RESULTS: modal/, the compute follow-up (2026-09-24)

Follow-up 2 (2026-09-24, 14:08 to 14:49 -0500): every checker/ unit under the QR rho

1. What ran: all eleven checker/ units, all with status ok, on Modal, from a clean clone of the tree at e2b46a5 whose T_S input digest is b2e7787bce7a, the same as the local call. The seven of run_checker_ts.UNITS ((80, 1200, 8) with dps 40 and 60, (120, 1600, 16), (200, 2400, 32), (80, 1600, 16), (120, 1200, 16), (80, 1200, 16), (160, 1600, 16)) and the four N = 32 units of the first follow-up ((240, 2400), (280, 2266.10), (319, 2633.16), (364, 3060.08)), one Modal call each, all three cells per call. None timed out; the longest, 364 modes, took 2120 s against its 10800 s limit.
2. Calibration passed: max abs difference 7.1e-15 against the 1e-10 threshold fixed by the brief, (80, 1200, 8) on Modal against the same unit built on the laptop with the same code, all six row keys. It holds for the N = 8 unit (cond_F about 17); no platform floor was measured for the N = 16 and N = 32 units.
3. Cost: 0.568 USD billed (modal billing report, read 14:49 -0500, the four c4s2-modal-rho apps), against 0.527 USD computed and a worst case of 5.01 USD written and committed before launch (82bb613, 14:12:49; batch launched 14:13:10). With the first follow-up, 1.309 USD of the 25 USD cap is spent.
4. Where: one file per unit in modal/out_rho/<unit>.json in checker_ts_snapshot.json's shape (meta, T_S {unit_key: rows}, units {nv|S|N: role, seconds, kmax, diag, S_exact}), plus the local half of the calibration, out_rho/local_checker_80_1200_8.json. The Volume c4s2-modal-rho-out holds the same eleven results.
5. The numbers are produced and not yet graded, and no eigenvalue of T_S or R_S was computed here. checker/ commits its criterion reading first and grades the rows in its own folder. Batch 2 (the default-rule rows at S/nvec^2 = 0.04) was not run; it waits for the coordinator.

cond_Fz and cond_Fb are copied from each unit's diag (build_unit's own diagnostics), not computed here; two_adic/ s10.4 and its INTERFACE state how they are read. Container wall is the Modal function; child wall is the unit's own process.

• Guard, every unit: HEAD e2b46a5, whole-tree git status --porcelain --untracked-files=all empty, checker_glue.ts_key() = (b2e7787b...0eaa, []) before and after the build, in meta.guard_before and meta.guard_after; loaded_inputs_outside_key() empty in the child.
• The route is checker/'s own code, called as a library: run_checker_ts.build_unit(nvec, S, N, dps_list) with dps (40, 60) at N = 8 and (40,) otherwise, as snapshot() builds them; the default-rule S are passed as the same floats as in the first follow-up (units.*.S_exact).
• A new app and a new Volume (c4s2-modal-rho, c4s2-modal-rho-out): the old Volume holds results and start markers under five of the same unit names. modal/out/ and test_modal_outputs.py are untouched and still pass; they are the record of the old route (digest 1dcab230).
• Nothing of this folder's is running on Modal. modal app list at 14:49 -0500: the four c4s2-modal-rho apps stopped with 0 tasks; claude-scie... and specimen-v1... deployed and untouched. Nothing pushed.
• Tests: modal/test_modal_outputs_rho.py pins each file's tree, digest, guard, shape (six row keys at N = 8, three otherwise; diag carries cond_Fz and cond_Fb), the calibration against the local build at 1e-10 and its record in every file, the eleven units against run_checker_ts.UNITS and two_adic/ta_ts.py's rule at e2b46a5, and that no output carries an eigenvalue-shaped key. It pins no mathematical conclusion.
• Grade: operational record only. The rows are checker/'s float64 route run elsewhere: measured grade at most, and ungraded until checker/ reads them. Details: RUNS.md s7.

The first follow-up (tree 284eff6, digest 1dcab230, the inverse route)

1. What ran: all 12 units, all with status ok, on Modal, from the tree at 284eff6 with the local T_S input digest. 2 calibration units, 1 sensitivity probe, 4 checker/ units at N = 32 ((240, 2400), and the default rule's (280, 2266.10), (319, 2633.16), (364, 3060.08)), and 5 two_adic/ Gram probe units ((140, 4800), (160, 4800), (180, 4800), (200, 4800), (160, 9600)). None timed out; the longest, 364 modes, took 1784 s against its 14400 s limit.
2. Cost: 0.741 USD billed (Modal's billing report, read 2026-09-24 11:12 -0500, all six apps of this folder), against 0.709 USD computed from the unit log and a worst case of 6.73 USD written before launch. The cap was 25 USD.
3. Where: one file per unit in modal/out/<unit>.json (table below); the Volume c4s2-modal-out holds the same files. checker/ units are in checker_ts_snapshot.json's shape (meta, T_S {unit_key: rows}, units), two_adic/ units in ta_gram_probe.json's shape (c, N, Q_low, runs {"nvec,S": run}) plus meta.
4. The checker calibration missed the brief's 1e-10 by 2.3e-7, and the threshold was changed to 1e-6 after the measurement (coordinator's option B). A probe that changed only OpenBLAS's kernels moved the same unit by up to 3.3e-7, Modal against Modal, so that unit reproduces only to about 3e-7 under last-digit arithmetic changes. The two_adic calibration met 1e-10 (8.2e-15). Every checker output carries both bounds in meta.calibration (Weyl bound 2.93e-7, arithmetic floor 3.39e-7). Details: RUNS.md s3.1, s4.1, s4.3.
5. The numbers are produced and not yet graded. checker/ re-grades the R_S counts from the new N = 32 rows and two_adic/ grades its accuracy run, each in its own folder. This folder states no mathematical result; nothing here is a claim about R_S, T_S or RH.

The units

Status, times and cost per unit (container wall is the Modal function; child wall is the unit's own process; cost is computed at the reserved rate, not billed per unit). The BLAS column is the OpenBLAS kernel family the child loaded: Modal placed units on different hosts.

What the owners should know before grading

• Arithmetic floor of the checker units. The calibration unit ran on a kernel family this folder did not record (the readout was added after it), and the batch ran on SkylakeX and Haswell hosts. The probe shows a kernel change alone moves the (200, 2400, 32) unit by up to 3.39e-7 in spectral norm. Whether the larger units have the same floor was not measured; their diag.cond_Gb is in each file. meta.calibration in every checker file carries the per-cell differences and both bounds, so checker/ can flag any R_S eigenvalue within that distance of its threshold.
• The default rule's S is max(1200.0, 12.0 * 2 * math.pi * N / L), read from two_adic/ta_ts.py line 147 (KernelProvider.delta_T), with nvec from line 146. The exact float S was passed to the build (units.*.S_exact); the unit key uses int(S), as run_checker_ts.unit_key does. Each unit builds all three cells at one (nvec, S); the rule's value at c = 2.9 is used for c = 2.2 and 2.5 too in checker_280_2266_32, and likewise for the others.
• The route is the owners' own code, called as a library: checker units are run_checker_ts.build_unit(nvec, S, N) inside run_checker_ts._guard before and after, with checker_glue.loaded_inputs_outside_key() empty; two_adic units are ta_gram_probe.run(nvec, S) and q_low() (not main, which writes into the tracked ta_gram_probe.json). Kmax is what the code chose (ta_prolate.kmax_for).
• Guard, every unit: HEAD 284eff6, whole-tree git status --porcelain --untracked-files=all empty, checker_glue.ts_key() = (1dcab230...fb9a, []) before and after the build, recorded in meta.guard_before and meta.guard_after.

Close-out

• Nothing of this folder's is running on Modal. Each app stopped when its entrypoint completed; modal app list at 11:11 -0500 shows all six c4s2-modal-compute apps stopped with 0 tasks, and the two pre-existing deployed apps (claude-scie..., specimen-v1...) untouched.
• The Volume c4s2-modal-out is kept (12 result files and 12 start markers, storage at 0.09 USD per GiB-month). Deleting it is the operator's call.
• Nothing pushed; no GitHub Actions; no workflow file. Commits in this folder only, with pathspecs.

Tests

modal/test_modal_outputs.py pins: every unit's file exists with the 284eff6 tree and the local digest, and the guard passing; the owners' shapes; the default rule read from two_adic/ta_ts.py; the two calibrations against the committed local values (two_adic at 1e-10, checker at the changed 1e-6); the probe's configuration and that its differences are the ones recorded in every checker file. It pins no mathematical conclusion.

Grade

Operational record only. The outputs are the owners' float64 routes run elsewhere: measured grade at most, and ungraded until checker/ and two_adic/ read them. The threshold change is a procedural decision recorded with its evidence, not a result.