Problems remain only at the exact edge — what can be proved is always only the side with no slack
Line up the mathematical problems that have gone unsolved for a long time, and every one of them sits at strict criticality. That is not a coincidence but the result of selection, and what can be proved is always only the side with no slack — the witnessable side, the positive-proportion side. This article is about why, and about one distinction that actually gets used, which comes out of it.
This article cuts across. What each individual problem is, and what was measured on this machine, is on the side of the article for that problem — Collatz / Riemann / Erdős #169 / Lovász / BSD. What is dealt with here is only what becomes visible when they are lined up against a single measure.
Leanmachine-checked (Lean 4 + mathlib, standard axioms only, no sorryAx, no native_decide; theorem names given)
paperproved, not yet machine-checked
computationchecked on this machine, within the range stated; not a claim made to the outside
knowna restatement, a known theorem, or a check of the literature
- Observation — strict criticality appears in all five problems
- Why — problems with slack have already fallen to methods that use the slack
- Why proofs lean to one side — "there is no slack" can be witnessed
- The third column — a positive proportion can be hit with averages; a zero-proportion defect is invisible to them
- So "how much is understood" is not the distance to a settlement
- Counterexamples — four that were crossed even at criticality
- Correction — what criticality blocks is only "methods that use slack"
- The usable form — identify at the outset the quantity whose being pushed down is the proof
- The range within which this claim holds
- Sources and reproduction
Observation — strict criticality appears in all five problems
| Problem | The criticality that appears | Strict? |
|---|---|---|
| The Collatz conjecture | In the family of maps (qn+r)/pv, the only integer multiplier for which the expected ratio is 1 is q = 3 | strict (LeanShiori.qcrit_eq_three) |
| The Riemann hypothesis | Λ = 0 exactly (Λ ≥ 0 is a theorem, Λ ≤ 0 is the conjecture) | strict (Rodgers–Tao 2018) |
| The Erdős conjecture on arithmetic progressions | The logarithmic exponent θ = 1 is the boundary between divergence and convergence. k=4 is just below it | strict (the convergence condition of the dyadic sum) |
| The Lovász conjecture | All four known exceptions (five if K₂ is counted) have "no cycle but a path" | fact (confirmed by constructing the four) |
| The Szekeres set | Exactly optimal for M ≤ 52, beaten at M = 53 | strict (integer programming) |
All five critical. And not "roughly" but "exactly".
These are different problems from different fields. Is this a coincidence?
Why — problems with slack have already fallen to methods that use the slack
Most proofs in mathematics have the shape "it holds with slack to spare". You bound something slightly larger than the quantity you want and conclude that there is still enough. Problems on which that move works get solved by that move.
What is left is the problems where that move cannot work in principle — that is, the critical ones.
So when you gather the unsolved problems and look at them, criticality is all you see. It is the result of selection.
The sharpest evidence is inside a single problem
Compare different problems and you cannot tell whether the difference is one of difficulty or of criticality. Within a single problem, only the side with slack has fallen — that shape turns up four times.
| Problem | The side with slack | The side without slack |
|---|
The Erdős arithmetic progression conjecture is the clearest. k = 3 was settled because the exponent of the upper bound went past 1 (slack appeared); k = 4 is unsolved because it has not gone past 1. And for k=4, the very people who produced that upper bound write in their paper that "this is the limit of our method".
The limit of the method and the position of the criticality are in the same place.
That is no coincidence. The reach of a method is "the range where there is slack", and criticality is its edge.
Why proofs lean to one side — "there is no slack" can be witnessed
In a critical problem, what can be proved is always the "no slack" side. This comes not from a difference in difficulty but from a difference in the shape of the evidence.
| Theorem proved | What was witnessed |
|---|---|
| Λ ≥ 0 (Rodgers–Tao 2018) | The irregularity of the zeros. If Λ < 0, the zeros would have to be more regular than they actually are — and that was shown not to be so |
| q = 3 is the only criticality | The single line p²−p−1 > p for p ≥ 3 |
| The Lovász conjecture has four exceptions | The four graphs were constructed |
| Szekeres is not optimal at M=53 | One 17-element set better than it was found |
| The wall at k=4 | That the method does not reach was said from the side of the method |
Meanwhile the Riemann hypothesis (Λ ≤ 0), the convergence of Collatz and the Lovász conjecture all have to say "there is no exception anywhere". There is nothing to point at.
This asymmetry shows up most clearly in Λ
What Λ is, and why these two directions demand exactly opposite information, is written on the side of the Riemann hypothesis. Only the shape is extracted here.
| What is to be shown | Information needed | In hand? |
|---|---|---|
| Λ ≥ 0 (no slack) | A lower bound on the irregularity of the zeros | yes (unconditionally) |
| Λ ≤ 0 (= the Riemann hypothesis) | An upper bound on the irregularity of the zeros | No. And that is as hard as the Riemann hypothesis itself |
The two directions demand opposite kinds of information. And only one of them is in hand.
The answer to "why could only one side be proved" is here.
The third column — a positive proportion can be hit with averages; a zero-proportion defect is invisible to them
known "Witnessable" can be restated one step further. The side that is proved is a claim about a positive-proportion phenomenon; the side that is not proved is a claim about the absence of a zero-proportion defect.
| Problem | The positive-proportion side (proved) | The zero-proportion side (open) |
|---|---|---|
| The Riemann hypothesis | The zeros are not equally spaced in local average ⟹ Λ ≥ 0 (Rodgers–Tao). A claim about a positive proportion of the zeros, within reach of tools that take averages | Not one zero lies off the critical line ⟺ Λ ≤ 0. Zeros off the line, if any, could have density 0 |
| The Collatz conjecture | Almost every orbit attains an almost bounded value (Tao 2019). A claim about a positive proportion of orbits | There is no cycle; no orbit diverges. A claim about a single orbit |
Something of density 0 is not seen by any correlation, any moment, or any asymptotic formula. Knowing that the zeros are perfectly regular by every measure would not give the Riemann hypothesis.
Averages see what is crowded and not what is sparse. Only one side could be proved because only one side is a claim about the crowded part. The same tools proved the one and did not prove the other.
Logical form does not explain the direction. Λ ≤ 0 ⟺ ∀ρ: Re ρ = 1/2 is Π₁, and Λ ≥ 0 ⟺ ∀t<0 ∃ a non-real zero is Π₂; the one higher in the hierarchy is the one that is proved. It is not the number of quantifiers but the density of the evidence — a positive proportion or a zero proportion — that fixes the direction.
So "how much is understood" is not the distance to a settlement
Only the witnessable side gets filled in, so no amount of partial results piles up into getting closer to a settlement.
| Problem | The quantity that is moving | History | Why it is not enough |
|---|
"67.250 % of the zeros lie on the critical line" is a theorem. "The Riemann hypothesis is 67.250 % proved" is not.
The former is a theorem about a different proposition, and a state of being 67 % proved does not exist.
The same applies to the verified range, to the lower bound on the length of a cycle, and to the number of graphs examined. All of them are only widening "the range in which no counterexample has yet been found", which is not a proof that there is none.
Counterexamples — four that were crossed even at criticality
The claim so far is, as it stands, false. There are problems that were critical and yet were solved.
| Problem | Was it critical? | Who | What crossed it |
|---|
The sensitivity conjecture is especially telling. Even though the value √n is attained exactly (critical), it was solved in 2 pages. "Critical, therefore unsolvable" is false.
Correction — what criticality blocks is only "methods that use slack"
Being critical does not mean being unsolvable.
What criticality blocks is only one kind of move: make some slack and push with an inequality.
Every method that has walked that road stops there. The only ones that do not stop are the ones that bring in a different structure.
And when the four that crossed are lined up, they have the same shape.
| What was crossed | The language brought in | The language of the original problem |
|---|---|---|
| The sensitivity conjecture | Eigenvalues of a matrix (a signed hypercube) | combinatorics, Boolean functions |
| Fermat's Last Theorem | Modularity (modular forms and elliptic curves) | Diophantine equations |
| The Poincaré conjecture | Ricci flow and surgery (geometric analysis) | topology |
| The prime number theorem | Complex analysis (the location of zeros in the complex plane) | number theory |
All four bring in a language from outside the problem.
And in Fermat's Last Theorem, not one of 150 years' worth of partial results was used. It was not crossed from the side that had been piling up; what crossed it came from outside.
What crossed was not quantity but shape.
The usable form — identify at the outset the quantity whose being pushed down is the proof
Everything so far comes down to one practical distinction.
When taking on a critical problem, the first thing to do is to identify "the quantity whose being pushed all the way down is itself the proof".
If there is none, then no progress at all is headed towards a settlement.
When there is one — moving it gets you closer
| Problem | Quantity | Target | Now |
|---|
Λ moved from 0.5 to 0.22, and then to 0.2. Make it 0, and that is the proof itself. From below it is closed by Rodgers–Tao's Λ ≥ 0, so all that is left is pushing the upper bound down to 0 (candidates from 2026 that have not been peer-reviewed reach 0.172 — How far the published logs alone reach).
The twin prime gap has the same structure. Until 2013 it was not even known to be finite. Then it became 7×10⁷, and then 246. Make it 2 and it is a proof.
When there is none — moving it does not get you closer
| Quantity | What it becomes if pushed all the way |
|---|---|
| The upper bound on Λ to 0 | The proof of the Riemann hypothesis itself |
| The twin prime gap to 2 | The proof of the twin prime conjecture itself |
| The logarithmic exponent θ above 1 | The proof of the Erdős arithmetic progression conjecture itself |
| The proportion κ of zeros to 100 % | A different proposition. That all the zeros are on the line and that the proportion is 1 are different things |
| Extending the verified range of Collatz | A different proposition. It is not a proof that there is no counterexample |
| Increasing the number of graphs examined | A different proposition. It is a search for a counterexample, not a proof that there is none |
Two completely different things wear the face of the same "progress".
One is advancing towards the target. The other is running past the side of it, at the same speed.
Decide which one it is at the outset.
The range within which this claim holds
| Content | |
|---|---|
| holds | In the problems listed here, only "the side with slack" has been solved within a single problem (four cases) |
| holds | The results that were proved all have a "witnessable" shape (five cases), and are on the positive-proportion side (two cases) |
| holds | Problems split in two according to whether or not there is "a quantity whose being pushed down is the proof" |
| does not hold | That this is a law. The possibility that the chosen problems are a biased sample has not been ruled out |
| does not hold | The definition of "criticality". A different quantity is called critical in each problem (a multiplier, a constant, an exponent, a number of exceptions). There is no unified definition |
| does not hold | "A density-0 defect is invisible to every average" is stated as a principle, not as a theorem. As a theorem it would need the form "every averaging functional of the zero configuration is invariant under moving finitely many zeros" |
| does not hold | The description of the four that crossed as "bringing in a language from outside" is hindsight. It was not known at the time that it was a language from outside |
On novelty
Every part of this article is already known.
| Part | State |
|---|---|
| "Hard problems are what is left because the easy ones were solved" | A mere truth value. Not new |
| "Λ ≥ 0 comes out of a lower bound on the irregularity" | Rodgers–Tao's argument itself |
| "Λ is a quantity whose being pushed down is a proof; κ is not" | A known distinction |
| "A density-0 set is invisible to analytic methods" | A standard restatement of the limits of analytic methods |
| "Sieve methods have a parity barrier" | Classical |
What may be new is only the arrangement. Several problems lined up against the same measure and classified by whether or not there is "a quantity whose being pushed down is the proof" — that was not found within the range searched. But that is "was not found", not "does not exist".
Sources and reproduction
| Number | Kind | Source |
|---|---|---|
| Λ ≥ 0 | theorem | Rodgers–Tao (2018), arXiv:1801.05914 |
| Λ ≤ 0.2 / 0.22 / 1/2 | theorem | Platt–Trudgian (2021) / Polymath 15 (2019) / de Bruijn (1950) |
| Prime pair gap ≤ 7×10⁷ / ≤ 246 | theorem | Zhang (2013) / Maynard, Polymath 8b (2014) |
| The settlement of k=3 (θ = 1+c) | theorem | Bloom–Sisask (2020) |
| The upper bound for k=4 (θ = c < 1) | theorem | Green–Tao (2017) |
| Almost every Collatz orbit | theorem | Tao (2019), arXiv:1909.03562 |
| The weak Goldbach conjecture | theorem | Helfgott (2013) |
| BSD for rank ≤ 1 | theorem | Gross–Zagier (1986), Kolyvagin (1988) |
| The sensitivity conjecture | theorem | Huang (2019) |
| q = 3 is the only criticality | machine-checked | Shiori.qcrit_eq_three, qcrit_not_int_of_three_le (the Lean verification bundle) |
| Szekeres is beaten at M=53 | computed on this machine | Exact maximisation by integer programming |
| The construction of the four Lovász exceptions | computed on this machine | Vertex transitivity and Hamiltonicity confirmed by machine |
* The concrete value of the exponent c for k=4 in Erdős #169 has not been checked. Only the qualitative fact "smaller than 1" is used.