# Mixed divided-Frobenius jets and covariance bounds for quadratic determinants Mira Cairn ## 1. Results and relation to earlier work A polynomial jet matrix over an imaginary quadratic order can lose most of its rank when reduced modulo an inert prime. We retain successive divided Frobenius layers before reducing, and normalize the higher Hasse columns so that the retained information remains integral. A coefficient determinant for an explicit differential connection then proves full row rank. The criterion permits derivative orders larger than the auxiliary prime and approaches the necessary column count. Let $\mathcal O$ be the maximal order of an imaginary quadratic field and write $\overline z$ for conjugation. For positive integers $K,S$, put $R=KS$ and $$ \Lambda=\{(r,s)\in\mathbb Z_{\ge0}^2:r+KsB_e$ for every $0\le e\le d$, then (1.1) has full row rank over the quadratic field. No generic-position assumption is used. In particular, $J\le p$ is not required. A simpler sufficient condition, proved in Section 3, is $$ n\ge2S,\qquad n\left[J-\frac{RJ}{p^2}-\frac Rp\right]>\frac{(R+p)^2}{2K}. \tag{1.3} $$ With fixed $K$, $J/R\to\epsilon\in(0,1/4)$, $p/R\to0$ and $R/p^2\to0$, its least prescribed node count satisfies $nJ/M\to1$, subject to the inertness and distinct-residue hypotheses. **Theorem 2 (an explicit quadratic-node family).** For every negative fundamental discriminant $-D$ with $h(-D)=1$ and $D\ge10000$, define $$ \begin{aligned} K&=43,& S&=\left\lfloor\frac{29(D-3)}{172}\right\rfloor,& R&=43S,\\ J&=\left\lfloor\frac{13R}{200}\right\rfloor,& A_0&=\left\lceil\frac{11}{10}D^{3/4}\right\rceil,& H&=J-\frac{RJ}{A_0^2}-\frac R{A_0},\\ n&=\max\left(2S,1+\left\lfloor\frac{(R+2A_0)^2}{86H}\right\rfloor\right). \end{aligned} \tag{1.4} $$ Then $H>0$, $n\le\lceil3D/2\rceil$, and (1.1) has full row rank at the $n$ nearest quadratic integers to the origin. The parameter rule has $nJ/M=1+O(D^{-1/4})$. This asymptotic concerns the explicit rule, not the existence of infinitely many class-number-one fields. **Theorem 3 (complete discriminant application).** The rank theorem, published local determinant and disk-Sobolev lemmas, and the directed certificates described in Sections 5–7 prove $$ h(-D)=1\quad\Longrightarrow\quad D<9500. \tag{1.5} $$ Thus one tenth-discriminant bound is $B=9499$, with $\log B=9.158941808890182446528155948973\ldots$. The proof uses neither a global discriminant bound nor a census, known classification or finite elimination. Its local inputs remain explicitly imported. The endpoint is weaker than the reviewed $D\le499$ in Severi Jet [1]; the contribution is the rank mechanism and its complete application. Section 8.1 additionally compares the new and published node rules at the same large support. Its positive comparison at $D=499$, combined separately with [1]’s global bound, gives the qualified corollary $D\le498$. **Theorem 6 (closed Charlier application).** The rank theorem, the local arithmetic lemmas and the closed complement estimate in Section 8 prove $$ h(-D)=1\quad\Longrightarrow\quad D<7000, \qquad B=6999,\quad\log B=8.853522560689539660215591928953\ldots. \tag{1.6} $$ A closed scalar tail for $D\ge10000$ meets four finite interval certificates. This route uses no global discriminant theorem, census, finite class-number elimination, spectral grid or optimized price table. Its mathematical ingredients are the same rank and local arithmetic mechanisms as above, together with exact Charlier inverse diagonals and elementary radial integration. The stronger $D\le498$ corollary remains conditional on the explicitly imported global $D\le499$ theorem. **Finite Hermite covariance and correlation estimates.** Sections 8.4--8.6 add a common unit-triangular complex-Hermite metric, a closed omitted-column penalty, an analytic minimizer for its core, and two complementary grouping results. Whole horizontal fibers have an explicit vanishing normalized-gain ceiling in the stated scaling. Macroscopic two-dimensional clusters instead admit a positive correction for every selected minor after the total omission charge. A prescribed actual-lattice family has correction greater than $1.19178330873866\times10^{-8}$ at every integer scale. This small lower bound establishes a mechanism; it does not establish a new discriminant cutoff. The $D<7000$ proof and the qualified imported $D\le498$ consequence retain exactly their preceding meanings. The local divisibility tower and nearest-node covering estimate originate in Severi Jet's framework [1]. Earlier lower-support Wronskians [2] prove independence by a coefficient determinant in characteristic zero, with an overhead depending on the conjugate degree. The complementary-rank argument [3] combines independent subspaces when full ambient rank is unavailable. Our theorem instead proves the entire triangular row space independent. The complete disk-Sobolev estimates and exact degree counting used in Sections 5–7 are imported from [4], using the classical disk-polynomial identities of Figueroa [5]. Section 8.2 gives a closed Charlier complement estimate. Section 8.3 uses it to prove the stronger complete bound $D<7000$ from local inputs, with no spectral or row grids. The earlier disk-Sobolev route remains as a separate comparison. The inert Frobenius congruence and its use in determinant nonvanishing are classical. Van Dijk [6] uses an auxiliary inert prime and a polynomial root count; twinheterodox [7, Section 5] uses the confluent version in explicit discriminant comparisons. Our additional step retains the divided layers of the evaluated matrix and the higher derivative columns, producing a mixed local interpolation problem. Positive-characteristic connections, $p$-curvature and cyclicity are classical as well; Bostan–Caruso–Schost [8] provide relevant background. We give the elementary connection argument in full. Bertrand's postulate [9] supplies the uniform auxiliary prime. None of these classical tools is claimed as new. The general proof and exact finite controls were developed by the author. The controls use separately assembled algebraic expressions but are not independent implementations by another researcher. The claimed numerical inequalities are directed certificates, not floating optimization results. ## 2. Proof of the mixed rank theorem ### 2.1 Integral triangular layers Put $g=z^p-w$. The polynomials $$ g^m z^a w^s,\qquad 0\le a0$. Each original monomial occurs exactly once as a leading monomial, by Euclidean division of its $z$ degree by $p$. Since $p$ is inert, $\mathcal O/p\mathcal O=k=\mathbb F_{p^2}$ and $\overline\lambda\equiv\lambda^p\pmod p$. At a fixed node write $$ g(\lambda+t,\overline\lambda)=pU(t)+t^p,\qquad\deg Um$. All scalings are invertible in characteristic zero. They will be used only for rank, not as changes to the later metric or determinant valuation normalization. ### 2.2 A row relation gives mixed local jets Suppose the scaled matrix has a nonzero row relation over $k$. Collect the coefficients in layer $m$ into $$ H_m(z)=\sum_{a,s}c_{m,a,s}z^{a+ps},\qquad F(z,u)=\sum_{m=0}^dH_m(z)u^m. \tag{2.4} $$ The exponents $a+ps$ in each layer are distinct because $0\le aK$, the maximum has $s=\lfloor(R-mp-1)/K\rfloor$ and $a=(R-mp-1)\bmod K$. Write $u_i(z)$ for the reduced local polynomial expressed in the coordinate $z-\xi_i$ at node $i$. Formula (2.3) is equivalent to $$ (\partial_u^{[\ell]}F)(z,u_i(z)) =O((z-\xi_i)^{L_\ell}), \qquad L_\ell>0. \tag{2.5} $$ Here $\partial_u^{[\ell]}$ denotes the Hasse derivative. The replacement of the fixed factor $\xi_i^{ps}$ by $z^{ps}$ is legitimate because $(\xi_i+t)^{ps}\equiv\xi_i^{ps}\pmod{t^p}$ and every $L_\ell\le p$. Differentiating (2.2) in characteristic zero and reducing gives the exact local identity $$ u_i'(z)=z^{p-1}-(z-\xi_i)^{p-1}. \tag{2.6} $$ Thus the error in replacing the derivative along the local curve by the connection below begins at order $p-1$. ### 2.3 The cyclic differential connection Set $E=k(z)$ and $$ \Delta=\partial_z+z^{p-1}\partial_u \quad\text{on }E[u]. $$ Its $p$th iterate is a derivation, since all intermediate binomial coefficients in the iterated Leibniz rule vanish. It sends $z$ to zero, sends $u$ to $(p-1)!=-1$, and kills $k$. Consequently $$ \Delta^p=-\partial_u. \tag{2.7} $$ Let $e=\deg_uF\le dK$. The recurrence $$ [u^m]\Delta F=H_m'+(m+1)z^{p-1}H_{m+1} $$ then gives $\deg([u^m]\Delta^jF)\le D_m-j$ by induction. Thus $$ \deg W_F\le\sum_{m=0}^eD_m-\frac{e(e+1)}2=B_e. \tag{2.9} $$ At node $i$, translate the coefficient matrix in (2.8) by $u\mapsto u+u_i(z)$. This has determinant one. Its row-$j$, column-$\ell$ entry is $$ (\partial_u^{[\ell]}\Delta^jF)(z,u_i(z)). $$ The two operators commute. Equations (2.5)–(2.6) imply that this entry has vanishing order at least $\max(0,L_\ell-j)$. To justify the truncation, one differentiation of a polynomial along $u_i$ differs from its $\Delta$ derivative by a multiple of $(z-\xi_i)^{p-1}$. After $j$ differentiations the cumulative error starts at order at least $p-j$. Since $L_\ell\le p$, it cannot affect the claimed order $L_\ell-j$. This also covers $L_\ell=p$. The capacities $L_\ell$ are nonincreasing. For $a\ge b$ and $x\ge y$, $$ (a-x)_++(b-y)_+\le(a-y)_++(b-x)_+. $$ Exchanging out-of-order pairs shows that the minimum determinant-term order pairs $L_0,L_1,\ldots,L_e$ with row indices $e,e-1,\ldots,0$. Therefore $$ \operatorname{ord}_{\xi_i}W_F\ge \sum_{\ell=0}^e\max(0,L_\ell-e+\ell)=\mu_e. \tag{2.10} $$ The $n$ residues are distinct, so (2.9)–(2.10) require $n\mu_e\le B_e$, contrary to the hypothesis for the actual degree $e$. The reduced scaled matrix has full row rank, and so does the original matrix. This proves Theorem 1. When $J\le p$ and $ep$; replacing it by a first-channel count would discard those conditions. Checking only $e=d$ is not part of the general theorem. ### 2.5 Colliding first residues and unequal capacities The retained Fermat quotient also allows first residues to collide. Let the nodes be distinct modulo $p^2$, and allow node $i$ to have a capacity $1\le J_i\le R$. Group nodes by their first residue $\xi_i\in k$. Within each group their local series have the form $$ u_i(z)=c_i+v_\xi(z),\qquad v_\xi(\xi)=0, $$ since every nonconstant coefficient in (2.2) depends only on $\xi$. The constants $c_i$ in a group are distinct. Indeed, if $\lambda_i=\lambda_0+p\eta$, then $$ \frac{\lambda_i^p-\overline\lambda_i}{p} \equiv\frac{\lambda_0^p-\overline\lambda_0}{p}-\overline\eta \pmod p. \tag{2.11} $$ Thus equality of both coordinates is equivalent to equality modulo $p^2$. The coordinate interpretation is classical [10]; we use it in the following extension of the coefficient-determinant proof. For each actual degree $e\le d$ and first residue $\xi$, form the multiset $$ \left\{L_{i,\ell}:=\min(p,\max(0,J_i-\ell p)): \xi_i=\xi,\ 0\le\ell\le e\right\}. $$ Let $a_{\xi,0}\ge\cdots\ge a_{\xi,e}$ be its largest $e+1$ members. Set $$ \mu_{e,\xi}=\sum_{k=0}^e\max(0,a_{\xi,k}-e+k). \tag{2.12} $$ **Corollary 4 (grouped mixed jets).** Under $KB_e\qquad(0\le e\le d). \tag{2.13} $$ To prove it, choose the largest-capacity columns as prefixes of Hasse orders at each center $c_i$. This is possible because $L_{i,\ell}$ decreases with $\ell$; break ties in favor of lower orders. Let the chosen lengths be $t_i$, with $\sum t_i=e+1$. Evaluation of polynomials of $u$ degree at most $e$ in these Hasse prefixes has confluent Vandermonde determinant $$ \prod_{iB_e$. If $e\ge Q$, all nonzero capacities occur among $L_0,\ldots,L_e$, whence $$ \mu_e\ge J-(Q+1)e\ge J-\frac{RJ}{p^2}-\frac Rp. \tag{3.1} $$ Also $D_m\le p(R-mp)/K$. Writing $R=dp+s_0$, $02$. Using $M=R(R/K+1)/2$ gives $$ \frac{nJ}{M}=1+O\left(\frac pR+\frac R{p^2}+\frac1p+\frac1R\right). \tag{3.3} $$ This is a conditional rank consequence; it does not by itself supply inert primes, distinct residues or a discriminant bound. At $p\asymp R^{3/4}$ its overhead is $O(R^{-1/4})$; the next section makes that scale available explicitly for the quadratic nodes. If distinct residues are available at $p\asymp R^{2/3}$, (3.3) instead gives $O(R^{-1/3})$. That smaller-prime geometry is not assumed in our application. ## 4. The explicit nearest-node family ### 4.1 Inertness and a prime below the norm threshold Use the parameters (1.4). Bertrand's postulate [9] supplies $A_0\frac{377D}{800}-\frac{4167}{800}. $$ Consequently $$ \frac HD\ge h_-(D):= \left(\frac{377}{800}-\frac{4167}{800D}\right) \left(1-\frac{725}{121\sqrt D}\right) -\frac{145}{22D^{3/4}}, \tag{4.4} $$ where $h_-$ is positive and increasing for $D\ge10000$. Moreover, $$ \frac{(R+2A_0)^2}{86D^2}\le b_+(D):= \frac1{86}\left(\frac{29}{4}+\frac{11}{5D^{1/4}}+\frac2D\right)^2, \tag{4.5} $$ which decreases. Directed evaluation proves $$ \frac32h_-(10000)-b_+(10000)>0.0050176383940755. \tag{4.6} $$ Thus $NH>(R+2A_0)^2/86$. Since $N$ is integral, the second term in the maximum defining $n$ is at most $N$. The first also is, because $2S\le29D/86<3D/2$. This proves $n\le N$ without presupposing the node geometry. By (4.3), $2\rho_n\le2\rho_NR$, only the primary credit $\lfloor r/q\rfloor$ is lost. Under class number one, the principal-norm overcount in [3, (2.8)] gives $$ \ell_D(r)=\log R \sum_{\substack{t,y\ge1\\ty^2D\le4r}} \left(1+\sqrt{r/t-Dy^2/4}\right). \tag{5.3} $$ For fixed $r$, it decreases with $D$, including the removal of boundary terms. It counts composite norms and repeated representations as well as primes, so it is an upper bound rather than a prime census. Replacing the inert sum in (5.2) by the sum over all primes and subtracting $\ell_D(r)$ is valid whenever the displayed non-inert-prime conditions hold. ### 5.2 Physical disk-Sobolev upper bound Let $\rho$ contain the nodes, let $T>1+1/(2\rho)$, and put $\mathcal R=\rho T$. Distinct integral nodes are separated by at least one, so disjoint sampling disks of radius $1/2$ are available. Set $$ \sigma_Q=\sum_{\substack{q^\ell\le1000\\q\ \mathrm{prime},\ \ell\ge2}} \frac{\log q}{q^\ell},\qquad H_Q=\sum_{\substack{q^\ell\le1000\\q\ \mathrm{prime},\ \ell\ge2}}\log q. $$ Legendre's formula implies $C_j/j!\le e^{H_Q-\sigma_Qj}$. The monic continuous disk basis and weighted sampling theorem [4, (3.7)–(3.9)], scaled to physical monomials as in [3, (2.5)], give $$ \log\Gamma\le P+\sum_{(r,s)\in\Lambda}\log h_{r,s}, \tag{5.4} $$ where $$ \begin{aligned} h_{r,s}&=4S^2e^{2H_Q}\kappa\mathcal R^{2r+2s+2} \widetilde h_{r,s}^{\alpha} \sum_{j=0}^{\min(J-1,r)}e^{-a_j} \left(\frac{e^{-\sigma_Q}}{\mathcal R}\right)^{2j} (r)_{\underline j}(s+\alpha+1)_{\overline j},\\ \widetilde h_{r,s}^{\alpha} &=\frac{r!s!\Gamma(r+\alpha+1)\Gamma(s+\alpha+1)} {(r+s+\alpha+1)\Gamma(r+s+\alpha+1)^2},\\ \kappa&=\left(\frac{T^2-1}{T^2-(1+1/(2\rho))^2}\right)^{\alpha+J-1},\\ P&=\sum_i\sum_{0\le jB_e$ of Theorem 1. For each candidate interval, $J\le p$ and the node count satisfies the special-case inequality for every candidate; all smaller actual-degree inequalities follow or are checked. This pays the largest required node count over the candidate set. ### 6.2 A single geometry for the entire interval Put $$ b_+=\sqrt{\frac{n\sqrt{D_+}}{2\pi}},\qquad \delta_+=\frac{\sqrt{D_++4}}4, \qquad\eta_+=\delta_+/b_+. $$ Choose the rational $\rho$ by rounding $b_++\delta_+$ upward to a multiple of $2^{-40}$. For every $D\in[D_-,D_+]$, (4.2) implies $$ |\lambda_i|\le\rho,\qquad \left|\frac{\lambda_i}{\rho}\right|^2 \le q_{\eta_+}(i/n),\qquad q_\eta(v)=\left(\frac{\sqrt v+\eta}{1+\eta}\right)^2. \tag{6.3} $$ The diameter upper bounds for the eight intervals respectively are below $950,968,997,1016,1050,1099,1170,1338$. Each is below the smallest candidate prime, so all node residues are distinct at the selected inert prime. Theorem 1 supplies full rank throughout each interval. This same $\rho$ enters every term of (5.5), including the derivative scale $L=\log(R/\rho)-\sigma_Q$, normalized sampling radius $1/(2\rho)$, and restored price profile. There is no change of physical row normalization between rank and norm estimates. ### 6.3 Directed costs and interval coverage For each row-cell the certifier uses exact integer cell counts and the upper disk-Sobolev phase from [4, Sections 3–4]. The arithmetic credits (5.1) use exact integer floors, with $\log q$ rounded downward. Formula (5.3) is evaluated at $D_-$ and the cell's upper $r$ endpoint. In all intervals, $$ \frac{D_--3}{4}>\max(K,S),\qquad \left(\frac{D_--3}{4}\right)^2>R, $$ so every non-inert prime satisfies the hypotheses of Section 5.1. The restored price uses (6.3), the increasing radial integrand, its exact primitive from [4, (3.5)], and the endpoint correction for the node sum. Convex chords account for all integer derivative orders. All transcendental calculations use 256-bit Arb enclosures. Row tables are rounded outward to exact multiples of $2^{-40}$ and then summed with integer counts. The floating optimizer that proposed the finite-interval price witness reached its iteration cap. Its convergence and optimality are not proof premises: the retained rational prices are verified directly. The resulting exact dyadic gaps are positive and imply the table. Each statement excludes the entire closed interval, not merely sampled endpoints. The intervals meet and cover $[9500,18000]$. To finish Theorem 3, we supply this method's own uniform tail next. ## 7. A uniform full-triangle tail for every $D\ge18000$ Set $D_0=18000$, $X=(D-3)/4$, $X_0=17997/4$ and $$ K=43,\quad S=\lfloor29X/K\rfloor,\quad R=KS,\quad J=\lfloor13R/200\rfloor,\quad\alpha=\lfloor S/2\rfloor, \quad n=\lceil3D/2\rceil,\quad T=129/128. \tag{7.1} $$ Theorem 2 gives full rank at an initial subset of these nodes. Adding the remaining columns preserves it. At the anchor, $S_{\min}=3034$, $R_{\min}=130462$, $J=8480$, $n=27000$, and $M=197976085$. ### 7.1 Uniform geometry and all physical costs The integer node rule satisfies $$ c_-^n:=\frac6{29}\le\frac nR\le c_+^n:=\frac6{29}+\frac{6K/29+11/2}{KS_{\min}} =\frac{1566379}{7566796}. \tag{7.2} $$ Indeed, $R\le29(D-3)/4$ gives the lower bound, while $D<4K(S+1)/29+3$ and $n<3D/2+1$ give the upper bound. Define $$ \begin{aligned} R_*&=29X_0-K,& \nu_-&=c_-^n(29-K/X_0),&\nu_+&=29c_+^n,\\ \eta_0&=\frac14\sqrt{\frac{2\pi(D_0+4)}{\nu_-X_0\sqrt{D_0}}},\\ C_+&=\frac12\log\frac{\nu_+}{\pi} +\frac14\log\frac{D_0}{D_0-3}+\log(1+\eta_0),\\ \rho(X)&=e^{C_+}X^{3/4},&\rho_0&=\rho(X_0). \end{aligned} \tag{7.3} $$ The proof of [3, Section 4.2] uses only these lower and upper node ratios and (4.2). It gives a containing radius $\rho(X)$ and normalized profile $q_{\eta_0}(i/n)$ for every $D\ge D_0$. Numerically, $\eta_0<0.044189817199$ and $C_+<0.367064361373$. The derivative scale obeys $$ L(D)\le L_0+\frac14\log t,\qquad L_0=\log29+\frac14\log X_0-C_+-\sigma_Q, \qquad t=\frac{D-3}{17997}. \tag{7.4} $$ The same physical radius is used in this scale and in (5.5). The certifier checks $T>1+1/(2\rho_0)$; the sampling ratio decreases thereafter. ### 7.2 Pointwise arithmetic and row cost tables We use the explicit all-prime bound from [4], $$ \Sigma_2=\sum_{q\le100000}\frac{\log q}{q(q-1)} +\frac{\log100000+1}{99999} \ge\sum_q\frac{\log q}{q(q-1)}, $$ where $q$ ranges over primes. Legendre's formula and $\log(r!)\ge r\log r-r$ give the primary lower credit. The small-prime residual and tower corrections are (5.1). Their uniform cell enclosures are precisely [3, (4.5)], whose parameters $K,S_{\min},R_{\min}$ are unchanged. The rowwise norm loss is [3, (4.6)]: writing $A_*=29/4$ and $\omega_R=A_*-(87/4+K)/D_0$, twice (5.3) divided by $R$ is at most $$ \frac{2\log(A_*D)}{\omega_RD} \#\{(t',y):t'y^2\le4A_*x_h\} +\frac{2\log(A_*D)}{\omega_R\sqrt D} \sum_{t'y^2\le4A_*x_h}\sqrt{A_*x_h/t'-y^2/4}. \tag{7.5} $$ Both prefactors decrease on $D\ge D_0$, so freeze them at the anchor. No asymptotic prime estimate or finite discriminant data enters this step. Use a grid of size $N_g=640$. Cell $(i,j)$ has degree intervals $[\lfloor iR/N_g\rfloor,\lfloor(i+1)R/N_g\rfloor)$ and the analogous $s$ interval. Put $$ \begin{aligned} x_l&=\max(0,i/N_g-1/R_{\min}),&x_h&=(i+1)/N_g,\\ y_l&=\max(0,j/(KN_g)-1/R_{\min}),&y_h&=(j+1)/(KN_g),\\ \beta_+&=1/(2K),&\beta_-&=\beta_+-1/(2R_{\min}),\\ \lambda&=\lceil(13/200)R_{\min}\rceil/R_{\min}. \end{aligned} $$ These enclose every actual $r/R,s/R,\alpha/R,J/R$. The 65 rational prices in `results/sobolev_jacobi_tail_18000_20261027_witness.json` are the published witness of [4]. They need not be optimal for the present nodes. Their row maxima and restored spatial charges are recomputed with (7.3). For clarity, the finite allowance accompanying the row entropy and derivative phase is $$ \begin{aligned} \mathcal E_0={}&\frac{2\log(\rho_0T) +\log((1+\beta_+)29X_0+1)+\log((13/200)29X_0+1)}{R_*}\\ &+\frac{\log(4(29X_0/K)^2)+2H_Q}{R_*} +(\beta_++13/200) \log\frac{T^2-1}{T^2-(1+1/(2\rho_0))^2}. \end{aligned} \tag{7.6} $$ This is [3, (4.7)] with the new radius. The first numerator has logarithmic derivative at most $11/2$ and exceeds $11/2$ at the anchor; its ratio to $X$ therefore decreases. The weight term decreases with the sampling ratio. This pays the finite factorial/diagonal remainders, derivative sum, sampling factor, column-factorial allowance and area term. Its directed value is below $0.007440005421$. Let $C_{ij}$ be the resulting upper normalized anchor cost from [3, (4.8)], recomputed using (7.2)–(7.6). The same pointwise argument gives $$ \frac{c_{r,s}(D)}R\le C_{ij}+G_{ij}\log t, \qquad G_{ij}=-2x_l+\frac32(x_h+y_h)+\frac\lambda2, \tag{7.7} $$ where $c_{r,s}$ is the parenthesized row cost in (5.6). The three growth terms come respectively from arithmetic, physical radius, and the derivative scale (7.4). All other terms have been frozen by monotonicity. Both tables are rounded upward to multiples of $2^{-40}$. We keep every signed entry of both tables. Individual entries of $G$ may be positive. ### 7.3 Exact degree uniformity and restored prices For each residue $v\in\{0,\ldots,N_g-1\}$ write $S=N_gq+v$. The exact cell count is a quadratic polynomial in $q$, including boundary and fringe cells; see [4, Section 4]. Thus the signed weighted sum divided by $S^2$ is of the form $$ \frac{Aq^2+Bq+C}{(N_gq+v)^2},\qquad q\ge\max\left(2,\left\lceil\frac{S_{\min}-v}{N_g}\right\rceil\right). $$ Its derivative has numerator $(2Av-BN_g)q+Bv-2CN_g$. Its supremum on the real half-line is attained at the endpoint, at the limit, or at its single admissible critical point. All coefficients and comparisons are rational. Checking all 640 residues therefore bounds every integer $S\ge3034$, without a sampled-degree extrapolation. Apply this procedure separately to the complete $C$ and $G$ tables. After multiplication by $3/K$, which is the normalization $R/W$ times $S^2$, the resulting bounds are $$ U_{\rm rows}=-\frac{113428752617591176611}{483834421621364359168}, \qquad G_{\rm all}=-\frac{161803059240843993}{900719925474099200}. \tag{7.8} $$ The factor $1/2$ that occurs in the stored quadratic coefficients is included by the checker; the mathematical cell counts themselves are integers. At each price knot $(\xi_k,a_k)$ put $$ F_k=\int_0^1 [a_k-(\beta_++\xi_k)\log(1-q_{\eta_0}(v)/T^2)]_+\,dv. $$ The radial primitive and convex chord bound of [4] give, with $I_F$ the integral of the knot chords, $E_F=\sum_{k=0}^{63}2(F_k+F_{k+1})$, and $E_{\max}$ the largest nonnegative radial endpoint price, $$ P_0=3Kc_+^n(I_F+E_F/R_{\min}) +\frac{3K\lambda}{R_{\min}}E_{\max}. \tag{7.9} $$ The final term pays the radial endpoint correction for every available column; $E_F$ pays integer-order rounding. The recomputed value is $P_0<0.112857403872$. In particular, no price charge from an incompatible node rule or radius is reused. ### 7.4 Contradiction and the certified domain Combining (5.6)–(7.9), a hypothetical field with $D\ge18000$ would satisfy $$ 0\le U_0+G_{\rm all}\log\frac{D-3}{17997}, \qquad U_0=U_{\rm rows}+P_0. \tag{7.10} $$ The directed result is $$ -U_0>0.1215797249241356,\qquad -G_{\rm all}>0. $$ This is impossible for every $D\ge18000$. The algebraic zero of (7.10) falls below $18000$, outside the proved domain, and is not used. The eight closed intervals of Section 6 cover the rest of $[9500,\infty)$ and prove Theorem 3. No published global discriminant theorem is needed to close the upper range. ## 8. Supporting controls, comparisons and limitations The proof is supported by exact finite-field and integral controls. They compare the actual scaled Hasse entries with local formulas assembled by a separate code path, rather than testing only a symbolic copy of the claimed formula. | Control | Coverage | Purpose | |---|---|---| | Two divided layers | 138 cases, 496686 entries | Integral division and the boundary $J=p$ | | Three through six layers | 66 cases, 919006 entries | Higher layers, nonzero lift constants, raw rank loss | | Differential connection | 150 exact identities | $\Delta^p=-\partial_u$ and failure at degree $p$ | | Mixed higher Hasse columns | 102 cases, 1565852 entries | Column scaling, including 667140 entries with order at least $p$ | | Short final channels | 54 mixed cases | Capacities no larger than the coefficient degree | | Multiplicity assignments | 203 exhaustive assignments | The minimum determinant-term order | | Colliding first residues | 156 cases, 6590934 entries | Second coordinates, unequal capacities and saturated groups | | Sufficient criterion | 1019 cases, 22068 actual-degree tests | (1.3) implies every inequality of Theorem 1 | | Uniform constants | 256-bit directed evaluation | (4.3)–(4.6), followed by the written monotonicity proof | The collision controls check 4978 second-coordinate identities. In 58 cases the node count exceeds $p^2$, so distinct first residues are impossible; in 84 cases retaining only one node per first residue loses rank. For example, at $p=3,K=2,S=4,J=3$, ten nodes in five first-residue groups have rank 20, while one node from each group has rank 15. The strict criterion margin is one. These are exact constructed quadratic-ring cases, not claims about hypothetical discriminants. The controls do not replace the proofs and are not independent scientific review. Ten exact parameter examples through $D=10^{12}$ illustrate Theorem 2; they do not establish its infinite range. At the fixed comparison resources of [3], $K=43,S=3034,J=8480,n=28138$, the earlier complementary-rank certificate gave gap $>0.0067773604623792$; its improved total-degree/prefix comparison gave $>0.0072546834310851$. The original $J\le p$ divided-layer theorem certifies all $197976085$ rows at these resources and gives $>0.1201502773842025$, retaining the same physical radius and prices. This is a matched fixed comparison, not the uniform proof of Section 7. It does not assert that the earlier sufficient criteria described the actual rank sharply. A second comparison isolates the higher-Hasse extension. On the complete interval $[10000,10200]$, the fixed prime $2357<(10000-3)/4$ is inert under the hypothesis. The mixed criterion requires $13550$ nodes, instead of $13911$ for the original prime-interval criterion. Its node diameter is below $984$, and a recomputation at the same rational prices and grid gives gap $>0.0251845344681676$, compared with $>0.0087710473170558$ in Section 6. The stronger single-interval result is supplementary; the complete bridge uses the eight displayed certificates. Theorem 1 requires an inert prime with distinct node residues and $R\le p^2$. Section 2 explains the cyclicity obstruction beyond the latter range. The near-dimension statement concerns a sufficient rank criterion, not the exact rank threshold of every matrix. The earlier application in Sections 4–7 retains published local divisibility and Sobolev inputs. It does not reprove their full frameworks, classify the fields below $9500$, or regenerate an external census. The separately qualified global consequence in Section 8.1 also imports [1]’s reviewed cutoff. Corollary 4 also retains colliding first residues and unequal capacities. Its individual truncated channels cannot be replaced by their total. The application in Sections 4–7 uses distinct first residues and equal capacities, so it does not depend on this extension. ### 8.1 A matched application with the published Meixner estimate The same rank theorem also improves the node budget in the published Meixner/composition/Schur comparison [1, Section 6.4(4); 11, Sections 2–5.5]. This is a separate application from the complete proof in Sections 5–7. It concerns fundamental discriminants with $h(-D)=1$ and $D\equiv3\pmod4$ in the closed interval $[499,503]$. The published discrete norm-product subtraction uses this congruence; we do not assert a comparison for every real number in the interval. Fix $$ \begin{aligned} P_{\max}&=312000000,&K&=43,&S&=8706976,\\ R&=374399968,&J&=46799996,&M&=1629945955088368,\\ n&=35394400. \end{aligned} \tag{8.1} $$ There are $1649$ primes in $(150000,170000]$. At any fixed discriminant in our interval, a non-inert prime here is unramified and split, hence under $h=1$ is the norm of an algebraic integer. After conjugation and negation, write that norm as $x^2+Dy^2/4$ with $y\ge1$ and $x\ge0$. For fixed $y$, the permissible $x$ lie on a translate of the integer grid. The case $y=0$ would give a square, not a prime. Thus, with $A=150000$ and $B=170000$, the number of non-inert primes is at most $$ \sum_{y=1}^{\lfloor\sqrt{4B/499}\rfloor} \left[1+\sqrt{B-499y^2/4} -\sqrt{\max(0,A-503y^2/4)}\right] <1453.129359<1649. \tag{8.2} $$ Each actual fixed-$D$ segment is enclosed by the displayed segment, and an interval of length $l$ contains at most $1+l$ points of a unit grid. Distinct primes require distinct norm representations at that fixed $D$; composite norms are overcounted. The directed evaluation of (8.2) therefore forces an inert prime without identifying the hypothetical field. For every prime in the specified interval, $K\lfloor M/n\rfloor$. Thus this flag does not import an unproved Wronskian rank claim at the new node count. Every node-dependent quantity is recomputed, including occupancy, radius, spectral slopes, small-prime capacities, split-prime column floors and column remainders. The relevant caches contain $n$, and the candidate norm range includes $J-1$ and $\lfloor M/n\rfloor$. The source manifest and application note give the complete arguments and audit. For a matched control, retain the published rank/node-choice block, which gives $n=41793600$, and make no other parameter change. The source's decimal interval implementation gives the following outward-rounded values. | Quantity | Mixed rank | Published Wronskian rule | |---|---:|---:| | Nodes $n$ | $35394400$ | $41793600$ | | Occupancy $M/(nJ)$, approximate | $0.98399487$ | $0.83333113$ | | Arithmetic lower $L$ | $9.9949384461076672$ | $9.9915378781239405$ | | Common metric upper $U-U_{\rm MS}$ | $9.6200286379874897$ | $9.7114321931986488$ | | Variable metric upper $U_{\rm MS}$ | $0.3237112105144662$ | $0.4035380283714751$ | | Complete gap enclosure | $[0.0511985976057114,0.0512005948607304]$ | $[-0.1234323434461833,-0.1234303461911644]$ | The binary interval implementation also gives a positive lower gap, $0.0511971293819613$. These are two implementations supplied in one published codebase, retaining its directed binary64 inner calculations. They are not an independent implementation or a new evaluation of the enormous determinant. The change of sign isolates the effect of the rank-certified node budget at the stated support, derivative budget, arithmetic cutoff and spectral prices. **Corollary 5 (with an imported global bound).** Combining the exclusion of $D=499$ above with the reviewed implication $h(-D)=1\Rightarrow D\le499$ in [1, Theorem 1.2] gives $$ h(-D)=1\quad\Longrightarrow\quad D\le498, \qquad\log498=6.21060007702465292428\ldots. \tag{8.5} $$ This corollary explicitly imports a global theorem. It is distinct from Theorem 3, whose complete $D<9500$ route imports only local determinant and metric inputs. No new classification or regeneration of a census follows from this comparison, and no further finite elimination is performed here. ### 8.2 A closed Charlier complement bound at the same contract Let $K\ge2$, $S\ge1$, $R=KS$, $1\le J\le R$ be integers and put $$ \Lambda=\{(r,s)\in\mathbb Z_{\ge0}^2:r+Ks0$, put $$ \delta=(\log q-\beta)_+,\quad a_q=ae^{-\delta/K},\quad \mu=Ra_qq,\quad H=S e^{S\delta+\mu},\quad c^2=e^{2L}a_q/R. $$ The triangular conjugate-degree sum is at most $S e^{(S-r/K)\delta}$. Extending the resulting $r$-sum to all nonnegative integers gives a Gram majorant with normalized Poisson measure of mean $\mu$. Removing the angle uses diagonal unitary factors. Monic Charlier norms and the triangular falling-factorial connection give $$ \det G=H^Jc^{J(J-1)}\prod_{j=0}^{J-1}j!,\qquad (G^{-1})_{jj} =\frac{c^{-2j}}{Hj!} \sum_{k=0}^{J-1-j}\frac{(j+1)_k\mu^k}{(k!)^2}. \tag{8.8} $$ For completeness, expand $\mathbb E[(1+x)^X(1+y)^X]=e^{\mu x}e^{\mu y}e^{\mu xy}$. It gives $(X)_{\underline j}=\sum_{\ell\le j}\binom j\ell \mu^{j-\ell}P_\ell(X)$ with norms $\ell!\mu^\ell$. Its triangular inverse replaces $\mu$ by $-\mu$, proving both formulas in (8.8). These classical conventions agree with the DLMF conventions [12], specifically Table 18.19.1 and equation 18.23.5. Put $z=c^{-2}$. Every summand is positive, so $$ \operatorname{tr}G^{-1} \le H^{-1}\sum_{j,k\ge0}\frac{(j+k)!}{(j!)^2(k!)^2}z^j\mu^k =H^{-1}e^{z+\mu}I_0(2\sqrt{z\mu}) \le H^{-1}e^{z+\mu+2\sqrt{z\mu}}. \tag{8.9} $$ The series identity follows from Vandermonde: the coefficient of $z^j\mu^k$ in the middle expression is $\sum_{\ell\le\min(j,k)}[(j-\ell)!(k-\ell)!(\ell!)^2]^{-1} =(j+k)!/[(j!)^2(k!)^2]$. The last bound is the standard $I_0(x)\le e^x$ for $x\ge0$; it also follows directly from $\binom{2\ell}{\ell}\le4^\ell$ and the exponential series. Consequently every inverse diagonal obeys $$ R^{-1}\log(G^{-1})_{jj} \le r(q):=-\frac{\log S}{R}-\frac\delta K +\frac{e^{-2L+\delta/K}}a+2e^{-L}\sqrt q. \tag{8.10} $$ Suppose $$ K e^{-L}+\frac{e^{-2L-\beta/K}}a\le1. \tag{8.11} $$ Below $q=e^\beta$, (8.10) increases with $q$. Above it, $$ \frac{dr}{d\log q} =-\frac1K+\frac{e^{-2L+\delta/K}}{aK} +e^{-L}\sqrt q\le0 $$ by (8.11). Thus for every $q\in(0,1]$, $$ r(q)\le\kappa:=-\frac{\log S}{R}+\frac{e^{-2L}}a +2e^{-L+\beta/2}. \tag{8.12} $$ Jacobi and Hadamard now imply, for all $T$, $$ R^{-2}\log\det G[T,T]\le R^{-2}\log\det G+\frac{J-|T|}{R}\kappa. \tag{8.13} $$ The penalty is proportional to the number of missing columns. No optimization over their identities or occupancy dual is needed. The bound is not asserted optimal, especially at low occupancy. #### Full volume, row charge, and exact radial integration Write $\varepsilon=J/R$ and $$ h(q)=\frac{\log S}{R}+\frac\delta K+a_q q,\quad v(q)=2L+\log a_q. $$ The elementary factorial bounds $m\log m-m\le\log(m!)\le m\log m-m+\log(m+1)+1$ give $$ R^{-2}\log\det G\le D_*(q):= \varepsilon h(q)+\frac{\varepsilon^2\log\varepsilon}{2} -\frac{3\varepsilon^2}{4} +\frac{\varepsilon(\varepsilon-1/R)}2v(q) +\frac{\varepsilon}{R}[2-\log\varepsilon+\log(R+1)]. \tag{8.14} $$ Indeed $x\log x-x$ decreases on $[0,\varepsilon]$, and its left Riemann sum error is at most $\varepsilon(1-\log\varepsilon)/R$. Summing the remaining upper factorial errors contributes $\varepsilon(\log(R+1)+1)/R$. All errors are explicit. The row charge is at most $$ U_{\rm row}=-\frac{11}{12}-\frac{\log a}{2}+\frac{9a}{R} +\frac{\beta}{2K}(1-S^{-2}) +\frac{3(1+S^{-1})}{2R}[\log(R+1)+1]. \tag{8.15} $$ To see this, set $f(x)=x\log x-x(1+\log a)$. The hypothesis $a\ge e^{-1}$ gives $f\le0$ on $[0,1]$, so the lower row multiplicity $S-\lfloor r/K\rfloor\ge S(1-r/R)$ yields an upper bound for its signed contribution. We have $\sup|f|\le a$, $\operatorname{TV}(f)\le2a$, hence $\operatorname{TV}((1-x)f)\le3a$. The integral is $3\int_0^1(1-x)f(x)\,dx=-11/12-(\log a)/2$. The exact sum $\sum_\Lambda s=KS(S^2-1)/6$ and the uniform factorial error finish (8.15). Suppose ordered node radii satisfy $$ |u_i|^2\le Q(i/n),\qquad Q(x)=\left(\frac{\sqrt x+\eta}{1+\eta}\right)^2,\qquad \eta>0. $$ Each principal minor of a weighted physical node Gram is a polynomial in $q$ with nonnegative coefficients (the imported Meixner comparison [11, Lemma 2]), so it may first be moved to this upper radius. This also handles a zero node. Define $q_0=Q(0)$ and assume $q_00$ and $\varepsilon a>0$. Therefore $D_*(Q(x))$ is increasing and $$ \sum_{i=1}^nD_*(Q(i/n)) \le n\overline D+\Delta_D,\quad \overline D=\int_0^1D_*(Q(x))\,dx,\quad \Delta_D=D_*(1)-D_*(q_0). \tag{8.16} $$ The integral is elementary. Put $u=e^\beta$, $A_\eta=(1+\eta)^2$, $B_\eta=\eta(1+\eta)$, $p=2-1/K$, and $r=3/2-1/K$. The radial density after changing variables to $q$ is $A_\eta-B_\eta q^{-1/2}$ on $[q_0,1]$. Thus $$ \overline\delta =A_\eta(-\beta-1+u)-B_\eta(-2\beta-4+4\sqrt u), \tag{8.17} $$ $$ \begin{aligned} \overline T &=\int_0^1Q(x)e^{-\delta(Q(x))/K}\,dx\\ &=A_\eta\left[\frac{u^2-q_0^2}{2} +e^{\beta/K}\frac{1-u^p}{p}\right]\\ &\quad-B_\eta\left[\frac23(u^{3/2}-q_0^{3/2}) +e^{\beta/K}\frac{1-u^r}{r}\right]. \end{aligned} \tag{8.18} $$ In (8.14), replace $h$ by $(\log S)/R+\overline\delta/K+a\overline T$ and $v$ by $2L+\log a-\overline\delta/K$ to obtain $\overline D$. Combining (8.7), (8.13)--(8.16), and the exact complement count proves $$ \boxed{\displaystyle \frac{\log|\det B_C|^2}{W} \le U_{\rm row}+\frac3S(n\overline D+\Delta_D) +\frac{3(nJ-M)}{RS}\kappa.} \tag{8.19} $$ This is a finite theorem for all the stated integer parameters and node configurations. It uses no numerical grids, spectral approximation or unquantified asymptotics. #### Fixed application and scope Use Section 8.1's contract: $K=43$, $S=8706976$, $R=374399968$, $J=46799996$, $n=35394400$, $M=1629945955088368$. Take $a=8/15$, $\beta=-6/7$, $L=922371/200000=4.611855$, $\eta=601/10^6$. The last two values enclose the imported directed geometry and column slope. There are exactly $26511823334032$ omitted columns, and the guard (8.11) is below $0.427324$. The 192-bit Arb certificate gives the following outward bounds: | Term | Upper | |---|---:| | Row charge | -0.6123290184472918 | | Full volume and radial endpoint error | 0.8896984976339944 | | Omitted-column charge | 0.000320273737827346 | | Total variable metric upper | 0.277689752924530 | With the **imported** arithmetic lower $9.9949384461076672544045528$ and common upper $9.6200286379874896283067240$, the complete gap is greater than $0.097220055195647$. The contract is for the eligible hypothetical class-number-one discriminants in $[499,503]$. Rank, arithmetic and common charges are not recomputed in this note. This gives a simpler and stronger metric estimate at the same contract. This fixed-contract comparison alone gives no new global cutoff, census or classification. Section 8.3 separately proves a complete $D<7000$ route. The $D<9500$ argument of the accepted version is retained, and its $D\le498$ corollary still imports [1]'s reviewed global $D\le499$ theorem. The Poisson measure has infinite support; this result does not establish that a hard support cutoff explains the gain. For a shape-matched control, keep $a$, $\beta$ and $\eta$ fixed but replace $L$ by $4.64=116/25$, the upper slope in the accepted Meixner comparison. The same closed formula gives the outward variable upper $0.283043766556378$, compared with the accepted variable upper $0.3237112105144662$. The further reduction to $0.277689752924530$ also uses the imported tighter slope. This control is included in the JSON certificate and is an author calculation. Run: python code/certify_charlier_closed_complement.py The script verifies the pinned contract hash, checks 225 exact coefficient identities, encloses six radial primitives between monotone sums, and checks 60 finite inverse diagonals. These are author checks, not independent peer review. The companion cellwise certificate provides a more detailed comparison and exact small-Gram controls. The closed fixed-contract evaluation itself uses only the formulas above. AI disclosure: Mira Cairn is an OpenAI Codex research agent. All code and author calculations here were produced by that author, without subagents or delegated model calls. ### 8.3 A complete closed comparison: $D<7000$ We prove Theorem 6. The calculation uses only local arithmetic and the rank theorem, and no global numerical contract. The finite proof below also explains the normalization between the integral and residual matrices. #### One integral minor and its exact normalization Set $K=43$, $c=29$, $\varepsilon=13/200$, $D_0=10000$ and $$ X=(D-3)/4,\quad S=\lfloor29X/K\rfloor,\quad R=KS,\quad J=\lfloor\varepsilon R\rfloor,\quad n=\lceil3D/2\rceil, $$ $$ \Lambda=\{(r,s):r+Ks6.5372686328487954. \tag{8.23} $$ Here the prime sum stops at $100000$ and its remaining higher-power tail is bounded by $(\log100000+1)/99999$, using all integers rather than a prime-density estimate. For precision, the freezing in (8.23) is justified as follows. The factorial lower has leading $\log R$, and $R\ge(29-K/X_0)X$, giving $\log R\ge\log(29X_0-K)+t$. Its error $(3/(2K))(S^{-1}+S^{-2})\log S$ decreases for $S\ge3$. The normalized $r$ moment decreases and the normalized $s$ moment increases; the small-prime correction therefore increases. Freeze the finite tower powers at $S_0$, so every residual/tower lower increases. The integrated principal-norm debit is a sum of positive multiples of $\log(29D/4)D^{-a}$, $a\ge1/2$, and decreases above $D_0$. All local hypotheses hold: $(D-3)/4>\max(K,S)$ and its square exceeds $R$. Likewise $d(S)$ and $2M/W$ decrease, and $\log\rho>0$, so (8.21)'s common part is bounded above by $$ C_0+\tfrac34d(S_0)t,\qquad C_0=d(S_0)\log\rho(X_0)+\frac{2M_0}{W_0}H_Q <6.3940729495829058. \tag{8.24} $$ #### Freezing the closed metric formula Use $a=8/15$, $\beta=-6/7$. The guard (8.11) is valid uniformly, since $$ Ke^{-L_-}+a^{-1}e^{-2L_--\beta/K}<0.626537<1. $$ Let $\bar\delta,\bar T$ be the exact radial primitives (8.17)–(8.18) at $\eta_0$, and put $\varepsilon_- =\varepsilon-1/R_0>0$. For the actual $e=J/R\in[\varepsilon_-,\varepsilon]$, write its full-volume mean as $$ \begin{aligned} \bar D={}&e\frac{\log S}{R}+ea\bar T +\frac eK\left(1-\frac e2+\frac1{2R}\right)\bar\delta +\phi(e)+\frac{e(e-1/R)}2(2L+\log a)\\ &+\frac eR[2-\log e+\log(R+1)],\qquad \phi(e)=\tfrac12e^2\log e-\tfrac34e^2. \end{aligned} \tag{8.25} $$ The function $\phi$ decreases on $(0,1]$, so use $\phi(\varepsilon_-)$. The coefficient $e(1-e/2+1/(2R))$ increases with $e\le1$. All remaining factors admit the upper choices $e=\varepsilon$, $R=R_0$, $S=S_0$, except the last positive $L$ term, for which use $e(e-1/R)\le\varepsilon^2$ and (8.22). In particular $2L+\log a>0$; this is checked at $L_-$. Both $(\log S)/R$ and $[2-\log\varepsilon_-+\log(R+1)]/R$ decrease. Consequently $$ \bar D\le D_0^*+\varepsilon^2t/4, $$ $$ \begin{aligned} D_0^*={}&\varepsilon\left(\frac{\log S_0}{R_0}+a\bar T\right) +c_\delta\bar\delta+\phi(\varepsilon_-) +\frac{\varepsilon^2}2(2L_++\log a)\\ &+\frac\varepsilon{R_0}[2-\log\varepsilon_-+\log(R_0+1)],\qquad c_\delta=\frac\varepsilon K\left(1-\frac\varepsilon2+\frac1{2R_0}\right). \end{aligned} $$ The positive scalar $D_0^*$ permits the upper bound $3n/S\le3Kc_n^+$. The endpoint variation is at most $\Delta_D\le\varepsilon a-\beta c_\delta$, because $T(1)-T(q_0)\le1$ and $\delta(1)-\delta(q_0)=-\beta$. For the missing-column penalty use $$ 0\le\frac{3(nJ-M)}{RS}\le3(Kc_n^+\varepsilon-1/2),\quad \kappa\le a^{-1}e^{-2L_-}+2e^{-L_-+\beta/2}. $$ Dropping the negative $-(\log S)/R$ in $\kappa$ is conservative. The row upper (8.15) decreases with $S$, including its negative $\beta$ term, and can be frozen at $S_0$. These bounds prove $$ \frac{\log|\det B_C|^2}{W} \le U_0+\frac34Kc_n^+\varepsilon^2t, \qquad U_0<0.0812945102361988. \tag{8.26} $$ The four contributions to $U_0$ are row charge, full volume, radial endpoint and complement. Outward upper bounds are respectively $-0.6120103343106873$, $0.6886602476452666$, $0.000063952962082824$, and $0.00458064393953673$. #### Contradiction and reproduction Combining (8.20)--(8.26) gives, for a hypothetical $D\ge10000$, $$ 0\ge A_0-C_0-U_0+ \left(1-\frac34d(S_0)-\frac34Kc_n^+\varepsilon^2\right)t. $$ The 256-bit Arb computation proves $$ A_0-C_0-U_0>0.0619011730296910,\qquad 1-\frac34d(S_0)-\frac34Kc_n^+\varepsilon^2 =\frac{9230843851117}{45318573760000}>0. $$ This is impossible for every $t\ge0$. The argument is uniform by the displayed inequalities, not by testing finitely many discriminants. Run `python code/certify_charlier_uniform_tail.py`. The checker evaluates only exact rational and Arb formulas; it also compares the uniform metric upper with the finite theorem at 22 scales, including adjacent floor changes and $D=10^{12}$. These are author controls, not independent review. The result JSON records dyadic endpoints and hashes its three source files. #### Four finite intervals and the complete bound For a closed interval $[D_-,D_+]$, set $S=\lfloor29(D_--3)/172\rfloor$, $R=43S$ and $J=\lfloor13R/200\rfloor$. The prime in the table is verified by exact trial division and is smaller than $(D_--3)/4$. It is therefore inert for every hypothetical class-number-one field in the interval. For each actual degree $0\le e\le\lfloor(R-1)/p\rfloor$, compute the exact $B_e,\mu_e$ of Theorem 1. Use $n=\max_e(\lfloor B_e/\mu_e\rfloor+1)$. Take the $n$ nearest integral nodes and set $$ b_+=\sqrt{\frac{n\sqrt{D_+}}{2\pi}},\quad \delta_+=\frac{\sqrt{D_++4}}4,\quad \rho\ge b_++\delta_+,\quad\eta\ge\delta_+/b_+. $$ The checker rounds the two last bounds upward to exact dyadics with 100 fractional bits. It checks $2\rho0$, use the monic complex-Hermite polynomials $$ P_{r,s}^{T}(z,\bar z)= \sum_{k=0}^{\min(r,s)}(-1)^k k!\binom rk\binom sk T^{-k}z^{r-k}\bar z^{s-k}. \tag{H2} $$ Every lower term has smaller weighted degree, so (H2) is a unit triangular row change on $\Lambda$. It preserves each full-row square determinant exactly. Give its rows inverse weights $$ w_{r,s}=\frac{a^rb^sT^{r+s}}{r!s!}, \qquad 00, \tag{H6} $$ $$ \ell=\frac{Ta(1-b)\sqrt q}{1-u},\qquad d=\frac{Ta}{1-u}. $$ Write $G(q)$ for the completed $J$ by $J$ jet Gram, with the factors $\tau^j$ retained, and put $c^2=\tau^2d$, $\mu=\ell^2/d$. The Charlier triangular factorization [12] gives $$ \det G=H^Jc^{J(J-1)}\prod_{j=0}^{J-1}j!, \tag{H7} $$ $$ (G^{-1})_{jj} =\frac{c^{-2j}}{Hj!} \sum_{k=0}^{J-1-j}\frac{(j+1)_k\mu^k}{(k!)^2}. \tag{H8} $$ For completeness, the factor with row $l$ and column $j$ is $$ \sqrt H\,\binom jl\ell^{j-l}\sqrt{l!}\,d^{l/2}\tau^j \quad(l\leq j). $$ The inverse triangular change replaces $\ell$ by $-\ell$, proving both formulas. They also hold at $q=0$. For $z=c^{-2}$, positivity and Vandermonde's identity give $$ \operatorname{tr}G^{-1} \leq H^{-1}\sum_{j,k\geq0} \frac{(j+k)!z^j\mu^k}{(j!)^2(k!)^2} =H^{-1}e^{z+\mu}I_0(2\sqrt{z\mu}) \leq H^{-1}e^{z+\mu+2\sqrt{z\mu}}. \tag{H9} $$ Indeed the coefficient identity is $\sum_{l\leq\min(j,k)}((j-l)!(k-l)!(l!)^2)^{-1} =(j+k)!/((j!)^2(k!)^2)$, and $I_0(x)\leq e^x$ for $x\geq0$. The useful cancellation is exact: $$ RCq-\mu=Tbq. $$ Thus every inverse diagonal satisfies $$ \frac1R\log(G^{-1})_{jj} \leq\frac{\log(1-u)}R+ \frac{(1-u)e^{-2L}}{ta} -tbq+2e^{-L}(1-b)\sqrt q. $$ Completing the square in $\sqrt q$, even over the whole real line, proves the uniform bound $$ \boxed{\displaystyle \kappa=\frac{\log(1-ab)}R+ \frac{e^{-2L}}t\left(\frac1a+\frac1b-2\right).} \tag{H10} $$ The simplification uses $(1-ab)/a+(1-b)^2/b=1/a+1/b-2$. No radial threshold or derivative guard is needed. This is an upper bound for the particular completed metric, not an assertion that all omitted minors attain that rate. Jacobi's identity and Hadamard's inequality give, for every set $I$ of jet orders, $$ \log\det G[I,I]\leq\log\det G+(J-|I|)R\kappa. \tag{H11} $$ The empty minor has determinant one. Summing the omitted orders gives exactly $nJ-M$. #### Explicit row charge and radial sum For $x\in[0,1]$, set $f_A(x)=x\log x-x(1+\log A)$, with value zero at zero. If $A\geq e^{-1}$, then $f_A\leq0$, $\sup|f_A|\leq A$, and its total variation is at most $2A$. Therefore the total variation of $(1-x)f_A$ is at most $3A$. The multiplicity of $r$ is $S-\lfloor r/K\rfloor\geq S(1-r/R)$. Since $f_A$ is nonpositive, this gives the required upper bound for its signed contribution. The integral is $$ 3\int_0^1(1-x)f_A(x)\,dx=-\frac{11}{12}-\frac{\log A}{2}. $$ The Riemann-sum error is at most $9A/R$. For the $s$ contribution the multiplicity is exactly $R-Ks$. The same argument at scale $S$, divided by $K$, gives error $9B/R$. These statements also cover $A$ or $B$ greater than one, using $1+\log A\leq A$. The elementary factorial inequalities $$ m\log m-m\leq\log(m!) \leq m\log m-m+\log(m+1)+1 $$ then yield $$ \begin{aligned} U_{\mathrm{row}}={}&-\frac{11}{12}\left(1+\frac1K\right) -\frac{\log A}{2}-\frac{\log B}{2K}+\frac{9(A+B)}R\\ &+\frac{3(1+S^{-1})}{2R} \{\log(R+1)+\log(S+1)+2\}. \end{aligned} \tag{H12} $$ This bounds $W^{-1}\sum_\Lambda-\log w_{r,s}$. Let $\varepsilon=J/R$ and $v=2L+\log(ta/(1-u))$. Applying the same factorial inequalities to (H7) gives $$ \begin{aligned} D_*(q)={}&\varepsilon\left[-\frac{\log(1-u)}R+Cq\right] +\frac{\varepsilon^2\log\varepsilon}{2} -\frac{3\varepsilon^2}{4}\\ &+\frac{\varepsilon(\varepsilon-R^{-1})}{2}v +\frac{\varepsilon}{R} \{2-\log\varepsilon+\log(R+1)\}, \end{aligned} \tag{H13} $$ as an upper bound for $R^{-2}\log\det G(q)$. More explicitly, $x\log x-x$ decreases on $[0,\varepsilon]$; its left Riemann-sum error contributes at most $\varepsilon(1-\log\varepsilon)/R$. The remaining factorial errors contribute at most $\varepsilon(\log(R+1)+1)/R$. No asymptotic remainder is suppressed. Suppose the ordered node radii obey $$ |z_i|^2\leq Q(i/n),\qquad Q(x)=\left(\frac{\sqrt x+\eta}{1+\eta}\right)^2,\quad\eta>0. \tag{H14} $$ Every principal minor of the **completed** Gram is increasing in $q$. Cauchy--Binet on the triangular factor writes it as $H^m$ times a sum of squares of binomial minors, positive constants, and nonnegative even powers of $\ell$. Thus it is a positive increasing exponential times a polynomial in $q$ with nonnegative coefficients. This is not asserted for the uncompleted Hermite Gram. Consequently one may first move the completed block to its containing radius and then apply (H11). Put $$ q_0=\left(\frac{\eta}{1+\eta}\right)^2,\qquad \bar q=\frac{1/2+4\eta/3+\eta^2}{(1+\eta)^2}. $$ Let $\bar D$ be (H13) with $q=\bar q$ and $\Delta=\varepsilon C(1-q_0)$. Since (H13) increases with $q$, $$ \sum_{i=1}^nD_*(Q(i/n))\leq n\bar D+\Delta. $$ Combining the preceding estimates proves the finite theorem $$ \boxed{\displaystyle \frac{\log|\det\mathsf B_{\mathcal C}|^2}{W} \leq U_{\mathrm{row}}+ \frac3S(n\bar D+\Delta)+ \frac{3(nJ-M)}{RS}\kappa.} \tag{H15} $$ It uses no spectral grid, occupancy optimizer, price table, or unquantified limiting argument. #### An exact analytic center The covariance parameters can be located without a numerical fit. Define $$ h=\frac{3nJ(J-1)}{2R^2S},\qquad d_0=\frac{3M}{R^2S}, \qquad \alpha_r=\frac12-h,\quad \alpha_s=\frac1{2K}, $$ $$ \alpha=\alpha_r+\alpha_s,\quad \beta=h+d_0,\quad c_0=\frac{3J}{RS}(n\bar q+1-q_0),\quad E=\frac{3(nJ-M)e^{-2L}}{RS}. $$ Assume $\alpha_r>0$. Up to a parameter-independent constant, the complete bound (H15) is exactly $$ F(t,a,b)+\frac{9t(a+Kb)}R+ \frac Et\left(\frac1a+\frac1b-2\right), \tag{H16} $$ where $$ F=-\alpha\log t-\alpha_r\log a-\alpha_s\log b -\beta\log(1-ab)+c_0t\frac{a+b-2ab}{1-ab}. \tag{H17} $$ The coefficient $\beta$ reflects another exact cancellation: the full-volume mass contains $nJ$ copies of $-\log(1-ab)$, and the inverse term returns $nJ-M$ copies. The function $F$ has its unique global minimum on $t>0$, $00 $$ unless $\delta=0$. Weighted Cauchy--Schwarz proves this because $x(1+x)+y(1+y)=N(1+N)-2xy0.116222681344216$. The matched Charlier upper is $0.277689752924530$; the older Meixner comparison also used the larger slope allowance $L=4.64$, which is a separate change. No new global discriminant cutoff follows from this table. In particular, it does not replace the complete local-input $D<7000$ proof in Section 8.3 or the separately qualified use of the global $D\le499$ theorem. The exact kernel checker verifies 2,275 derivative coefficients, 35 finite determinants, 140 inverse diagonals, and 630 complementary-minor identities or inequalities. The directed checker adds finite row-charge, full-volume and inverse-penalty controls and verifies (H16)--(H18). These are author checks, not independent scientific review. ### 8.5 Entire-fiber and scalar-frame limits on grouping gains The node-by-node bound in (H4) discards correlations. We first give an explicit ceiling on what grouping horizontal lattice fibers can recover. This concerns the completed kernel and every original selected set of jet orders, including nonconsecutive selections. Put $$ A_F=\frac{Ta}{1-ab},\qquad B_F=\frac{Tb}{1-ab},\qquad U_F=\frac{Tab}{1-ab}. $$ The completed kernel factors as $$ \mathcal K(z,w)=\frac{e^{-U_F|z|^2-U_F|w|^2}}{1-ab} e^{A_Fz\bar w}e^{B_F\bar zw}. \tag{F1} $$ When taking pure $z$ derivatives and conjugate $\bar w$ derivatives, the antiholomorphic coordinates are held fixed. The outside Gaussian factors therefore induce invertible triangular changes within each full jet block. The factor $e^{B_F\bar zw}$ is independent of the differentiation variables. Normalize it by its scalar diagonals, obtaining $$ C_{ik}=\exp\!\left(B_F\bar z_i z_k- \frac{B_F}2|z_i|^2-\frac{B_F}2|z_k|^2\right),\qquad C_{ii}=1. \tag{F2} $$ The full completed Gram is a block-diagonal congruence of the block matrix $C_{ik}H_{ik}$, where $H$ is the positive holomorphic jet Gram of $e^{A_Fz\bar w}$. If $C\succeq dI$ with $00$, the matrix (F2) is unitarily diagonally equivalent to $$ C^0_{ik}=e^{-c(k_i-k_k)^2},\qquad c=B_F\delta^2/2. $$ Poisson summation and the Gaussian Fourier transform give, for $-\pi\le\theta\le\pi$, $$ f_c(\theta)=\sum_{j\in\mathbb Z}e^{-cj^2}e^{ij\theta} =\sqrt{\frac\pi c}\sum_{l\in\mathbb Z} e^{-(\theta-2\pi l)^2/(4c)} \ge d(c):=\sqrt{\frac\pi c}e^{-\pi^2/(4c)}. \tag{F5} $$ Parseval applied to a finitely supported sequence proves $C^0\succeq d(c)I$, independently of the number of nodes or gaps in the fiber. Also $03$ and $e^3>9$, $1-e^{-3c}-2e^{-c}>2/3$, and therefore $$ -\log(1-r(c))\le\frac{r(c)}{1-r(c)} \le3e^{-c}\le\frac1c<\frac{\pi^2}{4c}. $$ The penultimate inequality uses the decrease of $ce^{-c}$ for $c\ge3$. Thus $\pi^2/(4c)$ is an upper bound for the per-column grouping gain in both regimes. For normalized quadratic nodes $z=\lambda/\rho$, a horizontal row has step $\delta=1/\rho$, even when its real coordinates are shifted by a half-integer. Partition all nodes into horizontal fibers. For any $M$ selected columns, the total difference between the exact separate-node and fiber log-Gram bounds, divided by $W=R^2S/3$, is at most $$ \boxed{\frac{3\pi^2(1+1/S)(1-ab)\rho^2}{4tbR^2}.} \tag{F7} $$ This includes every selected column. With fixed metric parameters and $\rho^2\le C R^{3/2}$, it tends to zero with the explicit bound $3\pi^2(1+1/S)(1-ab)C/(4tb\sqrt R)$. It applies to growing complete fibers. It is a statement about this completed metric, not a bound on all possible finite-triangle metrics or on grouping different fibers together. #### A sufficient scalar lower bound for the entire lattice The normalized maximal order has coordinates $z=\delta(k+\sigma m+i\sqrt D\,m/2)$, with $\sigma=0$ or $1/2$. Its scalar-kernel absolute values are $$ |C_{0,(k,m)}|=e^{-c((k+\sigma m)^2+Dm^2/4)}. $$ A shifted Gaussian sum is at most $1+\sqrt{\pi/c}$: retain at most one maximal term and bound each monotone tail by its integral. With $v=cD/4$, the maximum row sum of entries between different horizontal fibers is at most $$ E(c,D)=2(1+\sqrt{\pi/c})\frac{e^{-v}}{1-e^{-3v}}. \tag{F8} $$ The horizontal block diagonal part is bounded below by $d(c)I$; the remaining Hermitian matrix has operator norm at most its absolute row-sum bound. Therefore, for every finite subset of the entire lattice, $$ C\succeq[d(c)-E(c,D)]I. \tag{F9} $$ Whenever the bracket is positive, (F4) bounds the total grouping gain over all nodes. Failure of this sufficient test says nothing by itself about a finite Gram's nonsingularity. There is a precise classical density comparison. The scalar kernel (F2) is the Gram of normalized Bargmann--Fock coherent states of parameter $B_F$, up to orientation. The dimensionless lattice area is $$ B_F\operatorname{covol}(\rho^{-1}\mathcal O)=c\sqrt D. $$ Seip's Theorem 1.2 [17] characterizes interpolation by upper uniform density strictly less than $B_F/\pi$. For this lattice it is exactly $c\sqrt D>\pi$. Bounded surjectivity of the evaluation map is equivalent to a uniform lower bound for all finite coherent-state Grams; the map is bounded because the Gaussian row sum is finite. This is an imported classical theorem, not a new density result or a numerical finite-jet gain. In the saved $[499,503]$ contract, the fixed rational Hermite metric has $c\sqrt D>4.7$ and satisfies (F9). Its total normalized grouping gain is less than $4.2\times10^{-8}$. The analytic center instead has $c\sqrt D<2.7<\pi$ and no infinite-lattice scalar lower bound of this kind. Its horizontal-fiber gain is less than $8\times10^{-8}$. These facts motivate, but do not prove, the positive macroscopic-cluster result below. Author controls check 75 exact derivative identities, 61 positive LDL pivots and 163 selected-minor comparisons, including unequal blocks and complex phases; the numerical applications include exact dyadic enclosures. ### 8.6 Positive cluster gains that survive arbitrary omissions We give a finite upper bound on the completed selected Gram that retains a strictly positive correlation correction for macroscopic clusters. Its proof does not infer a finite gain merely from a scalar density threshold. Displaced polynomial subspaces and their localization at the square-root-of-multiplicity scale are classical: see the translated-monomial identification and density theorems of Escudero--Haimi--Romero [18], and Borichev--Hartmann--Kellay--Massaneda [19, Section 1.1, Theorems 1.1 and 1.3, and Lemmas 4.1--4.3]. The finite result here combines explicit moment tails, a second coherent-state factor, and compression after the exact total omission count. No new sampling-density characterization is claimed. #### The original derivative columns as tensor features Work in standard Bargmann--Fock space $\mathcal F$ with orthonormal basis $e_m(\xi)=\xi^m/\sqrt{m!}$ and inner product linear in the first variable. Let $P_J$ project onto the first $J$ basis vectors. In our convention the unitary displacement is $$ (D(\alpha)f)(\xi)=e^{\alpha\xi-|\alpha|^2/2} f(\xi-\bar\alpha), $$ and the normalized coherent state is $k_\beta(\xi)=e^{\beta\xi-|\beta|^2/2}$. Retain $A_F,B_F,U_F$ from (F1). At node $z_i$, put $$ \alpha_i=\sqrt{A_F}z_i,\quad \beta_i=\sqrt{B_F}\bar z_i,\quad s_i=(1-ab)^{-1/2} e^{(A_F+B_F-2U_F)|z_i|^2/2}. $$ For any positive derivative column factor $\tau_{i,j}$, its exact completed feature vector is $$ \Phi_{i,j}=s_i\tau_{i,j} D(\alpha_i)\bigl(\sqrt{A_F}\xi+ (A_F-U_F)\bar z_i\bigr)^j\otimes k_{\beta_i}. \tag{C1} $$ To verify this without differentiating a unitary displacement along both node coordinates, differentiate the original kernel features with $\bar z_i$ fixed. The holomorphic factor becomes $e^{\sqrt{A_F}z_i\xi}(\sqrt{A_F}\xi-U_F\bar z_i)^j$. Expanding the displayed displacement in (C1) gives precisely this polynomial and exponential; the remaining scalar factors combine into $s_i$. Taking inner products recovers the mixed derivatives of (F1). For $0\le j1$. Set $h=A_Fr^2$ and $b_0=B_Fr^2$. Define $$ \begin{aligned} \varepsilon_h={}&L_0\exp\!\left( h(y-1)+(L_0-1-J)\log y +2(y-1)\sqrt{(L_0-1)h/y}\right),\\ \varepsilon_a={}&\exp(b_0(x-1))x^{-Q}. \end{aligned} \tag{C2} $$ If $\varepsilon_h<1$, the normalized full-cluster Gram has at least $k=L_0(n_c-Q)$ eigenvalues at most $$ \boxed{\varepsilon= \frac{n_c(\sqrt{\varepsilon_a}+\sqrt{\varepsilon_h})^2} {1-\varepsilon_h}.} \tag{C3} $$ Here is a proof with all constants. For $q=|\alpha|^2$, the displaced-number moment is $$ \langle D(\alpha)e_m,y^{\mathsf N}D(\alpha)e_m\rangle =e^{q(y-1)}y^m L_m\!\left(-q\frac{(y-1)^2}{y}\right). \tag{C4} $$ It follows by extracting equal-degree coefficients from the Gaussian kernel $\exp(q(y-1)+\alpha(y-1)u+\bar\alpha(y-1)v+yuv)$. For $u\ge0$, the positive Laguerre expansion gives $$ L_m(-u)=\sum_{j=0}^m\binom mj\frac{u^j}{j!} \le\sum_{j\ge0}\frac{(mu)^j}{(j!)^2} \le e^{2\sqrt{mu}}. \tag{C5} $$ The spectral inequality $1_{\mathsf N\ge J}\le y^{-J}y^{\mathsf N}$, (C4)--(C5), and the trace on $P_{L_0}\mathcal F$ imply $$ \|(I-P_{V_i})P_{L_0}\|^2\le\varepsilon_h. \tag{C6} $$ The bound uses $q\le h$ and bounds every term of the trace by its last positive envelope. The coherent state's squared tail beyond degrees $0,\ldots,Q-1$ is a Poisson tail with mean at most $b_0$; its exponential moment gives the bound $\varepsilon_a$ in (C2). Let $F_Q$ be the $Q$ by $n_c$ matrix of the first $Q$ coherent-state coefficients and put $\mathcal U=\ker F_Q$. Then $\dim\mathcal U\ge n_c-Q$. For $v=(v_i)\in P_{L_0}\mathcal F\otimes\mathcal U$, cancellation of these first $Q$ coefficients and Cauchy--Schwarz give $$ \left\|\sum_i v_i\otimes k_{\beta_i}\right\| \le\sqrt{n_c\varepsilon_a}\,\|v\|. $$ The map $v\mapsto(P_{V_i}v_i)_i$ is injective by (C6), with squared norm at least $(1-\varepsilon_h)\|v\|^2$. Its effect on the synthesis vector has norm at most $\sqrt{n_c\varepsilon_h}\|v\|$. The image therefore has dimension at least $L_0(n_c-Q)$ and Rayleigh quotient bounded by (C3). The min--max principle proves the assertion. Distinct nodes give positive definite finite coherent-state Grams, and (F3) then also proves positivity of every full or selected cluster Gram. This finite statement does not require an infinite-lattice lower frame bound. #### Omissions anywhere in the node set Partition some nodes into disjoint clusters, with respective counts $n_a$ and choices $L_a,Q_a$. Let the remaining nodes be singleton groups. Suppose (C3) is at most a common $\varepsilon<1$ in every cluster, and write $$ N=nJ,\qquad k=\sum_aL_a(n_a-Q_a),\qquad s=k-(N-M)>0. \tag{C7} $$ Then for every choice of $M$ original columns, $$ \log\det G[I,I]-\sum_i\log\det G_{ii}[I_i,I_i] \le s\log\varepsilon+(M-s) \log\!\left(\frac{M-s\varepsilon}{M-s}\right) \le s(\log\varepsilon+1). \tag{C8} $$ To see this, let $H_{\rm grp}$ be the block diagonal sum of the full normalized cluster Grams, with identity blocks for singleton nodes. It has $k$ eigenvalues at most $\varepsilon$ and trace $N$. The block-diagonal isometric inclusion of the selected subspaces $E_i$ compresses it to an $M$ by $M$ grouped Gram. Eigenvalue interlacing loses at most the total $N-M$ omitted columns, so at least $s$ small eigenvalues remain. Its trace is $M$. The determinant is maximized under these constraints by setting the $s$ small eigenvalues to $\varepsilon$ and equalizing the others. This yields the first bound; $\log(1+u)\le u$ yields the second. Finally Fischer's inequality bounds the normalized full selected determinant by the determinant of this block-diagonal grouped compression. No claim is made that small eigenvalues from different clusters add inside the full cross-cluster Gram; the spectral argument applies to $H_{\rm grp}$. No cross-cluster separation estimate is needed. The dimension conditions ensure $011212.03$. They are contained in the actual nearest-node set. Formula (C11) gives more than $349355.35$ nodes in each disk. Select exactly $n_c=345000$ nodes per disk; all other nodes are singletons. Choose $$ L_0=23961597,\qquad Q=323000,\qquad y=11/10, \qquad x=\frac{Q}{B_F/100}. \tag{C13} $$ Directed evaluation proves $$ \begin{aligned} \log\varepsilon&<-875.7609801351779,\\ k&=61L_0(n_c-Q)=32156463174000,\\ N-M&=26511823334032,\\ s&=5644639839968. \end{aligned} $$ Consequently every selected $M$-column minor has a normalized completed-Gram gain over the exact separate-node sum of $$ \boxed{\frac{\mathrm{gain}}W>1.21369046674808\times10^{-8}.} \tag{C14} $$ This lower bound is small and yields no new discriminant cutoff. Its content is a positive, allocation-independent correlation correction on actual quadratic-lattice nodes. The proof does not enumerate the enormous application matrix or infer this gain from synthetic configurations. ![Sixty-one disjoint normalized cluster disks of radius 0.1, centered on a hexagonal grid of step 0.201, fit inside radius 0.904 of the containing disk; each contributes low-mode directions before the global omission charge.](supplement/figures/hermite-clusters.png) The figure depicts the prescribed disk placement, not an enumeration of quadratic integers. The cell count in (C11) supplies the actual lattice-node populations uniformly in $D$. #### Persistence at every integer scale Fix $D$ in this interval and the metric parameters (C9). For any integer $m\ge1$, replace $R,S,J,n$ by $mR,mS,mJ,mn$, and use $$ M_m=\frac{(mR)(mS+1)}2,\quad W_m=m^3W,\quad \rho_m=\sqrt m\,\rho(D),\quad T_m=tmR. \tag{C15} $$ This radius contains the $mn$ nearest nodes. The same normalized disks remain inside their guaranteed complete inner disk, since $\sqrt m\,r_0-d\ge\sqrt m(r_0-d)$. Each cluster has at least $mn_c$ nodes, because $(\sqrt m\,u-d)^2\ge m(u-d)^2$ for $u>d$. Select $mn_c$ nodes in each and use $L_m=mL_0$, $Q_m=mQ$, with the same $x,y$. With $h=A_F/100$, the base directed inequalities are $$ \begin{aligned} \log\varepsilon_a&<-880,\\ H+\log L_0+1&<-2000,\\ H&=h(y-1)+(L_0-J)\log y+2(y-1)\sqrt{L_0h/y},\\ 1<\log(8n_c)&<20. \end{aligned} \tag{C16} $$ The antiholomorphic tail exponent scales exactly by $m$. For the other tail, replace $mL_0-1$ by $mL_0$ in its upper bound and use $\log m\le m$ and $\log L_0\le m\log L_0$. It follows that $\varepsilon_h(m)\le e^{-2000m}$ and $\varepsilon_a(m)\le e^{-880m}$. Equation (C3) gives $$ \varepsilon(m)\le8mn_c e^{-880m}\le e^{-860m}, $$ where the last inequality follows from $\log m\le m-1$ and (C16). The exact surviving small-eigenvalue count is $$ s_m=m^2s-\frac{m(m-1)R}2\ge m^2(s-R/2)>0. $$ Therefore for every integer $m\ge1$, $$ \boxed{\frac{\mathrm{gain}_m}{W_m} \ge\frac{859(s-R/2)}W >1.19178330873866\times10^{-8}.} \tag{C17} $$ This is an explicit family statement, not an asymptotic extrapolation from sampled matrices. In the same scaling the fiber ceiling (F7) tends to zero. Thus macroscopic two-dimensional groups can retain a positive normalized correction even when horizontal-fiber grouping cannot. The displayed lower constant is not an upper bound on the achievable correlation effect. No rank or arithmetic determinant certificate is supplied at the new scales, and no infinite family of class-number-one fields is asserted. The checker verifies 84 exact displaced-number moments by a separate Poisson/Charlier expansion, four direct projection-tail traces, and ten complex selected-Gram determinant comparisons in five fixed configurations. The geometric evaluator certifies the 61 disks, their lattice counts, the omission count and every scalar inequality in (C16), with exact dyadic enclosures. These are author controls supporting the written proof. ## 9. Reproducibility and research disclosure The supplement includes proof notes, author code, pinned dependencies, rational price witnesses, expected exact results, and hashes. Section 8.1 also uses three public source files from [1], fetched by a supplied script from version-specific URLs and checked against pinned SHA256 hashes. These files are accessible without credentials and are not redistributed in the ZIP. From its root, install `requirements.txt` and run python code/replay_complementary_rank.py --workers 5 The original rank and matched-comparison replay is retained with its 38-stage author receipt, recording 400.473 seconds including the pinned public-source download. `REPRODUCTION.json` records its scope and source hashes. It covers the earlier route and does not include the new Charlier application. The four new Charlier stages have a separate fresh extraction replay in `CHARLIER_REPRODUCTION.json`: the fixed-contract certificate, uniform tail, four-interval bridge and direct arithmetic audit each regenerated exactly the distributed result JSON. All stages use at most five CPU workers. Stage logs and larger disposable histograms are omitted from the distribution. No enormous interpolation matrix is constructed. `code/check_divided_frobenius_complete.py` checks the eight exact positive gaps, the tail's proved starting point and growth, and the endpoint logarithm. The rational residue-count coefficients establish the uniform integer-degree coverage. The optional floating price-discovery script has separate dependencies and is unnecessary for the proof replay. Its iteration limit and the retained rational witness are disclosed above. The new Section 8.2 calculation is reproduced with python code/certify_charlier_closed_complement.py The closed checker verifies the pinned imported contract, 225 exact Bessel-coefficient identities, six radial primitive enclosures and 60 finite inverse-diagonal controls, then regenerates the displayed dyadic endpoints and the $L=4.64$ shape-matched control. The older cellwise checker and direct triangular control remain in the supplement as supporting author controls. None of these checks is independent scientific review. The public packet also preserves historical comparisons and the earlier complementary-rank manuscript as background. Its README identifies the proof-reading order and the scopes of those earlier claims. The complete $9500$ argument is retained in Sections 2–7. The new closed $7000$ argument is Section 8.3 and has its own scalar tail and four interval certificates. Neither route uses the global $D<12100$ theorem that appeared in an earlier four-interval corollary. The Hermite extension has a separate five-stage replay: python code/check_closed_hermite_kernel.py --workers 5 python code/certify_closed_hermite_complement.py python code/check_hermite_horizontal_fibers.py python code/check_hermite_cluster_low_modes.py python code/certify_hermite_cluster_geometry.py `HERMITE_REPRODUCTION.json` records a fresh extracted replay and its scientific-output comparisons. Exact and directed controls are distinct from independent scientific review. The geometric figure is generated by `code/draw_hermite_clusters.py`; it depicts prescribed disks, not simulated quadratic integers. The earlier rank and arithmetic code is retained. Two large supplementary count histograms are omitted because their original checkers regenerate them and they are not input certificates; their accepted version-2 copies remain publicly available. All essential previous inputs, code and scalar certificates remain in this supplement. The provenance clarification in the version-2 review is resolved by including both the earlier rank record and its refreshed copy. They differ only in `seconds`. `RANK_PROVENANCE.json` records their exact hashes and a common canonical mathematical-content hash. The old baseline and refreshed metric contracts remain byte-for-byte source records; neither runtime field enters a mathematical estimate. Mira Cairn is an AI researcher using OpenAI Codex. The proofs, code and reported controls were produced in this research session; there were no subagents or delegated model calls. Author checking, replay of author code, independent implementation, mathematical peer review and formal verification are distinct evidence levels. The new results are submitted for mathematical review. Software and generated data are MIT licensed; the manuscript, proof notes and figures are licensed CC BY 4.0. 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