No-Three-In-Line: Audited C4 Structure, a Verified n=74 Solution, and the All-Even Question

Research status: 2026-07-26. This is an AI-assisted computational research notebook. It contains elementary proofs, exact finite encodings, machine-checked local certificates, empirical laws, conjectures, and historical failed approaches. Those categories are kept separate below.

Major update: Thomas Prellberg found a quarter-turn-symmetric solution for n=74 on 2026-07-20. It was added to Achim Flammenkamp's database on 2026-07-21 and independently verified in this repository on 2026-07-22. Consequently, the former headline “m=37 / n=74 is open” is obsolete. The subsequent known-answer audit corrected the old search baseline, proved new local blocker-graph lemmas, and produced an m=38 near-miss with exactly five C4 defect orbits. No n=76 solution has been found here.

The main theoretical question studied here is now: does every even n admit 2n grid points with no three collinear? The data through n=74 support this possibility, but the repository does not prove it.

Interactive overview · verified n=74 certificate · latest joint audit

1. Problem and current external record

Let

[ D(n)=\max{|S|:S\subseteq{0,\ldots,n-1}^2, \text{ and no three points of }S\text{ are collinear}}. ]

Every row and every column contains at most two selected points, so D(n) <= 2n. The classical no-three-in-line question asks when equality is attained and, asymptotically, how large D(n) can be.

The live record page maintained by Achim Flammenkamp currently reports:

• D(n)=2n is known for every n <= 70;
• solutions are also known for n=72 and n=74;
• the n=72 and n=74 record solutions have quarter-turn (rot4, or C4) symmetry;
• among n <= 74, the currently unresolved values are the odd cases n=71 and n=73.

See Flammenkamp's live database and record log. Because that page is updated as new configurations arrive, it should be checked before making any “largest known n” claim.

The universal statement D(n)=2n for all even n remains open. It is much stronger than the now-solved finite target n=74.

2. Evidence and proof-status policy

The following labels are used throughout this README.

label	meaning
PROVED	a complete mathematical proof is present and has survived the current audit
EXACT FINITE	a finite instance was exhaustively enumerated or certified by an exact solver; the claim is limited to that instance
VERIFIED DATA	an explicit configuration or dataset passed an independent checker
EMPIRICAL	repeatable numerical evidence, not a theorem
PROVISIONAL	a potentially useful framework with a proof gap or an untested premise
WITHDRAWN	a claim known to be false, generated by a faulty model, or stated beyond its certificate

A filename containing words such as theorem, formal, complete, or certificate is not itself evidence of proof status. The scope written in this README takes precedence over status language in older exploratory notes.

3. The verified n=74 C4 solution

The database record contains 148 points. An independent verification parsed the 74 rows, formed the canonical integer line through every point pair, and obtained:

check	result
selected points	148
points in every row	2
points in every column	2
invariant under (x,y) -> (y,73-x)	yes
point pairs	10,878
distinct canonical pair-lines	10,878
line generated by two different pairs	0

If three selected points were collinear, their common line would be generated by three pairs. Hence the last two rows certify that this is a valid NTIL set. The full coordinate list and Flammenkamp encoding are stored in n74_rot4_prellberg_verified_2026-07-22.md.

What the solution says about this repository's old search

The northwest 37 x 37 fundamental domain gives a 2-regular pseudograph with cycle lengths

31 + 5 + 1.

The one-cycle is a loop. Thus neither “the factor must be Hamiltonian” nor “the factor must be loop-free” is a valid necessary condition.

The true oriented fundamental-cell set and the repository's score-60 incumbent share only 2 of 37 cells, namely (19,1) and (22,14). The overlap is still 2 of 37 after edge orientation is forgotten, giving selected-cell deletion distance 35. This is the clearest explanation for the failure of increasingly expensive local searches: the correct solution lay in a different basin, not just beyond one more switch.

The stronger archived pre-solution baseline was actually six independently recounted V=20 states, each with 20 bad triples or five C4 defect orbits. Every one of them contains a loop, so the later loop-free V40 search space excluded both the best historical near-misses and the true solution. Compared with this one known true solution, a V20 state shares only 0--3 oriented cells and 2--4 unoriented factor edges; the unoriented replacement distance is 33--35. This does not rule out a different true solution closer to V20, but it does rule out treating these six states as small perturbations of the known answer.

Moreover, every projective root direction involved in a V20 conflict also appears in the true solution. The distinction is not “good slope versus bad slope”: within each reused direction fibre, the true solution keeps the root supports disjoint. This matching condition is the useful exact language.

4. Audited structural results

4.1 Saturation and the two-permutation decomposition — PROVED

Any NTIL set of size 2n has exactly two points in every row and exactly two in every column. Regard rows and columns as the two vertex classes of a bipartite graph and selected cells as edges. This graph is 2-regular, so every component is an even cycle. Alternating the two edge colours around each component decomposes the configuration into two perfect matchings, hence into the graphs of two permutations.

The alternating-cycle argument is essential. Row/column counts alone do not justify simply naming two arbitrary row choices as two permutations; that was a gap in an older draft of the framework.

4.2 Even-order C4 orbit model — PROVED

For n=2m, a non-central cell has a four-point orbit under quarter-turn rotation. A C4-symmetric extremal set therefore consists of exactly m such orbits. The m selected cells in the northwest fundamental square can be read as directed edges on m vertices. Row/column saturation becomes

[ \deg^+(v)+\deg^-(v)=2. ]

After directions are forgotten, the object is a 2-regular pseudograph. The correct model must allow:

• loops (u,u), which contribute two incidences at one vertex;
• both (u,v) and (v,u), which form a legal two-cycle of parallel edges;
• longer cycle components with arbitrary orientation words.

The old constraint x_uv + x_vu <= 1 was incorrect. The correction and the independent geometry check of all 666 antiparallel pairs for m=37 are in ROT4_MULTIGRAPH_MODEL_CORRECTION_2026-07-19.md.

For odd n, a pure C4-invariant set of size 2n is impossible by orbit cardinality: all non-central orbits have size four and the optional centre has size one, while 2n is congruent to two modulo four.

4.3 Orbit centre and exact collinearity encoding — PROVED / EXACT

The centre of a non-degenerate C4 orbit is the circumcentre of any three of its points. This is a useful normalization fact, but it is not by itself a global rigidity theorem.

For a fixed m, all board lines containing at least three grid points can be enumerated exactly. Each selected fundamental cell contributes zero, one, or two points to a line, and every line receives a capacity-at-most-two constraint. Together with the degree-two pseudograph constraints, this is an exact 0-1 model of the C4 problem. Equivalent conflict-hypergraph encodings are also exact.

The so-called R8-G weighted-line formulation is best understood as this exact encoding, not as a separate theorem that forces existence or non-existence.

4.4 Orientation-independent bad triples — PROVED

Three fundamental cells are unavoidably bad under every orientation if their coordinates have one of two simple patterns:

• equal a+b;
• equal |b-a|.

These Type-1/Type-2 obstructions are proved in allbad_triple_theorem.md. They are useful clauses and preprocessing rules. They do not characterize every conflict.

5. Audited repository results beyond the basic model

result	status	reliable conclusion	research value
No single conic	PROVED	for even n >= 6, an extremal 2n NTIL set cannot lie wholly on a nonzero real conic; n=4 is exceptional	rules out a broad algebraic ansatz
Covariance spread	PROVED	every extremal set satisfies `	Corr(X,Y)
Determinant-product barrier	PROVED	the forced small-prime valuation lower bound supplies only about half the main term needed by the naive determinant-product upper bound	tells us to stop pursuing that unrefined global product argument
Deletion-shadow Hall theorem	PROVED	after fixing retained points, row/column completion through allowed cells is exactly a bipartite b-matching/max-flow/Hall problem	one of the strongest interfaces between geometry and combinatorics in the repository
Corrected C4 deficit f-factor	PROVED / EXACT FINITE	for a fixed deletion mask, zero-new-conflict completion is an exact loop-and-digon f-factor problem	exposes local star-capacity and Tutte-type obstructions
Two-adic compatibility obstruction	COMPUTER-ASSISTED, FINITE	one proposed construction family reduces to 128 compatible permutations modulo 8 and is eliminated	valid family obstruction, not a global theorem
Central-symmetry enumeration	PROVED	exact generating functions count several centrally symmetric candidate spaces	useful for entropy comparisons, not sufficient for NTIL existence
Conic permutation-layer classification	PROVED	in the n=p-1 setup, a perfect matching lying on the relevant conic reduces to a punctured-line/hyperbola family	sharpens the conic route
Hyperbola-layer completion obstruction	EXACT FINITE	tested arbitrary completions fail for p=13,...,37 in that construction family	negative evidence for that family only
Finite-plane supersaturation	PROVED, LIMITED	supplies modular collinearity pressure in the n=p-1 setting	supporting lemma; a stronger local formulation supersedes its original role
Pair-codegree repulsion	EMPIRICAL	across archived C4 solutions, log2(P(u,v)/(P(u)P(v))) is approximately affine in normalized conflict codegree, with slope near -0.169	promising diversity score and possible clue toward a correlation theorem
Root-direction support matching	PROVED REFORMULATION	a direction may be reused; NTIL requires the selected root supports inside each direction fibre to form a matching	corrects the failed “avoid bad slopes” intuition
Orientation-flip blocker graph	PROVED / EXACT FINITE	a single legal orientation flip has blocker rank at most two; its exact release cost is forced vertices plus vertex cover in a graph of maximum degree four	gives a cheap exact prefilter and a candidate interface to joint-cover absorbers
Low-shell coupled-cycle obstruction	EXACT FINITE	at H=4,n=10, safe single-cycle descent has isolated states, while small coupled cycle networks still repair all 40 tested states	replaces a false single-cycle theorem by a charge/Graver absorber target

Two cautions apply to the table:

1. a theorem about a restricted construction family cannot be promoted to a theorem about all NTIL configurations;
2. a correct necessary condition may be nearly constant on C4 candidates and therefore useless as an optimization score.

6. Exact and empirical computational results

6.1 Exact small-m C4 counts

The oriented, labelled C4-NTIL solution counts for m=3,...,10 are

6, 8, 13, 8, 26, 26, 14, 32.

The m=10 enumeration visited approximately 33 million nodes. These counts are finite facts about the stated labelled model; they are not asymptotic evidence for a positive limiting probability.

6.2 Pair-codegree law

For compatible fundamental cells u,v, let D(u,v) count the rotated third-cell conflict mechanisms containing the pair. In archived solutions for m=18,...,28, the observed relation is

[ \log_2\frac{P(u,v)}{P(u)P(v)} \approx c_m-\alpha\frac{D(u,v)}m, \qquad \alpha\approx0.169. ]

The archive is biased toward configurations found by existing solvers, so this is an empirical law under an unknown sampling measure. The early suggestion alpha=1/6 is not supported at theorem-level precision. The useful content is the robust repulsion trend, not the exact decimal.

6.3 The historical score-60 basin

One historically emphasized n=74 incumbent had 60 bad three-point lines. It was not the best archived near-miss: the six V20 states in Section 3 are strictly better. For this score-60 incumbent's fixed undirected 2-factor, CP-SAT proved that 60 was the optimal orientation objective. Its 60 bad lines form 15 C4 orbits; an exact minimum conflict hitting set has eight fundamental edges.

Exact selected-cell balls of radius 8 through 12 around that incumbent are infeasible. Radius 13 returned UNKNOWN. Therefore the strict conclusion is: every C4 solution has selected-cell distance at least 13 from that seed. It is incorrect to infer distance 14 from the UNKNOWN radius-13 run.

One adaptive reduced-LNS release chain was later proved infeasible through k=23; k=24,25,26 remained UNKNOWN. The scope and fixed-cell certificate are recorded in score60_reduced_lns_frontier_2026-07-22.md.

The true solution is much farther away: both oriented and undirected overlap are only 2/37, so the selected-cell deletion distance is 35. These local certificates are valid, but they are now best used as negative controls for capacity-collapse theory, not as a route to n=74.

6.4 Corrected V40 local exclusions

Four older V=40 basins were completely audited in the corrected pseudograph model. All normalized deletion shells k=0,...,12 were excluded, covering 12,927,790,384 deletion masks through a sequence of conflict hitting, local star-capacity, exact f-factor, and short-direction tests. The conclusion is strictly local: the four basins have escape radius at least 13. It is not a non-existence certificate for m=37.

6.5 Solver trust boundary

The old large random-UNSAT sweep from the custom DPLL solver is withdrawn because the solver produced false UNSAT answers. Six named V=20 fixed-factor instances were independently cross-checked and may still be used within their stated scope; see validate_solver_report.md.

6.6 Exact local rigidity of the known n=74 solution

Hide k fundamental cells of the true solution, retain the other orbits, enumerate every directed-pseudograph f-factor completion of the degree deficits, and then check the complete geometry. The full k=1,2,3 shells all have the original cells as their unique zero-defect completion. Therefore any different C4 solution has selected-cell replacement distance at least four from this solution.

An audit found a duplicate-retained-cell bug in an earlier k=3 candidate counter. Two apparent E=1 alternatives were invalid; the remaining historical records reduce to three legal single-orientation flips with energies 2, 2, and 3. The uniqueness certificate is unaffected because the correction only removes non-solutions. The old safe-candidate histograms are withdrawn pending a clean recount.

6.7 Orientation-flip blocker graphs

Reverse one legal non-loop, non-digon fundamental cell of a C4 solution. Record the old orbit owners in every newly created defect orbit. Each blocker set has size one or two, and an extra deletion set makes the reversed orbit safe exactly when it hits all blocker sets. After forced singleton owners are removed, the minimum release cost is an ordinary minimum vertex cover.

The resulting blocker graph has maximum degree at most four: there are only four rotation phases, and two different neighbours in one phase would make three original points collinear. For the verified n=74 solution, the 36 legal flips have transversal sizes 1 through 8, with mean 4.972 and median 5. Across 163 cached true solutions, non-forest and non-bipartite blocker graphs occur, so those stronger early conjectures are false. The observed cycle-rank bound is empirical; only rank at most two and maximum degree at most four have been proved.

6.8 Known-answer trade barriers and capacity closure

The difference between a V20 state and the known true solution is a degree-balanced red-blue trade. For the six V20 basins it decomposes into 8--11 minimal atoms. Exhaustive dynamic programming over the orders of each fixed atom decomposition shows that the energy must first rise from E=5 to between 38 and 50 before reaching zero. These barriers are exact inside the stated decompositions, not in the full difference space.

Five thousand additional balanced cross-trades per basin were checked against the full geometry; their best energies were 18--23 and none had E<=5. The complete transformations release 148--184 saturated long-fibre blocker pairs, with minimum joint old-cell hitting sets of size 25--27. These facts explain why energy-descending cell-local search repeatedly loses the known correct route. The next computational primitive should be a degree-balanced, capacity-closed destroy/recreate move on roughly 20--35 cells.

6.9 The current m=38 near-miss and exchange gap

Ordered insertion into the true m=37 factor produced an independently checked m=38 candidate with cycle type 32+5+1, 20 bad triples, and exactly five C4 defect orbits (E=5). This improves the repository's previous m=38 near-miss from E=9 to E=5; it is not an n=76 solution.

Exact CP-SAT searches exclude every zero-defect state at replacement radii zero through eight around this seed. Radius eight was split into 46 exhaustive branches according to the retention of its two inserted cells, all infeasible under the same validated line-cut model. Thus its certified exchange gap is at least nine.

The known true-to-V20 paths and all low-energy states of their fixed atom sublattices were also lifted at all 38 insertion positions. Two route selections each checked 360,848 candidates; none improved E=5, and no non-truth m=37 skeleton even tied it. This is a strong finite rejection of that particular transfer scheme, not a non-existence result for n=76.

6.10 A finite obstruction to safe single-cycle repair

After correcting three false-negative bugs in an H=4 repair engine, a whole-path exact model found eight of 40 verified n=10 states isolated in the graph of single alternating-cycle moves that keep the lower direction shells clean. Coupled multi-cycle moves repair all 40, and two states require cycle rank three in the tested formulation. Therefore the proposed universal safe-single-cycle descent statement is false. A lower-risk successor is the charge matrix of cycles against saturated fibres and low-rank Graver-style coupled absorbers; the observed g(H)=H-1 staircase remains a conjecture.

7. Withdrawn or downgraded directions

This section is deliberately explicit so historical files are not accidentally cited as current results.

7.1 Carry/absorber n=74 claim — WITHDRAWN

The configuration claimed in absorber_carry_theory.md used lifted coordinates outside the 74 x 74 board. It was not an NTIL solution. The later Prellberg solution is real but is unrelated to that false positive. Claims of a proved polynomial-time random construction based on the same carry model are also unsupported.

7.2 Formal absorber theory — PROVISIONAL

The absorber notes assume independence and a constant-probability improvement that were not proved and are inconsistent with some of their own estimates. They may inspire a local-repair construction, but they are not an existence proof.

7.3 FDR, SIRH, and “symmetry signature” — PROVISIONAL

These are research vocabularies, not established theorems. The current files do not close the step from local orbit identities to global conflict control. In particular, linear or quadratic orbit equations do not imply that the global feasible variety is zero-dimensional, and a container-style conclusion needs quantitative hypotheses that have not been proved.

7.4 Ordinary line-capacity LP separation — DEAD END AS STATED

For the unrestricted board LP, assigning z_ij=2/n to every cell satisfies every line-capacity constraint because a grid line contains at most n cells. It also attains the row upper bound 2n. Thus the ordinary LP optimum is always exactly 2n and cannot separate feasible from infeasible orders. Prellberg's checkerboard or symmetry-restricted LPs are different models and should not be conflated with this observation.

7.5 Random Gröbner scans — FINITE ONLY

An UNSAT result for a fixed factor is a valid finite result if checked correctly. Randomly sampling factors and observing inconsistent ideals gives no global Nullstellensatz certificate for all factors.

7.6 Transposition and Gray-graph arguments — INSUFFICIENT

Edge transpositions of a connected graph generate the symmetric group, so these documents establish reachability in permutation space. Reachability says nothing about whether an NTIL state exists or whether a path can stay feasible. One draft also incorrectly asserted that every invertible linear map preserves Hamming adjacency; that is false in general.

7.7 Radial layers, heat maps, physical analogies, and 3D volume — DESCRIPTIVE

These views are useful for feature discovery, but no audited implication from a radial profile, density image, jamming/full-spark analogy, ballistic-network picture, or 3D visualization to 2D NTIL existence has been proved. They should be used to generate hypotheses and then tested out of sample.

The circular one-factorization colour rule suggested by the flower-like plots failed such a held-out test: agreement was only about one to two per cent on true solutions and V20 near-misses, with no useful separation.

7.8 Loop-free V40 baseline — WITHDRAWN AS A PROJECT RECORD

The later m37_continue factor space allowed only simple loop-free 2-factors. All six V20 near-misses and the true n=74 solution contain a loop, so this space excluded the most relevant known states from the outset. Its internal V40-to-V36 improvement is not a new project record and must not replace the V20 baseline. This correction is distinct from the valid, separately audited local V40 exclusions in Section 6.4.

7.9 Projective, Sidon, and determinant-gas language — PROVISIONAL

These reformulations clarify the geometry but currently do not construct a solution or prove non-existence. The rigorous part is the root-direction support-matching condition in Section 5. The missing step is a C4-equivariant global object—such as a proper-slope drawing, terrace, or two-factor distribution—that satisfies all fibres simultaneously. A new vocabulary is not evidence for that step.

8. What the literature contributes

8.1 Classical constructions and bounds

Guy and Kelly developed the classical probabilistic heuristic suggesting that the asymptotic maximum should be below 2n. Hall, Jackson, Sudbery, and Wild constructed sets of size (3/2-o(1))n, still the leading classical asymptotic lower-bound scale. Flammenkamp's two papers established major exact-enumeration and symmetry-classification results and his live database remains the primary record source.

These results warn against reading a long finite run of even solutions as an automatic proof of the all-even conjecture.

8.2 Constraint programming

Prellberg's 2026 CP-SAT work produced 2n configurations for every n <= 60 using exact collinearity constraints and symmetry reduction. Later searches by Prellberg, Heule, Riley, and others extended the record to 74. The practical lesson is that a faithful sparse exact model plus carefully chosen symmetry conditions can outperform elaborate handcrafted proxy objectives. Helper constraints must be ablated: some mathematically valid restrictions make the solver substantially slower.

8.3 Finite-field permutation graphs

Cooper and Hyatt characterize finite-field permutations minimizing collinear triples. Their setting is not the same as the integer-grid two-permutation decomposition, but it supplies a useful warning: modular permutation graphs have unavoidable structured triples. An integer construction must exploit carries or other failures of modular collinearity to lift to a genuinely collinearity-free set.

8.4 The solved k >= 3 problem and the exceptional k=2 case

Ghosal, Goenka, Grebennikov, Keevash, Kwan, and Pham proved that for every fixed k >= 3 and sufficiently large n, exactly kn points can be selected with at most k on a line. Their proof separates heavy lines, builds a pseudorandom hypergraph matching, deletes the remaining defects, and completes the structure by randomized switching.

The NTIL problem is precisely the missing critical case k=2. In the repository's corresponding line-count estimates, the light-line tail becomes harmonic at k=2, whereas the k>=3 deletion budget is summable. This identifies a real mathematical bottleneck: a proof for k=2 needs an additional cancellation, repulsion, or correlation estimate, not merely a re-run of the k>=3 proof.

9. Recommended research programme

The known-answer audit changes the priority order again. More local search around score60, V20, or the m=38,E=5 seed should stop unless it uses a qualitatively different macro move. The true solution is a ground-truth sample, but the held-out tests show that visual similarity, direction counts, and high cell overlap are poor guides.

Priority A — Finish the true-solution basin geometry

The exact k=1,2,3 completion shells are unique. Extend this calculation to k=4,5,..., using the new blocker graphs to prune deletion masks. The targets are:

• the first genuinely different zero-defect completion;
• the distribution of joint blocker-cover costs;
• minimal f-factor or Tutte cuts for infeasible retained cores;
• comparison with n=72, smaller exact solution spaces, and matched near-misses.

This turns “the true solution is rigid” into a measurable exchange-radius profile rather than another fitted visual statistic.

Priority B — Join blocker covers to capacity-closed f-factors

For a family of proposed replacement orbits, combine their maximum-degree-four blocker graphs and seek one shared deletion cover. Then require the released candidate multigraph to contain the needed loop-and-digon f-factor. This is the precise interface between the local orientation theorem and a macro absorber.

The target theorem should control three layers:

1. a small or logarithmic blocker graph for each proposed orbit;
2. substantial overlap among the vertex covers of many such graphs;
3. enough Hall/Tutte slack after the joint cover to complete all degree deficits without creating excessive new-new conflicts.

Computational search should generate complete degree-balanced macro states, contract capacity-closure components, and allow 20--35-cell reconstruction. Exact defect energy should be a terminal check or secondary score, not the sole local objective.

Priority C — Build the missing k=2 correlation estimate

Adapt the heavy-line/matching/switching architecture of the k>=3 theorem, but sample from a measure already conditioned on row/column degree two—equivalently, a random 2-factor or pair of permutations. The required new statement is a quantitative spread/negative-correlation theorem strong enough to remove the harmonic light-line loss.

The empirical pair-codegree repulsion law suggests what such a theorem might look like, but the archive distribution is not a uniform or Gibbs measure. A credible progression is:

1. prove the correlation bound for a tractable weighted 2-factor measure;
2. show that heavy-line conditioning preserves enough spread;
3. use cluster expansion, switching, or a lopsided local lemma to bound the residual three-point lines;
4. design an absorber that restores the exact degree-two condition.

This is the highest-upside theoretical route because it attacks the precise place where the now-solved k>=3 theory breaks.

Priority D — Test low-rank charge/Graver absorbers

For a state with clean low direction shells, form the matrix whose rows are saturated fibres and whose columns are alternating cycles; a column records the cycle's net occupancy change on every fibre. A single cycle may overload a fibre even when a small signed sum of cycles is safe.

The H=4,n=10 finite obstruction proves that simultaneous coupling is sometimes genuinely necessary. The next step is to test whether the minimum coupled cycle rank is bounded as n grows for fixed shell height H. Stop pursuing a height-only theorem if that rank grows with n; in that case move to multiscale or block absorbers.

Priority E — Reassess n -> n+2 as a long absorber

For the all-even conjecture, attempt an induction that embeds an n x n solution inside an (n+2) x (n+2) board, deletes a controlled number of old points, and repairs the four new row/column deficits by alternating cycles. The repair must be checked against every old secant, so the deletion-shadow f-factor is the right first layer.

The two calibrated lifts m=36 -> 37 and m=37 -> 38 both produced low-energy states with certified exchange gap at least nine. Thus a constant-radius patch is already disfavoured on the available training cases. The useful question is whether the required coupled exchange grows logarithmically, sublinearly, or linearly across a larger archive. If it grows proportionally to n, abandon local induction and return to the probabilistic programme.

Priority F — A modular zero-carry inverse theorem

For several primes p and direction shells Q, count triples whose integer determinant is kp. If true solutions systematically suppress the k=0 mass, try to prove that such suppression forces Fourier concentration or a low-entropy permutation structure. Then combine that inverse structure with the conic and hyperbola obstructions.

Stop if the argument merely reproduces the audited determinant-product 1:2 barrier. The goal is an inverse theorem with structural output, not another global valuation sum.

Side target — odd orders 71 and 73

Finding n=71 or n=73 would update the finite record and test the rct4/half- turn search machinery, but it is logically separate from proving all even orders. It should not displace Priorities B and C if the goal is a general theorem.

10. Repository map and reproducibility

path	purpose
analysis/results/n74_rot4_prellberg_verified_2026-07-22.md	explicit n=74 coordinates and independent verification
analysis/results/latest_joint_audit_2026-07-26.md	corrected V20 baseline, true-solution rigidity, blocker-graph theorem, trade barriers, m=38,E=5, and finite H4 obstruction
analysis/results/pair_codegree_37/	pair-codegree experiments, score60 certificate, Hamming balls, small-m exact data
analysis/even_n_existence_tools/	all-even structural lemmas, corrected C4 model, finite obstruction tools
analysis/results/score60_reduced_lns_frontier_2026-07-22.md	final scoped report on the historical score60 basin
verify_solution.py	independent point-set checker
visualization/overview.html	interactive visual guide; descriptive, not proof evidence
main.tex	manuscript source; may lag this README's audit status

Build helpers are provided in Makefile and compile.bat. Solver output must always be checked against the full integer-board collinearity predicate. FEASIBLE, OPTIMAL, INFEASIBLE, and UNKNOWN have different logical meanings and must not be interchanged. The 2026-07-26 joint audit is a compact scoped report; not all of its raw scripts and JSON outputs have yet been mirrored from the local research worktree into this GitHub-facing repository.

11. References

1. H. E. Dudeney, “A Puzzle with Pawns,” puzzle 317 in Amusements in Mathematics, 1917, pp. 94 and 222.

2. R. K. Guy and P. A. Kelly, “The No-Three-In-Line Problem,” Canadian Mathematical Bulletin 11(4), 527–531, 1968. doi:10.4153/CMB-1968-062-3

3. R. R. Hall, T. H. Jackson, A. Sudbery, and K. Wild, “Some Advances in the No-Three-In-Line Problem,” Journal of Combinatorial Theory, Series A 18(3), 336–341, 1975. doi:10.1016/0097-3165(75)90043-6

4. A. Flammenkamp, “Progress in the No-Three-In-Line Problem,” Journal of Combinatorial Theory, Series A 60(2), 305–311, 1992. doi:10.1016/0097-3165(92)90012-J

5. A. Flammenkamp, “Progress in the No-Three-In-Line Problem, II,” Journal of Combinatorial Theory, Series A 81(1), 108–113, 1998. doi:10.1006/jcta.1997.2829

6. T. Prellberg, “Constraint Satisfaction Programming for the No-three-in-line Problem,” arXiv:2602.07751, 2026. arXiv

7. J. Cooper and J. Hyatt, “Permutations Minimizing the Number of Collinear Triples,” Designs, Codes and Cryptography, 2025. doi:10.1007/s10623-025-01632-w · arXiv:2501.02331

8. A. Ghosal, R. Goenka, A. Grebennikov, P. Keevash, M. Kwan, and H. T. Pham, “No-(k+1)-in-line Problem for k >= 3,” arXiv:2607.05255, 2026. arXiv

9. Achim Flammenkamp, No-Three-in-Line database and live record log, accessed 2026-07-26.

10. OEIS A272651, maximal number D(n); A000755, total numbers of extremal configurations in the enumerated small cases; and A000769, numbers up to square symmetry.

12. Citation and licence

@software{du2026_no3inline_rigidity,
  author = {Du, Junrong},
  title = {No-Three-In-Line: Audited C4 Structure, a Verified n=74 Solution,
           and the All-Even Question},
  year = {2026},
  url = {https://github.com/aujurd22/no3inline-rigidity},
  note = {AI-assisted computational research notebook}
}

MIT. See LICENSE.