# Reading roadmap: Manuscript The manuscript expresses curvatures in tangent-circle subpackings using shifted binary quadratic forms. It selects a logarithmic-scale family of shifts, lower-bounds the total number of represented integers, and upper-bounds overlaps using a quaternary-form count, local densities and congruence estimates. Combining single-set mass with the overlap bound yields a positive-density lower bound. Source SHA-256: `803e26afc4c48a289769131a5b512a8d371607181bdef83b156a5b3ea4e444a8`. Indexed 51,870 characters. Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates. ## Internal claim index ### Theorem 1.1 — `iteratedens` Source lines 113–118. For an integer Apollonian circle packing $P$, let $\kappa(P, X)$ denote the number of distinct integers up to $X$ occurring as curvatures in the packing. Then for $X$ large we have $$\kappa(P, X) \gg X $$ where the implied constant depends on the packing $P$. *Extractive statement; proof gist not independently reconstructed.* ### Proposition 2.1 — `lowbound` Source lines 183–187. For an integer Apollonian circle packing $P$, let $\kappa(P, X)$ denote the number of distinct integers less than $X$ occurring as curvatures in the packing. Then we have $$\kappa(P, X) \gg {{X}\over{\sqrt{\log X}}}.$$ *Extractive statement; proof gist not independently reconstructed.* Proof opening: We fix a circle $C_0$ of non-zero curvature $a_0$ in the packing $P$, and count the integers which occur as curvatures of circles tangent to $C_0$ in $P$. This is identical to considering the orbit of $A_1$ acting on a quadruple $\mathbf v$ of mutually tangent circles $(a_0,b,c,d)$, since $A_1$ fixes the first coordinate of $\mathbf v$ and its orbit represents all of the circles tangent to $C_a$. Note that $A$ generates all possible Descartes configurations in the packing $P$, and there can only be finitely many circles of curvature $a_0$ in the packing since the total area of all the inscribed circles is bounded by the area of the outside circle. Therefore it is reasonable to count the circ Candidate references: `descartes` (candidate_proof_dependency); `firstchange` (candidate_proof_dependency); `secondchange` (candidate_proof_dependency); `intersection` (candidate_proof_dependency); `density` (candidate_proof_dependency); `relatekappa` (candidate_proof_dependency); `external:Elstrodt` (external_reference); `external:james` (external_reference). ### Lemma 2.2 — `intersection` Source lines 263–266. Let $\tilde\Gamma$ and $\rho$ be as before. Let $\Lambda(2)$ be the congruence $2$-subgroup of $PSL_2(\mathbb Z)$. Then the preimage of $\tilde{\Gamma}$ in $PGL_2(\mathbb Z)$ under $\rho$ is $\Lambda(2)$. *Extractive statement; proof gist not independently reconstructed.* Proof opening: We can extract from the generators of ${\Gamma}'$ as well as the formula in (\ref{homo}) that the preimage of $\tilde{\Gamma}$ under $\rho$ contains \[ \left( {cc} 1&0\\ -2&1\\ \right) \mbox{ and } \left( {cc} 1&-2\\ 0&1\\ \right), \] and so $\tilde{\Gamma}$ contains the congruence subgroup $\Lambda(2)$ of $SL_2(\mathbb Z)$. Recall that the area of $A_1\backslash \mathbb H^2$ is $\pi$, and note that $SO_{\Delta}(\mathbb Z)\bigcap \Gamma'$ contains exactly those elements of $\Gamma'$ which have even word length when written via the generators of $\Gamma'$, making up half of the whole group. Therefore the area of $\tilde{\Gamma}\backslash \mathbb H^2$ is $2\pi$, which is equal to the area of $ Candidate references: `homo` (candidate_proof_dependency). ### Lemma 2.3 — `density` Source lines 306–312. (James): Let $f$ be a positive definite binary quadratic form over $\mathbb Z$ of discriminant $-D$, where $D$ is a positive integer. Denote by $B_D(X)$ the number of integers less than $X$ represented by $f$. Then $$B_D(X)= {{c\cdot X}\over{\sqrt{\log X}}}+O\left({X\over{\log X}}\right),$$ where $$\pi c^2 = \prod_{\stackrel{q\equiv 3 \; (4)}{q\not| D}}\left(1-{1\over {q^2}}\right)^{-1}\prod_{p|D}\left(1-{1\over p}\right)\sum_{n=1}^{\infty}\left({{-D}\over n}\right)n^{-1}.$$ *Extractive statement; proof gist not independently reconstructed.* ### Lemma 3.1 — `sumsa1` Source lines 365–371. Let $\mathcal A$ and $f_a$ be as before. Then we have \sum_{a\in \mathcal A}|S_a|\quad\gg\quad\eta X *Extractive statement; proof gist not independently reconstructed.* ### Lemma 3.2 — `blomgran` Source lines 386–397. (Blomer, Granville): Let $f$ be a binary quadratic form of discriminant $-D$, and let $U_f^0(X)$ be as before. Let $\mathcal G$ be the group of genera of binary quadratic forms of discriminant $-D$. Denote by $s$ the smallest positive integer that is represented by $f$, and by $u$ the smallest positive integer represented by some form in the coset $f\mathcal G$. Then U_f^0(X) = \pi\cdot \left(1-\frac{1}{2u}\right)\cdot \frac{X}{\sqrt D} + E_0(X, D) where E_0(X,D) \ll \sqrt{\frac{X}{s}} + \tau(D)\cdot \left(\frac{X\log X}{D}+\frac{X}{D^{\frac{3} *Extractive statement; proof gist not independently reconstructed.* ### Proposition 3.3 — `intersectionlemma` Source lines 446–452. Let $\mathcal A$, $S_a$, $S_{a'}$, $\eta$, and $X$ be as before. Then there exists $c>0$ depending only on $a_0$ such that \sum_{a\not=a'\in\mathcal A}\left|S_a\cap S_{a'} \right|\leq c\eta^2X *Extractive statement; proof gist not independently reconstructed.* ### Lemma 3.4 — `niederheath` Source lines 461–483. Let $F$ be as before, and let $(\log X)^2\leq a,a'\leq (\log X)^3$. Let $\chi_{a,a'}$ denote the characteristic function on the region $\mathcal{B}_{a,a'}$, and let $$R_{\chi_{a,a'}}(a-a') = \sum_{\stackrel{\mathbf{x}\in \mathbb Z^4}{F(\mathbf x)=a-a'}}\chi_{a,a'}(\mathbf x).$$ Let $\Delta$ be as above. Then we have &&R_{\chi_{a,a'}}(a-a')\nonumber \\ &=&|I_{\chi_{a,a'}}(a-a')|\cdot| \mathfrak S(a-a')| + O\left(\frac{X\cdot \Delta^{100}}{(\log X)^{\lambda}}\right) where the first factor is the singular integral I_{\chi_{a,a'}}=\int_{-\infty}^{\ *Extractive statement; proof gist not independently reconstructed.* ### Lemma 3.5 — `singseries` Source lines 493–497. Let $\mathfrak S(a-a')$ be the singular series defined in (\ref{singser}). We have $$\mathfrak S(a-a')\ll \prod_{\stackrel{p|aa'(a-a')}{p\not| (a,a')}}\Bigl(1+\frac{1}{p}\Bigr)\cdot 2^{\omega((a,a'))}$$ *Extractive statement; proof gist not independently reconstructed.* Proof opening: We compute an upper bound for the expression in the limit in (\ref{sigma}) by letting $k=1$ since the expression in the limit decreases with $k$. Note that if $p|(a,a')$, we have $F$ is not degenerate modulo $p$ by the primitivity of the packing and the definition of the coefficients of $f_a$ in (\ref{coefficients}). Therefore $$\sigma_p