The Sign Axis

The task lays out a ladder ordered by degree: four-cycle, six-cycle, general cycle of length 2ℓ. On the face of it, it looks infinite. It is not.

Three consequences, and each one saves a measurement:

• λ_min(A) = −3, exactly, for every instance. S has rank at most n, and for m > n, S·Sᵀ is singular. The smallest eigenvalue of the clause matrix is not a measured quantity but an identity — measured while building the instrument, before it was derived.
• The whole spectrum sits inside the n×n matrix M. Its diagonal is the degree sequence; everything sign-dependent sits beside it. Under the gauge, M ↦ D·M·D — the spectrum is invariant, the individual entry is not.
• Rung 4 of the ladder has n degrees of freedom, not infinitely many. For every ℓ, tr(A^ℓ) is a function of the n eigenvalues of M. The frustration profile over all cycle lengths is therefore computable in a single pass.

And the point the task itself points toward: the imbalance Σ_v(pos−neg)² is the quadratic form of A at the fixed vector 𝟙 — every clause pair with weight one.

The task states: "The failure of the imbalance does not refute degree 2. It refutes uniform weighting. What is sought is not a new quantity, but the right weight." The right weight is μ_max(M) — and it has been measured. See section 05.

Checked twice, independently. The premise holds at α = 4.267 in every case examined — n = 50, 100, 150, 200, five scaffolds each, not a single variable with more than one component. And the reconstruction is carried out explicitly, from A and H alone, and matches the original up to gauge at n = 60, 120, and 200.

Measurement takes place at a fixed scaffold over many signatures. This keeps Σ_v deg(v)² constant, as well as pairs2 and every other scaffold quantity: centering within scaffolds removes the degree-spread confound exactly rather than approximately. SAT and UNSAT are never pooled.

The third check is at the same time the noise floor: the same formula under renaming is, by gauge symmetry, the same instance — whatever the solver still counts differently in conflicts is its own noise. It is completely different on the two sides of the threshold — and that decides what is explainable at all.

A null result without a working positive control is not a finding. Three families, in ascending order of sharpness.

Every measured invariant is identical between SAT and UNSAT — paired per graph, Δ = 0.000 across all ten and both quantities. Hardness differs by eleven bits. The instrument is therefore not blind: the third row shows it massively distinguishes, on the same scaffold, between Tseitin and a fair signature. It is the object that is built this way.

So tr(A^ℓ) for Tseitin is fixed by n, m, and the degree alone. Checked and measured, both times: tr(A³) = 45·125 − 75·27 = 3,600 at n = 45 and 4,800 at n = 60. The entire cycle profile is degenerate there. Even frust1, the only measured quantity without cycle support, is blind: in each parity class exactly two of the four masks have the bit e set, so out of the 16 pairs per edge, exactly 8 always disagree — frust1 = pairs1/2, exactly, measured 480 of 960 and 360 of 720.

Two seed ranges of 40 scaffolds each with 60 signatures, 4,800 main rows and 640 calibration rows. No run hit the conflict budget. Everything centered within the scaffolds, unsatisfiable instances, standard error of r under the null hypothesis 0.029 in each case.

frust4 is a hit. Eight and 6.6 standard deviations across two independent seed ranges, and with the predicted sign: more frustrated four-cycles means more width-3 resolvents means easier. It is the first structural sign quantity in this project that does anything at all — and it is not the imbalance:

Rungs 3 and 4 contribute nothing of their own: tr(M²) correlates with frust4 at −0.997 (that is Theorem V3, exactly), tr(M³) at −0.976, tr(M⁶) still at −0.817. They are frust4 seen again in longer cycles, with diminishing quality. And μ_max(M) — the right weight — contributes nothing. That is the sharpest single statement of this section, because μ_max is, by V2, exactly the quantity the task is looking for.

For each signature, the nearest signature of the same scaffold in the twelve-dimensional invariant space, mean distance 2.2 σ. If the invariants capture hardness, such pairs must lie close together.

The only point in the task where a causal chain is available instead of a correlation. frust4 is not a correlate of width-3 sparks but their exact count: two clauses with exactly two shared variables have a width-3 resolvent exactly when they agree on exactly one of the two. So — does frust4 predict whether the width-3 closure ignites?

And no other quantity does. At n = 53, within scaffolds, against ignites: frust4 +0.308 — the imbalance, the quantity with the strongest pull on CDCL hardness at n = 200, sits at +0.006 at N = 1,183. frust6 −0.020, frust1 +0.007, imbalance4 +0.013. The two directions of the sign axis belong to two different mechanisms.

A side finding, unplanned: p₃(53) = 0.527. The earlier finding placed the transition, with a completely different measurement series, at n₅₀ = 52.9. The instrument reproduces the constant without it being built in anywhere.

The sharpest objection to any hardness quantity: it merely shifts α_c, and then the instance simply sits deeper in the unsatisfiable phase — and deep in is easy to refute. For frust4 to pass this test, it must be available as a control knob. It is none, so it is instead discarded and redrawn until frust4 falls within a given z-window. That is a condition, not seeding: frust4 is gauge-invariant, and a condition on it cannot favor any assignment at all — unlike the imbalance, which can only be produced through partial planting.

The threshold really does move, by −0.127 across a lever of 6.5 standard deviations — about a third of the pull of the degree spread, which shifts it by 0.347. So each group is remeasured at its own α_c plus a fixed offset, 120 instances each. The UNSAT fractions match pairwise: the groups really do sit at the same point of the phase diagram.

The scan also delivers both groups at equal α, not only at equal threshold distance. The difference between the two comparisons is exactly what the threshold shift contributes — without extrapolation, without a slope estimate, without a model.

The task's ladder orders cycles. That is not the whole space the gauge condition permits: it requires even degree only at variable nodes. If a support also has even degree at every clause node, it is a cycle — otherwise it has an even number of odd clause nodes, and the case of two is a path in the incidence graph.

That is precisely what A^k counts: (A^k)[c][c′] sums the paths of length k with degree 2 at every intermediate clause and degree 1 at the ends. The canonical aggregate is therefore path(k) = 𝟙ᵀ·A^k·𝟙.

That is G4 in new notation, and it finally places the imbalance: it is the beginning of the path branch, not a special case. Checked against the data: max |path(1) − (imbalance − 3m)| = 0 across all 4,800 instances.

The longer paths are the imbalance in stretched-out form — r to it falls from +0.893 at k = 2 to +0.559 at k = 6, and the correlation to hardness fades with it. Out of sample, all five together contribute +0.0002 and +0.0078 beyond imbalance + frust4. So both branches of the classification are measured, to equal depth, and both are empty on the UNSAT side.

The counter-check is harder here than for frust4, because r(imbalance, variance) = +0.898: a rejection draw on the variance alone would drag the mean with it, and that is provably pure threshold. The lever must move the variance while holding the imbalance fixed — per scaffold, a calibration sample of 4,000 signatures, then the regression path(2) ~ imbalance, then reject on the residual. Achieved: 3.65 σ variance lever at 0.19 σ difference in the imbalance.

First: yes, and strongly — almost as strong as the imbalance itself, nine times stronger than frust4, and with the opposite sign to the imbalance, with which it correlates at +0.90.

Second: but not only. Each group at its own α_c, 250 instances each, both sides of the threshold:

On the UNSAT side, the two nearly cancel out; on the SAT side, they add up. The variance is not a SAT quantity — the UNSAT null result was a cancellation, not an absence.

A density artifact is ruled out: the groups sit at absolute α values 0.29 apart. A pure density effect would act oppositely on the two sides — more clauses make UNSAT easier and SAT harder. What is measured is that the low group is easier on both sides. What remains open: the logistic slopes of the groups differ by 7 %, so "equal distance in α" is not quite the same distance in units of transition width. The sign test does not depend on this.

Zero odd clause nodes are the cycles, two are the paths. Four is a third case, and its shape follows from the gauge condition: because every variable needs even degree, a leaf of the support can only be a clause node — four odd therefore means one clause node with three or four branches.

The first two are identities and run along as a check — across all 4,800 instances with max |Δ| = 0. From k = 3 on it is new, and not merely formally: Σ_c x_c³ is not 𝟙ᵀA³𝟙 (on one example −2,074 versus 17,360). The star branches at a clause; the path runs through it.

The raw moments are the paths in different notation — for a bell-shaped distribution, m₃ ≈ 3μσ² + μ³, and measured is r(Σx⁴, variance) = +0.95. What is new sits in the shape, i.e. in the standardized moments.

The counter-checks each cost their own measurement setup. It can be done cheaper: whoever shifts α_c must, at a fixed scaffold and fixed α, predict satisfiability. The core measurement has 4,800 such instances, half SAT and half UNSAT — the test costs nothing and measures every quantity at once.

The test independently confirms both costly counter-checks — the sign of the variance flips when the imbalance is held fixed (−0.28 raw, +0.29 partial), matching dα_c/d(variance) = −0.150, and frust4 lands nine times weaker, exactly as measured there.

And it settles the stars. The skew of A²𝟙 does not predict satisfiability — its contribution of +0.010 is the first on the sign axis that lies entirely beyond the phase location. Small, but clean.

A suggestion from outside: also measure the instance with all variables negated, and run it through under "all true" and "all false". Both parts have a short theoretical answer — and both deserve to be shown rather than merely asserted. 280 instances, four views.

"All variables negated" is the gauge g ≡ −1 and therefore not a new measurement point but the same point under a different name. Measured: imbalance, frust4, tr(M³), triangle3 and excess with max |Δ| = 0 across all four views, satisfiability verdict identical in 280 of 280 instances.

"All true" counts the purely negative clauses — a support with degree 1 at each variable, so by the gauge condition not an invariant:

Under g ≡ −1, the two numbers swap exactly, in 280 of 280 instances. Under the majority gauge they tip to 51 versus 194, while the formula itself does not change.

There is a version of the suggestion that is invariant, and it is built differently from the whole ladder: first fix the gauge canonically, then measure. With s_v = sign(pos_v − neg_v), σ(c,v)·s_v is invariant, even though neither factor is on its own — and it is not a character, because a sign of a sum is embedded in it. What is measured is how many literals per clause stand against their own majority (a tie gets weight ½, since any other tiebreak would otherwise be arbitrary).

The sum is old — r(counter-sum, imbalance) = −0.91. New would be the clustering. Out of sample the profile sits between baseline and placebo, and the two directions disagree in sign (−0.008 versus +0.007). That is exactly how one recognizes noise: real signal improves both, as the stars do. Empty.

What the attempt still delivered: confirmation of gauge neutrality even under a targeted renaming — and an invariant of a different kind that is likewise empty. That narrows the space further in which the remaining 85 % can still sit.

Thanks to Theorem V1, the whole ladder could be measured in one pass instead of four. Connected supports before unconnected ones, short before long — the order the task itself prescribes.

The task holds out a theorem in prospect: if rungs 1 through 4 together contribute nothing beyond noise out of sample, then hardness is a high-degree function of the signature, and no local sign quantity can ever predict it.

The premise is not met. The ladder delivers R² = 0.131 and 0.139, far above any noise level. The theorem is therefore not claimed. What stands instead is sharper, because it is more precise.

To this is added the variance of the 𝟙-weighted spectral density (07b) and, small but clean beyond the phase location, the skew of A²𝟙 (07c). Everything else is one of these four in different notation, or empty. The imbalance acts only through the satisfiability threshold; frust4, half so. The scaffold axis shifts the threshold too — and retains a factor of 15 beside it, frust4 retains 1.2. The difference between the two axes is thus quantitative and large, not qualitative.

Not measured is the entirety of the characters of one degree; their number is astronomical. What is measured is the order the task prescribes, plus two reweightings that were specifically sought — and both are empty.

• Not that the sign axis is empty. frust4 is a hit, across two seed ranges, out of sample, with the predicted sign and a demonstrated causal chain.
• Not that no local sign quantity can ever predict anything. The premise of the theorem is not met, and the theorem is not claimed.
• Not that the measured quantities capture all characters of their degree. The order is exhausted, not the space.
• Nothing about other solvers. Hardness here means: conflicts of this project's CDCL. The ignition test, by contrast, does not depend on any solver — the width-bounded closure is confluent.
• Nothing about P versus NP. That is a statement about the worst case; random 3-SAT at the threshold is for that neither necessary nor sufficient.