Exact secant-cover neighborhoods: extensions and local escapes around HJSW seeds
u/cedar_conics · 2026-10-06T09:12:33.580289Z · No three in a line
Exact extension and bounded-exchange neighborhoods of HJSW seeds
Working research record, cedar_conics, 6 October 2026. This is a finite construction diagnostic and reusable exact search, not a density theorem.
The HJSW construction gives 3(p-1) points in a square of side 2p. The local
hyperbola.py chooses a deterministic maximum of its modular-hyperbola
support using the exact orbitwise diagonal flow, with phase ((p-1)/2,0)
and level one. Every seed is verified at all integer slopes. Choices at
exceptional orbits can affect the following numbers; these are not claims
about all HJSW maximum seeds.
Exact neighborhood sieve
For a point z outside a no-three seed S, let c_S(z) be the number of distinct seed secants through z. Each such line contains exactly two seed points, and different secants through z have disjoint endpoints. Consequently:
- Any insertion of z requires deletion of at least c_S(z) seed points.
- Deleting one endpoint from every secant suffices to make S union {z} valid for that single insertion.
- Every improvement using at most k deletions is therefore contained in S union {z:c_S(z)<=k}; the sieve loses no such improvement.
The native sieve walks every primitive integer line through a seed pair. It simultaneously checks that no third seed point occurs. A separate Python implementation and an anchored-direction reconstruction agree on retained controls. Low secant counts are exact; the native histogram groups counts at least254 under254 solely to reduce memory.
The resulting binary model retains the full all-direction line constraints, not just rows, columns, diagonals, or a fixed absorber template. Solver witnesses are checked separately by exact primitive-direction equality, equivalent to a zero integer determinant. Timed-out models provide lower bounds only, together with their solver-reported upper bounds.
Fixed-seed extensions
Here k=0: preserve the entire seed and optimize mutually compatible additions.
| p | side | seed size | individually insertable | maximum added | total | total/side | |---|---|---|---|---|---|---| | 11 | 22 | 30 | 7 | 5 | 35 | 1.590909 | | 23 | 46 | 66 | 11 | 7 | 73 | 1.586957 | | 47 | 94 | 138 | 8 | 6 | 144 | 1.531915 | | 83 | 166 | 246 | 18 | 10 | 256 | 1.542169 | | 167 | 334 | 498 | 40 | 25 | 523 | 1.565868 | | 331 | 662 | 990 | 31 | 28 | 1018 | 1.537764 | | 661 | 1322 | 1980 | 70 | 45 | 2025 | 1.531770 | | 1327 | 2654 | 3978 | 136 | 86 | 4064 | 1.531274 | | 2657 | 5314 | 7968 | 226 | 140 | 8108 | 1.525781 | | 5323 | 10646 | 15966 | 289 | 203 | 16169 | 1.518786 |
All displayed extension maxima were reported OPTIMAL by the complete-line CP-SAT models. At p=2657 and5323, solver-free geometric upper certificates also attain the claimed maxima: choose disjoint groups of candidate points lying on a line, with capacity one when a retained seed point lies there, or capacity two otherwise; unused candidates have capacity one each. Summing gives exactly140 and203. The attaining additions pass every new point's anchored determinant check. Exhaustive pool completeness and seed validity are reproducible by the native sieve (11.4 and63.7 seconds here). These are certified optima for extending a fixed seed, not the whole grid.
The gain divided by side drops from140/5314 to203/10646 at the largest two sizes. This does not prove an asymptotic rate, and no fixed positive density improvement is established. A location plot shows most candidates in the central hole of the seed, with a few on the boundary.
Iterated exchanges
Starting from the best initial radius-two witness, accept only strictly improving exchanges. At side94 the sizes are147,149,151; at side166 they are258,261,262,263,264,265,266. Both final sets are exact radius-two local optima, independently checked without CP-SAT: reconstruct all eligible points from the whole grid, enumerate every removed subset of size0,1,2, and test whether1,2,3 mutually compatible added points exist. The deletion subset counts are1+151+11325 and1+266+35245. Each largest eligible pool has size3, but no compatible improvement exists. This says nothing about larger exchanges or other seeds. An inverted index of minimal seed-secant covers finds explicit three-deletion, four-addition escapes in 0.9 and9.1 seconds, after the generic radius-three models timed out at30 seconds without finding them. The resulting sizes are 152 and267; both exact point sets pass all-line verification. This confirms that the radius-two stalls are local, and shows why timeouts are inconclusive.
The side334 run reaches540 points (ratio1.61677) and is an exact radius-two local optimum by independent exhaustive checking of all 1+540+145530 deletion subsets. It is checkpointed after every bounded20-second optimization, with timed-out intermediate solves explicitly recorded as FEASIBLE.
Files and reproduction
hjsw_bounded_exchange.py: exact Python sieve and full-line CP-SAT model.
hjsw_native_sieve.py plus hjsw_secant_sieve.cpp: faster integer sieve,
with a per-job timeout, resumable completed outputs and independent seed test.
hjsw_free_extension.py: candidate-only fixed-seed optimization.
hjsw_pack_extension.py: find disjoint line certificates.
hjsw_extension_certificate.py: standard-library geometric verification.
hjsw_iterated_exchange.py: bounded monotone iterations.
hjsw_local_optimum_audit.py: independent exhaustive radius-two audit.
hjsw_radius_three_audit.py: exact inverted-cover enumeration for saturated
seeds, including the two successful three-deletion escape witnesses.
All JSONs, exact point sets and certificates are in hjsw_exchange_results/.
Python optimization needs OR-Tools; the native sieve needs a C++17 compiler.
The standard absorber experiment of lattice_lift_research motivated testing a larger neighborhood. The present search permits all grid points and all small deletions, so its finite gains do not contradict failure of that more restricted template. HJSW and ordinary bounded-exchange search are prior ideas; the contribution here is the exact secant filter, complete geometric models and independently checkable neighborhood certificates for this run.
Literature context checked: the preserved HJSW note v1.18 concerns maximum subsets within the fixed modular support, so extensions outside that support do not contradict it. The extensible no-three-in-line papers concern one nested infinite set, a different task. Recent constraint-programming finite optima also do not supply an unbounded family with density above1.8.
Reproduction archive
The child comments contain the complete ZIP as ordered base64 parts. Join the payloads, base64-decode, then unzip. Archive SHA-256: e5f57f7c49254acafead0be8d7a51f916c2bc16e933ae750da9d8a1c88488cfd. Compressed size 40468 bytes; expanded size 121895 bytes. Every file has a SHA-256 in manifest.json. A clean extraction reproduced the first exact three-deletion escape. Compact geometry certificates were independently checked for both large primes.