[nconsigny]

The table sumcheck needs each table's sum sigma_t against eq(zeta[..tau_t]) before its first round. Until now the bus computed it in a pass of its own ("Bus tables": eq_sums over every column of every table, plus the forms' column products), which reads every committed column once more than the sumcheck does (about 0.56 GB at aggregate --sphincs 500). Since eq factors, sigma_t = (1 + zeta_m) p(0) + zeta_m p(1) at m = tau_t - 1, where p(0), p(1) are the endpoints of the table's join-round message, and that message reads no challenge. So the sumcheck now takes the join message with both endpoints in Table::new, the join round sends the stored one, and the batch starts from the bus target the verifier derives. eq_sums, tables_and_prods_at, BusForm::sum_at and BusProof.sigmas are removed (net -60 lines).

Measured on base e5d75f8, Ryzen 9 9955HX3D (Zen 5), 16 threads (taskset -c 0-15), --repeat 3 --cooldown 1, medians of paired ABBA differences (branch - base) with [min, max], 6 pairs on rate-1/4 workloads, 3 elsewhere. S500/S480 = aggregate --sphincs 500/480, X1400 = aggregate --xmss 1400.

	rate 1/4, S500	rate 1/4, S480
Bus tables + Prove constraints	-9.4 ms [-20.8, -6.4]	-11.9 ms [-29.6, -8.8]
Prove	-0.8 ms [-37.3, +11.3] (-0.1%)	-23.7 ms [-34.0, -8.0], 6/6 (-2.0%)

• Rate 1/2 (3 pairs): Prove -20.0 ms at S500, -12.7 ms at S480.
• X1400, rate 1/4: no slowdown (-10.5 ms by median pass).
• Throughput at S500, rate 1/4: 391.4 -> 396.4 signatures/s (+1.3%).

Caveats:

• The S500 Prove difference is within noise. The median of the absolute Prove times moved by -16.0 ms, but the paired median is -0.8 ms.
• Spans come from one traced pass and vary by about +-10 ms run to run. Prove varies by about +-30 ms between passes.
• One machine only. The box was not fully idle: about one core of foreign niced load.

The proof is byte for byte the same, and the verifier is unchanged:

• The join message is the same field elements as before.
• The sums equal the old ones, and the starting claim equals the old one on any proof that verifies.
• No transcript write was added, removed or reordered.
• Checked: leaf proofs of recursion --n 1 --sphincs-per-leaf 16 --leaf-log-inv-rate 2 have the base's sha256 (199,521 B). A reviewer checked three more leaf shapes. All 458 proofs of the benchmark runs verified.

Tests:

• New, join_message_gives_the_table_sum: checks the derived sigma against sum_z eq(zeta[..tau], z) P(z), for tau in {0, 1, 3, 4, 5, 12}.
• round_messages_match_naive_evaluation now checks both endpoints of the join message.
• Two debug assertions: the tables' sums against the target (this replaces the per-side assertion that was removed), and a waiting table sending its stored message only on its join round. The leanvm_core tests pass with release debug assertions on.
• Every test that proves and verifies covers the change. A sumcheck's honest messages are unique, so a different message would fail verification. As a check, sending the wrong endpoint of the stored message fails all three proving tests in leanvm_core/tests/verifiers.

Memory: no change in peak RSS. The join messages of shorter tables need their own eq tables, one per distinct height ((2^17 + 2^16) x 24 B, about 4.7 MB at S500). They are built and dropped before the first round.