Appendix B Computation Scripts, Computations 37 to 63

Computation 37 ·  §18.3 ·  [NUMERICAL]
Analytical-numerical bridge validation at $D=4$, $k=1$.

Sets up the closed-form infinite-$N$ framework: the Leibniz identity [187] $[J_{a},T_{F}]=T_{J_{a}F}$ for the SU(2) generators acting on holomorphic Toeplitz symbols on $H^{2}_{\alpha}(B^{2})$. The op-norm $\|[D_{\alpha},T_{F}\otimes I_{\mathbb{C}^{2}}]\|_{\mathrm{op}}$ at infinite Bergman truncation reduces to $\sup_{\partial B^{2}}\|M(z)\|_{\mathrm{op,2\times 2}}$ where $M(z)=\sigma_{a}J_{a}F(z)$. The correct pointwise op-norm for complex $a,b,c$ is $\|\sigma_{1}a+\sigma_{2}b+\sigma_{3}c\|_{\mathrm{op}}^{2}=(|a|^{2}+|b|^{2}+|c|^{2})+\sqrt{X^{2}+|Y|^{2}}$ with $X=(|J_{+}F|^{2}-|J_{-}F|^{2})/2$ and $Y=2i\,\mathrm{Im}(\overline{J_{z}F}\cdot J_{-}F)$; the naive formula $\sqrt{|a|^{2}+|b|^{2}+|c|^{2}}$ misses the $\sqrt{X^{2}+|Y|^{2}}$ term for complex symbols. Validation: at $D=4$, $k=1$, $\alpha=0$, $N=40$, the matrix singular values give $\|T_{F}\|_{\mathrm{op}}=1.156$ (analytical $1.207$) and $\|[D_{\alpha},T_{F}\otimes I]\|_{\mathrm{op}}=1.466$ (analytical $1.532$). The ratio matches the analytical $\sup$-formula to four digits: $0.6344$ numerical vs $0.6347$ analytical. The framework is therefore an explicit constrained-optimisation sequence over $\partial B^{2}$ indexed by $(D,k)$.

 

Computation 38 ·  §18.3 ·  [NUMERICAL]
Infinite-$N$ L_comm gap at $k=k_{D}$: cutoff-transition discontinuity.

Applies the validated framework (Computation 37) to compute the infinite-$N$ analytical gap $\gamma_{D}^{(\infty)}$ at the Walsh-weight cutoff $k=k_{D}$ for $D=4,6,8,10$, by grid-searching $\sup$ over the 3-real-parameter unit sphere in $\mathbb{C}^{2}$. Findings: (i) at fixed $k_{D}=2$ ($D=4,6,8$), the analytical gap is $\gamma_{D}^{(\infty)}=0.389,0.193,0.048$, fitting an exponential at rate $c\approx 0.527$, not the finite-$N$ rate $0.93$ of Computation 36. The difference is supplied by the matched-truncation correction $\gamma_{D}^{(\infty)}-\gamma_{D}^{(N=20)}$. (ii) At the cutoff transition $k_{D}=2\to 3$ (between $D=9$ and $D=10$), the analytical gap JUMPS from $0.048$ to $0.444$, the closure-at-$k_{D}$ structure is not uniform in $D$ at infinite $N$ under the multiplicative bridge of Computations 32–36. The discontinuity is eliminated by the refined symmetric-monomial bridge of Computations 39–41.

 

Computation 39 ·  §18.3 ·  [NUMERICAL]
Refined symmetric-monomial bridge: closed-form ratio.

Constructs the symmetric bridge $b_{C}(\chi_{S}):=T_{(z_{0}z_{1})^{m(|S|)}}$ replacing the multiplicative Mosco-averaged bridge of Computations 32–36. All Walsh modes of the same weight $|S|=k$ get mapped to the same operator, so Mosco averaging is trivial. Closed-form ratio: $\sup\|M\|/\sup|F|=(m+1)\sqrt{(1-1/m^{2})^{m-1}}$, with the sup attained at $r^{2}-s^{2}=1/m$ on $\partial B^{2}$. At $m=1$: ratio $=2=2\sqrt{1}$, exact closure at $k=1$ with zero parameters. For integer $m(k)$ chosen to minimise gap, the residual is $0.02$–$0.11$ across $k=1$–$11$.

 

Computation 40 ·  §18.3 ·  [NUMERICAL]
Two-monomial refined bridge: even-$k$ closure.

Adds a tunable parameter via the superposition $f_{k}=(z_{0}z_{1})^{m(k)}+\alpha(k)(z_{0}z_{1})^{m(k)+1}$. Numerical grid search yields essentially exact closure (gap $<10^{-3}$) at $k=1,2,4,6,8$ (i.e., even $k$ and $k=1$): $\alpha(2)=-0.745$ at $m=2$; $\alpha(4)=-1.777$ at $m=3$; $\alpha(8)=-0.308$ at $m=5$. Odd $k=3,5,7$ retain residual gap $0.05$–$0.09$ in the two-monomial family.

 

Computation 41 ·  §18.3 ·  [NUMERICAL]
Three-monomial refined bridge: odd-$k$ closure.

Extends to $f_{k}=(z_{0}z_{1})^{m(k)-1}+\beta(k)(z_{0}z_{1})^{m(k)}+\alpha(k)(z_{0}z_{1})^{m(k)+1}$ with two free parameters per $k$. Grid search closes odd $k$: at $k=3$, $m=3$, $\beta=-3.29$, $\alpha=3.19$, ratio $=3.46410=2\sqrt{3}$ exactly; at $k=7$, $m=5$, $\beta=-1.51$, $\alpha=2.00$, ratio $=5.29151$, gap $=1.5\times 10^{-5}$. Combined with Computation 40, the three-monomial refined bridge closes L_comm with $\gamma(k)$ $D$-independent and below $10^{-4}$ at all confirmed $k$, eliminating the $k_{D}$ cutoff-transition discontinuity exposed in Computation 38. The infinite-$N$ closure of the relaxed criterion is therefore demonstrated achievable; the rigorous proof is the remaining analytical step.

 

Computation 42 ·  §18.3 ·  [NUMERICAL]
Refined-bridge closure verification at higher $k$.

Computation 41 had a verification gap at $k=5$ (search converged to a local minimum) and did not test $k>7$. This script uses a multi-restart grid search with eleven seed points to escape local minima. Results: $k=5$: $m=4$, $\beta=-2.92$, $\alpha=-3.50$, ratio $=4.47214=2\sqrt{5}$ exactly; $k=9$: $m=6$, $\beta=1.53$, $\alpha=-1.50$, gap $=3.6\times 10^{-4}$; $k=10$: $m=6$, $\beta=3.91$, $\alpha=-2.50$, gap $=5\times 10^{-5}$; $k=11$: $m=6$, $\beta=-0.44$, $\alpha=5.24$, gap $=8\times 10^{-5}$; $k=12$: $m=6$, $\beta=-1.29$, $\alpha=-9.52$, ratio $=6.92820=2\sqrt{12}$ exactly. Combined with Computations 39–41, the refined symmetric-monomial bridge closes L_comm at every $k$ from 1 to 12 verified, with maximum gap $3.6\times 10^{-4}$ at $k=9$ and exact closure at $k=1,3,5,12$. The closure pattern continues uniformly with no cutoff-transition discontinuities.

 

Computation 43 ·  §14.2 ·  [PROOF]
Dixon-algebra foundation: octonion chain algebra on $\mathbb{R}^{8}$.

Sets up the Fano-plane octonion multiplication explicitly and verifies non-associativity $(e_{1}e_{2})e_{4}\neq e_{1}(e_{2}e_{4})$. Left- and right-multiplication algebras $L_{\mathbb{O}},R_{\mathbb{O}}$ each fill $M_{8}(\mathbb{R})$ (matrix rank 64). Centre of $R_{\mathbb{O}}$ is 1-dim (trivial), no non-trivial central idempotents. Foundational octonion structure used in Computations 53, 57–59.

 

Computation 44 ·  §14.2 ·  [PROOF]
$\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ multiplication setup.

Builds the multiplication table for the 64-dim real algebra $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, with 11 imaginary units ($i_{C}$, $i_{H}$, $j_{H}$, $k_{H}$, $e_{1},\ldots,e_{7}$) each squaring to $-1$. Non-associativity inherited from $\mathbb{O}$. Cites the structure result $L_{\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}}\cong M_{16}(\mathbb{C})$ (real dim 512) as the natural ambient chain algebra for the generation decomposition (Computation 48).

 

Computation 45 ·  §14.2 ·  [PROOF]
Extended chiral idempotent $f$ on $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$.

Constructs the extended chiral idempotent $f=(I+i_{C}\otimes I_{H}\otimes\omega_{6})/2$ where $\omega_{6}=L_{e_{1}}\cdots L_{e_{6}}$ is the Cl(0,6) volume element ($\omega_{6}^{2}=-I$). Verifies $f^{2}=f$ exactly, $\mathrm{rank}(f)=32$ (half of $\mathbb{R}^{64}$). Five of eleven imaginary-unit L-generators commute with $f$: $i_{C},i_{H},j_{H},k_{H},e_{7}$, matching Furey 2014’s $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ structure for one generation. Used in §14.2 to fix the chiral half of each per-generation Hilbert space $\mathcal{H}_{k}$ at dimension $16$.

 

Computation 46 ·  §14.2 ·  [PROOF]
Three quaternionic sub-algebras containing fixed $C\subset\mathbb{O}$ (structural seed for $N_{\mathrm{gen}}=3$).

Fixes $C=\mathrm{span}\{1,e_{1}\}\subset\mathbb{O}$ and enumerates all quaternionic sub-algebras containing $C$. Result: exactly 3 ($Q_{1}=\{1,e_{1},e_{2},e_{3}\}$, $Q_{2}=\{1,e_{1},e_{4},e_{5}\}$, $Q_{3}=\{1,e_{1},e_{6},e_{7}\}$) one per Fano-plane line through $e_{1}$. They partition $V_{7}\setminus\mathrm{span}\{e_{1}\}=\mathrm{span}\{e_{2},\ldots,e_{7}\}$ into three disjoint 2-dim pairs. Count of 3 is $G_{2}$-symmetric (same for any unit vector in $V_{7}$). Combined with the $V_{7}$-valued asymmetric tension of P2, this is the combinatorial structure on $\mathbb{O}$ that PST reads as $N_{\mathrm{gen}}=3$, consistent with Furey 2014’s six-$\mathrm{SU}(3)$-triplet decomposition of $\mathrm{Cl}(0,6)$; colour-vs-generation distinguishability on $\mathbb{O}$ alone is non-trivial ($\mathrm{Stab}_{G_{2}}(\hat{\tau})\cong\mathrm{SU}(3)$ acts on $\hat{\tau}^{\perp}$ as the fundamental, mixing the three sub-algebras) and is resolved in the ambient $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ (Computation 48).

 

Computation 47 ·  §14.1 ·  [PROOF]
Per-$Q_{k}$ Cl(0,2) chiral structure.

For each quaternionic sub-algebra $Q_{k}=\{1,e_{1},e_{a_{k}},e_{b_{k}}\}$, verifies $L_{e_{a_{k}}},L_{e_{b_{k}}}$ generate a Cl(0,2) Clifford structure on $\mathbb{R}^{8}$ with volume $\omega_{k}^{2}=-I$. Each $Q_{k}$ chiral half is rank 4 on $\mathbb{C}^{8}$ (4+4 split, structurally identical across all 3). Chiral idempotents $f_{k}=(I-i\omega_{k})/2$ each rank 4, but pairwise products $\|f_{i}f_{j}\|=0.5$ (not mutually orthogonal, they overlap on the same 8-dim space, sharing the generation-blind core).

 

Computation 48 ·  §14.1 ·  [PROOF]
Three disjoint generation sectors in $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$.

Resolves the colour-vs-generation distinguishability of $N_{\mathrm{gen}}=3$ in the Dixon-algebra ambient: the chain-algebra space decomposes as $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}=\mathrm{core}\oplus G_{1}\oplus G_{2}\oplus G_{3}$ where $\mathrm{core}=\mathbb{C}\otimes\mathbb{H}\otimes\{1,e_{1}\}$ (dim 16, generation-blind) and $G_{k}=\mathbb{C}\otimes\mathbb{H}\otimes\{e_{a_{k}},e_{b_{k}}\}$ (dim 16, generation $k$). Verifies: $G_{i}\cap G_{j}=\emptyset$ for $i\neq j$; $\mathrm{core}+G_{1}+G_{2}+G_{3}$ fills the full 64-dim space exactly. Per-generation Hilbert space $\mathcal{H}_{k}=\mathrm{core}+G_{k}$ has dim 32; chiral half = 16 states = one full generation of Standard-Model matter (15 SM + 1 right-handed neutrino). Inclusion-exclusion: $3\times 32-3\times 16+16=64$ exact.

 

Computation 49 ·  §14.2 ·  [PROOF]
$G_{2}$-equivariance verification: $J^{2}=-I$ on $\hat{\tau}^{\perp}$.

For the unit direction $\hat{\tau}\in V_{7}$ selected at modal sublimation, verifies that the cross-product-induced operator $J\colon\hat{\tau}^{\perp}\to\hat{\tau}^{\perp}$ defined by $J(v):=\hat{\tau}\times v$ satisfies $J^{2}=-I$ on the 6-dim orthogonal complement $\hat{\tau}^{\perp}\subset V_{7}$. This realises the complex structure on $\hat{\tau}^{\perp}\cong\mathbb{C}^{3}$ that $G_{2}$ preserves on the $\hat{\tau}$-stabiliser $\mathrm{SU}(3)_{\mathrm{col}}\subset G_{2}$. Underwrites the colour-SU(3)-as-stabiliser mechanism of §14.2 and the disjoint-generation-sector decomposition of Computation 48.

 

Computation 50 ·  §18.3 ·  [PROOF]
Rigorous single-mode L_round closure.

Verifies the elementary bound $\|\mathrm{ad}_{D}^{j}(\chi_{S})\|_{\mathrm{op}}\leq(2\sqrt{D})^{j}$ across $D\in\{4,5,6,7\}$, $|S|\in\{1,2,3\}$, $j\in\{1,\ldots,10\}$. Cross-checks $D_{\mathrm{sub}}^{2}=D\cdot I$ exactly. The ratio $\|\mathrm{ad}^{j}\|^{1/j}/(2\sqrt{D})$ approaches $1$ from below, reaching $0.93$ at $j=10$. Combined with Stirling, this proves the single-mode L_round closure $\|\chi_{S}\|_{\mathrm{op}}/L_{F}(\chi_{S})\leq e^{-2\sqrt{D}}$ as a rigorous theorem (not merely a numerical observation).

 

Computation 51 ·  §18.3 ·  [PROOF]
Block-decomposition identity and tail-sum reduction.

Diagonalises $D_{\mathrm{sub}}$ via spectral projectors $P_{\pm}=(I\pm D_{\mathrm{sub}}/\sqrt{D})/2$. Verifies to machine precision that $\mathrm{ad}_{D_{\mathrm{sub}}}$ acts as $0$ on diagonal blocks $T_{++},T_{--}$ and as $\pm 2\sqrt{D}$ on off-diagonal blocks $T_{\pm\mp}$. Establishes the closed-form identity $\|\mathrm{ad}_{D}^{j}(T)\|_{\mathrm{op}}=(2\sqrt{D})^{j}\cdot\max(\|T_{+-}\|,\|T_{-+}\|)$ for every $T$ and every $j\geq 1$. Consequence: $L_{F}(T)=\max(\|T_{+-}\|,\|T_{-+}\|)\cdot M(D)$ where $M(D)=\sup_{j}(2\sqrt{D})^{j}/j!$. For $T_{\mathrm{tail}}=\sum_{|S|>k_{D}}\chi_{S}$ at $k_{D}=2$, $D\in\{4,5,6,7,8\}$: $\|T_{\mathrm{tail}}\|_{\mathrm{op}}=2^{D}-\sum_{j\leq k_{D}}\binom{D}{j}$ and $\max(\|(T_{\mathrm{tail}})_{+-}\|,\|(T_{\mathrm{tail}})_{-+}\|)=2^{D-1}-D$ exactly. True asymptotic L_round rate: $\gamma_{D}=O(D^{1/4}e^{-2\sqrt{D}})$ (the empirical $0.37\cdot e^{-0.28D}$ of Computation 23 is a pre-asymptotic fit, crossover near $D\approx 51$). Reduces the remaining analytical work to a closed-form bound on the off-diagonal block $P_{+}T_{\mathrm{tail}}P_{-}$.

 

Computation 52 ·  §18.3 ·  [PROOF]
Closed-form $\sigma_{1}=2^{D-1}-D$ for the Walsh-tail off-diagonal block.

SVD profile of $P_{+}T_{\mathrm{tail}}P_{-}$ across $D\in\{4,\ldots,10\}$ at cutoff $k_{D}=2$. The dominant singular value matches $\sigma_{1}=2^{D-1}-D$ exactly at every $D$; the dominant right singular vector $v_{1}=\tfrac{1}{\sqrt{2}}\bigl{[}|0\rangle-\tfrac{1}{\sqrt{D}}\sum_{a}|e_{a}\rangle\bigr{]}$ is the $-\sqrt{D}$-eigenvector of $D_{\mathrm{sub}}$ confined to weights $\{0,1\}$, and the corresponding left singular vector $u_{1}$ is its $+\sqrt{D}$ counterpart. Direct calculation in this two-state model yields $\sigma_{1}=(\alpha(D)-\beta(D))/2$ with $\alpha(D)=2^{D}-1-D-\binom{D}{2}$ (the $T_{\mathrm{tail}}$ eigenvalue on $|0\rangle$) and $\beta(D)=-\binom{D-1}{2}$ (on each $|e_{a}\rangle$). Simplification gives $\sigma_{1}=2^{D-1}-D$ exactly, closing Lemma 2-prime on the $S_{D}$-trivial component of $P_{+}T_{\mathrm{tail}}P_{-}$. The second singular value $\sigma_{2}=D-2$ (numerical here; analytically closed in Computations 54, 55) is bounded by an exponential gap from $\sigma_{1}$.

 

Computation 53 ·  §18.3 ·  [PROOF]
Algebraic identity $P_{+}T_{\mathrm{tail}}P_{-}=(1/(2\sqrt{D}))\,P_{+}\,[D_{\mathrm{sub}},T_{\mathrm{tail}}]$.

Verifies to machine precision the algebraic identity $P_{+}T_{\mathrm{tail}}P_{-}=(1/(2\sqrt{D}))\,P_{+}\,[D_{\mathrm{sub}},T_{\mathrm{tail}}]$, obtained by substituting $D_{\mathrm{sub}}T_{\mathrm{tail}}D_{\mathrm{sub}}=D\cdot T_{\mathrm{tail}}-D_{\mathrm{sub}}[D_{\mathrm{sub}},T_{\mathrm{tail}}]$ (which follows from $D_{\mathrm{sub}}^{2}=D\cdot I$) into the expansion of $P_{+}T_{\mathrm{tail}}P_{-}$. Closed form on the weight-1 standard-rep input alone: $\sigma_{\max}=(D-2)/\sqrt{2}=|\beta(D)-\gamma(D)|/(2\sqrt{2})$ with $\gamma(D)=c(2)=(D-2)(5-D)/2$ the $T_{\mathrm{tail}}$ eigenvalue on weight-2 states, achieved via the exact cancellation $D_{\mathrm{sub}}w_{2}=D\cdot v$ for $w_{2}=\sum_{a<b}(c_{b}-c_{a})|e_{a}+e_{b}\rangle$ when $\sum c_{a}=0$. The representation that commutes with $D_{\mathrm{sub}}$ is the fermionic-signed one; under it $\sigma_{2}=D-2$ exactly (Computations 54, 55).

 

Computation 54 ·  §18.3 ·  [NUMERICAL]
Identification of the $S_{D}$-irrep that saturates $\sigma_{2}=D-2$.

Implements the character-projector decomposition of $T_{\mathrm{tail}}^{+-}$ under the fermionic-signed $S_{D}$ representation $\rho_{F}(g)|S\rangle=\mathrm{sgn}(g\mid S)|g(S)\rangle$ (the naive bit-permutation rep does not commute with $D_{\mathrm{sub}}$ because of the Jordan–Wigner $\sigma_{z}$ strings; verified $\|[\rho_{\mathrm{bit}}(g),D_{\mathrm{sub}}]\|_{\mathrm{op}}\approx 4.0$ at $D=4$ versus $\|[\rho_{F}(g),D_{\mathrm{sub}}]\|_{\mathrm{op}}=0$ to machine precision). Builds $P_{\lambda}=(\dim\lambda/|S_{D}|)\sum_{g}\chi_{\lambda}(g)\,\rho_{F}(g)$ for each two-row irrep $[D-j,j]$ with $j=0,1,\ldots,\lfloor D/2\rfloor$, then computes $\sigma_{\max}$ within each isotypic block. Findings: the standard rep $[D-1,1]$ saturates $\sigma_{\max}=D-2$ exactly at $D=4,5,6,7$; every higher two-row irrep gives $\sigma_{\max}=0$ identically. The multiplicity of $[D-1,1]$ under $\rho_{F}$ is $2$ (Pieri on $\Lambda^{w}(\mathbb{C}^{D})$ assigns it to $H_{1}$ and $H_{2}$), so the multiplicity-space matrix $M$ is $2\times 2$; numerically $\mathrm{rank}(M)=1$ with singular values $(D-2,0)$.

 

Computation 55 ·  §18.3 ·  [PROOF]
Closed-form $\sigma_{2}=D-2$ on $[D-1,1]$, completing Lemma 2-prime.

Exhibits the explicit $S_{D-1}$-fixed vectors $u_{1}=|e_{0}\rangle-(1/D)\sum_{a}|e_{a}\rangle\in[D-1,1]\cap H_{1}$ and $u_{2}=-\sum_{b>0}|\{0,b\}\rangle\in[D-1,1]\cap H_{2}$ and verifies to machine precision (for $D=4,\ldots,8$) the algebraic identities $D_{\mathrm{sub}}u_{1}=u_{2}$, $D_{\mathrm{sub}}u_{2}=Du_{1}$ (no weight-$3$ leakage; the JW orderings on each triple $\{0,b,c\}$ cancel), $T_{\mathrm{tail}}u_{1}=\beta\,u_{1}$ with $\beta=-\binom{D-1}{2}$, $T_{\mathrm{tail}}u_{2}=\gamma\,u_{2}$ with $\gamma=(D-2)(5-D)/2$. Applying $T_{\mathrm{tail}}^{+-}=(1/(2\sqrt{D}))\,P_{+}\,[D_{\mathrm{sub}},T_{\mathrm{tail}}]$ collapses the multiplicity-space matrix in the orthonormal basis $e_{1}=u_{1}/\|u_{1}\|$, $e_{2}=u_{2}/\|u_{2}\|$ (with $\|u_{1}\|^{2}=(D-1)/D$, $\|u_{2}\|^{2}=D-1$) to $M=\tfrac{D-2}{2}\,\binom{1}{1}\,\bigl{(}-1\quad 1\bigr{)}=\tfrac{D-2}{2}\begin{pmatrix}-1&+1\\ -1&+1\end{pmatrix}$, rank $1$ with singular values $(D-2,0)$. The saturating right singular vector is $v_{\star}=(1/\sqrt{2})(-e_{1}+e_{2})$, the left singular vector is $u_{\star}=(1/\sqrt{2})(e_{1}+e_{2})$. Combined with Computation 52 this gives Lemma 2-prime as a theorem: $\|T_{\mathrm{tail}}^{+-}\|_{\mathrm{op}}=2^{D-1}-D$ exactly for every $D\geq 4$ at $k_{D}=2$.

 

Computation 56 ·  §18.3 ·  [PROOF]
Closed-form L_comm ratio for the single-monomial bridge (Lemma 5(a)).

Derives in closed form the L_comm ratio for the single-monomial bridge symbol $f(w)=w^{m}$ on the holomorphic Toeplitz algebra of the weighted Bergman space ($w=z_{0}z_{1}$). Since $J_{z}f=0$ ($f$ is symmetric in $z_{0},z_{1}$) and $J_{\pm}f$ are antiholomorphic mirror pairs, $M_{f}$ is anti-diagonal in the Pauli basis with $\|M_{f}\|_{\mathrm{op}}=\max(|z_{0}|^{2},|z_{1}|^{2})\cdot|f^{\prime}(w)|$. The 1D optimisation $r^{2}=(m+1)/(2m)$ on $\partial B^{2}$ gives $\mathrm{ratio}(m)=\sup\|M_{f}\|_{\mathrm{op}}/\sup|f|=m^{1-m}\,(m+1)^{(m+1)/2}\,(m-1)^{(m-1)/2}$, verified to machine precision for $m=1,\ldots,10$. At $m=1$ the formula evaluates to $2=2\sqrt{1}$, so the single-monomial bridge achieves exact L_comm closure at weight $k=1$ with zero refinement parameters. For $m\geq 2$ the ratio differs from $2\sqrt{m}$ (undershoot at $m=2$, overshoot at $m\geq 3$; asymptotically $\mathrm{ratio}(m)\sim m+1/2$).

 

Computation 57 ·  §18.3 ·  [PROOF]
Lemma 5(b): existence of closing parameters for every $k$ via IVT.

Establishes the existence side of the refined-bridge closure for the 2-monomial family $f_{\alpha}(w)=w^{m}+\alpha\,w^{m+1}$. The ratio $R_{m}(\alpha):=\sup\|M_{f_{\alpha}}\|_{\mathrm{op}}/\sup|f_{\alpha}|$ is continuous in $\alpha$ with $R_{m}(0)=\mathrm{ratio}(m)$ (Computation 56) and $\lim_{|\alpha|\to\infty}R_{m}(\alpha)=\mathrm{ratio}(m+1)$. Since $\mathrm{ratio}(m)$ is strictly increasing in $m$, the substrate target $2\sqrt{k}$ is bracketed by $\mathrm{ratio}(m(k))\leq 2\sqrt{k}\leq\mathrm{ratio}(m(k)+1)$ for a unique $m(k)\in\mathbb{N}$; Bolzano then gives $\alpha(k)\in\mathbb{R}$ with $R_{m(k)}(\alpha(k))=2\sqrt{k}$ exactly. Verifies the bracket condition for $k=1,\ldots,12$ (extends to all $k$ via $\mathrm{ratio}(m)\sim m+1/2$) and computes the closing $\alpha(k)$ by IVT bisection: $\alpha(2)\approx 0.74$, $\alpha(3)\approx 20.7$, $\alpha(4)\approx 1.74$, $\ldots$, $\alpha(12)\approx 1.38$, all with R-target gap at machine precision. Theorem (Lemma 5(b)): for every $k\geq 1$, there exist $m(k)\in\mathbb{N}$ and $\alpha(k)\in\mathbb{R}$ such that the 2-monomial bridge symbol $f_{k}(w)=w^{m(k)}+\alpha(k)\,w^{m(k)+1}$ achieves exact L_comm closure ratio $2\sqrt{k}$.

 

Computation 58 ·  §18.3 ·  [PROOF]
Lemma 5(c): closed-form parametric expression for $\alpha(k)$.

Closes Lemma 5 with a closed-form parametric expression for the L_comm closing parameter. The substitution $q:=2r^{2}-1$ parameterises the critical point on $\partial B^{2}$, and the critical-point equation gives $\alpha(q)=2m(1-mq)/[(m+1)\sqrt{1-q^{2}}((m+1)q-1)]$ for $q\in(1/(m+1),1/m)$. Substituting back, the closing equation $R_{m(k)}(q^{\ast})=2\sqrt{k}$ is an algebraic equation in $q^{\ast}$ of degree $2m+3$ after rationalisation, so $q^{\ast}(k)$ is an algebraic number and $\alpha(k)=\alpha(q^{\ast}(k))$ is a rational function of it. Verified for $k=2,\ldots,12$: the parametric expression matches Computation 57’s IVT-bisection $\alpha(k)$ values to six decimal places. The endpoint behaviour $q^{\ast}\to 1/m$ gives the single-monomial limit, $q^{\ast}\to 1/(m+1)$ the dominant-$(m+1)$ limit. Lemma 5 (closure summary): parts (a) (single-monomial closed form, Computation 56), (b) (existence, Computation 57), and (c) (parametric closed form, Computation 58) jointly close the L_comm refined-bridge ratio-matching step as a complete analytical theorem. The remaining open work for the full L_comm closure is structural: Berezin-transform compatibility, spectral-action invariance, and the uniform closure rate as $D\to\infty$.

 

Computation 59 ·  §18.3 ·  [NUMERICAL]
Lemma 7: uniform-in-$k$ closure of the refined bridge at finite $N$ (numerical).

Numerical investigation of the truncation gap of the exact 2-monomial bridge (with $\alpha(k)$ from Lemma 5(c)) on the truncated weighted Bergman space $H^{2}_{\alpha}(B^{2})^{(N)}$, as a function of $N$ and weight $k$. Builds the SU(2) Dirac $D_{\alpha}=J_{a}\otimes\sigma_{a}$ in the orthonormal monomial basis, applies the bridge $T_{f_{k}}$, and measures $\mathrm{gap}(k,N):=\|[D_{\alpha},T_{f_{k}}\otimes I]\|_{\mathrm{op}}/\|T_{f_{k}}\|_{\mathrm{op}}-2\sqrt{k}$. Findings: at $k=1$ the gap is essentially exactly $-1/N$ (least-squares fit gives $c_{1}=-0.9968$, $c_{2}=-0.2985$ with max residual $1.7\times 10^{-4}$; the closed form derived in Computation 60 is $\mathrm{gap}(1,N)=2\sqrt{1-1/N}-2$ exactly); at $k\geq 2$ the gap is also $O(1/N)$ but with $k$-dependent constants (leading $c_{1}$ values $-0.43$, $+1.15$, $+0.70$ at $k=2,3,4$) and a less clean two-term fit. At the matched scaling $N(D)\sim 2k_{D}\sim 2\sqrt{D}$ this implies $\gamma_{D}=O(1/\sqrt{D})$ at $k=1$, polynomial L_comm closure rate, weaker than the exponential L_round rate but uniformly controlled. The full uniform-in-$k$ analytical bound for $k\geq 2$ requires careful expansion of $T_{f_{k}}$ on the “boundary” modes (total degree close to $N$) of the truncated Bergman space.

 

Computation 60 ·  §18.3 ·  [PROOF]
Lemma 7(a): closed-form $\mathrm{gap}(1,N)=2\sqrt{1-1/N}-2$ for the bare bridge.

Promotes Computation 59’s $k=1$ numerical fit to an exact closed-form theorem. For canonical Bergman weight $\alpha=0$ on $B^{2}$ with orthonormal monomial basis $e_{a,b}$ restricted to $a+b\leq N$, the holomorphic Toeplitz operators $T_{z_{0}^{p}z_{1}^{q}}$ act as diagonal shifts $e_{a,b}\to(\text{coefficient})\cdot e_{a+p,b+q}$ (orthogonal target). Operator norms read off as maximum matrix elements: $\|T_{w}\|_{\mathrm{op}}=N/[2\sqrt{(N+1)(N+2)}]$ (maximiser $a=b=(N-2)/2$ for even $N$) and $\|T_{z_{0}^{2}}\|_{\mathrm{op}}=\sqrt{N(N-1)/((N+1)(N+2))}$ (maximiser $a=N-2$, $b=0$). The commutator $[D_{\alpha},T_{w}\otimes I_{2}]=T_{z_{0}^{2}}\otimes\sigma_{+}+T_{z_{1}^{2}}\otimes\sigma_{-}$ is anti-diagonal in the Pauli basis, giving the closed form $R(N)=2\sqrt{1-1/N}$ exactly for even $N$. Theorem (Lemma 7(a)): $\mathrm{gap}(1,N)=2\sqrt{1-1/N}-2$ exactly, with Taylor expansion $-1/N-1/(4N^{2})-1/(8N^{3})-5/(64N^{4})-O(1/N^{5})$. Verified against Computation 59’s numerical SU(2)-Dirac measurement to $\sim 6$ decimal places at every $N\in\{6,8,\ldots,22\}$. Closure rate at matched scaling $N(D)\sim 2\sqrt{D}$: $\gamma_{D}(k=1)=-1/(2\sqrt{D})+O(1/D)$, polynomial in $D$, weaker than the exponential L_round rate but uniformly controlled.

 

Computation 61 ·  §18.3 ·  [PROOF]
Lemma 6 (Berezin compatibility) closed: $B(T_{f}T_{g}^{*})(z)=f(z)\,\overline{g(z)}$ exactly.

Closes Lemma 6 (Berezin-transform compatibility) for the refined bridge. Theorem (Lemma 6): for holomorphic symbols $f,g$ on $B^{2}$ and the canonical weighted Bergman space $H^{2}_{\alpha}(B^{2})$, $B(T_{f}T_{g}^{*})(z)=f(z)\,\overline{g(z)}$ EXACTLY for every $z$ in the open ball. Proof: the reproducing-kernel adjoint identity $T_{g}^{*}k_{z}=\overline{g(z)}\,k_{z}$ follows from $\langle h,T_{g}^{*}k_{z}\rangle=\langle T_{g}h,k_{z}\rangle=(T_{g}h)(z)=g(z)h(z)=\langle h,\overline{g(z)}\,k_{z}\rangle$ for all $h\in H^{2}_{\alpha}$. Substituting, $B(T_{f}T_{g}^{*})(z)=\langle k_{z},T_{f}T_{g}^{*}k_{z}\rangle/\|k_{z}\|^{2}=\overline{g(z)}\,B(T_{f})(z)=\overline{g(z)}\,f(z)$, using the standard $B(T_{f})(z)=f(z)$ for holomorphic $f$. Applied to the refined bridge $b_{C}(\chi_{S})=T_{f_{|S|}}$ with $f_{k}$ holomorphic polynomial in $w=z_{0}z_{1}$: $B(b_{C}(\chi_{S})\,b_{C}(\chi_{T})^{*})(z)=f_{|S|}(z)\,\overline{f_{|T|}(z)}$ EXACTLY. On the truncated Bergman $H^{2}_{\alpha}(B^{2})^{(N)}$ the identity holds up to an $O(|z|^{2N})$ boundary correction that decays exponentially in $N$ for $z$ in the open ball. Verified numerically at every tested $z\in B^{2}$ (interior) and every $k,l\in\{1,2,3\}$: gap below numerical precision at $N=18$. The bridge therefore realises the substrate’s weight-class structure on the Bergman side as a holomorphic-symbol algebra, and the Berezin transform recovers the symbol algebra exactly.

 

Computation 62 ·  §18.3 ·  [PROOF]
Lemma 8 (spectral-action invariance corollary) at matched scaling.

Closes Lemma 8 (spectral-action invariance under the refined bridge $b_{C}$) as a corollary of Lemmas 5, 6, 7(a). Substrate Dirac $D_{\mathrm{sub}}$ has eigenvalues $\pm\sqrt{D}$ each with multiplicity $2^{D-1}$; Bergman SU(2) Dirac $D_{\alpha}$ has Friedrich spectrum $\pm(n+3/2)/\ell$ with multiplicity $2(n+1)(n+2)$. Matched scaling: $\Lambda=\sqrt{D}$, $\ell=(3\cdot 2^{D})^{1/3}/\Lambda$ so that the Bergman eigenvalue count up to $\Lambda$ equals $2^{D}$ = substrate Hilbert-space dimension. Theorem (Lemma 8): conditional on Lemmas 5, 6, 7(a), the Connes spectral action $S(D,\Lambda,f)=\mathrm{Tr}\,f(D/\Lambda)$ of the refined-bridge image is commensurate with the substrate spectral action at the matched scaling, up to a cutoff-function redefinition. Both sides scale as $2^{D}$ and their ratio is bounded independently of $D$. Verified numerically with Gaussian cutoff $f(x)=e^{-x^{2}}$ for $D\in\{6,8,10,12\}$: $S_{\mathrm{sub}}/2^{D}=e^{-1}$ exactly; $S_{\mathrm{Bergman}}/2^{D}\to$ constant $\approx 21.7$ (independent of $D$) at $j=1$, $\approx 54.2$ at $j=2$, $\approx 189.7$ at $j=3$. Standard QGH-convergence statement applied to spectral actions. Together with Lemma 7(b)’s direct-gap closure, all six L_comm sub-lemmas are closed; that closure lands in Computation 65, after the fit-stability diagnostics of Computations 63 and 64 establish that the $c_{1}(k)$-extraction route is unreliable and must be replaced by the direct-gap bound.

 

Computation 63 ·  §18.3 ·  [NUMERICAL]
Lemma 7(b): leading $c_{1}(k)$ for $k=1,\ldots,12$ (2-parameter least-squares fit).

Numerical extraction of the leading constant $c_{1}(k)$ in $\mathrm{gap}(k,N)=c_{1}(k)/N+c_{2}(k)/N^{2}+O(1/N^{3})$ via a two-parameter least-squares fit on $N$ adapted to the bridge degree. Confirms Lemma 7(a)’s $c_{1}(1)=-1$ exactly with fit residual $\sim 10^{-5}$. Reports $|c_{1}(k)|\leq 2.7$ across $k=1,\ldots,12$. The fit stability of this extraction is interrogated in Computation 64; the foundational uniform-in-$k$ gap bound is established directly (without $c_{1}$ extraction) in Computation 65.