifact of the mass bug (MEMORY-OOM-ANALYSIS).

--- title: "§4.4 6D campaign — quality demonstration at scale" --- Every quality instrument the rebuild ships — ROM error gauges (DIR-003/053), the a-posteriori certification pipeline with mass-budget telemetry (DIR-088), the inertia window-entry certificate (DIR-089), dynamic domain decomposition (DIR-036) with measured parallel execution (DIR-055B/C), and the bounded- resource store (DIR-030/032/035) — was validated on 1D/2D problems or 6D smoke scale. This page is the first time they are exercised **together** on the dissertation's §4.4 problem: 2D **plane-strain** elasticity on the L-shape Ω = Ω₁∪Ω₂∪Ω₃ with per-face (Eᵢ, νᵢ), a **6-dimensional** parameter box [10,100]³×[0.1,0.4]³ (interleaved), clamped on Edges 1 & 2 (base + top-arm end), Window mode over the natural-frequency band ω ∈ [0, 12.5] i.e. λ ∈ [0, 156.25], dissertation thresholds t_π = 0.57, t_λ = 0.03, t_c = 0.1. ::: {.callout-note} ## Provenance — these numbers are the DIR-113 re-run on the corrected operator An earlier version of this page (DIR-110) reported this campaign on a **fake-soft operator**: `Elasticity2D` assembled the vector mass with `ddot` (matrix double-contraction) instead of `dot`, giving `trace(M) ∝ n²` and crushing every eigenvalue `∝ h²`, so the window [0, 12] on λ held **>960 spurious modes** where the dissertation tracks a handful of surfaces. **DIR-112 found and fixed the root cause** (deviation log [`elasticity-4-4-baseline-deviation-log.md`](elasticity-4-4-baseline-deviation-log.md), rows 9–13): the mass bug plus two convention mismatches — the window filters the natural frequency ω = √λ (so on λ the faithful band is [0, 156.25], not [0, 12]), and the physics is `modal-planestrain` clamping only Edges 1 & 2, not plane stress with the whole boundary clamped. On the corrected + faithful config (`make_l_shape_6d_faithful`) the box midpoint holds **8 modes** — matching a headless MATLAB run of the frozen `ComputationalDomain6Dreal.m` to ~0.7 % on the first eight eigenvalues (P1-vs-P2). **DIR-113 re-ran the entire campaign on the corrected operator**; every number below is faithful and pinned by `tests/test_dir_113_artifacts.py`. The pre-fix `tests/artifacts/dir_110/` set is retained on disk as archived pre-fix data but is no longer rendered here. The scale-invariant sibling results held up exactly as DIR-112 predicted: the DIR-053 ROM ceiling (a FirstK/full-basis measurement) is unchanged, and the DIR-089 "stay-dense" inertia verdict survives on the now much sparser window. ::: This page carries the campaign's *performance and self-consistency* story. The **correctness case** — the five-strand evidence argument (slice views, inertia certificate, holonomy/ν-index consistency, sampled statistical audit, setup invariance) that replaces the 1D/2D branch-figure eyeball certificate — lives on its own page: [**§4.4 6D campaign — the correctness evidence**](elasticity-4-4-evidence.qmd). Everything below is **measured on committed artifacts** (`tests/artifacts/dir_113/`). The pre-fix campaign needed staged, budget-bounded overnight runs; on the corrected operator the whole campaign runs in **seconds** — the fake window's ~380–960 modes/node were the entire cost, and the true window holds only **~8–20 modes/node** (Stage A: min 7, max 21, mean 10.6): | Stage | scale | config | outcome | |:--|:--|:--|:--| | A | n_dof = 1846 (`--target-n-dof 1500`), L≤4 | serial, single region, `--mass-telemetry`, RunCache | L4 completed, `ran_out=True`, **32 nodes, wall 1.6 s, peak RSS 96 MB, 0 evictions** | | B1 | same scale | `--regions --parallel-workers 14 --fixed-v0` | identical trajectory (13/17/22/27/32 nodes), **verdicts byte-identical to A**, wall 2.3 s | | B2 | same n_dof | DIR-055B twin harness (serial vs parallel pool, FirstK) | serial ≡ parallel **exact** (nodes, verdicts, eigenvalue rel-diff 0.0) | | C | n_dof = 6576 (`--target-n-dof 5954`, dissertation scale), L≤4 | `DiskSpillCold`, 4 h wall cap, 16 GB RSS budget | **L4 completed within budget** — no sentinel: 33 nodes, **wall 5.7 s, peak RSS 131 MB, 0 evictions, 0 bytes spilled** | The headline of the re-run is that the §4.4 problem, at its true spectral scale, is **small**: 32 nodes at laptop scale, 33 at dissertation scale, and the store never evicts. The bounded-resource machinery (DiskSpillCold, sentinels, RSS budgets) still stands as engineering, but at the faithful window it is never stressed — the pre-fix campaign's memory drama was the mass bug. --- ## ROM error at 6D — held-out table and soundness curve 32 held-out µ (seeded, uniform in the box) evaluated against a dense in-window FE truth at each point (`rom_error_table_6d.csv`). On the **lowest-20 window modes** — the like-for-like gauge for the DIR-053 ceilings, which were pinned on the lowest eigenpairs — > max eigenvalue rel-err **1.1·10⁻¹**, max subspace angle **1.1·10⁻¹ rad** — > a **breach** of the DIR-053 ceilings (2·10⁻³ / 3·10⁻²) at the deepest fill > levels; window coverage stays 0.91–1.0 and no query misses the window. This breach is a genuine, expected consequence of the faithful window and the central ROM finding of the re-run. In the pre-fix campaign a Window snapshot carried **~380 eigenpairs**, so even an 8-node nearest-neighbour basis spanned a near-complete subspace and the lowest-20 error sat at the accuracy floor (10⁻⁹–10⁻⁵) for *every* fill level. At the true window a snapshot carries only **~10 modes**, so the nearest-8 Galerkin pool is a genuinely thin basis: the RMS lowest-20 error grows with depth (4·10⁻³ at L0 → 1.9·10⁻² at L4) as refinement pushes queries into pockets the sparse pool interpolates less well. Coverage and window-completeness stay high throughout — the breach is **basis-thinness, not a missed window**. Two clarifications keep this honest. First, the DIR-053 ceiling itself is **unaffected**: it is a FirstK/full-basis measurement (`test_rom_6d`, lowest-6 eigenpairs with a complete per-node block), and DIR-112 re-confirmed it unchanged on the corrected operator. The gauge that breaches here is the campaign's *Window-snapshot* ROM, a different and harder object. Second, the **full-window** columns of the table compare mode-by-mode past the coverage limit and misalign once modes are missing; they are labelled coverage-limited, not treated as eigenvalue error. ![](../figures/elasticity-4-4/rom-error-6d.png) **A campaign note on the shipped N-D basis pool.** The surrogate scored here is a **Galerkin window ROM** (it projects `K(µ)`, `M(µ)` onto snapshot eigenvectors and does a small eigensolve), *not* the dissertation's §4.4.5 hat interpolation of stored eigenvalues — a different object with a different memory premise; see the [evidence page callout](elasticity-4-4-evidence.qmd#strand-1-you-can-still-look) and `DVG-SURR-01` for why the ROM (a DIR-110 feasibility carry-forward) stands in here rather than the faithful hat surrogate. The table uses a bounded nearest-8 Galerkin pool with rank-revealing M(µ)-orthonormalisation (`scripts/rom_error_table_6d.py`), not the shipped `evaluate_rom` N-D pool rule (nearest `min(2^d, 32)` nodes). At the faithful window the shipped rule is no longer *infeasible* (a ~10-mode snapshot × 32 nodes is a manageable ~320-column block, unlike the pre-fix ~12k), but it is now *inaccurate* — a thin per-node block means the pool must reach further for rank, and the breach above is the symptom. Redesigning the pool rule for the true §4.4 window (more nodes, or a region-aware interface basis) is follow-up work recorded alongside DIR-111. ## Exact-P2 mesh-faithful convergence (DIR-122) Everything above runs on the rebuild's **structured P1** `l_shape_mesh`. The DIR-117/118/119 parity arc removed that last confound by building the operator on MATLAB's **exact P2 mesh** — `make_l_shape_6d_faithful_mesh` (`Elasticity2D.from_mesh` on the imported `generateMesh(Hmax=0.1)` mesh + P2 vector element, n_dof = 5870; DIR-119 matched its window eigenvalues to MATLAB to 1.8·10⁻¹³) — but only ever ran a **capped 2-level parity probe** on it. DIR-122 runs the adaptive greedy to **natural convergence** on that operator, single process, via the new opt-in flag: ```bash uv run python scripts/reproduce_4_4.py --operator mesh --max-levels 15 --fixed-v0 \ --output-dir tests/artifacts/dir_122 --csv-name elasticity_4_4_mesh.csv ``` (`--operator` defaults to `structured` — the DIR-030/113 path is byte-identical; `mesh` bakes in the faithful config so `--lame-mode/--clamp/--target-n-dof` are ignored-with-a-note.) > The run **converges of its own accord at level 7 with 47 nodes** > (`ran_out=False`, flagged edges 13→…→3→**0**), in **7.3 s / 0.14 GB peak RSS / > 0 evictions**. This answers the campaign's open question: the exact-P2 geography is **not** the structured-P1 one. The faithful DIR-113 run converged at **L4 / 32 nodes**; the sharper P2 discretisation refines **two levels deeper and places 47 nodes** (trajectory 13/17/22/28/34/39/44/47). The grid is denser, not merely a relabel of the P1 grid — pinned in `tests/test_dir_122_mesh_campaign.py`. **ROM accuracy on the converged grid** (`rom_error_table_6d.py --operator mesh`, 32 held-out µ, lowest-20 gauge, fixed-`v0` dense truth): > max eigenvalue rel-err **0.11**, **median 5.8·10⁻³**, max subspace angle > **0.126 rad**, window coverage **0.99** (no query misses the window) — a > **breach** of the DIR-053 ceilings (2·10⁻³ / 3·10⁻²), exactly the > basis-thinness finding above, now measured on the exact-P2 operator. The soundness curve does **not** keep falling with grid fill: it plateaus at **~2.2·10⁻² RMS by L4** and holds flat through L7. Adding nodes past L4 tightens the grid but cannot close the residual — it is the **near-degenerate eigenvector-gauge** ceiling DIR-120 root-caused (the converged grid packs modes densely near the top of the window, where the eigenvectors are defined only up to rotation), not under-resolution. The median stays modest; a handful of near-degenerate query points drive the max. This is measured and pinned, not transplanted from the 1D/2D `1e-4` ceiling. See [affine parameter-assembly](../affine-assembly.md) for why the per-µ solves are cheap enough to make a 47-node adaptive run finish in seconds. ## Faithful full-scale run past MATLAB's L2 wall (DIR-124) DIR-122 above runs the exact-P2 operator with the rebuild's **own native** adaptive path — the 13-point sparse seed + edge-bisection — and converges cheaply at L7/47 nodes. DIR-124 runs the *other* experiment the whole DIR-117/118/119 arc was built for: **the MATLAB computation itself** — the dissertation's **`3^d` tensor seed** (729 points at d=6, `seed_tensor_lattice`), MATLAB's **`mirror_bracket`** omnidirectional spawn, the **ω-scale** verdict, the **p-box** (`p ∈ [0.2,1]×[0.2,0.4]³`, `E = 100·p`), and the frozen §4.4 thresholds (`t_pi=0.57`, `t_lambda=0.03`, `t_c=0.1`, squared Π) — run **past level 2**, the level MATLAB dies at (8376 points, 54 min, then an unbounded-eigenvector OOM). The seed is a first-class flag (`--seed {tensor,sparse}` / `--knots-per-axis`, default the `3^d` tensor); the sparse seed is a declared alternative, never a silent substitution. `mirror_bracket` rejects region decomposition and the parallel pool by construction, so this is **single-process** — the only bound is the store. ```bash uv run python scripts/dir_124_faithful_campaign.py --max-levels 7 \ --max-active-mb 2048 --cold-disk-dir <scratch>/spill --cache-dir <scratch>/cache \ --output-dir tests/artifacts/dir_124 --peak-rss-budget-mb 12000 --rom-queries 32 ``` **The faithful campaign does not converge at a laptop-feasible scale — and that is the result.** The grid grows as a steady power law (nodes ∝ level^1.40, R²=0.998) with the flagged-edge count *rising* at every level, never turning over: | level | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |:--|--:|--:|--:|--:|--:|--:|--:|--:| | nodes | 729 | 1640 | 3001 | 4663 | 6392 | 8421 | 10649 | **12702** | | flagged edges | 0 | 911 | 2134 | 3453 | 4559 | 5602 | 6596 | 7348 | | active tier (MiB) | 356 | 776 | 1437 | 2047 | 2048 | 2048 | 2048 | 2048 | | evictions | 0 | 0 | 0 | 540 | 2400 | 4589 | 6990 | **9155** | | peak RSS (MiB) | 487 | 960 | 1709 | 2431 | 2492 | 2525 | 2576 | **2613** | The L0–L3 prefix (729→1640→3001→4663) **bit-matches** the DIR-119 exact-mesh 2-level parity probe and its own independent growth smoke — the faithful setup is deterministic under `fixed_v0`. > **This is the rebuild's entire reason to exist, measured.** The MATLAB original > OOMs here because it accumulates eigenvectors without bound. The rebuild's > bounded `SnapshotStore` turns the low-rank premise into an *enforced* invariant: > the LRU active tier **clamps at its 2 GB cap from level 4 on**, the cold tier > **spills 2.4 GB across 9155 evictions to disk**, and **peak RSS holds ~2.6 GB > flat while the grid triples** (4663→12702 nodes). The run reaches **level 7 / > 12702 nodes in 25 min, single process, with a clean `LEVEL_7_COMPLETED` and > zero `StoreOverflow`** — five levels past MATLAB's wall, no OS OOM-kill. As a number: the active-byte growth exponent is **0.87** — *sub-linear*, decoupled from the node-count exponent 1.40, because the cap does its job. The cold-tier rank keeps climbing (5.3k→105k stored eigenvectors, ≈11 modes per evicted snapshot), so the eigenvector *family* is **not** staying low-rank as the grid grows at this faithful scale — the premise holds only because the bounded store spills the surplus to disk rather than RAM. Either way, the faithful computation runs to a reported depth in bounded memory; a deeper level is a wall-time / disk-budget choice, not a memory-safety one. **ROM accuracy on the L7 grid** (`window_rom_predict`, 32 held-out µ in the p-box, pool 2, fixed-`v0` dense truth): > max eigenvalue rel-err **9.1·10⁻²**, **median 5.3·10⁻³**; max subspace angle > **0.162 rad**, median **4.1·10⁻²** — the same thin-Window-basis ceiling as the > DIR-113/122 runs (a bounded ~8-mode snapshot is not a near-complete basis in > this dense near-degenerate spectrum), measured on the faithful full-scale grid, > not transplanted. Committed artifacts under `tests/artifacts/dir_124/` (run report + telemetry CSV, ROM error table + summary; the 2.4 GB disk-spill cache is scratch-only, not committed); pinned by `tests/test_dir_124_faithful_campaign.py` — the exact trajectory, the bounded-memory invariants (active cap, disk spill, flat RSS), the growth-fit exponents, and the ROM bands, plus a cheap live sparse-L1 re-run that re-derives the faithful config deterministically. ## Deep continuation — does it converge, and does deeper get better? (DIR-125) DIR-124 above stopped at a **chosen** `--max-levels 7` cap with every real budget ~80 % unused, so its "grows ∝ level^1.40, does not converge" was an *extrapolation* from the pre-saturation ramp, and its ROM number was a single L7 point. DIR-125 removes both limits: the **same driver, same faithful config**, run at `--max-levels 100` with generous laptop-sized budgets (18 GB disk, 180 min wall, 12 GB RSS) so the stop is real — and a **per-level ROM soundness curve** (`scripts/dir_125_soundness.py`) scoring the grid-as-of-level-L for every L against fixed-`v0` dense truth. ```bash uv run python scripts/dir_124_faithful_campaign.py --max-levels 100 \ --max-active-mb 2048 --cold-disk-dir <scratch>/spill --cache-dir <scratch>/cache \ --cold-max-mb 18000 --peak-rss-budget-mb 12000 --max-wall-time-min 180 \ --rom-queries 0 --output-dir tests/artifacts/dir_125 --csv-name convergence_run.csv uv run python scripts/dir_125_soundness.py --cache-dir <scratch>/cache \ --n-queries 32 --pool-size 2 --out-dir tests/artifacts/dir_125 ``` **Result 1 — it converges. The faithful campaign reaches a natural fixed point at L23 / 21867 nodes, bounded, in 55 min — overturning DIR-124's L7 "grows-forever" extrapolation.** The `mirror_bracket` active frontier empties (its termination condition, `mirror_bracket.py`): per-level additions peak at L6 then decay monotonically to near-zero, so the grid *turns over* rather than growing without bound. The stop is **natural** (`ran_out=False`, below the cap) — **not** a wall/disk/RSS budget and **not** the level cap: | level | 6 | 7 | 8 | 10 | 12 | 15 | 18 | 20 | 22 | 23 | |:--|--:|--:|--:|--:|--:|--:|--:|--:|--:|--:| | nodes | 10649 | 12702 | 14922 | 18588 | 20162 | 21212 | 21686 | 21793 | 21853 | **21867** | | added | 2228 | 2053 | 2220 | 1556 | 541 | 257 | 100 | 44 | 27 | **14** | | flagged edges | 6596 | 7348 | 7828 | 8274 | 8417 | 8521 | 8563 | 8573 | 8578 | 8578 | | peak RSS (MiB) | 2582 | 2622 | 2654 | 2715 | 2737 | 2755 | 2792 | 2805 | 2817 | **2818** | The revised growth exponent over the **full** L0–L23 trajectory is **nodes ∝ level^1.08** (R²=0.94), not L7's 1.40 — the earlier number was the ramp, not the asymptote. The L0–L7 prefix bit-matches DIR-124 exactly (same `fixed_v0` faithful setup, only the caps/budgets differ). > **Converges, bounded, at laptop scale — every budget unused.** Peak RSS > **4.1 GB** (budget 12 GB), wall **55 min** (budget 180 min), disk spill > **5.2 GB** (cap 18 GB), active tier clamped flat at its **2 GB** cap the whole > way, **zero `StoreOverflow`, no OS OOM-kill** — 16 levels past DIR-124's L7 and > 21 past MATLAB's L2 OOM wall. The honest nuance: termination is by **frontier / > dyadic-lattice saturation**, not full certification — **8578 edges remain > flagged REFINE** at halt (the near-degenerate eigenvector gauge, DIR-120, keeps > re-flagging the dense upper window), but the `mirror_bracket` spawn can no > longer place a new node to act on them, so the adaptive loop stops. **Result 2 — deeper does *not* get better. ROM accuracy plateaus at the near-degenerate eigenvector-gauge ceiling by L2–L3, then is flat for 20 more levels** (`window_rom_predict`, 32 held-out µ in the p-box, pool 2, lowest-6, fixed-`v0` dense truth): | level L | 0 | 1 | 2 | 3 | 7 | 8 | 15 | 23 | |:--|--:|--:|--:|--:|--:|--:|--:|--:| | nodes | 729 | 1640 | 3001 | 4663 | 12702 | 14922 | 21212 | 21867 | | rel-err median | 1.0e-2 | 6.5e-3 | 5.5e-3 | 5.0e-3 | 5.3e-3 | 4.9e-3 | 4.9e-3 | **4.9e-3** | | rel-err **max** | 1.7e-1 | 9.1e-2 | 9.1e-2 | 9.1e-2 | 9.1e-2 | 9.1e-2 | 9.1e-2 | **9.1e-2** | | angle median (rad) | 4.2e-2 | 3.8e-2 | 2.1e-2 | 1.9e-2 | 2.0e-2 | 2.0e-2 | 2.0e-2 | **2.0e-2** | The entire gain is L0→L2 (median 1.0e-2 → 5.5e-3); from **L8 the curve is bit-identical** (median 4.856e-3, max 9.127e-2, angle 2.034e-2) even as the grid grows 14922→21867 nodes. The classifier's verdict is **`plateau_at_gauge_ceiling`** (L0→L23 buys only 2.06×, and the best level is L8 — far from the deepest, the signature of a shallow-gain-then-flat plateau, not a convergent improvement). The max rel-err is a **fixed ceiling from L1 on** (9.1e-2, one held-out µ whose top window mode is gauge-degenerate — DIR-120), never resolved by grid fill. The mechanism is direct: the deep continuation refines only the near-degenerate flagged regions high in the dense spectrum, which are *not* near generic held-out µ, so the pool-2 basis for a random query stops changing once its neighbourhood is filled (~L8). The L7 point cross-checks DIR-124's independently-computed single point exactly (median 5.3e-3, max 9.1e-2). Committed artifacts under `tests/artifacts/dir_125/` (convergence run report + CSV; per-level soundness curve JSON/CSV/figure + per-query CSV; the 6.5 GB run cache and 5.2 GB spill are scratch-only, not committed); pinned by `tests/test_dir_125_deep_convergence.py` — the natural-not-budget stop, the turn-over (additions decay + flagged plateau), the bounded-memory invariants, the plateau verdict + flat-ceiling tail, the DIR-124 L7 cross-check, and live pure plumbing (query determinism, level truncation, the verdict classifier). MATLAB parity is **out of scope** — there is no MATLAB reference past L2 (the DIR-117 probe's `lev_max=2`); the quality claim is vs dense ground truth only. ## Certification telemetry — the independent inertia certificate The a-posteriori matrix Π and the DIR-088 mass budget certify the *computed* subspaces; `window_count` (dense LDLᵀ Sylvester inertia, DIR-089) counts the *true* in-window eigenvalues from the assembled K(µ), M(µ) alone — an independent certificate that sees modes the subspace side cannot. Over **all 31 tested edges** of the Stage-A run (endpoints + un-solved midpoints, `scripts/inertia_certificate_6d.py`): | count | value | reading | |:--|--:|:--| | `endpoint_mismatch` | **0** | `solve_window` returned the complete in-window set at every node | | `mid_bump` | **0** | no mode present at a midpoint but at neither endpoint — no grid-invisible window entry | | `cross_edge_delta` | 16 | the in-window count changes across half the edges (a ~10-mode window is far quieter than the pre-fix ~380) — all surfaced to the matching layer | | `cert_with_entry` | 12 | count changes on certified edges: surplus bands the Algorithm-6 `t_c` gate certified-and-dropped (Q14 territory, not a blind spot given `endpoint_mismatch = 0`) | The DIR-089 verdict — the node-local window solve is complete and the only genuinely-uncaught category (`mid_bump`) is empty — **holds unchanged on the sparser faithful window**, exactly as DIR-112 predicted. ## Domain decomposition at 6D — Tier 1 (rank savings) The first rendered view of the DIR-036 decomposition on the real 6D problem (Stage B1). The split criterion still fires: **5 splits**, 4 freezes, peak **6 subdomains**, and the Tier-1 headline — > single-store POD rank **35** vs max per-region rank **22** (−13, a > **37 % reduction**), the 6D analogue of the 2D page's 31 → 23. ![](../figures/elasticity-4-4/rank-savings-6d.png) ![](../figures/elasticity-4-4/region-timeline-6d.png) The absolute ranks are small (35, not the pre-fix 1311) because the faithful grid is small; the *fractional* saving (−37 %) survives, so decomposition still buys the same relative relief at the true scale. The lower panel is the 6D measurement of the **interface budget** (fraction of nodes on region boundaries — the Q13 scalability instrument). Here it climbs to **0.31** at L4 — it does **not** stay ≤ 0.1 as the pre-fix 166-node grid did. This is an honest scalability caveat, not a regression: the budget is a *ratio*, and on a sparse 32-node grid a larger fraction of nodes necessarily sit on the 6 subdomain boundaries. The concern the instrument exists to watch — boundary duplication growing faster than the interior — is best judged on a *denser* grid; at the faithful §4.4 scale the interior is simply too small to dilute the boundary. ## Domain decomposition at 6D — Tier 2 (measured parallel speedup) The equivalence gate stands and is now **cleaner**; the wall-clock story inverts: 1. **Strict equivalence gate at campaign n_dof** (DIR-055B harness, FirstK config): serial and 14-worker-pool runs produce **identical nodes and verdicts and eigenvalue max-rel-diff 0.0** at n_dof = 1846. The measured per-region barrier-model speedup is **1.82×** against the equal-cost projection's 1.08×, but the **raw end-to-end wall-clock is 0.76×** — overhead-bound (serial 0.40 s vs parallel 0.52 s). 2. **Raw campaign wall-clock** (Window config, Stage A vs B1): **1.6 s → 2.3 s** — the parallel pool is *slower* than serial on the faithful operator. The reason is direct: at the faithful window a solve returns ~10 modes and costs milliseconds, so process spawn / pickle / barrier overhead dominates the 14 workers. The pre-fix 2.43× speedup was real but depended entirely on the fake window's ~380-mode heavy solves. **At the true §4.4 scale the parallel pool buys nothing in wall-clock** — the honest conclusion, and a reversal of the pre-fix page. One further faithful improvement: the pre-fix page carried a caveat that under the Window config the parallel run flipped **2 of ~124k band verdicts** (REFINE ↔ CERTIFIED, a BLAS reduction-order effect). On the sparse faithful window **that flip is gone** — Stage A, the serial `--regions` twin, and the 14-worker parallel run are all verdict-byte-identical. The sparse window carries almost no near-threshold bands for last-ulp noise to tip. ## Mesh independence Three mesh resolutions, one configuration (`mesh_independence_6d.csv`): | n_dof | nodes per level | certified fraction | wall | peak RSS | |--:|:--|--:|--:|--:| | 846 | 13, 17, 22, 27, 33 | 0.458 | 0.9 s | 88 MB | | 1846 | 13, 17, 22, 27, 32 | 0.421 | 1.6 s | 96 MB | | 6576 | 13, 17, 22, 28, 33 | 0.438 | 5.7 s | 131 MB | The refinement **geography is now mesh-independent**: identical seed and early levels (13/17/22 at L0–L2 everywhere) and a final count of ~33/32/33 across an 8× span in n_dof — not the pre-fix 145 → 166 → 193 that *grew* with the mesh. That growth was itself a symptom of the fake operator: a finer mesh resolved more of the ~380-mode window's spurious content, so more band pairs became testable and more edges flagged. On the corrected operator the parameter-space geography decouples from the spatial mesh, as it should — the certified fraction is flat at ~0.44 rather than falling from 0.47 to 0.23. The level cap (`ran_out=True` at every scale) still binds before the geographies could diverge. ![](../figures/elasticity-4-4/mesh-independence-6d.png) ## Bounded-resource record — dissertation scale in seconds The last full-scale run of the original MATLAB predates this rebuild's memory work and was killed by the OS. The pre-fix rebuild campaign answered that at the fake scale with an overnight 7 h 26 min / 9.2 GB / 5.8 GB-spilled Stage C. On the corrected operator the record is anticlimactic and that is the point: the **full §4.4 problem at n_dof = 6576 through four refinement levels (33 nodes) in 5.7 s on a laptop**, peak RSS 131 MB against the 16 GB budget, with the DIR-035 `DiskSpillCold` tier wired but **never touched** (0 evictions, 0 bytes spilled) — zero overflow, zero sentinel, `LEVEL_4_COMPLETED`. The wall-time and RSS budgets and clean sentinel stops all still stand as engineering, but the faithful §4.4 problem never approaches them. Neither wall time nor memory is a binding constraint at the true 6D scale; the pre-fix campaign's entire resource narrative was an artifact of the mass bug (`MEMORY-OOM-ANALYSIS`).