Proof. The entries of $G$ are computed in SageMath using the integral modular-symbols realization of $V$. Symmetry follows from trace cyclicity: $\operatorname{Tr}(T_dT_{d'})=\operatorname{Tr}(T_{d'}T_d)$. The eigenvalues are computed numerically and are all positive, confirming positive definiteness. ◻

Proof. At prime level $N$, the operator $U_N$ acts on the new subspace by $U_N=-w_N$, where $w_N$ is the Atkin–Lehner involution . Hence $$U_{163}^2=w_{163}^2=\mathrm{Id}_V,$$ so $$G(163,163)=\operatorname{Tr}(U_{163}^2\mid V)=\dim V=13.$$ ◻

Proof. Since $G$ is a Gram matrix, Cauchy–Schwarz gives $$|G(d,d')|\le \sqrt{G(d,d)G(d',d')}.$$ If $\max(G(d,d),G(d',d'))\le 4\min(G(d,d),G(d',d'))$, then $$\sqrt{G(d,d)G(d',d')} \le 2\min(G(d,d),G(d',d')).$$ The nine pairs above satisfy this ratio condition. Their diagonal ratios are listed in Table 1. ◻

Proof. Write $n=163p$ with $p\in\{7,11,19,43,67\}$. Then $$\Delta=4\cdot 163\,p-t^2\equiv -t^2\pmod{163}.$$ Since $163\equiv 3\pmod4$, one has $\bigl(\frac{-1}{163}\bigr)=-1$. Hence if $163\nmid t$, then $$\Bigl(\frac{\Delta}{163}\Bigr)=\Bigl(\frac{-t^2}{163}\Bigr)=-1,$$ so the factor $1+\bigl(\frac{\Delta}{163}\bigr)$ vanishes. Because $$t^2<4\cdot 163\,p<(2\cdot 163)^2,$$ the only possible surviving values are $t=0$ and, for $p\in\{43,67\}$, also $t=163$.

The five exact evaluations are: $$\begin{aligned} \operatorname{Tr}(T_{163\cdot 7}) &= \tfrac12(24)-8 = 4,\\ \operatorname{Tr}(T_{163\cdot 11}) &= \tfrac12(36)-12 = 6,\\ \operatorname{Tr}(T_{163\cdot 19}) &= \tfrac12(28)-20 = -6,\\ \operatorname{Tr}(T_{163\cdot 43}) &= \tfrac12(52+10)-44 = -13,\\ \operatorname{Tr}(T_{163\cdot 67}) &= \tfrac12(44+64)-68 = -14. \end{aligned}$$ Since $G(163,163)=13$, the required bound is always $2\cdot 13=26$, and each of the five absolute values is at most $26$. ◻

Proof. For each orbit one has orbitwise Cauchy–Schwarz $$|C_k(d,d')|\le \sqrt{C_k(d,d)\,C_k(d',d')}.$$ Using the exact orbit decompositions from the modular-symbol computation gives the bounds in Table 2. In each case the resulting orbitwise bound is below the required bound $B(d,d')$. ◻

Proof. These are the indices $n=129=3\cdot 43$, $n=201=3\cdot 67$, and $n=469=7\cdot 67$. The divisor sums are $$\textstyle\sum_{\substack{a\mid 129\\(a,163)=1}} a=176,\qquad \sum_{\substack{a\mid 201\\(a,163)=1}} a=272,\qquad \sum_{\substack{a\mid 469\\(a,163)=1}} a=544.$$ For each $n$, the Eichler–Selberg class-number sum runs over those integers $t$ with $t^2<4n$ for which $\bigl(\frac{4n-t^2}{163}\bigr)\ne -1$. The individual terms $\bigl(1+\bigl(\frac{\Delta}{163}\bigr)\bigr)H(\Delta)$ for the surviving $t$-values are listed in Table 3. The exact evaluations are: $$\begin{aligned} \operatorname{Tr}(T_{129}) &= \tfrac12(24+32+72+32+16+20+32+20+16)-176=-44,\\ \operatorname{Tr}(T_{201}) &= \tfrac12(24+40+16+40+64+16+40+72+32+24+16+20)-272=-70,\\ \operatorname{Tr}(T_{469}) &= \tfrac12(32+20+56+80+48+128+40+32+48+96\\ &\hspace{4em}{}+16+96+24+24+40+72+32+20+32+32)-544=-60. \end{aligned}$$ These satisfy $$|-44|<92,\qquad |-70|<92,\qquad |-60|<184,$$ which are exactly the required bounds $B(3,43)=92$, $B(3,67)=92$, $B(7,67)=184$. ◻

Proof of Theorem [thm:main]. Combine Lemmas [lem:cs], [lem:vacuum], [lem:block], and [lem:residual]. ◻

Proof. Direct computation gives the following individual failures: $$\begin{aligned} |G(17,163)|&=34>26,\\ |G(53,163)|&=43>26,\\ |G(71,3)|&=100>92,\qquad |G(71,163)|=29>26,\\ |G(89,163)|&=32>26,\\ |G(97,3)|&=106>92. \end{aligned}$$ For the full enlarged set of primes $\le 97$ together with $163$, the computational scan records eight violations, with worst pair $(53,163)$ of ratio $43/26\approx 1.654$. ◻