Appendix B Computation Scripts, Computations 1 to 36

Computation 1 ·  §14.3 ·  [PROOF]
Yukawa hierarchy.

Generation-symmetry structural-scope theorem for a structural Yukawa hierarchy: substrate-level $S_{3}$ invariance forces Yukawa eigenvalues to be $S_{3}$-symmetric in leading order, so the observed hierarchy lives in the contingent $T(C)$ rather than the structural layer (occupying $\approx 31\%$ of the $2^{D}=64$ bit budget at $D=6$).

 

Computation 2 ·  §4.12 ·  [PROOF]
Lorentzian signature.

Lorentzian signature $(+,-,-,-)$ from Derrick-style hyperbolic well-posedness of the Goldstone wave equation; the macroscopic dimension $n=4$ is fixed by the minimal Connes-spectral-triple split given the verified substrate KO-6 (Computation 3) and the total KO-2 of the product.

 

Computation 3 ·  §1.5 ·  [PROOF]
Explicit product spectral triple of KO-dimension 2.

Builds the explicit product spectral triple $(A,H,D,J,\gamma)$ for $M\times F$ where $M$ is the 4D Lorentzian spacetime (KO-4) and $F$ is the substrate internal factor (KO-6), verifying total KO-dim $4+6=10\equiv 2$ (mod 8), the Connes Standard-Model value whose sign $J^{2}=-1$ makes the Euclidean fermionic action well-defined. Five checks: (K1) explicit KO-4 representative triple for $M$ with signs $(-1,+1,+1)$; (K2) two substrate carriers of the KO-6 internal factor $F$, the octonion $\mathbb{C}^{8}$ picture ($\gamma_{F}=i\,L_{e_{1}}\cdots L_{e_{6}}$, $J_{F}=K$) and the bit-space complementation/parity triple at $D=3$, both deliver KO-6 $(+1,+1,-1)$, so the two $F$-pictures agree at the KO level; (K3) $F$ is even and carries colour: $[\gamma_{F},\mathfrak{su}(3)]=0$; (K4) the explicit 32-dim $(\gamma,J,D)$ for $M\times F$ with a generic admissible internal $D_{F}$ is verified to be a genuine real spectral triple of KO-2, so $4+6=10\equiv 2$ is a real product of spectral triples, not just mod-8 bookkeeping; (K5) the two open inputs ($D_{M}$ from spin structure, $D_{F}$ as Yukawa/order-one selection) are explicitly located. Everything else ($A$, $H$, $\gamma$, $J$, the product law) is in hand.

 

Computation 4 ·  §7.8 ·  [PROOF]
Spin structure on $\mathbb{R}\times S^{3}$.

Closes the spin-structure step for $M=\mathbb{R}\times S^{3}$, splitting into a topological part (closed rigorously) and an analytic part (scoped). Topological closure: every orientable 3-manifold is parallelisable (Stiefel), in particular $S^{3}=\mathrm{SU}(2)$; the spin obstruction $w_{2}(S^{3})\in H^{2}(S^{3};\mathbb{Z}/2)=0$ vanishes; spin structures form a torsor over $H^{1}(S^{3};\mathbb{Z}/2)=0$, so the spin structure is unique; globally hyperbolic $M=\mathbb{R}\times\Sigma$ is spin iff $\Sigma$ is spin, so $\mathbb{R}\times S^{3}$ carries a canonical Dirac operator. Analytic part (scoped, residual): Mosco / strong-resolvent convergence is naturally a statement about Dirichlet forms / Laplacians; on a spin manifold the Dirac operator is the canonical first-order square root of (a curvature-shifted) $\Delta$ (Lichnerowicz: $D^{2}=\nabla^{*}\nabla+R/4$). With the spin structure fixed and unique, $D_{M}$ is determined; the remaining task is to show the substrate’s Boolean$\to$CAR/Clifford structure converges to this canonical $D_{M}$, not merely that $\Delta$ converges. Numerical verification of the canonical $S^{3}$ Dirac spectrum and its Weyl law (spectral dimension 3, eigenvalues $\pm(n+3/2)/\ell$) is included.

 

Computation 5 ·  §8.5 ·  [NUMERICAL]
Einstein–Hilbert and Newton’s $G$ from the spectral action.

Derives $G$ structurally from the Chamseddine–Connes spectral action $S=\mathrm{Tr}\,f(D/\Lambda)$, replacing the Kaluza–Klein dimensional-reduction identity (moot in the 10-foundational reading). The 4D heat-kernel asymptotic expansion (Chamseddine–Connes 1996; Chamseddine–Connes–Marcolli 2007) reads $S\sim 2f_{4}\Lambda^{4}a_{0}+2f_{2}\Lambda^{2}a_{2}+f_{0}a_{4}+O(\Lambda^{-2})$ with moments $f_{2}=\int_{0}^{\infty}uf(u)\,du$, $f_{4}=\int_{0}^{\infty}u^{3}f(u)\,du$, $f_{0}=f(0)$. The three terms: $\Lambda^{4}a_{0}\to$ cosmological constant + volume; $\Lambda^{2}a_{2}\to$ Einstein–Hilbert $(1/16\pi G)\int R\sqrt{g}\,d^{4}x$; $\Lambda^{0}a_{4}\to$ Yang–Mills + Higgs + Weyl² (the SM bosonic sector). So gravity is neither put in by hand nor obtained via Kaluza–Klein reduction; the EH term is the $\Lambda^{2}$ coefficient. Verifies the cutoff moments $f_{0},f_{2},f_{4}$ are finite positive for a sample $f$ ($f(u)=e^{-u}$); the closed-form scaling $1/(16\pi G)=(f_{2}\Lambda^{2}/\pi^{2})\,\kappa_{\mathrm{grav}}\,N_{\mathrm{dof}}$ with $\kappa_{\mathrm{grav}}$ an $O(1)$ scheme constant and $N_{\mathrm{dof}}$ the fermionic multiplicity, solving for $\Lambda$ given observed $G$ and the SM fermion count places $\Lambda$ at a unification/Planck-ish scale.

 

Computation 6 ·  Chapter 13 ·  [PROOF]
Gauge group from $A_{F}$ via unimodular unitaries; gauge bosons and Higgs as inner fluctuations.

Derives the Standard-Model gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ as the unimodular unitary group of $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ via the Chamseddine–Connes machinery, with the gauge fields arising as inner fluctuations of the Dirac operator $D\to D+A+\varepsilon^{\prime}JAJ^{-1}$, $A=\sum_{i}a_{i}[D,b_{i}]$ for $a_{i},b_{i}\in A_{F}$. This replaces the $G_{2}$-holonomy route to $\mathrm{SU}(3)$ (which dies in the 10-foundational reading) with the standard Connes derivation applied to the substrate-supplied $A_{F}$. Four checks: (N1) Lie-algebra dimensions $u(A_{F})=13$; unimodular condition removes one dim $\to 12=\dim(\text{SM group})$; (N2) $\mathfrak{su}(2)$ from imaginary quaternions, $\mathfrak{su}(3)$ from Gell-Mann generators, both closure checks exact; (N3) inner-fluctuation count: gauge-boson count $=$ adjoint $=12$; spacetime $[D_{M},a]$ gives the vector bosons $A_{\mu}$, the finite $[D_{F},a]$ gives the Higgs; (N4) $\mathrm{U}(1)$ bookkeeping: which $\mathrm{U}(1)$ is removed by unimodularity, which survives as $\mathrm{U}(1)_{Y}$.

 

Computation 7 ·  §14.4 ·  [PROOF]
Order-one condition selects the Yukawa form of $D_{F}$ on the chiral bimodule.

Constructs an explicit one-generation lepton-sector finite spectral triple (8-dim: $\nu_{L},e_{L},\nu_{R},e_{R}$ + 4 antiparticles) and verifies the chirality and order-one structure. Track J established that the naive substrate ladder $\mathrm{SU}(2)$ is, under either natural chirality grading, vector-like or chirality-mixing, not the SM’s chiral $\mathrm{SU}(2)_{L}$. The Connes resolution uses the CHIRAL representation: $\mathbb{H}$ acts on left-handed doublets only, so $\mathrm{SU}(2)_{L}$ is chiral by construction; the order-one condition $[[D_{F},a],Jb^{*}J^{-1}]=0$ then SELECTS the Yukawa form of $D_{F}$. Four checks: (O1) the representation is even ($[\gamma,a]=0$) and the $\mathrm{SU}(2)_{L}$ action of $\mathbb{H}$ is chiral, supported on left-handed particles only; (O2) order-zero $[a,Jb^{*}J^{-1}]=0$ for all $a,b\in A_{F}$; (O3) order-one $[[D_{F},a],Jb^{*}J^{-1}]=0$ for the Yukawa $D_{F}$, and FAILURE for a generic admissible $D_{F}$, so order-one is a genuine constraint that selects the Yukawa form; (O4) the KO-6 sign of the finite real structure ($\varepsilon=+1$), consistent with KO-2 product. Honest residual: this uses the chiral representation as CCM input; whether the substrate furnishes it is settled by Computations 8 and 11.

 

Computation 8 ·  §14.4 ·  [PROOF]
Chirality bridge: $\gamma_{M}$ vs the directed modal threshold.

Tests the two candidate substrate mechanisms for furnishing the chiral bimodule of Computation 7, locating chirality at the directed modal threshold rather than at $\gamma_{M}$. (a) $\gamma_{M}$ does not furnish weak chirality: the physical chirality of an internal generator $1\otimes T$ is governed entirely by $[T,\gamma_{F}]$; the $\gamma_{M}$ factor tensors along and cannot change it (negative finding, structurally located). (b) The substrate DOES offer $\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$: quaternion left- and right-multiplications give two commuting $\mathfrak{su}(2)$’s $\cong\mathfrak{so}(4)$, the $S^{3}$ isometry algebra. Chirality selection is the choice of one factor, and the directed modal threshold $\varepsilon=|T(C)|-\tau>0$ (book Ch. 13 parity-violation mechanism) is the physical principle that selects between the parity-conjugate choices. (c) Residual: the selected $\mathrm{SU}(2)_{L}$ must act on the $\gamma_{F}=+1$ block only (bimodule support); bridged here to the threshold mechanism and closed in Computation 11.

 

Computation 9 ·  §18.3 ·  [NUMERICAL]
Dirac convergence scoping.

Dirac-convergence scoping in spectral propinquity; numerical companion verifies $\mathrm{Cl}(0,2D)$ anticommutation at every $D$, Walsh-to-spherical multiplicity flow, and exact low-mode commutator-gap on Walsh-weight $\leq 1$ modes; residual reduced to one uniform Sobolev-tail estimate (256) (the full first-order route; the committed closure is the relaxed criterion of §18.3, whose all-$(k,N)$ uniform bound is the surviving quantitative refinement).

 

Computation 10 ·  §15.10 ·  [EMPIRICAL]
Empirical $M_{*}$ bounds.

SMEFT operator-by-operator survey of $M_{*}=4\pi m_{h}\sqrt{2/3}$ at PST’s tree level vs LHC/EWPO bounds; $\Delta S_{\mathrm{PST}}=\Delta T_{\mathrm{PST}}=0$ structurally. Forward reference: the script’s $M_{*}=1.573$ TeV predates the off-by-one correction, and its Wilsonian reading with $O(1)$ coefficients is the universal-kernel reading Computation 205 excludes; its structural argument is the Higgs-doublet reading that survives there.

 

Computation 11 ·  §14.4 ·  [PROOF]
Bimodule support: threshold-selected $\mathrm{SU}(2)_{L}$ acts on the $\gamma_{F}=+1$ block only.

Closes the chiral-bimodule residual left open by Computation 8. Constructs a minimal but representative chiral $H_{F}$: one weak doublet $((\nu_{L},e_{L}),(\nu_{R},e_{R}))$ with $\gamma_{F}=\mathrm{diag}(+1,+1,-1,-1)$. Two representations of $A_{F}$ are mathematically possible a priori: Choice A ($\mathbb{H}$ acts on $H_{F}^{(+)}$ left block, trivial on $H_{F}^{(-)}$, the parity-violating observed SM); Choice B ($\mathbb{H}$ acts on $H_{F}^{(-)}$ right block, trivial on $H_{F}^{(+)}$, the parity-conjugate model, never observed). Both give consistent spectral triples in the Connes–Chamseddine sense; the chirality of $\mathrm{SU}(2)_{L}$ therefore reduces to “which of A or B is realised in nature?”. PST’s claim is that the directed modal threshold (book Ch. 13) is the physical principle that selects Choice A. Verifies (i) both representations of $\mathbb{H}$ commute with $\gamma_{F}$ (necessary condition); (ii) both have clean block support; (iii) the substrate-side $\mathfrak{su}(2)_{L}$ and $\mathfrak{su}(2)_{R}$ operator-norm signatures under the threshold $\varepsilon>0$ break the L$\leftrightarrow$R symmetry as required, selecting Choice A.

 

Computation 12 ·  §13.6 ·  [PROOF]
Consistency of octonionic-$\mathrm{SU}(3)$ embedding with Computations 1–11.

Examines whether the octonionic-$\mathrm{SU}(3)$ embedding ($\mathrm{SU}(3)\subset G_{2}=\mathrm{Aut}(\mathbb{O})$ acting as automorphisms preserving the octonion product) is consistent with the rest of PST’s structure, vs. the canonical block embedding ($\mathrm{SU}(3)\subset\mathrm{SU}(4)\cong\mathrm{Spin}(6)$ acting as block $\mathrm{diag}(\mathrm{SU}(3),1)$ on the half-spinor). If the octonionic interpretation is consistent with Computations 1–11, Furey’s three-generation construction ports directly into PST without changing framework assumptions. Six structural checks: (§1) two $\mathrm{SU}(3)$ subgroups of $\mathrm{Spin}(6)$ compared; (§2) Computation 6’s gauge group derivation does not specify which $\mathrm{SU}(3)$; (§3) Computation 7’s order-one transfers under either embedding; (§4) inner fluctuations and the Higgs sector are unaffected; (§5) Computation 4’s spin structure and Computation 9’s convergence are independent of the embedding; (§6) verdict: octonionic interpretation is structurally consistent with PST’s existing results.

 

Computation 13 ·  §13.6 ·  [PROOF]
Canonical vs octonionic $\mathrm{SU}(3)$ side-by-side comparison.

Computes the consequences of canonical vs octonionic $\mathrm{SU}(3)$ embedding by re-deriving Computation 1, section 7 of the script (inner-fluctuation extension of the Yukawa structural-scope theorem) and Computation 7 (order-one condition selecting Yukawa form) under both, then compares for “best PST fit”. Best-fit criteria in priority order: (C1) does $N_{\mathrm{gen}}=3$ emerge structurally? (C2) do existing PST results survive, chirality, Yukawa contingency, empirical bounds, gauge group? (C3) is the construction algebraically clean, no new posits? (C4) consistency with the Connes–Chamseddine framework? Seven sections culminate in the recommendation of the octonionic embedding as the natural source of $N_{\mathrm{gen}}=3$ via Furey’s six-$\mathrm{SU}(3)$-triplet decomposition of $\mathrm{Cl}(0,6)$, with the open work being the explicit chain-algebra construction (Computations 14–17).

 

Computation 14 ·  §13.6 ·  [PROOF]
Inner-fluctuation structural-scope theorem extension (Computation 1, section 7 of the script) under octonionic embedding.

Step B of the octonionic-$\mathrm{SU}(3)$ re-derivation programme identified by Computation 13. Original Computation 1, section 7 of the script (canonical embedding) assumed $H_{F}=H_{F}^{(1)}\otimes\mathbb{C}^{N_{\mathrm{gen}}}$ tensor-factor structure for generations, with $A_{F}$ acting as $a^{(1)}\otimes I_{\mathrm{gen}}$ (generation-blind), so $[D_{F},b]$ inherits generation structure of $D_{F}$ unchanged. Under the octonionic embedding, $H_{F}$ is not in tensor-factor form for generations; the three “generation copies” are internal states of the 64-complex-dim algebra $\mathbb{C}\otimes\overleftarrow{\mathbb{O}}$, with $A_{F}$’s $M_{3}(\mathbb{C})$ acting via the octonionic $\mathrm{SU}(3)$, mixing generation copies by $\mathrm{SU}(3)$ rotations. The canonical §7 argument “$A_{F}$ is gen-blind” does not transfer. The substantive question is whether the structural-scope STATEMENT itself still holds: can inner fluctuations turn a gen-symmetric $D_{F}^{(0)}$ into one with generation-asymmetric Yukawa eigenvalues? Five sections: (§1) toy octonionic-$\mathrm{SU}(3)$ model; (§2–§3) inner fluctuations under each embedding; (§4) substrate-level $S_{3}$ constraint as the deeper invariant; (§5) verdict: individual fluctuations can mix generations but substrate $S_{3}$ invariance constrains physical observables; Yukawa contingency preserved.

 

Computation 15 ·  §13.6 ·  [NUMERICAL]
Order-one Yukawa selection (Computation 7) under octonionic embedding.

Step C of the octonionic re-derivation programme. Verifies Computation 7’s result “order-one condition selects the Yukawa form” under the octonionic $A_{F}$ action and determines the specific Yukawa form selected. Original Computation 7 (canonical): $H_{F}=8$-dim lepton-sector finite triple, $A_{F}$ acts via canonical lepton-sector representation ($M_{3}(\mathbb{C})$ trivial on leptons, $\mathbb{H}$ on left doublet), order-one $[[D_{F},a],Jb^{*}J^{-1}]=0$ SELECTS the Yukawa form of $D_{F}$. Under the octonionic embedding, $A_{F}$’s action on $H_{F}$ changes (the chain-algebra structure of $\overleftarrow{\mathbb{O}}$), so the order-one condition is a different algebraic constraint. Two possible outcomes: (a) order-one still selects A Yukawa form (possibly different from canonical), supports the octonionic interpretation; (b) order-one is inconsistent with any Yukawa form, falsifies octonionic interpretation. Seven sections culminate in the verdict: order-one selecting condition is unitarily covariant; Yukawa residue $\sim 2\times 10^{-5}$ vs $\sim 10$ for generic $D_{F}$ under BOTH embeddings, the selecting condition transfers, and the octonionic interpretation passes this test.

 

Computation 16 ·  §13.5 ·  [NUMERICAL]
Furey foundation: $\overleftarrow{\mathbb{O}}\cong\mathbb{C}\otimes\mathrm{Cl}(0,6)\cong M_{8}(\mathbb{C})$, octonionic $\mathrm{SU}(3)$ as $G_{2}$-stabiliser.

The most ambitious computational physics in this track series: attempts to carry out Furey’s 2014 construction explicitly inside PST’s substrate Clifford algebra. Twenty-two sections; the core run: (§1) octonion multiplication table (Fano-plane convention); (§2) left-multiplication operators $L_{e_{a}}$ as $8\times 8$ complex matrices; (§3) verify the algebra structure of $\overleftarrow{\mathbb{O}}$ as 64 explicit $8\times 8$ matrices; (§4) identify the primitive idempotent $f$ (vacuum) of $\overleftarrow{\mathbb{O}}$; (§5) construct the octonionic $\mathrm{SU}(3)\subset G_{2}$ as stabiliser of $f$ and verify the 8 adjoint elements with exact commutation relations; (§6) decompose $\overleftarrow{\mathbb{O}}$ (64-complex-dim) under this $\mathrm{SU}(3)$ into irreducible representations; (§7) compare to Furey’s $16+48$ split; (§8) honest assessment of how rigorously this verifies the three-generation interpretation; (§9–§22) extended analysis, including the convention-aligned complex structure $J=L_{e_{7}}$ with Cayley pairing $(e_{1},e_{3}),(e_{2},e_{6}),(e_{4},e_{5})$ (§16) and the full $\mathrm{SU}(3)$ lift. Verifies $\overleftarrow{\mathbb{O}}\cong\mathbb{C}\otimes\mathrm{Cl}(0,6)=M_{8}(\mathbb{C})$ and octonionic $\mathrm{SU}(3)$ as $G_{2}$-stabiliser of $e_{7}$ to machine precision. The matter-sector multiplicity is off by 6 due to degenerate Cartan weights; resolved by Casimir-discrimination in Computation 17.

 

Computation 17 ·  §13.5 ·  [PROOF]
Definitive $\mathrm{SU}(3)$ decomposition of $M_{8}(\mathbb{C})$ via Casimir.

Closes the off-by-6 residue from Computation 16 by using the $\mathrm{SU}(3)$ quadratic Casimir $C_{2}=\sum_{a}(\Lambda_{a})^{2}$ as an irrep discriminator. Computation 16 identified the eight-generator octonionic $\mathrm{SU}(3)$ subgroup (8 adjoint elements verified, commutation relations exact), but the matter-sector multiplicity count was off because the joint $(\Lambda_{3},\Lambda_{8})$ eigenbasis is not unique on degenerate weights, singlets and $(0,0)$ members of triplets share Cartan weights and can only be told apart by the Casimir. Six sections: (§1) reconstruct the octonionic $\mathrm{SU}(3)$ generators (lifted from Gell-Mann via the $J=L_{e_{7}}$ complex structure on $(e_{1},e_{3}),(e_{2},e_{6}),(e_{4},e_{5})$, the correct Cayley pairing for the $G_{2}$-stabiliser of $e_{7}$); (§2) build $\mathrm{ad}(\Lambda_{a})$ on $M_{8}(\mathbb{C})$; (§3) build the Casimir $C_{2}$ as a $64\times 64$ Hermitian matrix; (§4) diagonalise $C_{2}$ and group eigenvalues by $\mathrm{SU}(3)$ irrep Casimir values (0 for $\mathbf{1}$, $4/3$ for $\mathbf{3}$, $3$ for $\mathbf{8}$, $10/3$ for $\mathbf{6}$); (§5) confirm the decomposition $6\cdot\mathbf{1}+5\cdot\mathbf{3}+5\cdot\overline{\mathbf{3}}+2\cdot\mathbf{8}+\mathbf{6}+\overline{\mathbf{6}}$ (total dim 64) matches the Clebsch–Gordan expectation to machine precision; (§6) status statement for the octonionic-$\mathrm{SU}(3)$ item: structurally derived, decomposition exact.

 

Computation 18 ·  §13.7 ·  [PROOF]
$\omega$ as the unique Spin(6) singlet.

Spin(6)-invariance in $\mathrm{Cl}(0,6)\otimes\mathbb{C}$: $\omega=L_{e_{1}}\dots L_{e_{6}}$ is the unique (up to scalar) Spin(6)-invariant element of positive grade. Hence $f=(I+i\omega)/2$ is the unique (up to chiral swap) non-trivial Spin(6)-invariant idempotent: the chirality projector, whose canonical rank-1 refinement is Furey’s primitive idempotent.

 

Computation 19 ·  §13.7 ·  [PROOF]
Substrate parity grading = $\mathrm{Cl}(0,6)$ chirality grading.

Schur on the irreducible chiral halves $H_{\pm}$ of $H_{F}=\mathbb{C}^{8}$ gives Hermitian Spin(6)-commutant $=\mathrm{span}\{I,\gamma\}$ (2-real-dim). In the JW Majorana representation, Computation 3’s parity grading $\gamma_{K}=Z^{\otimes D}$ equals $i\omega_{\mathrm{JW}}$ exactly ($\|\gamma_{K}-i\omega_{\mathrm{JW}}\|=0$ to machine precision). The chirality projector $f=(I+i\omega)/2$ is delivered directly by the substrate’s parity-selected real structure; Furey’s primitive idempotent is its canonical rank-1 refinement [121], and her acting-on-$\mathbb{C}\otimes\mathbb{O}$ chain-algebra construction then delivers $\mathrm{SU}(3)\times\mathrm{U}(1)$ on one generation under $\mathrm{SU}(3)_{c}$. The synthesis to the full $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ requires the $\mathbb{H}$ factor from the emergent Lorentzian Dirac sector $\mathrm{Cl}(1,3)\cong M_{2}(\mathbb{H})$ (Computation 66) combined via Dixon’s hyperspinor framework $T=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ [146].

Dirac-convergence step (§18.3): L_round and L_comm computations.

 

Computation 20 ·  §18.3 ·  [NUMERICAL]
L_round sweep.

PST analogue of [119] Lemma 3.2 across $D=4,6,\dots,20$ and three test classes under eleven candidate Lip-norms. All polynomial Lip-norms FAIL; exponential Lip-norms $e^{c|S|}$ with $c>\log 2$ succeed at rate $\gamma_{D}=O((2/e^{c})^{D})$. Uses fast Walsh–Hadamard transform.

 

Computation 21 ·  §18.3 ·  [PROOF]
Dirac commutator Lip-norm scaling.

Measures $\|[D,\chi_{S}]\|_{\mathrm{op}}$ for the discrete Dirac $D=\sum_{a}\chi_{a}$ on Walsh modes $\chi_{S}$; result $=2\sqrt{|S|}$ exactly at every $D$ and every $|S|$, matching the Friedrich Dirac eigenvalue scaling on round $S^{3}$ [113]. The first-order Dirac structure on the substrate is polynomial (not exponential) in Walsh weight.

 

Computation 22 ·  §18.3 ·  [NUMERICAL]
Higher-order Dirac commutators.

$\mathrm{ad}_{D}^{k}(\chi_{S})$, $k=1,\dots,10$. Result: $\|\mathrm{ad}_{D}^{k}(\chi_{S})\|^{1/k}\to 2\sqrt{D}$ (independent of $|S|$ at large $k$). The spectral radius of $\mathrm{ad}_{D}$ is $2\sqrt{D}$, so the Fréchet smooth Lip-norm $L_{F}(\chi_{S})=\sup_{j}\|\mathrm{ad}_{D}^{j}(\chi_{S})\|/j!\sim e^{2\sqrt{D}}$ by Stirling.

 

Computation 23 ·  §18.3 ·  [NUMERICAL]
Fréchet smooth L_round closure.

Measures $L_{F}$ from equation (259) on the constant-coefficient Walsh tail. Result: $\gamma_{D}\approx 0.37\cdot e^{-0.28D}$ at $D=4,6,8,10,12$, exponential decay matching the natural spectral-triple smooth subalgebra. With [116] Theorem 5.2 this gives QGH convergence of the Walsh substrate algebras to $C(S^{3})$.

 

Computation 24 ·  §18.3 ·  [PROOF]
$\chi_{a}$–$\chi_{S}$ discrete-side audit.

Confirms (i) $\|[\chi_{a},\chi_{S}]\|_{\mathrm{op}}=0$ if $a\notin S$, exactly $2$ if $a\in S$; (ii) the Hilbert–Schmidt overlap of $[\chi_{a},\chi_{S}]$ with every Walsh mode $\chi_{T}$ is zero to machine precision. The bridge defect lives entirely in the non-abelian $\sigma_{y}/\sigma_{x}$ Pauli sector.

 

Computation 25 ·  §18.3 ·  [PROOF]
Friedrich $S^{3}$ Dirac spectrum.

Eigenvalues $\pm(n+3/2)/\ell$, multiplicity $2(n+1)(n+2)$ ([113]). Cross-checks the naive scalar-spinor count $2(l+1)^{2}$, exhibiting the Pauli mixing factor $(n+2)/(n+1)$ from Friedrich’s eigenspinor decomposition ([114]). Continuum-side reference for the spinor extension of $b_{D}$.

 

Computation 26 ·  §18.3 ·  [PROOF]
Bijective-bridge no-go.

Structural no-go ruling out the entire bijective single-mode bridge class for L_comm. The discrete-side commutator norm is exactly $\|[\chi_{a}^{\mathrm{Cliff}},\chi_{S}]\|_{\mathrm{op}}=2$ for $a\in S$ (zero otherwise), independent of $D$ and of $|S|$ (verified at $D\in\{4,6,8\}$, $|S|\in\{1,2,3\}$, machine precision). The continuum-side norm scales as $\sqrt{l(l+1)}$ via the SU(2) Casimir. The L_round scaling on the same bridge (Computations 20, 25) fixes the weight-graded scalar, leaving no freedom to reconcile the two scales. Conclusion: no bijective single-mode bridge closes L_comm.

 

Computation 27 ·  §18.3 ·  [PROOF]
Bergman/Toeplitz framework on $H^{2}_{\alpha}(B^{2})$.

Full operator-theoretic implementation of [119]. The Toeplitz commutator $[T_{\bar{z}_{1}^{k}},T_{z_{1}^{k}}]$ is diagonal on the monomial basis with explicit entry $\Lambda(J,k,\alpha)=(m_{1}+1)_{k}/(|J|+\alpha+3)_{k}-(m_{1})_{(k)}/(|J|+\alpha+3-k)_{k}$. At $k=1$, $\|[T_{\bar{z}},T_{z}]\|_{\mathrm{op},\alpha}=1/(\alpha+3)$ exactly (machine-precision match against the analytical formula, J-maximiser $(0,0)$); at $k=2$ the commutator norm is $0.30$ at $\alpha=0$, $0.20$ at $\alpha=1$, $0.085$ at $\alpha=5$. The Toeplitz operator norm $\|T_{z_{1}^{k}}\|_{\mathrm{op},\alpha}$ approaches $1$ and is $\alpha$-INDEPENDENT at leading order; the ratio $R(k,\alpha)$ is $\alpha$-DEPENDENT ($R(1,-0.5)=0.40$ vs $R(1,10)=0.08$). With $\alpha=\alpha(D)$ fidelity and gap rate can be tuned simultaneously.

 

Computation 28 ·  §18.3 ·  [NUMERICAL]
Explicit substrate–Toeplitz bridge probe (D = 4).

First explicit instantiation of the candidate substrate $\leftrightarrow$ Toeplitz bridge at $D=4$, $N=5$. Sites $0,1$ of the substrate map to $T_{z_{0}},T_{z_{1}}$; sites $2,3$ map to $T_{\bar{z}_{0}},T_{\bar{z}_{1}}$. Single-mode bridge images have op-norm $\sim 0.84$ at $\alpha=0$, decreasing with $\alpha$ as predicted. Commutator-gap probe across six $(a,S)$ pairs from Computation 26: cases where $a\in S$ at single sites map to zero on the Toeplitz side (multiplicative bridge with holomorphic generators commutes); mixed holomorphic/anti-holomorphic cases produce non-zero bridge commutators ($0.193$ at $\alpha=0$ for $S=\{0,2\}$). Conclusion: the multiplicative bridge is a necessary scaffolding but not sufficient. It correctly maps algebraic generators but does not preserve the abelian Walsh structure: spurious cross-commutators appear at weight $k\geq 2$, scaling as $O(1/\alpha)$ at large $\alpha$ (consistent with Computation 27’s prediction). The next concrete step for closing the Dirac-aware lift is the Mosco-limit reweighting at each weight $k$: map a SUM of Walsh modes to a single degree-$l$ Toeplitz symbol $Y^{l}(z,\bar{z})$, with coefficients chosen to suppress the spurious cross-commutators while preserving the $\alpha$-dependent gap-rate scaling.

 

Computation 29 ·  §18.3 ·  [NUMERICAL]
Mosco-limit reweighting via holomorphic Toeplitz subalgebra.

Restricts the bridge image of Computation 28 to the holomorphic Toeplitz subalgebra of $H^{2}_{\alpha}(B^{2})$, where all operators commute (multiplication by holomorphic functions on $H^{2}_{\alpha}$ is just multiplication, no projection needed). Site-to-holomorphic mapping at $D=4$: sites $0,1\to T_{z_{0}},T_{z_{1}}$; sites $2,3\to T_{z_{0}^{2}},T_{z_{1}^{2}}$. Bridge of $\chi_{S}$ = product of individual-site images, in the abelian subalgebra. Sanity check: all bridge-image cross-commutators $[b(\chi_{a}),b(\chi_{b})]$ are zero to machine precision (the abelian Walsh structure is preserved by construction, eliminating the spurious cross-commutators of Computation 28). Bridge image norms scale with weight: $0.85$ at $k=1$, $0.38$ at $k=2$, $0.17$ at $k=3$ (truncation collapse at $k=4$ at $N=5$). Gap probe: substrate commutator norm $2$ (when $a\in S$) maps to Toeplitz commutator $\sim 0.71$ at $\alpha=0$, decaying with $\alpha$; cases with $a\notin S$ (substrate norm $0$) still give residual non-zero Toeplitz commutators ($\sim 0.43$ at $\alpha=0$). Conclusion: holomorphic Toeplitz subalgebra is the structurally right target (abelian preservation), but the simplest Clifford analogue $T_{\bar{z}_{a}}$ still produces residuals in the ”$a\notin S$” sector. The next concrete step is constructing the $\alpha$-twisted Berezin derivative $D_{\alpha}=\sum_{a}(T_{\bar{z}_{a}}+\alpha\text{-correction})$ designed to make $[D_{\alpha},b(\chi_{S})]$ vanish for $a\notin S$ while preserving the $\alpha$-dependent gap rate.

 

Computation 30 ·  §18.3 ·  [NUMERICAL]
Optimal-$\alpha$ closure diagnostic for the holomorphic bridge.

Tests whether a single $\alpha=\alpha(D)$ can match the substrate commutator across all $(a,S)$ test cases simultaneously, given the abelian-preserving holomorphic bridge of Computation 29. At $D=4$, $N=5$, sweep $\alpha\in[-0.5,30]$ across eight test cases. Finding: per-case optimal $\alpha$ values fall into two clusters: cases with substrate norm $=2$ ($a\in S$) want $\alpha=-0.5$ (boundary of the BS family), cases with substrate norm $=0$ ($a\notin S$) want $\alpha=30$ (deep into the ball). Spread of optimal $\alpha$ values is $30.5$, no single $\alpha$ closes both sectors. The aggregate optimal $\alpha=-0.5$ gives residual $\sim 8.9$. Structural observation: the substrate commutator is BIMODAL ($0$ or $2$), but the Toeplitz commutator under the simple $T_{\bar{z}_{a}}$ Clifford analogue is a CONTINUOUS function of $\alpha$; no scalar $\alpha$ can produce a bimodal output. Closing the Dirac-aware lift therefore requires additional internal degrees of freedom in the Clifford construction beyond a single $\alpha$: an $\alpha$-twisted Berezin derivative with selectivity per substrate site, or the explicit spinor extension on a tensor $\mathbb{C}^{2}$ factor.

 

Computation 31 ·  §18.3 ·  [NUMERICAL]
Spinor extension on $H^{2}_{\alpha}(B^{2})\otimes\mathbb{C}^{N}$ for L_comm.

Tests whether tensoring a spinor $\mathbb{C}^{2}$ or $\mathbb{C}^{4}$ fibre onto the Bergman space supplies the internal degrees of freedom identified as missing by Computation 30. Three constructions tested at $D=4$, $N=5$: (A) trivial $\mathbb{C}^{2}$ spinor with $\sigma$-factor on Clifford bridge, (B) $\mathbb{C}^{4}$ spinor with full $\mathrm{Cl}(0,4)$ generators tensored with HOLO Toeplitz, (C) mixed bridge with ANTI-HOLO Toeplitz tensored with $\mathrm{Cl}(0,4)$ generators. Findings: in (A) and (C) the tensor factor decouples in the operator norm (op-norm of $A\otimes B$ factorises as $\|A\|\cdot\|B\|$); the spinor factor multiplies the existing gap by a unitary, leaving the gap structure unchanged from Computations 29/36. In (B) all cross-commutators are exactly ZERO because both bridge images sit in the abelian holomorphic Toeplitz subalgebra. Conclusion: tensor spinor extensions do not close L_comm on their own. Closing it requires a NON-TENSOR coupling between the Bergman and spinor factors. The natural candidate is the round-$S^{3}$ Dirac $D=\sum_{a}\sigma_{a}\otimes X_{a}$, where $X_{a}$ are the SU(2) left-invariant vector fields on $H^{2}_{\alpha}$ acting as raising/lowering operators on the Wigner-D monomial basis.

 

Computation 32 ·  §18.3 ·  [NUMERICAL]
SU(2) Dirac operator on $H^{2}_{\alpha}(B^{2})\otimes\mathbb{C}^{2}$ for L_comm.

Implements the non-tensor coupling identified as missing by Computation 31: the round-$S^{3}$ Dirac $D=\sum_{a}\sigma_{a}\otimes J_{a}$, where $J_{a}$ are SU(2) generators acting on the Bergman monomial basis as raising/lowering operators ($J_{+}=z_{1}\partial_{z_{2}}$, $J_{-}=z_{2}\partial_{z_{1}}$, $J_{z}=\frac{1}{2}(z_{1}\partial_{z_{1}}-z_{2}\partial_{z_{2}})$). Verifies su(2) algebra to machine precision and Hermiticity of $D$ ($\|D-D^{*}\|<10^{-14}$); eigenvalue range $[-3.5,2.5]$ at $N=5$ approximates Friedrich’s $\pm(n+3/2)$ $S^{3}$ spectrum. Key finding: the L_comm gap $\|[D,b_{\mathrm{holo}}(\chi_{S})\otimes I]\|_{\mathrm{op}}$ is NON-ZERO, the non-tensor coupling finally produces a meaningful Dirac commutator. Normalised by bridge fidelity, the ratios are $1.02$ at $|S|=1$, $1.83$–$2.29$ at $|S|=2$, $2.45$–$3.04$ at $|S|=3$, approaching the substrate target $2\sqrt{|S|}$ but with variance across same-weight $S$. The ratios are $\alpha$-independent (the $J_{a}$ structure on the ONB is $\alpha$-canceling), meaning $\alpha$-dependence enters only through the bridge image norms. Conclusion: the SU(2) Dirac framework is operational and produces the right qualitative L_comm structure; quantitative closure requires (i) Mosco-limit averaging across same-weight Walsh modes to remove the variance, and (ii) $\alpha$-dependent rescaling to match $2\sqrt{|S|}$ exactly.

 

Computation 33 ·  §18.3 ·  [NUMERICAL]
Mosco averaging and $\alpha$ rescaling for L_comm closure.

Combines the two within-framework tunings identified by Computation 32. Mosco averaging: $b_{\mathrm{Mosco,k}}:=(1/\sqrt{C(D,k)})\sum_{|S|=k}b_{\mathrm{holo}}(\chi_{S})$, collapsing $C(D,k)$ weight-$k$ Walsh modes into one symmetric combination. $\alpha$ rescaling: $\mathrm{bridge}_{\mathrm{final,k}}:=b_{\mathrm{Mosco,k}}/\|b_{\mathrm{Mosco,k}}\|_{\mathrm{op}}$ (unit op-norm per weight class). Tested at $D=4$, $N=5$. Findings: Step 3 ratios $\|[D_{\alpha},\mathrm{bridge}_{\mathrm{final,k}}\otimes I]\|_{\mathrm{op}}/(2\sqrt{k})$ monotonically approach $1$ with weight: $0.60$ ($k=1$), $0.71$ ($k=2$), $0.87$ ($k=3$). The L_comm gap is closing asymptotically. $k=4$ is zero due to truncation collapse ($N=5$ cannot hold the full-weight product). Residual: Step 4 variance check shows per-$S$ ratios at fixed $|S|$ still range $1.02$–$1.45$ at $k=1$ even after rescaling, indicating the symmetric-sum Mosco is approximate. Exact uniformity would require a Wigner-D-component projection within each weight class. Conclusion: this is the FIRST computation demonstrating monotone convergence of the L_comm ratio towards closure in the operational framework. Closing the residual variance requires the proper Wigner-D-decomposed Mosco averaging; closing the truncation effect requires $N\geq 2D$. Both are within-framework refinements, not structural changes.

 

Computation 34 ·  §18.3 ·  [NUMERICAL]
L_comm convergence at larger $(D,N)$: asymptotic closure test.

Scales Computation 33 up to $D=4$, $N=10$ and $D=6$, $N=12$ to test whether the Mosco-averaged $\alpha$-rescaled bridge ratios converge to $1$. Findings at $\alpha=0$: ratios increase with $(D,N)$ at fixed $k$ (e.g., $k=3$: $0.86\to 0.81\to 0.96$ across $(4,5)$, $(4,10)$, $(6,12)$), and increase with $k$ at fixed $(D,N)$. The $k=3$ ratio at $D=6,N=12$ reaches $0.96$, within 4% of substrate target $2\sqrt{3}$. Crossover observation: at $D=6,N=12$, the ratio crosses $1$ around $k=3$–$4$ and continues growing: $1.12$ ($k=4$), $1.23$ ($k=5$), $1.32$ ($k=6$). This is structurally explained: the SU(2) Dirac has $3$ frame directions (giving $\sim\sqrt{3}\cdot\|J_{a}\|$ scaling), while the substrate has $D$ Clifford directions (giving $2\sqrt{|S|}$). These match at $k\sim 3$ but diverge at higher $|S|$. Important framing: this test compares $\|\mathrm{bridge}\,\mathrm{commutator}\|$ to $\|\mathrm{substrate}\,\mathrm{commutator}\|$, but the genuine L_comm criterion is the HOMOMORPHISM GAP $\|\mathrm{bridge}([D,\chi_{S}])-[D_{\alpha},\mathrm{bridge}(\chi_{S})]\|_{\mathrm{op}}\to 0$, which is a stronger condition. The norm-match closure at $k\sim 3$ is suggestive but the homomorphism-gap test is the next concrete deliverable.

 

Computation 35 ·  §18.3 ·  [NUMERICAL]
L_comm homomorphism-gap test (explicit construction).

Tests the genuine L_comm criterion,

$$\|\mathrm{bridge}([D_{\mathrm{sub}},\chi_{S}])-[D_{\alpha},\mathrm{bridge}(\chi_{S})]\|_{\mathrm{op}}.$$ (283)

Uses the natural multiplicative bridge extension $\mathrm{bridge}(\chi_{a}^{\mathrm{Cliff}}):=J_{a}\otimes\sigma_{a}$ (with cyclic Pauli assignment for the substrate-vs-Pauli site mismatch). Computes LHS $=2\sum_{a\in S}(J_{a}\,b_{\mathrm{holo}})\otimes\sigma_{a}$ and RHS $=[D_{\alpha},b_{\mathrm{holo}}\otimes I]$ directly, then their operator-norm difference. Negative finding: the gap/LHS ratio is large ($0.68$ to $1.01$ at $D=4,N=5$; $0.83$ to $1.00$ at $D=4,N=10$; $0.82$ to $1.00$ at $D=6,N=12$) and does not shrink with increasing $(D,N)$. The bridge with the natural multiplicative extension is therefore not a Dirac homomorphism, and the failure is uniform across truncations. Structural reading: the earlier norm-match successes (Computation 34 ratio $0.96$ at $k=3$) measured a WEAKER quantity than L_comm. Closing the true homomorphism gap requires either a different bridge construction (e.g., spinor fibre $\mathbb{C}^{2^{D}}$ matching the substrate Cl(0, 2D) representation rather than the round-$S^{3}$ $\mathbb{C}^{2}$) or a re-interpretation of L_comm in terms of norm-equivalence-up-to-rescaling rather than exact operator equality.

 

Computation 36 ·  §18.3 ·  [NUMERICAL]
L_comm finite-$N$ matched-truncation convergence at $D=8$, $N=20$.

Extends Computation 34 to substrate size $D=8$ with Bergman truncation $N=20$ (Bergman dim $231$, full Hilbert dim $462$). Measures the relaxed L_comm gap $\gamma_{D}(k)=|1-\mathrm{ratio}|$ where $\mathrm{ratio}=\|[D_{\alpha},\mathrm{bridge}_{\mathrm{final},k}\otimes I]\|_{\mathrm{op}}/(2\sqrt{k})$. Finding (finite-$N=20$): at the Walsh-weight cutoff $k=k_{D}=\lfloor\sqrt{D}\rfloor=2$, the ratio reaches $0.993$ ($\gamma_{D}=0.007$) at $\alpha=0$, within $0.7\%$ of the substrate target. Compared to $\gamma_{D}=0.285$ at $D=4$, $\gamma_{D}=0.203$ at $D=6$: linear fit $\log(\gamma_{D})=-0.93D+2.96$, i.e. $\gamma_{D}\sim 19.3\cdot e^{-0.93D}$, empirical finite-$N$ exponential decay with rate $c\approx 0.93$. Caveat established by Computations 37–38: the rate $0.93$ is specific to the matched truncation $N(D)\sim 2.5\,D$; the infinite-$N$ analytical rate at $k_{D}=2$ is $c\approx 0.527$. Ratios above the cutoff ($k>k_{D}$) diverge as expected. With the book-side commitment to the relaxed criterion and the finite-$N$ verification both done, the infinite-$N$ closure is supplied by the refined symmetric-monomial bridge of Computations 39–41.