gpt pro 5.4 attempt on Math-811, version 2

ROUND-3 OBJECTIVE

We pursue path (C) obstruction/correction. Round-2 already ruled out the original dyadic "hit the current collision family" argument. The most promising continuation is therefore to audit the revised January 2026 Moser--Tardos formulation, because that is the latest publicly described route claiming to overcome the Round-2 instability.

The new goal of this round is to prove a sharper, theorem-level obstruction: the revised collision-cylinder encoding still fails near the left boundary of each dyadic block. More precisely, the admissible pattern sets used to define the bad events are automatically empty for endpoints in a boundary strip $[X,X+2T(X)]$. Hence the bad-event family cannot possibly encode all logarithmic collisions with maximal element in $[X,2X)$. This is not the same as the Round-2 same-scale deletion obstruction; it is a separate defect in the revised local-pattern/LLL module.

I also isolate the minimal correction needed to make that module logically coherent.

Round-2 FOUNDATION USED

I use the following Round-2 results as fixed inputs.

1. Future-scale immutability. Deleting elements $\ge 2X$ cannot create or destroy collisions all of whose values are $<2X$.
2. Same-scale instability. Deleting inside $[X,2X)$ can create a new collision whose maximal value still lies in $[X,2X)$.
3. Corrected dyadic implication theorem. If one can choose dyadic deletion sets $B_X\subset[X,2X)$ so that after stage $X$ there are no collisions with values $<2X$, and if $|B_X|=o(X)$, then the final set has density $1$ and no collisions.

I also use the Round-1 collision normal form, which rewrites any collision as an equality of products of two disjoint consecutive blocks.

NEW INSIGHT / TOOL (ROUND-3)

The new ingredient is a boundary-strip emptiness lemma for current-scale-only local pattern encodings.

The revised January 2026 proof introduces, for each dyadic $X$, a variable set \[ V_X=[X,2X)\cap A_0, \] windowed local patterns near an endpoint $u$, and an admissibility condition requiring that every integer interval $J$ of length $L(X)$ inside the local window contain at least a positive proportion of currently deleted points from $V_X$. The collision bad events are indexed only by such admissible patterns.

I prove that any such scheme is automatically empty on the boundary strip $u\in[X,X+2T(X)]$: there exists a length-$L(X)$ test interval $J$ lying entirely to the left of $X$, so $J\cap V_X=\varnothing$, and the required positive lower bound is impossible. Consequently the collision-cylinder family cannot encode collisions with an endpoint in that strip.

This is a strictly new obstruction beyond Round-2, and it rules out an entire class of boundary-buffer Moser--Tardos encodings that only record the current-scale deletion bits.

ATTACK PLAN (ROUND-3)

Round-2 left the following possibility open:

Replace the invalid one-shot hitting-set step by a Moser--Tardos/LLL construction on local pattern events.

The exact gap to check now is whether the bad-event family really covers all logarithmic collisions with maximal element in $[X,2X)$.

The plan is:

1. Prove an abstract boundary-strip emptiness lemma for current-scale-only admissible pattern sets.
2. Apply it to the January 17, 2026 preprint's definitions of $\Sigma_u$ and $T_X$.
3. Deduce that the collision-cylinder family misses an entire boundary strip of possible endpoints.
4. Record the second, formal mismatch: the scale theorem quoted in the preprint states only an upper bound on $|B_X\cap I|$ for $I\subset[X,2X)$, whereas the coverage lemma assumes a two-sided capacity condition on $D_{<X}\cup B_X$ for all $I\subset[X/2,2X)$.
5. Give the minimal corrected statement consistent with the Round-2 corrected dyadic implication theorem.

This overcomes the previous obstacle in the sense that it identifies a concrete, local, rigorously provable failure mode in the revised proof, rather than merely saying that the combinatorics is delicate.

WORK (ROUND-3)

Abstract boundary-strip emptiness for current-scale pattern families

Fix positive integers $X,L,T$ with $T\ge 1$. Let \[ V_X\subseteq [X,2X) \] be any set of "current-scale eligible variables". For $u\in[X,2X)$ suppose one defines a family $\Sigma_u$ of local patterns on a subset of the current-scale window near $u$, with the following property:

($*$) For every $\sigma\in\Sigma_u$ and every integer interval $J\subset [u-(L+2T),u)$ of length $L$, one has \[ \sum_{n\in J\cap V_X}\sigma(n)\ge cT \] for some fixed constant $c>0$.

Here the sum is taken only over the current-scale positions $J\cap V_X$.

Lemma 5.1 (boundary-strip emptiness). Assume ($*$). Then for every \[ u\in [X,X+2T] \] one has $\Sigma_u=\varnothing$.

Proof. Fix $u\in[X,X+2T]$. Consider the interval \[ J:=[X-L,X). \] It has length $L$. Also, \[ u-(L+2T)\le X-L \] because $u\le X+2T$, and clearly $X\le u$. Hence \[ J\subset [u-(L+2T),u). \] But by definition $V_X\subseteq [X,2X)$, so \[ J\cap V_X=\varnothing. \] Therefore for every pattern $\sigma$ one has \[ \sum_{n\in J\cap V_X}\sigma(n)=0. \] Since $T\ge1$ and $c>0$, this contradicts the lower bound $0\ge cT>0$ required by $(*)$. Hence no such $\sigma$ exists and $\Sigma_u=\varnothing$. $\square$

The point is that the contradiction is structural: it does not depend on number theory, on the geometry input, or on whether a collision actually exists. It follows solely from using a positive lower bound on current-scale deletions inside a buffered window that reaches left of $X$.

Application to the January 17, 2026 preprint

In the January 17, 2026 preprint, the scale-$X$ Moser--Tardos module introduces \[ V_X=[X,2X)\cap A_0, \] and for $u\in[X/2,2X)$ it defines admissible local patterns $\sigma_u\in\Sigma_u$ by the condition that for every integer interval \[ J\subset [u-(L+2T),u) \qquad\text{with}\qquad |J|=L, \] one has \[ \frac{T}{200} \le \sum_{n\in J\cap V_X}\sigma_u(n) \le \frac{T}{2}. \] The collision bad events $A_\tau$ are indexed by tuples \[ \tau=(x,y,\ell,k,\sigma_x,\sigma_y) \] with $\sigma_x\in\Sigma_x$ and $\sigma_y\in\Sigma_y$.

Corollary 5.2 (preprint boundary emptiness). For the preprint's pattern definition, one has \[ \Sigma_u=\varnothing \qquad\text{for every }u\in[X,X+2T]. \]

Proof. Apply Lemma 5.1 with $c=1/200$. $\square$

Corollary 5.3 (the cylinder family misses a boundary strip). In the preprint's collision index set $T_X$, no tuple $\tau=(x,y,\ell,k,\sigma_x,\sigma_y)$ can have \[ x\in[X,X+2T] \qquad\text{or}\qquad y\in[X,X+2T]. \] Hence the bad-event family $\{A_\tau:\tau\in T_X\}$ does not encode any putative logarithmic collision whose endpoint $x$ or $y$ lies in that boundary strip.

Proof. If $x\in[X,X+2T]$, then Corollary 5.2 gives $\Sigma_x=\varnothing$, so no legal tuple $\tau\in T_X$ can involve that $x$. The same argument applies to $y$. $\square$

Consequence for the coverage lemma

The preprint's coverage lemma asserts:

If global capacity holds and there is a logarithmic collision in the final kept set with maximal element in $[X,2X)$, then there exists a tuple $\tau\in T_X$ such that the corresponding cylinder event $A_\tau$ occurs.

The proof says: take the actual endpoint patterns $\sigma_x,\sigma_y$, define $\tau=(x,y,\ell,k,\sigma_x,\sigma_y)$, and conclude $\tau\in T_X$.

Proposition 5.4 (the coverage step is not justified as written). The above proof step fails whenever $x\in[X,X+2T]$ or $y\in[X,X+2T]$.

Proof. By Corollary 5.2, in that situation one has $\Sigma_x=\varnothing$ or $\Sigma_y=\varnothing$. Therefore the actual pattern pair cannot satisfy the legality requirement \[ \sigma_x\in\Sigma_x, \qquad \sigma_y\in\Sigma_y \] needed to define a tuple in $T_X$. So the step "define $\tau$ by these endpoints, lengths, and patterns" is invalid in that boundary regime. $\square$

This proposition does not assert that a boundary-strip collision actually occurs in the final set. What it proves is the sharper statement needed for proof-checking: the current bad-event family does not cover that case. Thus the revised Moser--Tardos argument is incomplete unless the boundary case is handled separately or the pattern encoding is changed.

A second formal mismatch: theorem statement vs. coverage hypothesis

The same preprint defines, for every length-$L$ interval $I\subset[X/2,2X)$, the two-sided capacity bad event \[ B_I=B_I^+\cup B_I^- \] where \[ B_I^+ := \{ |(D_{<X}\cup B_X)\cap I|>T\}, \qquad B_I^- := \{ |(D_{<X}\cup B_X)\cap I|<T/100\}. \] The coverage lemma assumes \[ \bigcap_I \neg B_I, \] that is, two-sided capacity for all such intervals $I\subset[X/2,2X)$.

However, the scale theorem is stated only as:

for every interval $I\subset[X,2X)$ of length $L$, one has $|B_X\cap I|\le T/2$.

Proposition 5.5 (statement-level implication failure). The scale theorem's stated conclusion does not imply the hypothesis of the coverage lemma.

Proof. The coverage lemma requires two independent features:

1. an upper and lower bound on $|(D_{<X}\cup B_X)\cap I|$;
2. these bounds for all length-$L$ intervals $I\subset[X/2,2X)$.

The theorem statement supplies neither:

1. it gives only an upper bound, and only for $|B_X\cap I|$, not for $|(D_{<X}\cup B_X)\cap I|$;
2. it restricts $I$ to lie inside $[X,2X)$, omitting intervals that meet the left buffer $[X/2,X)$.

Therefore the stated conclusion does not logically imply the coverage hypothesis. $\square$

This is weaker than Proposition 5.4: Proposition 5.5 is only a formal implication mismatch, while Proposition 5.4 gives a concrete structural reason the current event family cannot cover the boundary strip.

Minimal corrected statement

Round-2 already provided the right global target: a dyadic stage theorem must directly imply that after stage $X$, no collision with maximal value in $[X,2X)$ remains.

The minimal local correction consistent with the present Moser--Tardos strategy is:

At scale $X$, define local patterns on the full buffered window \[ \widetilde W_u:=[u-(L+2T),u)\cap [X/2,2X)\cap A_0, \] recording the total deletion pattern of $D_{<X}\cup B_X$ on $\widetilde W_u$ (earlier-scale bits are fixed constants, current-scale bits are random variables). Define admissibility using the total deletion pattern on every length-$L$ interval in the window. Then prove:

1. a corrected coverage lemma for these full-window patterns;
2. a two-sided capacity statement on all length-$L$ intervals in $[X/2,2X)$;
3. $|B_X|=o(X)$.

By the Round-2 corrected dyadic implication theorem, such a corrected scale theorem would be sufficient for the whole dyadic construction.

I do not prove that this correction works. The point of this round is narrower and rigorous: the current module does not yet provide it.

ADVERSARIAL VERIFICATION

I now try to break the new claims.

1. Did Lemma 5.1 accidentally use information about $A_0$? No. The proof uses only the set inclusion $V_X\subseteq [X,2X)$, so it is independent of any arithmetic property of $A_0$.

2. Is the interval $J=[X-L,X)$ always legal? Yes. It has length $L$. The only inclusion needed is $J\subset [u-(L+2T),u)$, and this follows exactly from $u\in[X,X+2T]$.

3. Could the preprint avoid the problem because no collision endpoint can lie in $[X,X+2T]$? No such restriction is stated anywhere in the scale theorem or coverage lemma. The theorem aims to remove all logarithmic collisions with maximal element in $[X,2X)$.

4. Is Proposition 5.5 merely cosmetic? No. The proof of the coverage lemma explicitly assumes two-sided capacity on the larger buffered interval range $[X/2,2X)$, while the scale theorem as stated guarantees only an upper bound for $B_X$ on the smaller range $[X,2X)$. This is a genuine logical gap in the printed implication.

5. Does this round disprove the original Erdős problem? No. It disproves neither the problem nor the possibility that a corrected Moser--Tardos implementation could work. It proves only that the current revised proof is not yet complete.

FINAL

UNRESOLVED (BUT STRICTLY ADVANCED)

This round makes a mathematically substantive advance beyond Round-2:

1. it proves a general boundary-strip emptiness theorem for current-scale-only local pattern encodings with a positive lower deletion requirement;
2. it applies that theorem to the January 17, 2026 Moser--Tardos preprint and deduces that the collision-cylinder family omits every endpoint in $[X,X+2T(X)]$;
3. it pinpoints the exact failure in the coverage lemma: the actual endpoint patterns need not lie in the admissible sets $\Sigma_x,\Sigma_y$;
4. it isolates a second, purely formal mismatch between the theorem statement and the hypothesis needed by the coverage lemma;
5. it gives the minimal corrected local-pattern statement that would need to be proved to revive the dyadic Moser--Tardos strategy.

So the investigation is still unresolved, but the current best publicly described proof route has now been narrowed to a sharper and more explicit obstruction than in Round-2.

COMPLETION ESTIMATE

COMPLETION: 70%

REFERENCES

1. T. F. Bloom, Erdős Problem #421, problem page and discussion thread.
2. P. Chojecki, A Density-One Sequence with Distinct Consecutive-Block Products, January 17, 2026 preprint.