# Bourgain–Kontorovich: targeted source checks

18 October 2026. Review edition: arXiv 1205.4416v2, expanded native source already
frozen in the parent campaign. These are coordinating-assistant checks, separate
from the original Sol reports and the new independent arbitrations. A local
invalid calculation does not refute the density-one conclusion. No global color
or checked global repair is assigned here.

## 1. Vector versus projective stabilizers

Source: `lem:spec3`, expanded lines 2265–2307. The subgroup fixes the vector v0
modulo q, not its line. The source then asserts its index is asymptotic to q^2,
uses O(q) cosets for a fixed first coordinate and obtains a small-modulus cutoff
q<T2^((delta-Theta0)/2).

`exact_orbits.py` enumerates complete orbits of the actual three even-word
Descartes generators, not samples or partial conjugate sets. For the source's
root (-11,21,24,28), it also enumerates every nonzero Descartes null vector in
F_p^4 and checks equality of those two sets.

| p | Vector orbit / stabilizer index | Projective orbit | First-coordinate fiber sizes |
| --- | ---: | ---: | --- |
| 5 | 144 | 36 | 24 or 30 |
| 7 | 300 | 50 | 42 or 48 |
| 11 | 1,220 | 122 | 110 or 120 |
| 13 | 2,352 | 196 | 168 or 182 |
| 17 | 5,184 | 324 | 288 or 306 |
| 19 | 6,516 | 362 | 342 or 360 |

The printed q^2 and q fiber powers correspond to projective rather than vector
counting. Finitely many checks alone do not disprove an asymptotic. The all-prime
obstruction uses the cited arithmetic input: [Fuchs's Theorem 1.4](https://www.math.ucdavis.edu/~efuchs/FuchsStrongApprox.pdf) states that the
full Apollonian orbit is the nonzero null cone modulo every p>=5; the even-word
subgroup has index two, so its vector orbit has at least half that size.
For split primes p=1 mod 4 the form is linearly equivalent to xy+zt, whose
nonzero zero set has (p-1)(p+1)^2 points. Indeed x!=0 gives (p-1)p^2 solutions;
x=0 gives p(2p-1), and subtracting the origin gives the formula. Thus the actual
vector orbit has order p^3 along arbitrarily large split primes. This inference
uses Fuchs's orbit theorem, not the problematic full-SO identification below.

The theorem-bearing Fuchs author PDF and 2010 thesis are cached under references/;
[publication metadata](https://www.math.ucdavis.edu/~efuchs/research.html) records
the thesis as 2010 and the related article as JNT 131 (2011), 2282–2302. The author
snapshot is not asserted byte-identical to journal typesetting or a deposited
2010 file. Source acquisition and use of the theorem are not a full proof audit
of that external article.

**Possible local correction, not a checked repair:** With index of order q^3 and
fiber of order q^2, the intended main term is still of order T2^delta/q. The error
instead has order q^2 T2^Theta0, so it suffices to require q^3<=T2^(delta-Theta0).
Choosing q<T2^((delta-Theta0)/4) leaves a margin. Given the enormous printed
C=10^30/(delta-Theta0), the complementary range still starts above T1^(10^29).
This is encouraging exponent bookkeeping, not verification of every composite
modulus, lifted vector, spectral constant or subsequent use.

## 2. The full-SO quotient assertion needs correction

Source: `lem:GGq` part (2), expanded lines 1324–1344, together with the full
special-orthogonal-group definition in Section 2.1.

At p=7, exact matrix closure of the specified even-word generators has **58,800**
elements and does not contain -I. The latter preserves the Descartes form and has
determinant +1 in dimension four. Adjoining it gives a disjoint central coset and
therefore at least **117,600** elements of full SO_F(F_7). Consequently the printed
quotient cannot be isomorphic to full SO_F(F_7) under that ordinary definition.
This conclusion does not require a classification formula for the full group.

A spin-image/orthogonal-group convention or bounded-index replacement could repair
this formulation; the exact downstream subgroup and orbit facts must be retained.
This finite counterexample is not a counterexample to the density-one theorem.

## 3. Exact witness against the printed smoothing cancellation

Source: `lem:spec0`, expanded lines 2138–2191. The displayed argument claims that
if ||gamma1 gamma||<Y(1-10 eta), then both f(gamma1 gamma) and
f(gamma1 g^(-1) gamma h) equal one for small g,h.

`exact_smoothing.py` checks an explicit counterexample in exact rational matrices.
Write U=S1*S2, V=S2*S3, A=U-I, and

    g = I + t A + t(t-1) A^2/2,    t=1/100000.

The script verifies A^3=0, g^T F g=F and det(g)=1. Set eta=1/100, Y=100,
gamma1=V^700, gamma=gamma1^(-1), h=I. All lattice matrices are integral and
gamma1 is identity modulo 7. The exact squared Frobenius distance of g from I is
5000000001/1562500000000000000 < eta^2. The unperturbed product is I, with squared
norm 4<90^2. But the squared norm of gamma1*g^(-1)*gamma is

    14819373062498879323605601 / 1562500000000000000 > 100^2.

Thus the alleged pointwise cancellation fails: the perturbation is conjugated by
gamma1 and cannot be bounded uniformly as printed. This checks one step of the
provided proof, not the coset-counting lemma's conclusion.

**Repair lead, not a global repair certificate:** The paper's own cited template,
Bourgain–Kontorovich–Sarnak, arXiv:1001.4541v1 (25 January 2010), Section 4,
uses a different order. Its translated bump is based on psi(gamma0*g), and the
smoothing comparison on PDF page 25 is f(g^(-1)*gamma0*gamma*h), putting the
small perturbation outside the coset representative. The reviewed manuscript
instead defines Psi_q(g*gamma1). Correctly automorphizing the left-centered bump
is a plausible way to recover uniformity. Intermediate nonnormal subgroups,
base-eigenspace projection and Sobolev bounds require their own check.

The [dated arXiv source](https://arxiv.org/abs/1001.4541v1) and full PDF are cached.
Its statements concern SL2(R), so neither a dimensional transfer to SL2(C) nor the
weighted additional boundary in `lem:spec1` has been silently certified.

## 4. Weighted spectral estimate

Source: `lem:spec1`, expanded lines 2499–2571. The initial reparametrization gives
gamma2=gamma1^(-1)*gamma0*gamma2prime. For the newly defined f, the desired inner
sum is sum f(gamma0*gamma2prime). The unchanged kernel F_q(a,e)=sum f(a^(-1)*gamma)
therefore requires a=gamma0^(-1), whereas the printed replacement uses
a=gamma1*gamma0^(-1). Its later region tests also use incompatible matrix products.
This is a source-level ordering mismatch.

Correcting notation alone does not establish the new linear-functional boundary
strip count, smoothing losses in gamma1, or the uniform weighted spectral error.
The huge separation of T1 and T2 might absorb controlled T1-losses, but those bounds
remain obligations. Independent arbiters are examining the original source.

## 5. Quadratic-form normalization: the counting conclusion has a routine route

Source: `eq:cKzBnd`, expanded lines 3520–3556, and the earlier identity
AC-B^2=a^2. The ordinary discriminant is -4a^2; the proof instead writes
z1*C-B^2=-4a^2 and then changes its sign again in the root congruence.

The exact witness 5*x^2+4*x*y+y^2, obtained from x^2+y^2 by the determinant-one
substitution (x,y)->(x,2x+y), has A=5,B=2,C=1,a=1. Hence AC-B^2=1 and discriminant
-4. It has B^2=-a^2 mod 5, but not B^2=-4a^2 mod 5. This distinguishes the
normalizations without a numerical approximation.

The intended upper bound still has an elementary proof under the surrounding
positive-definite convention. A primitive representation of z1 puts A=z1;
then B^2=-a^2 mod z1. For p^e||z1 and alpha=v_p(a), if 2alpha>=e the number of
roots is p^floor(e/2). Otherwise, divide B by p^alpha; the unit equation has at
most two roots at odd p and at most four at p=2, with p^alpha lifts. Therefore
the local count is at most C_p*sqrt(gcd(p^e,a^2)), with C_2=4 and C_p=2 at odd p.
Multiplying and summing over square divisors z/z1 absorbs divisor factors into
z^epsilon and gives O_epsilon(z^epsilon*sqrt(gcd(z,4a^2))). One needs an upper
bound on classes by residue solutions; the printed equality is unnecessary.

`exact_local_roots.py` checks 100 local root-count cases with squared integer
inequalities (including powers of two). The proof above, rather than these finite
checks alone, justifies the uniform local estimate. This is a coordinating local
completion conditional on the surrounding form/class hypotheses, not a global
repair of the paper.

## Independent follow-up and budget

Five Gemini-high full-source arbitrations cover `lem:GGq`, `lem:spec0`, `lem:spec1`,
`lem:spec3` and `eq:cKzBnd`. Two Astra-max cases independently cover `lem:spec0`
and `lem:spec3`. The models receive the frozen manuscript and original accepted
referee evidence; the coordinating computations and repair leads in this note
were not inserted into their independent briefs or mislabelled as author responses.

The new seven requests reserve $12.301596. Together with $6.670553 already
accounted, the conservative campaign total is $18.972149 against the unchanged
$25 ceiling. No generation retries or paid repair search. Full Math Referee tests:
313 passed after adding an explicit arbiter effort option; Astra requests now
record max in the native payload and frozen ledger.