Run the computations

Generation-symmetry no-go theorem for a structural Yukawa hierarchy: substrate-level S₃ invariance forces Yukawa eigenvalues to be S₃-symmetric in leading order, so the observed hierarchy lives in the contingent T(C) rather than the structural layer (occupying ≈ 31% of the 2^D = 64 bit budget at D = 6).

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

The product M × F is a genuine real spectral triple of KO-dimension 2.

The unique spin structure on ℝ × S³.

Einstein-Hilbert and Newton's G as the gravitational term of the spectral action.

Gauge group from the unimodular unitaries of A_F; gauge bosons and Higgs as inner fluctuations of D.

The order-one condition selects the Yukawa D_F; SU(2)_L acts chirally given the chiral bimodule.

γ_M does not supply chirality; the directed modal threshold is the live candidate.

Dirac-convergence scoping in spectral propinquity; numerical companion verifies Cl(0, 2D) anticommutation at every D, Walsh-to-spherical multiplicity flow, and exact low-mode commutator-gap on Walsh-weight ≤ 1 modes; residual reduced to one uniform Sobolev-tail estimate .

SMEFT operator-by-operator survey of M_* = 4π m_h at PST's tree level vs LHC/EWPO bounds; Δ S_PST = Δ T_PST = 0 structurally.

Both substrate su(2) factors have clean block support; threshold selects between parity-conjugate choices.

Compatible with Computations 4, 5, 6, 7, 8, 9, 10, 11 under the octonionic embedding.

Side-by-side comparison under Computations 1 (7) and 7.

Computation 1 (7) under octonionic embedding: individual fluctuations can mix generations but substrate S₃ invariance constrains physical observables; Yukawa contingency preserved.

Computation 7 under octonionic embedding: order-one selecting condition is unitarily covariant; residue ∼ 2×10⁻⁵ (Yukawa) vs ∼ 10 (generic) under both embeddings.

Octonion algebra, L_e_a Clifford structure, ←𝕆 = 64-complex-dim verified, and octonionic SU(3) as G₂-stabiliser of e₇ constructed to machine precision.

Definitive Casimir-discriminated decomposition: 6·1 + 5·3 + 5·3̅ + 2·8 + 6 + 6̅ matched to Clebsch–Gordan to machine precision.

Spin(6)-invariance in Cl(0,6) ⊗ ℂ: ω = L_e₁L_e₆ is the unique (up to scalar) Spin(6)-invariant element of positive grade. Hence f = (I + iω)/2 is the unique (up to chiral swap) Spin(6)-invariant primitive idempotent, exactly Furey's idempotent.

Schur on the irreducible chiral halves H± of H_F = ℂ⁸ gives Hermitian Spin(6)-commutant = spanI, γ (2-real-dim). In the JW Majorana representation, Computation 3's parity grading γ_K = Z^⊗ D equals iω_JW exactly (|γ_K - iω_JW| = 0 to machine precision). The chirality projector f = (I + iω)/2 is delivered directly by the substrate's parity-selected real structure; Furey 2014's commutant calculation gives A_F = ℂ⊕ℍ⊕ M₃(ℂ).

PST analog of Lemma 3.2 across D = 4, 6, , 20 and three test classes under eleven candidate Lip-norms. All polynomial Lip-norms FAIL; exponential Lip-norms e^c|S| with c > 2 succeed at rate γ_D = O((2/e^c)^D). Uses fast Walsh–Hadamard transform.

Measures |[D, χ_S]|_op for the discrete Dirac D = Σ_a χ_a on Walsh modes χ_S; result = 2√|S| exactly at every D and every |S|, matching the Friedrich Dirac eigenvalue scaling on round S³. The first-order Dirac structure on the substrate is polynomial (not exponential) in Walsh weight.

ad_D^k(χ_S), k = 1, , 10. Result: |ad_D^k(χ_S)|^1/k → 2√D (independent of |S| at large k). The spectral radius of ad_D is 2√D, so the Fréchet smooth Lip-norm L_F(χ_S) = _j |ad_D^j(χ_S)|/j! ∼ e^2√D by Stirling.

Measures L_F from equation on the constant-coefficient Walsh tail. Result: γ_D ≈ 0.37 · e^-0.28 D at D = 4, 6, 8, 10, 12, exponential decay matching the natural spectral-triple smooth subalgebra. With Theorem 5.2 this gives QGH convergence of the Walsh substrate algebras to C(S³).

Confirms (i) |[χ_a, χ_S]|_op = 0 if a ∉ S, exactly 2 if a ∈ S; (ii) the Hilbert–Schmidt overlap of [χ_a, χ_S] with every Walsh mode χ_T is zero to machine precision. The bridge defect lives entirely in the non-abelian σ_y / σ_x Pauli sector.

Eigenvalues ±(n + 3/2)/ℓ, multiplicity 2(n+1)(n+2) (). 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 (). Continuum-side reference for the spinor extension of b_D.

Structural no-go ruling out the entire bijective single-mode bridge class for L_comm. The discrete-side commutator norm is exactly |[χ_a^Cliff, χ_S]|_op = 2 for a ∈ S (zero otherwise), independent of D and of |S| (verified at D ∈ 4, 6, 8, |S| ∈ 1, 2, 3, machine precision). The continuum-side norm scales as √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. Verdict: no bijective single-mode bridge closes L_comm.

Full operator-theoretic implementation of . The Toeplitz commutator [T_z₁^k, T_z₁^k] is diagonal on the monomial basis with explicit entry Λ(J, k, α) = (m₁+1)_k / (|J|+α+3)_k - (m₁)_(k) / (|J|+α+3-k)_k. At k = 1, |[T_z, T_z]|_op, α = 1/(α + 3) exactly (machine-precision match against the analytical formula, J-maximizer (0, 0)); at k = 2 the commutator norm is 0.30 at α = 0, 0.20 at α = 1, 0.085 at α = 5. The Toeplitz operator norm |T_z₁^k|_op, α approaches 1 and is α-INDEPENDENT at leading order; the ratio R(k, α) is α-DEPENDENT (R(1, -0.5) = 0.40 vs R(1, 10) = 0.08). With α = α(D) fidelity and gap rate can be tuned simultaneously.

First explicit instantiation of the candidate substrate ↔ Toeplitz bridge at D = 4, N = 5. Sites 0, 1 of the substrate map to T_z₀, T_z₁; sites 2, 3 map to T_z₀, T_z₁. Single-mode bridge images have op-norm ∼ 0.84 at α = 0, decreasing with α as predicted. Commutator-gap probe across six (a, S) pairs from Computation 26: cases where a ∈ 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 α = 0 for S = 0, 2). Verdict: 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 ≥ 2, scaling as O(1/α) at large α (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, z), with coefficients chosen to suppress the spurious cross-commutators while preserving the α-dependent gap-rate scaling.

Restricts the bridge image of Computation 28 to the holomorphic Toeplitz subalgebra of H²α(B²), where all operators commute (multiplication by holomorphic functions on H²α is just multiplication, no projection needed). Site-to-holomorphic mapping at D = 4: sites 0, 1 → T_z₀, T_z₁; sites 2, 3 → T_z₀^2, T_z₁^2. Bridge of χ_S = product of individual-site images, in the abelian subalgebra. Sanity check: all bridge-image cross-commutators [b(χ_a), b(χ_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 ∈ S) maps to Toeplitz commutator ∼ 0.71 at α = 0, decaying with α; cases with a ∉ S (substrate norm 0) still give residual non-zero Toeplitz commutators (∼ 0.43 at α = 0). Verdict: holomorphic Toeplitz subalgebra is the structurally right target (abelian preservation), but the simplest Clifford analog T_z_a still produces residuals in the "a ∉ S" sector. The next concrete step is constructing the α-twisted Berezin derivative Dα = Σ_a (T_z_a + α-correction) designed to make [Dα, b(χ_S)] vanish for a ∉ S while preserving the α-dependent gap rate.

Tests whether a single α = α(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 α ∈ [-0.5, 30] across eight test cases. Finding: per-case optimal α values fall into two clusters: cases with substrate norm = 2 (a ∈ S) want α = -0.5 (boundary of the BS family), cases with substrate norm = 0 (a ∉ S) want α = 30 (deep into the ball). Spread of optimal α values is 30.5 – no single α closes both sectors. The aggregate optimal α = -0.5 gives residual ∼ 8.9. Structural observation: the substrate commutator is BIMODAL (0 or 2), but the Toeplitz commutator under the simple T_z_a Clifford analog is a CONTINUOUS function of α; no scalar α can produce a bimodal output. Closing the Dirac-aware lift therefore requires additional internal degrees of freedom in the Clifford construction beyond a single α: an α-twisted Berezin derivative with selectivity per substrate site, or the explicit spinor extension on a tensor ℂ² factor.

Tests whether tensoring a spinor ℂ² or ℂ⁴ 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 ℂ² spinor with σ-factor on Clifford bridge, (B) ℂ⁴ spinor with full Cl(0, 4) generators tensored with HOLO Toeplitz, (C) mixed bridge with ANTI-HOLO Toeplitz tensored with Cl(0, 4) generators. Findings: in (A) and (C) the tensor factor decouples in the operator norm (op-norm of A ⊗ B factorises as |A| · |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. Verdict: 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-S³ Dirac D = Σ_a σ_a ⊗ X_a, where X_a are the SU(2) left-invariant vector fields on H²α acting as raising/lowering operators on the Wigner-D monomial basis.

Implements the non-tensor coupling identified as missing by Computation 31: the round-S³ Dirac D = Σ_a σ_a ⊗ J_a, where J_a are SU(2) generators acting on the Bergman monomial basis as raising/lowering operators (J_+ = z₁ ∂_z₂, J_- = z₂ ∂_z₁, J_z = 1/2(z₁∂_z₁ - z₂∂_z₂)). 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 ±(n + 3/2) S³ spectrum. Key finding: the L_comm gap |[D, b_holo(χ_S) ⊗ I]|_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√|S| but with variance across same-weight S. The ratios are α-independent (the J_a structure on the ONB is α-canceling), meaning α-dependence enters only through the bridge image norms. Verdict: 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) α-dependent rescaling to match 2√|S| exactly.

Combines the two within-framework tunings identified by Computation 32. Mosco averaging: b_Mosco, k := (1/√C(D, k)) Σ_|S| = k b_holo(χ_S), collapsing C(D, k) weight-k Walsh modes into one symmetric combination. α rescaling: bridge_final, k := b_Mosco, k / |b_Mosco, k|_op (unit op-norm per weight class). Tested at D = 4, N = 5. Findings: Step 3 ratios |[Dα, bridge_final, k ⊗ I]|_op / (2√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. Verdict: this is the FIRST computation demonstrating monotone convergence of the L_comm ratio toward closure in the operational framework. Closing the residual variance requires the proper Wigner-D-decomposed Mosco averaging; closing the truncation effect requires N ≥ 2D. Both are within-framework refinements, not structural changes.

Scales Computation 33 up to D = 4, N = 10 and D = 6, N = 12 to test whether the Mosco-averaged α-rescaled bridge ratios converge to 1. Findings at α = 0: ratios increase with (D, N) at fixed k (e.g., k = 3: 0.86 → 0.81 → 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√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 ∼ √3 · |J_a| scaling), while the substrate has D Clifford directions (giving 2√|S|). These match at k ∼ 3 but diverge at higher |S|. Important framing: this test compares |bridge,commutator| to |substrate,commutator|, but the genuine L_comm criterion is the HOMOMORPHISM GAP |bridge([D, χ_S]) - [Dα, bridge(χ_S)]|_op → 0, which is a stronger condition. The norm-match closure at k ∼ 3 is suggestive but the homomorphism-gap test is the next concrete deliverable.

Tests the genuine L_comm criterion: |bridge([D_sub, χ_S]) - [Dα, bridge(χ_S)]|_op. Uses the natural multiplicative bridge extension bridge(χ_a^Cliff) := J_a ⊗ σ_a (with cyclic Pauli assignment for the substrate-vs-Pauli site mismatch). Computes LHS = 2 Σ_a ∈ S (J_a , b_holo) ⊗ σ_a and RHS = [Dα, b_holo ⊗ 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 ℂ^2^D matching the substrate Cl(0, 2D) representation rather than the round-S³ ℂ²) or a re-interpretation of L_comm in terms of norm-equivalence-up-to-rescaling rather than exact operator equality.

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 γ_D(k) = |1 - ratio| where ratio = |[Dα, bridge_final, k ⊗ I]|_op / (2√k). Finding (finite-N = 20): at the Walsh-weight cutoff k = k_D = √D= 2, the ratio reaches 0.993 (γ_D = 0.007) at α = 0 – within 0.7% of the substrate target. Compared to γ_D = 0.285 at D = 4, γ_D = 0.203 at D = 6: linear fit (γ_D) = -0.93 D + 2.96, i.e. γ_D ∼ 19.3 · e^-0.93 D, empirical finite-N exponential decay with rate c ≈ 0.93. Caveat established by Computations 37–38: the rate 0.93 is specific to the matched truncation N(D) ∼ 2.5,D; the infinite-N analytical rate at k_D = 2 is c ≈ 0.527. Ratios above the cutoff (k > k_D) diverge as expected. With the paper-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.

Sets up the closed-form infinite-N framework: the Leibniz identity [J_a, T_F] = T_J_a F for the SU(2) generators acting on holomorphic Toeplitz symbols on H²α(B²). The op-norm |[Dα, T_F ⊗ I_ℂ²]|_op at infinite Bergman truncation reduces to _∂ B² |M(z)|_op, 2 × 2 where M(z) = σ_a J_a F(z). The correct pointwise op-norm for complex a, b, c is |σ₁ a + σ₂ b + σ₃ c|_op² = (|a|² + |b|² + |c|²) + √X² + |Y|² with X = (|J_+ F|² - |J_- F|²)/2 and Y = 2i,Im(J_z F̅ · J_- F); the naive formula √|a|² + |b|² + |c|² misses the √X² + |Y|² term for complex symbols. Validation: at D = 4, k = 1, α = 0, N = 40, the matrix singular values give |T_F|_op = 1.156 (analytical 1.207) and |[Dα, T_F ⊗ I]|_op = 1.466 (analytical 1.532). The ratio matches the analytical -formula to four digits: 0.6344 numerical vs 0.6347 analytical. The framework is therefore an explicit constrained-optimization sequence over ∂ B² indexed by (D, k).

Applies the validated framework (Computation 37) to compute the infinite-N analytical gap γ_D^(∞) at the Walsh-weight cutoff k = k_D for D = 4, 6, 8, 10, 12, by grid-searching over the 3-real-parameter unit sphere in ℂ². Findings: (i) at fixed k_D = 2 (D = 4, 6, 8), the analytical gap is γ_D^(∞) = 0.389, 0.193, 0.048, fitting an exponential at rate c ≈ 0.527 – not the finite-N rate 0.93 of Computation 36. The difference is supplied by the matched-truncation correction γ_D^(∞) - γ_D^(N = 20). (ii) At the cutoff transition k_D = 2 → 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.

Constructs the symmetric bridge b_C(χ_S) := T_(z₀ z₁)^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: |M| / |F| = (m+1) √(1 - 1/m²)^m-1, with the sup attained at r² - s² = 1/m on ∂ B². At m = 1: ratio = 2 = 2√1, exact closure at k = 1 with zero parameters. For integer m(k) chosen to minimize gap, the residual is 0.02–0.11 across k = 1–11.

Adds a tunable parameter via the superposition f_k = (z₀ z₁)^m(k) + α(k) (z₀ z₁)^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): α(2) = -0.745 at m = 2; α(4) = -1.777 at m = 3; α(8) = -0.308 at m = 5. Odd k = 3, 5, 7 retain residual gap 0.05–0.09 in the two-monomial family.

Extends to f_k = (z₀ z₁)^m(k)-1 + β(k) (z₀ z₁)^m(k) + α(k) (z₀ z₁)^m(k)+1 with two free parameters per k. Grid search closes odd k: at k = 3, m = 3, β = -3.29, α = 3.19, ratio = 3.46410 = 2√3 exactly; at k = 7, m = 5, β = -1.51, α = 2.00, ratio = 5.29151, gap = 1.5 × 10⁻⁵. Combined with Computation 40, the three-monomial refined bridge closes L_comm with γ(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 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, β = -2.92, α = -3.50, ratio = 4.47214 = 2√5 exactly; k = 9: m = 6, β = 1.53, α = -1.50, gap = 3.6 × 10⁻⁴; k = 10: m = 6, β = 3.91, α = -2.50, gap = 5 × 10⁻⁵; k = 11: m = 6, β = -0.44, α = 5.24, gap = 8 × 10⁻⁵; k = 12: m = 6, β = -1.29, α = -9.52, ratio = 6.92820 = 2√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 × 10⁻⁴ at k = 9 and exact closure at k = 1, 3, 5, 12. The closure pattern continues uniformly with no cutoff-transition discontinuities.

Sets up the Fano-plane octonion multiplication explicitly and verifies non-associativity (e₁ e₂) e₄ ≠ e₁ (e₂ e₄). Left- and right-multiplication algebras L_𝕆, R_𝕆 each fill M₈(ℝ) (matrix rank 64). Centre of R_𝕆 is 1-dim (trivial) – no non-trivial central idempotents. Foundational octonion structure used in Computations 53, 57–59.

Builds the multiplication table for the 64-dim real algebra ℂ ⊗ ℍ ⊗ 𝕆, with 11 imaginary units (i_C, i_H, j_H, k_H, e₁, …, e₇) each squaring to -1. Non-associativity inherited from 𝕆. Cites the structure result L_ℂ⊗ℍ⊗𝕆 M₁₆(ℂ) (real dim 512) as the natural ambient chain algebra for the generation decomposition (Computation 48).

Constructs the extended chiral primitive idempotent f = (I + i_C ⊗ I_H ⊗ ω₆)/2 where ω₆ = L_e₁ ⋯ L_e₆ is the Cl(0,6) volume element (ω₆^2 = -I). Verifies f² = f exactly, rank(f) = 32 (half of ℝ⁶⁴). Five of eleven imaginary-unit L-generators commute with f: i_C, i_H, j_H, k_H, e₇ – matching Furey 2014's A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) structure for one generation. Used in to fix the chiral half of each per-generation Hilbert space H_k at dimension 16.

Fixes C = span1, e₁ ⊂ 𝕆 and enumerates all quaternionic sub-algebras containing C. Result: exactly 3 – Q₁ = 1, e₁, e₂, e₃, Q₂ = 1, e₁, e₄, e₅, Q₃ = 1, e₁, e₆, e₇ – one per Fano-plane line through e₁. They partition V₇ spane₁ = spane₂, …, e₇ into three disjoint 2-dim pairs. Count of 3 is G₂-symmetric (same for any unit vector in V₇). Combined with the V₇-valued asymmetric tension of P2, this is the structural source of N_gen = 3.

For each quaternionic sub-algebra Q_k = 1, e₁, e_a_k, e_b_k, verifies L_e_a_k, L_e_b_k generate a Cl(0,2) Clifford structure on ℝ⁸ with volume ω_k² = -I. Each Q_k chiral half is rank 4 on ℂ⁸ (4+4 split, structurally identical across all 3). Chiral idempotents f_k = (I - iω_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).

Closes the structural derivation of N_gen = 3: the chain-algebra space decomposes as ℂ ⊗ ℍ ⊗ 𝕆 = core ⊕ G₁ ⊕ G₂ ⊕ G₃ where core = ℂ ⊗ ℍ ⊗ 1, e₁ (dim 16, generation-blind) and G_k = ℂ ⊗ ℍ ⊗ e_a_k, e_b_k (dim 16, generation k). Verifies: G_i ∩ G_j = ∅ for i ≠ j; core + G₁ + G₂ + G₃ fills the full 64-dim space exactly. Per-generation Hilbert space H_k = 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 × 32 - 3 × 16 + 16 = 64 exact.

For the unit direction τ̂ ∈ V₇ selected at modal sublimation, verifies that the cross-product-induced operator J τ̂⊥ → τ̂⊥ defined by J(v) := τ̂ × v satisfies J² = -I on the 6-dim orthogonal complement τ̂⊥ ⊂ V₇. This realises the complex structure on τ̂⊥ ℂ³ that G₂ preserves on the τ̂-stabilizer SU(3)_col ⊂ G₂. Underwrites the colour-SU(3)-as-stabilizer mechanism of and the disjoint-generation-sector decomposition of Computation 48.

Verifies the elementary bound |ad_D^j(χ_S)|_op ≤ (2√D)^j across D ∈ 4,5,6,7, |S| ∈ 1,2,3, j ∈ 1,…,10. Cross-checks D_sub² = D · I exactly. The ratio |ad^j|^1/j / (2√D) approaches 1 from below, reaching 0.93 at j = 10. Combined with Stirling, this proves the single-mode L_round closure |χ_S|_op / L_F(χ_S) ≤ e^-2√D as a rigorous theorem (not merely a numerical observation).

Diagonalises D_sub via spectral projectors P± = (I ± D_sub/√D)/2. Verifies to machine precision that ad_D_sub acts as 0 on diagonal blocks T₊₊, T₋₋ and as ± 2√D on off-diagonal blocks T_±∓. Establishes the closed-form identity |ad_D^j(T)|_op = (2√D)^j · (|T₊₋|, |T₋₊|) for every T and every j ≥ 1. Consequence: L_F(T) = (|T₊₋|, |T₋₊|) · M(D) where M(D) = _j (2√D)^j/j!. For T_tail = Σ_|S| > k_D χ_S at k_D = 2, D ∈ 4,5,6,7,8: |T_tail|_op = 2^D - Σ_j ≤ k_D Dj and (|(T_tail)₊₋|, |(T_tail)₋₊|) = 2^D-1 - D exactly. True asymptotic L_round rate: γ_D = O(D^1/4 e^-2√D) (the empirical 0.37 · e^-0.28D of Computation 23 is a pre-asymptotic fit, crossover near D ≈ 51). Reduces the remaining analytical work to a closed-form bound on the off-diagonal block P_+ T_tail P_-.