[Sudarsh3301]

Summary

Bounded investigation of the bit-side Problem X ordering question from #2278: is Boot(PG(2,16)−M) = 772 for the Hermitian Singer code? It builds on philippeVV's limits-1 (expXL) at ed21d624 and limits-2 at ae0cd3ed. No new bound on Boot and no κ claim. Boot stays in [772, 784].

New package research/pg16m-bootstrap-orders/:

• Independent order checker (stdlib Dinic in verify.py; scipy Dinic in search/indep_check.py). It agrees with limits-1/bootstrap/xl/check_order.py on positive and failure controls. It also re-checks the split identity cost = ½|∂S| + Σ_S(j−8)⁺ + Σ_{V∖S}(ℓ−8)⁺ at every prefix. Robustness note: check_order.py checks regularity and symmetry but not girth, so it would accept a degree-preserving rewired graph containing a 4-cycle; this checker also rejects that. No result is affected: an order's cost does not depend on girth, and the graph itself is pinned by the Singer rebuild.
• Graph identity. graph16_59.txt is the Singer incidence graph of PG(2,16) (difference set D′ ∪ {59}) minus the invariant matching point i — line i+59. ord784 is the alternating cyclic order with multiplier 118 and offset 81. Both rebuild byte for byte and are pinned by SHA256.
• Exact merge-family optimum. An exact DP over every interleaving of a cyclic point sequence and a cyclic line sequence, for all 144 × 144 multiplier pairs and 273 offsets (5,660,928 families), has minimum 784. Only the native multiplier 118 and its reverse attain it.
• Bounds. The spectral bound 751 is reproduced. o3_52 (max prefix cut 1544) shows that cut-width-type bounds (bisection, spectral, cut-width) cannot certify Boot > 772. No certified bound above 751.
• Bounded search (scoped negative, not a bound). Methods: partition annealing over ½|∂S| + peel(G[S]) + peel(G[V∖S]) (each value is the exact cost of a built order), beam search over prefix sets with exact rollout values, and exact-cost insertion annealing. None found an order below 784. Best non-cyclic order: 785 (flow-infeasible).

Evidence

• Registered replay (stdlib): python3.12 scripts/research_package.py --package research/pg16m-bootstrap-orders. It prints PASS_PG16M_BOOT_ORDERS, about 20 s. Metadata check: PASS_PACKAGE_METADATA_AND_PINS.
• The replay rebuilds the graph and ord784. It re-evaluates the 784 order, o3_52 and the best orders found here, replays the merge DP in pure Python on 47 recorded families (which agree with the numba sweep), and runs 4 corrupted-input rejections plus a flow-infeasible two-swap control.
• The full merge sweep and the searches are in search/ (numpy, scipy, numba). They are not part of the registered replay. The sweep takes about 27 min on 8 cores.
• Imported, not re-derived: dim C = 772 and the bootstrap theorem/corollary (limits-1, author-checked, not reviewed).
• Overlap check at the start of the session (gh, desktop): FINDING / HANDOFF: the complex budget is a* ≈ ν/Λ — a lever map for step changes #2278 (17 comments and replies), Handoffs and requests for independent checks #1166, Larger κ: proposals that change the current cost structure #1167 and FINDING: κ and undirected coding gaps (oblivious padded multiplier, AFKL with a gap, BH/BGS exponent) #2838 read through all pages; open, closed and merged PRs searched for Boot, PG(2,16), bootstrap, schedule, Problem X, Hermitian, cutwidth, ord784 and expXL; repository-wide Discussion search. Only FINDING / HANDOFF: the complex budget is a* ≈ ν/Λ — a lever map for step changes #2278 (philippeVV's own thread) touches this target, and nobody holds a claim on it. Audit scale-dependent seed averaging in the full recurrence #2881 (Ross Richardson) excludes Problem X explicitly.

Merge Danger

Low. The change adds one self-contained directory under research/ and touches no existing files, selected record, frontier or CI. The package registers one stdlib replay (300 s timeout). Scope note: dim C/E = 0.1767 < 0.193, so q = 16 is a test instance for the Hermitian route. It does not affect κ.

AI assistance: prepared by Sudarsh Chaturvedi with assistance from Claude (Anthropic), Ran separate Claude review agent tried to break the claims before submission.