BK and the Kontorovich–Zhang correction: reconciliation and a local lifting repair

20 October 2026. This update concerns the frozen arXiv 1205.4416v2 proof and a later, separately dated correction. The density-one theorem remains unresolved in this audit. The algebraic lifting Claim inside lem:spec3 now has a checked local replacement; the lemma's remaining counting and spectral obligations are distinct.

The literature connection we missed

Kontorovich and Xin Zhang, On Length Sets of Subarithmetic Hyperbolic Manifolds, arXiv:2201.10955v1, submitted 26 January 2022, explicitly identify an overlooked Nullstellensatz issue in §3.2, p. 14. Their bibliography identifies [BK14b] as the Apollonian paper and [BK15] as Thin Hypotenuses. The coordinate change can inflate coefficients from the small scale to the product of the small and large scales, defeating the intended effective height bound. Pages 14–15 give a modulus-factor separation and bounded-shear remedy in their two-coordinate setting. Frozen v1, pp. 14–15.

Our 19 October source adjudication identified the matching height defect but missed this published discussion. That literature omission is corrected here. The publication date precedes our audit by several years, while remaining later than the original paper's 24 May 2012 external-proof cutoff. Its correction is legitimate later evidence; it does not retrospectively make the reviewed text complete. Historical model outputs are preserved. This update relies on the public paper, not private correspondence or an inferred blanket author endorsement.

Objection-by-objection reconciliation

“Not addressed by the cited passage” is a scope statement. We have not searched the complete subsequent literature for a correction to every other finding, nor audited the whole 2022 paper.

A direct replacement for the four-coordinate lifting Claim

This is a new self-contained completion recorded in this follow-up, prompted by examination of the published correction. It uses the elementary adjugate identity and a determinant bound. No mathematical priority claim is made. The coordinating audit and a separate research agent checked the written argument; exact computations test its identities, edge cases and limits. It is not a formal proof-assistant certificate.

Let $S$ be a finite set of integer row vectors $a_\gamma\in\mathbb Z^4$, with

Suppose $q>120H^4$. Then there are integers $v_*\in\mathbb Z^4\setminus\{0\}$ and $z_*$ such that

If $S$ is empty the conclusion is vacuous. Otherwise form the integer matrix $A$ with rows $(a_\gamma,-1)$. Its modular kernel contains $w=(v,r)$.

If $A$ had rational rank five, choose a nonsingular five-row square submatrix $B$, with determinant $D$. Since $Bw\equiv0\pmod q$, the identity $\operatorname{adj}(B)B=DI$ gives $Dw\equiv0\pmod q$. Bézout and $\gcd(v_1,v_2,v_3,v_4,q)=1$ imply $q\mid D$. But expansion of the determinant has $5!=120$ terms. Every term contains one final-column entry of absolute value one and four entries bounded by $H$. Hence

a contradiction. Thus $A$ has rational rank at most four and has a nonzero rational kernel vector $(v_*,z_*)$. Clear denominators. Because $S$ is nonempty, a kernel vector with $v_*=0$ would also have $z_*=0$, so the obtained $v_*$ is nonzero. This proves the claim.

For BK take $a_\gamma=e_1^t\gamma$, so $H=O(T_1)$ in the fixed matrix-norm convention. The original $v=\gamma_2v_0$ is primitive as required in the source. The large-modulus hypothesis $q\ge T_1^{10^{29}}$ dominates $120H^4$ for sufficiently large $T_1$. The proof therefore establishes exactly the original Claim eq:wStar, for arbitrary composite moduli and prime powers as well as primes. It needs no unit coordinate and no bound on the large vector's height.

The conclusion is a new vector $v_*$ and scalar $z_*$, not necessarily the original $v$ and $r$. It does not assert that $v_*$ is Descartes-null. Let $g$ be the gcd of the coordinates of $v_*$. Any one equation $a_\gamma v_*=z_*$ gives $g\mid z_*$. Dividing both sides by $g$ makes $v_*$ primitive and hence nonzero modulo every prime. The separate prime-avoidance concern is thereby resolved without a height bound on denominators.

The bounded-shear mechanism in four coordinates

A direct analogue of the later source's normalization is also available. Suppose a modulus $q'$ has at most $L$ distinct prime divisors, all greater than $3L$, and $v$ is primitive modulo $q'$. For each such prime $p$ the polynomial

is nonzero modulo $p$ and has at most three roots. Among $t=0,\ldots,3L$, at most $3L$ choices in total are excluded by all the primes. Some remaining $t$ therefore makes $f_p(t)$ a unit modulo every prime dividing $q'$, hence a unit modulo $q'$.

Use the upper triangular unimodular matrix with first row $(1,t,t^2,t^3)$ and remaining rows $e_2^t,e_3^t,e_4^t$. The entries of both the matrix and its inverse are bounded in terms of $L$. Replacing $v$ by $Uv$ changes the equation row to $a_\gamma U^{-1}$, whose coefficients remain $O_L(T_1)$. This checks the four-coordinate height-preserving mechanism under the stated prime-factor premise. Establishing that premise by a divisor split and treating the complementary modulus ranges still requires the relevant small-modulus counting input. The determinant proof above avoids those extra steps for the algebraic lift itself.

The analytic finishing step remains conditional

Write $\Delta=\delta-\Theta_0>0$. Grant the needed good-prime orbit/fiber bounds and the uniform congruence-coset counting estimate. The corrected auxiliary-prime estimate would then be

Taking $p\asymp T_1^{\Delta/3}$ balances the two terms at scale $T_1^{\delta-\Delta/3}$, giving a conditional power saving. The printed choice with denominator two makes the error as large as $T_1^\delta$. This exponent calculation is checked algebra, not a verification of its analytic premises.

The statement of lem:spec0 promises uniformity for every intermediate congruence subgroup and coset; if established, it would permit the stabilizer of the newly lifted vector. The current dossier has not established that analytic statement independently: the smoothing proof has the separate defect already recorded. Orbit/fiber estimates must also cover the non-null possibilities for $v_*$, rather than silently reusing only a null-vector calculation. Composite moduli, bad primes, a uniform weakened $\eta_0$ and its uses throughout the minor arcs remain explicit obligations.

Reproduction and provenance

Exact checker · saved results · source and version record.

The checker uses integer and rational arithmetic. It tests 36 lifting systems over large prime-power and composite moduli, the empty and zero-row cases, and negative controls showing why the large-modulus and modular-primitivity hypotheses matter. It also checks 3,024 nonzero finite-field coefficient tuples and 150 composite-modulus bounded-shear instances. Finite tests support the implementation and scope; the written determinant and root-count arguments establish the general local claims.

No new paid review calls or retries were made. The 2022 source snapshot is cached with SHA-256 and native line locators; the public citation points to its versioned arXiv record. The frozen original BK manuscript and historical panel assessments are unchanged.