A counterexample to the uniform singular-series bound

20 October 2026 · Bourgain–Fuchs · Author-hosted manuscript, Lemma 3.6, pp. 20–21.

The uniform bound is false for the permitted strip-packing family below. Its right-hand arithmetic expression is always 4, whereas the singular series is

This is stronger than identifying an unsupported proof step. It does not refute the positive-density theorem. A corrected local bound and a checked averaging argument would be needed to restore this proof path; neither is supplied here.

Attribution, edition and checking scope

Igor Rivin supplied an argument generated in a separate Web GPT Pro conversation, following a related observation relayed by Nikolay Bogachev. The underlying Web model identity was not verified. The explicit family and proposed local computations came from that supplied argument, not from the original referee panel. This note records coordinating mathematical checks, a separate agent's independent source/arithmetic audit, and reproducible exact enumeration. The submitted-argument summary preserves the contribution and its checking scope. These are AI-assisted checks, not a human referee judgment or a formal certificate. No additional paid referee requests were made.

A subsequent response relayed by Nikolay Bogachev, described by the user as an Astra conversation, supplies the valuation-layer explanation and a more affirmative replacement-bound claim. We incorporate the checked explanation while retaining that general repair as unverified. Its assertion of a JAMS page location is attributed testimony, not a comparison of the journal PDF by this review. Neither the model identity nor the independence of the two conversations was audited; agreement between the responses is not counted as another native panel assessment.

The inspected source is the preserved 25-page author copy, SHA-256 10447d219eb8669db869c3099e3ea62a3b833f9b1e6b58e8616bca0928f56ddd, also available from the author's website. The displayed formula on p. 20 was checked visually as well as against extracted text. The corresponding native label is singseries (Lemma 3.5 in the original arXiv source index). The author copy corrects an older coefficient typo in the arXiv input. Its corrected form construction is used here; we do not silently replace the frozen arXiv source.

The JAMS typesetting was not inspected. An attempt to retrieve it did not yield the paper. This note therefore does not claim a verified JAMS page match. PDF creation metadata is not treated as a verified revision date. The calculation is self-contained and imports no later result to justify the original proof.

The bound being tested

For the manuscript's binary forms, put

and use the intended local density

This uses residues modulo $p^k$, correcting the manuscript's residue-domain typo. The counterexample addresses the intended mathematical estimate, rather than exploiting that typo. Lemma 3.6 asserts the uniform estimate

The implicit constant may depend on the fixed packing and initial curvature, but cannot depend on the varying pair or on $X$. The example keeps the packing and initial curvature fixed.

The permitted packing and forms

The author's introduction and Lemma 2.1 explicitly include the primitive strip packing with root $(0,0,1,1)$. Although the abstract speaks of bounded packings, Theorem 1.2 and the form construction in this inspected edition are not restricted to them. This example uses that permitted unbounded case; it does not separately establish a bounded-packing counterexample.

Fix the circle of curvature $a_0=1$. Its form is

For every $n\ge1$,

The vector $(n-1,1)$ is primitive and nonnegative, so these curvatures belong to the initial set $\mathcal A_0$ even with the manuscript's primitive-representation convention.

The Descartes configurations $(0,1,n^2,(n+1)^2)$ occur while retaining the same line and circle of curvature 1. Replacing $n^2$ by Descartes reflection gives

starting from the strip root. Ordering the resulting configuration as $(n^2,1,(n+1)^2,0)$ in the author's coefficient formula gives

The coefficients have greatest common divisor 1, the leading coefficient is positive, and the determinant is $n^4$. Thus these are primitive positive definite forms of discriminant $-4(n^2)^2$, not arbitrary substitute forms. Replacing them by integrally equivalent reduced forms preserves every local count, because the changes of variables are invertible modulo all prime powers.

Choose

Use the displayed form with $n$ and with $2n$, respectively. Then $\gcd(a,a')=5^{2t}$ and $aa'(a-a')=-12\cdot5^{6t}$, so

Exact calculation at the common prime

Complete the square:

The units $a+1$ and $a'+1$ are 1 modulo 5. Each has a square root in $\mathbb Z_5$: the root 1 modulo 5 lifts uniquely at every step because its derivative is a unit. Consequently invertible $\mathbb Z_5$ changes of variables, followed by $z=X-U$, $w=X+U$, give

For every $y,v$, the required right-hand side for $zw$ has valuation exactly $2t$. For $v_p(b)=r<k$, the number of solutions to $zw\equiv b\pmod{p^k}$ is

Indeed, for each $j=v_p(z)$ from 0 through $r$, there are $(p-1)p^{k-j-1}$ possible $z$ and $p^j$ possible $w$. Including the $5^{2k}$ free choices of $y,v$, for every $k>2t$ we obtain

The reduction modulo 5 is $x^2-u^2$, of rank two. Primitivity does not make this quaternary form nondegenerate. The normalized counts also need not decrease under lifting.

For $t=1$, independent enumeration of the original binary forms gives:

For $t=2$, the normalized counts at $5,25,125,625,3125$ are respectively $9/5,13/5,17/5,21/5,4$, agreeing with the eventual value $\tfrac45(2t+1)=4$. A single local factor above 2 would not refute an unspecified global constant; the unbounded family and the other local factors are essential.

Why the first-level count misses valuation layers

The response relayed by Nikolay Bogachev gives a useful way to read the product count: each possible valuation of the first factor contributes a separate layer of equal weight. In our family,

and each layer contributes $4/5$ to the normalized density. Thus

The printed factor $2^{\omega(\gcd(a,a'))}$ records that 5 is a common prime, but does not record its exponent or the growing number of valuation layers.

This also distinguishes two different steps in the disputed proof. In this example the finite-field count really is below $2p^3$: the equation modulo $p=5$ reduces to $x^2-u^2=0$, with $y,v$ free. The two lines $x=u$ and $x=-u$ have $2p-1$ points in total, so

That finite-field upper bound is not the error. The invalid inference is using it as an upper bound on the limiting density. Singular solutions can lift into valuation layers that the first-level count does not distinguish. The separate nondegeneracy assertion is also false: this reduction has rank two, despite primitivity of the binary forms.

The relayed response writes the mixed terms of the forms with positive signs. Substituting $y\mapsto-y$ and $v\mapsto-v$ gives our negative-sign forms. These are bijections modulo every prime power, so the local densities and counterexample are identical. This is the same mathematical family, rather than a second distinct counterexample.

The remaining factors and the Euler product

For an odd prime $p\ne5$, the quaternary form is nondegenerate and its determinant is a square. To see the relevant finite-field count directly, diagonalize and use additive-character orthogonality. For a nontrivial additive frequency, the four one-variable Gauss sums multiply to $p^2$, since the determinant is a square. Summing those frequencies gives $-1$ for a nonzero target and $p-1$ for target zero. Thus the counts are $p^3-p$ and $p^3+p^2-p$, respectively.

For $p\ne2,3,5$, the target is a unit and all solutions are nonsingular; each lifts to $p^3$ solutions at the next level. Hence $\sigma_p=1-p^{-2}$.

At $p=3$, the target has valuation exactly 1. The origin modulo 3 cannot lift modulo 9, since a quadratic form at a vector divisible by 3 is divisible by 9. All other zeros are nonsingular and lift uniformly. Therefore

At $p=2$, we have $n\equiv1\pmod4$ and $n^2\equiv1\pmod8$. The forms and target reduce to

Exact enumeration gives 1,024 solutions modulo 8, with normalized count 2. To justify stabilization, write $F(\mathbf z)=\mathbf z^TA\mathbf z$ with $A$ integral symmetric. At any solution $F(\mathbf z)$ is odd, so some coordinate of $A\mathbf z$ is odd. Partition the solution classes by the first such coordinate. For $k\ge3$, translation in that coordinate by $2^{k-1}$ is an involution on classes modulo $2^k$, preserves their parity partition and the equation modulo $2^k$, and flips the next bit of $F$. Exactly half the classes lift; each lifting class has 16 lifts. Thus $N_{k+1}=8N_k$ and $\sigma_2=2$.

The infinite product is absolutely convergent away from the three exceptional primes. Using $\zeta(2)=\pi^2/6$,

and therefore

No fixed implied constant can bound this by $R(a,a')=4$.

The displayed short intervals, and a selection qualification

The literal definition (3.6) in the author copy uses

Put $\alpha=\log_2 25$. It is irrational, since a rational relation would imply an equality between a positive power of 25 and a power of 2. Infinitely many continued-fraction convergents from below give positive integers $t,k$ with

For those pairs, $25^t=2^k(1+O(1/t))$ and $4\cdot25^t=2^{k+2}(1+O(1/t))$. Since $k\asymp t$, both lie in the displayed short intervals eventually, for every fixed $\eta>0$. Choose $\log X=a^{2/5}$. Then the full dyadic intervals at $k$ and $k+2$ lie between $(\log X)^2=a^{4/5}$ and $(\log X)^3=a^{6/5}$ for large $t$.

Qualification: the preceding manuscript argument actually obtains a sufficiently populated subinterval by pigeonhole, then writes its location as zero without loss of generality. The counterexample satisfies the intervals literally displayed; it does not show membership in every alternate shifted interval that a corrected selection argument might choose. The manuscript's universal local-density argument and literal uniform bound fail. A theorem restricted to some specially chosen different family would require its own statement and proof. This note does not rule out that possible strategy.

Consequences, possible repair and the earlier review

The manuscript uses this singular-series estimate in the pointwise intersection bound (3.27), which then feeds the summed-overlap estimate. The supplied justification of that pointwise bound must therefore be revisited. A false intermediate lemma does not make its downstream statements false automatically. The main positive-density theorem remains unresolved by this review, and its existing red badge continues to describe the supplied proof.

The mechanism suggests a weight retaining common-prime exponents. The suggested candidate $\tau(\gcd(a,a')^2)$ has the elementary identities

The first identity follows by factoring both sides prime by prime; the second follows from the Euler factor $(1+p^{-2})/(1-p^{-2})$. These identities suggest finite formal averages but do not establish the needed distribution estimates for the selected curvature set.

More explicitly, the response relayed by Bogachev proposes the replacement

This remains a proposed replacement, not a verified repair in this archive. The response sketches a singular-branch recurrence intended to prove $\sigma_p\le 2h+1$ at an odd common prime, where $h=\min(v_p(a),v_p(a'))$, and asserts an absolute adjustment at 2. Before adopting that conclusion, the following obligations remain:

• State the recurrence and prove its bounds uniformly for the permitted primitive binary forms, including split and nonsplit reductions and the different target valuations.
• Prove the corresponding uniform estimate at 2; an assertion that it costs an absolute constant is not its proof.
• Combine the local bounds into the claimed pointwise Euler-product estimate, with all hypotheses and constants explicit.
• Rework the overlap average for the actual selected curvatures using the heavier common-divisor weight, including the required prime-power distribution estimates.

Our checked calculation proves the necessity of retaining common-prime exponents in this family. It does not establish that the proposed weight is sufficient in general. Even a completed pointwise repair would leave the overlap-average obligation separate.

The first supplied response's claims about later applications and absence of published corrections were not audited and are not adopted.

Our earlier Sol response on singseries identified normalization, degeneracy and lifting objections, but explicitly stopped short of refuting the lemma. It failed quotation validation and did not receive completed independent arbitration. Its original delivery status remains unchanged. The accepted original red finding concerned the separate congruence-average/short-interval deduction. This supplement is new evidence and is separately attributed, rather than retrospectively counted as a successful original-panel finding.

Reproduction and limits

Run the standalone exact checker with Python 3; it requires no third-party packages. Its JSON output records 45 local-count cases and 24 direct product-count cases, all checked with integer arithmetic and exact rational normalization. It enumerates the two binary-form residue distributions independently and joins them using the required difference; it does not assume the predicted local density when counting.

The finite checks include $t=1,2,3$ at 2, 3, 7 and 11, and the transient/stable 5-adic counts at $t=1,2$. The formulas and lifting arguments above establish the infinite family; finite computation alone would not. The source and provenance record binds the report, checker and count output to the inspected source. No original referee output, source snapshot or neutral reading packet was rewritten.