[lee-phys]

The non-zero-sum demand (e,h,e+h+z) over F2[e,h,z]/(e,h,z)^2 was explicitly left open in #5259. This five-file package determines its exact receiving-row tariff for every six strictly positive prices, with matching constructions and a proof covering arbitrary finite interleavings.

For distinct i,j,k, the optimum is the minimum of the ten terms

D      = mu_0 + mu_1 + mu_2
T_i    = 2mu_i + 2c_j + 2c_k                       (3 terms)
R_ij;k = 2mu_i + mu_j + 2c_k                      (6 terms)

All ten terms have strict positive-price witnesses. The proof extends to any commutative unital local ring whose three demand classes in m/m^2 are linearly independent over R/m. A two-insertion solution must use two unit coefficients; if they enter different rows, incoming dirt restoration forces a bill strictly above min(T_i,T_j). This is a complete price envelope, not a complete componentwise Pareto-frontier classification.

Only research/r16-radical-triple-fanout/ is added. The contribution head is 4e08457f4d0b69c00a6ca62eca81c8d0a55f3e60, directly based on canonical main a5bff3bf133e482541c09eba4e773884bdb2b1b7. Entry and completion checks found no new same-scope result or active claim in the documented coverage. The original card and the donor-assigned scope change are retained in TASK.json. No new kappa or selected-frontier change.

Validation on CPython 3.12.13/macOS x86_64:

• All 22,152 applicable triples across rings of orders 16, 81 and 64; 221,520 constructions; 1,772,160 formal forward/inverse column evaluations. The order-64 fixture has m^2 != 0.
• All 65,536 scalar dirty/source inputs for each requested-ring construction, in both directions: 1,310,720 evaluations. Ten strict exposures and eleven hypothesis/sign/dirty-input controls pass.
• Four separate full-state bidirectional Dijkstra queries allow all 135 nonzero-coefficient gates. Optima 3, 6, 6 and 23/12 agree; 8,465 entries are settled. Exploratory searches that were interrupted or hit a guard are disclosed and not counted as certificates.
• Warning-strict isolated normal and optimized registered replay, metadata/source pins, staged whitespace, three optimized semantic mutations and exact committed-byte checks pass.
• Official five-file export integrity passes. Its optimized replay with a read-only filesystem tree and no Git on PATH also passes with all payload bytes unchanged.

python3.12 -B scripts/research_package.py --package research/r16-radical-triple-fanout
python3.12 -B scripts/check_research_commit.py --package research/r16-radical-triple-fanout

Finite checks corroborate the written theorem; they are not independent human or proof-kernel review. Hosted CI is pending at publication. The optional full-history preflight is not claimed in this shallow checkout; every source pin resolves at the direct parent. Directed-edge prices, dependent demand classes, nonlocal rings, helper rows, physical coefficient/source/routing/precision/stock costs and supplier admission remain outside scope. No selected-operator, Lean, Docker or historical suite was run.

Prepared by lee-phys with substantial OpenAI Codex assistance; no subagents. Retain #5259/#5514 interpreter/search credit, Ali Romman #5266/#5310 source-feasibility work, lee-phys #5139/#5163, Nick Petalas #5208/#4916, Krzysztof Wojt #5045, Ross Richardson #3063/#3121 and the retained lineage. Apache-2.0.

[lee-phys]

Could a maintainer please rerun workflow run 40926840924 for the unchanged PR head 4e08457f4d0b69c00a6ca62eca81c8d0a55f3e60? My rerun request was rejected because the current account lacks the required repository permissions.

The cancellation was caused by the shared plan job exceeding its configured 10-minute limit, before package selection or the mathematical verifier ran:

• Full-history fetch and checkout took about 6m59s.
• The onboarding helper suite then ran 707 tests in 190.197s, with the log summary OK (skipped=24).
• GitHub reported “The job has exceeded the maximum execution time of 10m0s”. The package-selection step was skipped, and the dependent replay job consequently did not run.

Evidence: plan job and logs, workflow configuration at the PR head.

Local validation has passed on CPython 3.12.13/macOS x86_64: warning-strict isolated normal and optimized registered replay, metadata/source pins, committed-byte checks, three rejecting semantic mutations, and official export integrity plus read-only Git-free optimized replay. The mathematical checks cover all 22,152 applicable demand triples in three finite rings, 221,520 constructions, all scalar R16 inputs for the ten requested-demand words, and four unrestricted full-state shortest-path queries.

Reproduction:

python3.12 -B scripts/research_package.py --package research/r16-radical-triple-fanout
python3.12 -B scripts/check_research_commit.py --package research/r16-radical-triple-fanout

These local results do not replace a successful hosted replay or independent review of the written proof. If a rerun also exceeds the limit, the shared planning job's time budget or checkout cost may need attention. Thank you.