Three places where the corpus contradicts itself — and a lever that carries further than any other and dies all the same.
This project's polynomial layer is width-bounded resolution: derive everything derivable, but discard every clause wider than w. For fixed w there are only O(n^w) of them, so the procedure is polynomial. The question is always the same — how far does it carry?
Two documents in the corpus answer differently, and the contradiction has a strategic consequence.
The other measures the same question directly, on 330 instances across eleven values of n, by two methodologically independent routes that agree:
| Level | n₅₀ | transition width |
|---|---|---|
| width 3 | 52.9 | 18.4 in n |
| width 4 | 108.2 | 12.3 in n |
Width 4 carries a good factor two further. That is not a small shift but the largest lever measured anywhere in this corpus — larger than extended resolution at factor 1.35.
Why the density argument fails is written in the very document that makes it: the obvious refinement, the branching factor λ as a threshold, is tested there and rejected — an instance with λ > 1 dies, one with λ < 1 ignites. If the round factor is no threshold, the static density certainly is not.
From the same two numbers the corpus draws a straight line, w*(n) ≈ 0.018·n. But those same two points also carry a geometric law, and the ratio 108.2 / 52.9 = 2.05 is conspicuously close to 2.
| Law | predicts n₅₀(5) | and about w* |
|---|---|---|
| linear | 163 | w* = Θ(n) already in the window |
| geometric | 222 | w* = Θ(log n) in the window |
They differ by 36 % and one measurement separates them. The difference is not cosmetic: geometric would mean that across the entire measurable range proofs of size n^O(log n) suffice — quasipolynomial rather than exponential. Asymptotically w* must become linear; where the crossover sits has never been asked.
The corpus holds width 5 to be unreachable because its space already has 10⁸ clauses at n = 55. That is the supply, not the closure. The same text measures elsewhere that the refuting width-4 closure stays cubic while its space grows quartically.
Here is that measurement again, with auxiliary variables and across five support points:
An earlier report concludes that random 3-SAT has neither of the two resources that carry short proofs: no small closure space and no symmetry. The automorphism group is provably trivial up to n = 400.
The obvious objection is that a proof might also lift across an approximate symmetry and pay only for the seam. The corpus takes it seriously, builds the measure, validates it against a positive control — and asks whether the quantity is Θ(1) or Θ(n). Then it says: running.
The run finished. Its output sat unexamined on disk for nine days.
| n | 100 | 150 | 200 | 300 | 400 |
|---|---|---|---|---|---|
| usable | 60 | 77 | 101 | 138 | 177 |
| usable / n | 0.600 | 0.513 | 0.505 | 0.460 | 0.442 |
Θ(1) is excluded. It would require usable/n to fall by a factor 4 from n = 100 to 400; the measurement gives 1.36. And the exclusion holds despite thin data, because every value is a lower bound: a lower bound growing faster than any constant rules out Θ(1) no matter how poor the search.
Extended resolution may invent a new variable. A definition
constrains the old variables not at all — z is defined, not restricted — and all three clauses fit inside level 3. This is the only direction that leaves the variable set, and therefore the only one the project's own counting argument cannot reach. For it, no lower bound is known for any family of formulas.
The standing verdict is nonetheless that there is no procedure for finding useful extension variables. The established method looks for recurring structure to factor out and finds exactly none on random instances — no surprise, since nothing occurs twice there.
The definition yields two binary clauses. One of them merges with every clause C containing ¬la into a width-3 clause over z. Two such descendants merge back into the level when their remainders share a position. Those are the new ignition points, and they are countable before any computation:
Measured on exactly those instances where level 3 fails without auxiliary variables — and provably fails, since the cap lies above the supply. At identical clause and variable counts, nothing distinguishes the runs but the choice of pairs.
| Comparison, k = n definitions | targeted | control |
|---|---|---|
| vs random variable pairs, n = 88 | 13/20 | 0/23 |
| vs random literal pairs, n = 88 | 5/9 | 0/9 |
| vs spread instead of concentrated choice, n = 80 | 6/6 | 0/6 |
| vs anti-selection, n = 64 (k = n/2) | 13/14 | 0/14 |
The third row was a refinement meant to be better, and it falls — which sharpens the mechanism: the definitions must build a hub, not a net. The fourth is the second side of the control: the anti-rule is worse than chance, not merely better than nothing.
As a curve over n, with k = n definitions:
Per unit of cost this is the best lever in the corpus. Width costs a factor n/2 and buys 2.05; the targeted definitions cost a fixed factor 8 and buy 1.72.
Whether this is more than one further lever is decided by the number of definitions required at size n. The cost is O((n+k)³) — for any polynomial k* the procedure would stay polynomial. That is measured as a budget response, across two complete curves.
The finding is now complete, and it is two-sided. Choosing the definitions is a real, large, mechanistically understood effect, controlled in both directions. And it is still no way out: the price grows exponentially, as with every other lever.
The unlikely reading would have been remarkable, since for extended resolution no theorem forbids a polynomial refutation of random 3-CNF. It is now measured and does not hold — at least not for this selection rule in this window.