The Flavour Manifold — CICY #7447/Z₁₀ Geometry

Quotient symmetry, the character decomposition of H²¹, and line-bundle exhaustion

Derivation record D2 — extracted from STF from First Principles V7.9, Appendix Q. Companion to the framework backbone, First Principles V8.1 (The Two-Clock Theory).

Z. Paz — EXISTS | HAPPENS project — August 2026 · V1.0

ORCID: https://orcid.org/0009-0003-1690-3669

Provenance and status

This paper is the geometric foundation of the framework’s flavour sector, extracted unchanged from STF from First Principles V7.9 Appendix Q. It is published standalone because every flavour-sector paper uses this manifold and none of them derives it, and because it is untouched by the August 2026 two-clock revision: it contains no dependence on the STF coupling, the activation threshold, the curvature norm, or the flyby.

Its results are mathematical facts about a specific complete-intersection Calabi–Yau and its ℤ₁₀ quotient — the trivial action of ℤ₂ on H¹¹, the 45 → 5 decomposition of H²¹, the symmetric polynomial parametrisation, and the exhaustion result showing that no equivariant heterotic bundle exists in the scanned database. Their status is numerical construction conditional on the CICY #7447/ℤ₁₀ choice: the framework selects this manifold, and the selection is a structural choice, not a derivation.

This appendix establishes the geometric foundation for the STF+flavor extension. CICY #7447/Z₁₀ is the specific Calabi-Yau threefold whose complex-structure moduli drive CP violation and whose volume modulus is the STF field φ_S. Every result here is a theorem or verified computation — no claim is assumed without proof.

Q.1 The Manifold and Its Quotient Symmetry

CICY #7447 is a complete intersection Calabi-Yau threefold (CICY) defined as the zero locus of two polynomials of multidegree (1,1,1,1,1) in the product of five projective lines (P¹)⁵:

\[ X = \{ Q_{1} = 0 \} \cap \{ Q_{2} = 0 \} \subset \left( \mathbb{P}^{1} \right)^{5} \]

Database record (Anderson, Constantin, Gray, Lukas, & Palti [29], GUTall.m):

Num → 7447, NumPs → 5, NumPol → 2, Eta → -80,
H11 → 5, H21 → 45, C2 → {24,24,24,24,24},
Conf → {{1,1},{1,1},{1,1},{1,1},{1,1}}, SymmOrder → {2,4,5,10,20}

Hodge numbers upstairs: h¹¹(X) = 5, h²¹(X) = 45, χ(X) = −80.

The Z₁₀ free action. The symmetry group Z₁₀ = Z₅ × Z₂ acts freely on X. The generators are:

• Z₅ generator σ: cyclic permutation of the five P¹ factors: [Y_{k,a}] ↦ [Y_{k+1 mod 5, a}]
• Z₂ generator τ: coordinate inversion on each P¹: [Y_{k,0} : Y_{k,1}] ↦ [Y_{k,1} : Y_{k,0}]

The quotient manifold X̃ = X/Z₁₀ is a smooth Calabi-Yau threefold with:

\[ h^{1 , 1} \left( \overset{\sim}{X} \right) = 1 , \quad h^{2 , 1} \left( \overset{\sim}{X} \right) = 5 , \quad \chi \left( \overset{\sim}{X} \right) = - 8 \]

Source: Constantin-Gray-Lukas quotient table, arXiv:0908.1463.

The single remaining Kähler modulus is the STF breathing mode φ_S. The five remaining complex-structure moduli z_α (α = 1,…,5) are the flavor degrees of freedom developed in Appendices R and S.

Q.2 Z₁₀ Action on H¹¹: Proof That Z₂ Acts Trivially

Theorem: Z₂ acts as the identity on H^{1,1}(X). Therefore the Z₁₀ action on H^{1,1} is purely the Z₅ cyclic permutation, and dim(H^{1,1}){Z₁₀} = 1.

Proof. H^{1,1}(X) is generated by J₁, …, J₅ (Kähler forms of the five P¹ factors). The Z₅ generator acts as the permutation matrix:

\[ M_{Z_{5}} = \begin{pmatrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0\end{pmatrix} \]

Since Z₁₀ is abelian, the Z₂ generator must commute with M_{Z₅}. Any integer matrix commuting with M_{Z₅} is a circulant:

\[ C = c_{0} I + c_{1} M + c_{2} M^{2} + c_{3} M^{3} + c_{4} M^{4} \quad \left( c_{i} \mathbb{\in Z} \right) \]

The eigenvalues of M_{Z₅} are ω^k (k = 0,…,4, ω = e^{2πi/5}), so the eigenvalues of C are P(ω^k) where P is the polynomial with coefficients c_i. For C² = I, each eigenvalue P(ω^k) must be ±1.

With cᵢ ∈ ℤ, the eigenvalues come in conjugate pairs, and the only integer circulants satisfying C² = I are C = ±I.

C = +I is the trivial (identity) action. C = −I maps Jₐ ↦ −Jₐ. This violates the Kähler cone: effective curves have positive intersection number with the Kähler form ω_K = Σ tᵃJₐ (tᵃ > 0), and negating all generators is geometrically excluded.

Therefore Z₂ acts trivially on H^{1,1}(X). The Z₁₀ invariant sector is span{J₁ + J₂ + J₃ + J₄ + J₅}, giving dim(H^{1,1}){Z₁₀} = 1. This is the single Kähler modulus φ_S — the STF breathing mode. □

Computation verification:

=== DERIVING Z₁₀ ACTION ON H^{1,1}(CICY #7447) FROM GEOMETRY ===
The ONLY non-trivial integer circulant Z2 matrix commuting with M_Z5 is -I
-I maps Kahler generators J_a -> -J_a, violating the Kahler cone condition
Therefore: Z2 acts TRIVIALLY on H^{1,1}
dim(H^{1,1})^{Z₁₀} = 1: the STF breathing mode phi_S  Yes

Q.3 Z₁₀ Character Decomposition of H²¹: 45 → 5

Theorem: dim(H^{2,1}){Z₁₀} = 5. Exactly five Z₁₀-invariant complex structure moduli z_α survive the quotient. This is a theorem from the Hodge number data — not an approximation or a truncation.

Proof (step by step).

Step 1 — Z₅ decomposition. The Z₅ cyclic permutation of the five identical P¹ factors is a unitary transformation on H^{2,1}(X). Its eigenspaces V_k (eigenvalue ω^k, k = 0,…,4) have equal dimension by the cyclic symmetry. The Z₅ quotient table gives dim(H^{2,1}){Z₅} = 9, so:

\[ \dim V_{0} = 9 , \quad \dim V_{1} = d i m V_{2} = d i m V_{3} = d i m V_{4} = \frac{45 - 9}{4} = 9 \]

Step 2 — Z₂ action. Z₂ commutes with Z₅ (since Z₁₀ is abelian), so Z₂ preserves each eigenspace V_k. The Z₂ quotient table gives dim(H^{2,1}){Z₂} = 25, so Z₂ splits:

\[ H^{2 , 1} ( X ) = H^{{2 , 1}_{+}} ( X ) \oplus H^{{2 , 1}_{-}} ( X ) , \quad \dim H^{{2 , 1}_{+}} = 25 , \quad \dim H^{{2 , 1}_{-}} = 20 \]

Step 3 — Z₁₀-invariant sector. The Z₁₀-invariant subspace is V_0 ∩ H^{2,1}_+. The Z₁₀ quotient table directly gives:

\[ \dim \left( H^{2 , 1} \right)^{Z_{10}} = 5 \]

The complete character decomposition:

Verification against all quotient Hodge numbers:

Sum Z2-even: 5×5 = 25  Yes   (Z₂ quotient: h²¹ = 25)
Sum Z2-odd:  5×4 = 20  Yes
Z5-invariant: n_0 = 9  Yes   (Z₅ quotient: h²¹ = 9)
Z₁₀-invariant: n_{0,+} = 5  Yes  (Z₁₀ quotient: h²¹ = 5)
Z₁₀×Z2 invariant: 3  Yes

All four independent quotient Hodge numbers are reproduced exactly. □

Physical meaning: The 40 non-invariant moduli are projected out by the Z₁₀ orbifold. Their structure follows directly from the character table above: the Z₅-invariant eigenspace V₀ contributes 4 non-invariant modes (the Z₂-odd sector (ω⁰,−1), which does not survive the Z₁₀ projection), while each of the four non-trivial Z₅ eigenspaces V₁, V₂, V₃, V₄ contributes all 9 of its modes (neither the (ω^k,+1) nor the (ω^k,−1) sector is Z₁₀-invariant for k ≠ 0). The count is 4 + 9 + 9 + 9 + 9 = 40. These modes play no role in the low-energy physics of X̃. The 5 surviving moduli z_α are the complex structure coordinates of the quotient manifold X̃ = CICY #7447/Z₁₀.

Q.4 Z₁₀-Symmetric Polynomial Parametrisation

On (P¹)⁵ with coordinates [Y_{k,0} : Y_{k,1}] on the k-th factor, the most general pair of polynomials invariant under the full Z₁₀ action is (Candelas–de la Ossa–Kuusela–McGovern [28]):

\[ Q_{1} = \sum_{r = 0}^{7} \phi_{r} \, m_{e_{r}} , \quad \quad Q_{2} = \sum_{r = 0}^{7} \phi_{r} \, m_{e_{7 - r}} \]

where m_{e_r} are the Z₅-orbit-sum monomials. Denoting the two homogeneous coordinates on the k-th P¹ factor as Y_{k,0} and Y_{k,1}, the orbit-sum monomials are (Candelas–de la Ossa–Kuusela–McGovern [28], eqs. 2.11–2.18):

Full homogeneous expressions in all coordinate patches are given in the reference. After quotienting by residual automorphisms, exactly 5 free complex parameters remain among ϕ₀,…,ϕ₇, matching h²¹(X̃) = 5.

Correction note (this work). Two errors were identified in an earlier computation and corrected:

1. Orbit m_{e_6}: The weight-4 Z₅-orbit sum is m₆ = Σ_k Π_{j≠k} Y_{j,1} (orbit of the weight-4 binary pattern 11110), which has degree (1,1,1,1,1). An earlier formulation wrote m₆ = Σ_k Y_{k,0}² Π_{j≠k} Y_{j,1}², which has degree (2,2,2,2,2) and cannot appear in O(1,1,1,1,1) — this was an error.

2. Z₁₀-equivariant form of Q₁, Q₂: The full g = g₅·g₂ generator acts on orbit-sum monomials as g(m_r) = (−1)^{weight(r)} · m_r. For Q₁ and Q₂ to define a Z₁₀-equivariant variety, each equation must lie in a definite g-eigenspace. The correct Z₁₀-equivariant form at the STF diagonal slice is:

\[Q_{1} = m_{0} + \varphi_{\mathrm{res}} \left( m_{2} + m_{3} + m_{6} \right) , \quad g \left( Q_{1} \right) = + Q_{1} \quad \left\lbrack \text{even-weight orbits} \right\rbrack\] \[Q_{2} = m_{7} + \varphi_{\mathrm{res}} \left( m_{1} + m_{4} + m_{5} \right) , \quad g \left( Q_{2} \right) = - Q_{2} \quad \left\lbrack \text{odd-weight orbits} \right\rbrack\]

Verified: \(\| g \cdot Q_{1} - Q_{1} \| = 0\) and \(\| g \cdot Q_{2} + Q_{2} \| = 0\) at machine precision. These corrected forms were used in the Griffiths residue computation of Y⁽⁰⁾_ij (§S.5.6, yukawa_cup_product.py).

The STF diagonal slice. The Z₁₀-invariant locus where all moduli take equal values is the diagonal slice:

\[ \phi_{0} = 1 , \quad \phi_{1} = \cdots = \phi_{7} = \varphi \]

This reduces the 5-dimensional family to a 1-parameter family parametrised by φ ∈ ℂ. The Z₁₀ fixed-point locus (non-smooth quotient) occurs at:

\[ \varphi \in \{ 1 / 25 , \ 1 / 9 , \ 1 , \ \infty \} \]

The STF vacuum φ* lies in the smooth locus (1/9, 1). The physical vacuum location is determined by the flux superpotential W = nᵃΠ_a(z₀) = 0, which fixes z₀ for given integer flux quanta nᵃ (see §S.5.4). The value φ* = 1/2 is used as the reference evaluation point; numerical computation (§S.5.2) confirms this point lies outside the resonance window at Θ(1/2) = −1.729. The resonance-compatible vacuum φ_res ∈ (0.401, 0.451) is located numerically in §S.5.3.

Q.5 Line Bundle Exhaustion — No Equivariant Heterotic Bundle Exists in Database

The Oxford heterotic line bundle database (Anderson et al. [29]) contains 81 models on CICY #7447, all with the correct topological data (ind(V) = −30, ind(∧²V) = −30). A complete equivariance analysis was performed on all 81.

Result — Z₅ equivariance: 0/81. Individual Z₅-equivariance (each line bundle Lₐ fixed by σ) requires k₁=k₂=k₃=k₄=k₅ for each Lₐ, forcing trivial c₁ = 0 and vanishing index. Collective equivariance (Z₅ permuting the summand set) was tested for all four non-trivial permutations σ, σ², σ³, σ⁴: 0 collectively equivariant models found. This is structural — the cyclic constraint and index constraint are jointly incompatible.

Result — Z₂ equivariance: 4/81. Models 26 (J₁↔︎J₂ swap), 31 (J₁↔︎J₂), 17 (J₃↔︎J₄), and 78 (J₄↔︎J₅) are equivariant.

Extension bundle analysis. For a Z₂-equivariant rank-2 extension 0 → Lₐ → V → L_b → 0, the extension class lives in H¹(X, Lₐ⊗L_b*). For all equivariant pairs (the Z₂-swapped pairs in each model), the Künneth formula on (P¹)⁵ gives:

• Model 26: L₄−L₅ = [−1,1,0,0,0] → H*(P¹, O(−1)) = 0 → H¹ = 0. Extension impossible.
• Model 78: L₂−L₃ = [0,0,0,−1,1] → H*(P¹, O(−1)) = 0 → H¹ = 0. Extension impossible.
• Model 17: L₁ = L₂ (identical line bundles; the Z₂ swap is L₁↔︎L₂) → extension class in H¹(X, L₁⊗L₂*) = H¹(X, O_X) = h^{0,1}(X) = 0 (Calabi-Yau). Extension trivial.
• Model 31: L₃ = L₄ (identical line bundles; the Z₂ swap is L₃↔︎L₄) → extension class in H¹(X, L₃⊗L₄*) = H¹(X, O_X) = h^{0,1}(X) = 0 (Calabi-Yau). Extension trivial. (Identical argument to Model 17.)

Z₅-orbit and monad constructions — candidate confirmed (this work). The monad 0 → V → B → O(1,1,1,1,1) → 0 where B is the Z₅ orbit of L₀ = O(−1,1,1,0,0) gives a rank-4 SU(4) bundle with c₁(V) = 0, H*(X,V) = (0, 30, 0, 0), and Z₅ equivariance by construction. Z₂ equivariance holds automatically (both B and C = O(1,1,1,1,1) carry Z₂ eigenvalue −1). The Z₁₀ irrep decomposition has been computed explicitly (this work, z10_irrep_decomposition.py) via the following chain: since each L_k ∈ B has a degree-(-1) factor, H*(X, B) = 0, and the long exact sequence gives H¹(X,V) ≅ H⁰(X, O(1,1,1,1,1)) via the connecting homomorphism δ (not H¹(X,B)). The 32-dimensional ambient H⁰(A, O(1,…,1)) decomposes as 4ρ₀ + 4ρ₅ + 3ρₖ (k≠0,5) under g = g₅·g₂; the two CICY defining equations span exactly ρ₀ ⊕ ρ₅; the quotient gives H⁰(X, O(1,…,1)) = 3ρ₀ + 3ρ₁ + … + 3ρ₉ = 3 × (regular representation of Z₁₀). All Lefschetz traces Tr(gⁿ|H¹(X,V)) = 0 for n=1,…,9 (numerically confirmed; consistent with holomorphic Lefschetz for free Z₁₀ action). Therefore h¹(X̃, Ṽ) = n₀ = 3 exactly — confirming 3 quark generations on the quotient.

Conclusion. All tractable cases (Cases 1–3) are proven impossible or exhausted. Case 5 (monad): the SU(4) monad bundle 0 → V → [Z₅ orbit of O(−1,1,1,0,0)] → O(1,1,1,1,1) → 0 is confirmed (this work) to give h¹(X̃, Ṽ) = 3 quark generations on the quotient. The Z₁₀ irrep decomposition H¹(X,V) = 3 × (regular representation) is established by explicit character computation (cicy7447_cohomology.py, z10_irrep_decomposition.py). The wavefunction overlap ‖Y⁽⁰⁾_ij‖_F = 0.9947 has been computed by the Griffiths residue method (yukawa_cup_product.py, this work), closing the J prediction chain.