\documentclass[11pt,a4paper]{article}
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\fancyhead[L]{GIFT v3.4: Supplement S1}
\hypersetup{pdftitle={GIFT v3.4 Supplement S1: Mathematical Foundations}}

\title{%
\LARGE\textbf{Supplement S1: Mathematical Foundations}\\[0.3em]
\large $E_8$ Exceptional Lie Algebra, $G_2$ Holonomy Manifolds, and $K_7$ Construction%
}
\author{Brieuc de La Fourni\`ere \\ \textit{Independent researcher, Beaune, France}}
\date{May 2026 \quad\textbar\quad v3.4}

\graphicspath{{../../../figures/}}

\begin{document}

\maketitle
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\thispagestyle{fancy}

\tableofcontents
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\emph{Complete mathematical foundations for GIFT, presenting E₈ architecture and K₇ manifold construction.}

\textbf{Lean Verification}: 213 certificate conjuncts, 4 axioms, 134 .lean files (Lean 4.29.0, zero \texttt{sorry})

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{Abstract}\label{abstract}

This supplement presents the mathematical architecture underlying GIFT. Part I develops the E₈ exceptional Lie algebra with the Exceptional Chain theorem. Part II introduces G₂ holonomy manifolds, including the correct characterization of the g₂ subalgebra as the kernel of the Lie derivative map. Part III discusses K₇ manifold topology: the pair (b₂,b₃)=(21,77) does not appear in the standard TCS catalogue, but the Joyce-Karigiannis (JK) Z₂³ orbifold construction realizes it via T³ × K3 / Z₂³ resolution (§8.4). A four-phase computer-assisted audit (V4 symplectic screen, anti-symplectic obstruction, K3 lattice existence via Mukai/Garbagnati-Sarti, Betti formula) closes the topological count exactly; the analytic torsion-free statement and an explicit polynomial Z₂³ realization are deferred. Part IV establishes the algebraic reference form determining det(g) = 65/32, with Joyce's perturbation theorem providing the existence criterion for a nearby torsion-free metric (subject to its hypothesis $\|T\| < \varepsilon_0 = 0.1$). PINN validation confirms near-G₂ holonomy with V₇ projection reduced by 97\%. All algebraic results are formally verified in Lean 4 (including \texttt{JoyceKarigiannisConstruction.lean}).

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\section{Part 0: The Octonionic Foundation}\label{part-0-the-octonionic-foundation}

\subsection{0. Why This Framework Exists}\label{why-this-framework-exists}

The GIFT framework emerges from a single algebraic fact:

\textbf{The octonions $\mathbb{O}$ are the largest normed division algebra.}

The derivation chain proceeds as follows:

\begin{verbatim}
$\mathbb{O}$ (octonions, dim 8)
    │
    ▼
Im($\mathbb{O}$) = ℝ⁷ (imaginary octonions)
    │
    ▼
G₂ = Aut($\mathbb{O}$) (automorphism group, dim 14)
    │
    ▼
K₇ with G₂ holonomy (explicit certified metric, Paper I)
    │
    ▼
Topological invariants (b₂ = 21, b₃ = 77)
    │
    ▼
95 observables (33(I) + 19(II) + 21(III) + 22(IV), 55 Lean-certified)
\end{verbatim}

\textbf{Status}: 95 observables across 4 types. 33 Type I algebraic, 19 Type II one-step, 21 Type III multi-step, 22 Type IV structural (incl.~6 metric block eigenvalues + Pinčák 2026 [42]). 55/95 Lean-certified (213 conjuncts, 4 axioms, 0 sorry). See §4 of main text and Supplement S3 for the complete dataset. The gauge breaking chain (§5 of main text) is certified in \texttt{TCSGaugeBreaking.lean} and \texttt{GaugeBundleData.lean}.

\subsubsection{0.1 The Division Algebra Chain}\label{the-division-algebra-chain}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}llll@{}}
\toprule\noalign{}
Algebra & Dim & Physics Role & Stops? \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
ℝ & 1 & Classical mechanics & No \\
ℂ & 2 & Quantum mechanics & No \\
ℍ & 4 & Spin, Lorentz group & No \\
\textbf{$\mathbb{O}$} & \textbf{8} & \textbf{Exceptional structures} & \textbf{Yes} \\
\end{longtable}
The pattern terminates at $\mathbb{O}$. There is no 16-dimensional normed division algebra. The octonions are \emph{the end of the line}.

\subsubsection{0.2 G₂ as Octonionic Automorphisms}\label{gux2082-as-octonionic-automorphisms}

\textbf{Definition}: G₂ = \{g ∈ GL($\mathbb{O}$) : g(xy) = g(x)g(y) for all x,y ∈ $\mathbb{O}$\}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Property & Value & GIFT Role \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
dim(G₂) & 14 = C(7,2) − C(7,1) = 21 − 7 & Q\_Koide numerator \\
Action & Transitive on S⁶ ⊂ Im($\mathbb{O}$) & Connects all directions \\
Embedding & G₂ ⊂ SO(7) & Preserves φ₀ \\
\end{longtable}
\subsubsection{0.3 Why dim(K₇) = 7}\label{why-dimkux2087-7}

The dimension 7 is a consequence of the octonionic structure, not an independent choice:
- Im($\mathbb{O}$) has dimension 7
- G₂ acts naturally on ℝ⁷
- A compact 7-manifold with G₂ holonomy provides the geometric realization

In this sense, K₇ is to G₂ what the circle is to U(1).

\subsubsection{0.4 The Fano Plane: Combinatorial Structure of Im($\mathbb{O}$)}\label{the-fano-plane-combinatorial-structure-of-imux1d546}

The 7 imaginary octonion units form the \textbf{Fano plane} PG(2,2), the smallest projective plane:
- 7 points (imaginary units e₁\ldots e₇)
- 7 lines (multiplication triples eᵢ × eⱼ = ±eₖ)
- 3 points per line

\textbf{Combinatorial counts}:
- Point-line incidences: 7 × 3 = 21 = C(7,2) = b₂
- Automorphism group: PSL(2,7) with \textbar PSL(2,7)\textbar{} = 168

\textbf{Numerical observation}: The following arithmetic identity holds:
\[(b_3 + \dim(G_2)) + b_3 = 91 + 77 = 168 = |{\rm PSL}(2,7)| = {\rm rank}(E_8) \times b_2\]

Whether this reflects deeper geometric structure connecting gauge and matter sectors, or is an arithmetic coincidence, remains an open question.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\section{Part I: E₈ Exceptional Lie Algebra}\label{part-i-eux2088-exceptional-lie-algebra}

\subsection{1. Root System and Dynkin Diagram}\label{root-system-and-dynkin-diagram}

\subsubsection{1.1 Basic Data}\label{basic-data}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Property & Value & GIFT Role \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Dimension & dim(E₈) = 248 & Gauge DOF \\
Rank & rank(E₈) = 8 & Cartan subalgebra \\
Number of roots & & Φ(E₈) \\
Root length & √2 & α\_s numerator \\
Coxeter number & h = 30 & Icosahedron edges \\
Dual Coxeter number & h∨ = 30 & McKay correspondence \\
\end{longtable}
\subsubsection{1.2 Root System Construction}\label{root-system-construction}

E₈ root system in ℝ⁸ has 240 roots:

\textbf{Type I (112 roots)}: Permutations and sign changes of (±1, ±1, 0, 0, 0, 0, 0, 0)

\textbf{Type II (128 roots)}: Half-integer coordinates with even minus signs:
\[\frac{1}{2}(\pm 1, \pm 1, \pm 1, \pm 1, \pm 1, \pm 1, \pm 1, \pm 1)\]

\textbf{Verification}: 112 + 128 = 240 roots, all length √2.

\textbf{Lean Status}: E₈ Root System \textbf{12/12 COMPLETE}. All theorems proven:
- \texttt{D8\_roots\_card} = 112, \texttt{HalfInt\_roots\_card} = 128
- \texttt{E8\_roots\_card} = 240, \texttt{E8\_roots\_decomposition}
- \texttt{E8\_inner\_integral}, \texttt{E8\_norm\_sq\_even}, \texttt{E8\_sub\_closed}
- \texttt{E8\_basis\_generates}: Every lattice vector is integer combination of simple roots (theorem)

\subsubsection{1.3 Cartan Matrix}\label{cartan-matrix}

\[A_{E_8} = \begin{pmatrix}
2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & -1 & 0 & 0 & 0 & 0 \\
-1 & 0 & 2 & -1 & 0 & 0 & 0 & 0 \\
0 & -1 & -1 & 2 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 2
\end{pmatrix}\]

\textbf{Properties}: det(A) = 1 (unimodular), positive definite.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{2. Weyl Group}\label{weyl-group}

\subsubsection{2.1 Order and Factorization}\label{order-and-factorization}

\[|W(E_8)| = 696,729,600 = 2^{14} \times 3^5 \times 5^2 \times 7\]

\subsubsection{2.2 Topological Factorization Theorem}\label{topological-factorization-theorem}

\textbf{Theorem}: The Weyl group order factorizes into GIFT structural constants:

\[|W(E_8)| = p_2^{\dim(G_2)} \times N_{gen}^{Weyl} \times Weyl^{p_2} \times \dim(K_7)\]

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}llll@{}}
\toprule\noalign{}
Factor & Exponent & Value & GIFT Origin \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
2¹⁴ & dim(G₂) = 14 & 16384 & p₂\^{}(holonomy dim) \\
3⁵ & Weyl = 5 & 243 & N\_gen\^{}(Weyl factor) \\
5² & p₂ = 2 & 25 & Weyl\^{}(binary) \\
7¹ & 1 & 7 & dim(K₇) \\
\end{longtable}
\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{weyl\_E8\_topological\_factorization}

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsubsection{2.3 Triple Derivation of Weyl = 5}\label{triple-derivation-of-weyl-5}

\textbf{Theorem}: The Weyl factor admits three independent derivations from topological invariants.

\subsubsection{Derivation 1: G₂ Dimensional Ratio}\label{derivation-1-gux2082-dimensional-ratio}

\[\text{Weyl} = \frac{\dim(G_2) + 1}{N_{gen}} = \frac{14 + 1}{3} = \frac{15}{3} = 5\]

\textbf{Interpretation}: The holonomy dimension plus unity, distributed over generations.

\subsubsection{Derivation 2: Betti Reduction}\label{derivation-2-betti-reduction}

\[\text{Weyl} = \frac{b_2}{N_{gen}} - p_2 = \frac{21}{3} - 2 = 7 - 2 = 5\]

\textbf{Interpretation}: The per-generation Betti contribution minus binary duality.

\subsubsection{Derivation 3: Exceptional Difference}\label{derivation-3-exceptional-difference}

\[\text{Weyl} = \dim(G_2) - \text{rank}(E_8) - 1 = 14 - 8 - 1 = 5\]

\textbf{Interpretation}: The gap between holonomy dimension and gauge rank, reduced by unity.

\subsubsection{Unified Identity}\label{unified-identity}

These three derivations establish the \textbf{Weyl Triple Identity}:

\[\boxed{\frac{\dim(G_2) + 1}{N_{gen}} = \frac{b_2}{N_{gen}} - p_2 = \dim(G_2) - \text{rank}(E_8) - 1 = 5}\]

\textbf{Status}: VERIFIED (algebraic identity from GIFT constants)

\subsubsection{Verification}\label{verification}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Expression & Computation & Result \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
(dim(G₂) + 1) / N\_gen & (14 + 1) / 3 & 5 \\
b₂/N\_gen - p₂ & 21/3 - 2 & 5 \\
dim(G₂) - rank(E₈) - 1 & 14 - 8 - 1 & 5 \\
\end{longtable}
\subsubsection{Significance}\label{significance}

The triple convergence suggests Weyl = 5 is structurally constrained by the E₈ x E₈/G₂/K₇ geometry. It enters:

\begin{enumerate}


\item
  \textbf{det(g) = 65/32}: Via Weyl x (rank(E₈) + Weyl) / 2\^{}Weyl = 5 x 13 / 32
\item
  \textbf{\textbar W(E₈)\textbar{} factorization}: The factor 5² = Weyl\^{}p₂ in prime decomposition
\item
  \textbf{Cosmological ratio}: sqrt(Weyl) = sqrt(5) appears in dark sector density ratios (see main paper, Section 4.8)
\end{enumerate}

\textbf{Status}: VERIFIED (three independent derivations)

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{3. Exceptional Chain}\label{exceptional-chain}

\subsubsection{3.1 The Pattern}\label{the-pattern}

A pattern connects exceptional algebra dimensions to primes:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lllll@{}}
\toprule\noalign{}
Algebra & n & dim(E\_n) & Prime & Index \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
E₆ & 6 & 78 & 13 & prime(6) \\
E₇ & 7 & 133 & 19 & prime(8) = prime(rank(E₈)) \\
E₈ & 8 & 248 & 31 & prime(11) = prime(D\_bulk) \\
\end{longtable}
\subsubsection{3.2 Exceptional Chain Theorem}\label{exceptional-chain-theorem}

\textbf{Theorem}: For n ∈ \{6, 7, 8\}:
\[\dim(E_n) = n \times prime(g(n))\]

where g(6) = 6, g(7) = rank(E₈) = 8, g(8) = D\_bulk = 11 (= dim(M₄) + dim(K₇) = 4 + 7, the total bulk dimension of M-theory).

\textbf{Proof} (verified in Lean):
- E₆: 6 × 13 = 78 ✓
- E₇: 7 × 19 = 133 ✓
- E₈: 8 × 31 = 248 ✓

\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{exceptional\_chain\_certified}

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{4. E₈×E₈ Product Structure}\label{eux2088eux2088-product-structure}

\subsubsection{4.1 Direct Sum}\label{direct-sum}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}ll@{}}
\toprule\noalign{}
Property & Value \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Dimension & 496 = 248 × 2 \\
Rank & 16 = 8 × 2 \\
Roots & 480 = 240 × 2 \\
\end{longtable}
\subsubsection{4.2 τ Numerator Connection}\label{ux3c4-numerator-connection}

The hierarchy parameter numerator:
\[\tau_{num} = 3472 = 7 \times 496 = \dim(K_7) \times \dim(E_8 \times E_8)\]

\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{tau\_num\_E8xE8}

\subsubsection{4.3 Binary Duality Parameter}\label{binary-duality-parameter}

\textbf{Triple geometric origin of p₂ = 2}:

\begin{enumerate}


\item
  \textbf{Local}: p₂ = dim(G₂)/dim(K₇) = 14/7 = 2
\item
  \textbf{Global}: p₂ = dim(E₈×E₈)/dim(E₈) = 496/248 = 2
\item
  \textbf{Root}: √2 in E₈ root normalization
\end{enumerate}

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{5. Exceptional Algebras from Octonions}\label{exceptional-algebras-from-octonions}

The foundational role of octonions is established in Part 0. This section details the exceptional algebraic structures that emerge from $\mathbb{O}$.

\subsubsection{5.1 Exceptional Jordan Algebra J₃(O)}\label{exceptional-jordan-algebra-jux2083o}

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\begin{longtable}[]{@{}ll@{}}
\toprule\noalign{}
Property & Value \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
dim(J₃(O)) & 27 = 3³ \\
dim(J₃(O)₀) & 26 (traceless) \\
\end{longtable}
\textbf{E-series formula}: The dimension 27 itself emerges from the exceptional chain:

\[\dim(J_3(\mathbb{O})) = \frac{\dim(E_8) - \dim(E_6) - \dim(SU_3)}{6} = \frac{248 - 78 - 8}{6} = \frac{162}{6} = 27\]

This shows the Jordan algebra dimension is derivable from the E-series structure.

\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{j3o\_e\_series\_certificate}

\subsubsection{5.2 F₄ Connection}\label{fux2084-connection}

F₄ is the automorphism group of J₃(O):
\[\dim(F_4) = 52 = p_2^2 \times \alpha_{sum}^B = 4 \times 13\]

\subsubsection{5.3 Exceptional Differences}\label{exceptional-differences}

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\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Difference & Value & GIFT \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
dim(E₈) - dim(J₃(O)) & 221 = 13 × 17 & α\_B × λ\_H\_num \\
dim(F₄) - dim(J₃(O)) & 25 = 5² & Weyl² \\
dim(E₆) - dim(F₄) & 26 & dim(J₃(O)₀) \\
\end{longtable}
\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{exceptional\_differences\_certified}

\subsubsection{5.4 Structural Derivation of τ}\label{structural-derivation-of-ux3c4}

The hierarchy parameter τ admits a purely geometric derivation from framework invariants:

\[\tau = \frac{\dim(E_8 \times E_8) \times b_2}{\dim(J_3(\mathbb{O})) \times H^*} = \frac{496 \times 21}{27 \times 99} = \frac{10416}{2673} = \frac{3472}{891}\]

\textbf{Prime factorization}:
- Numerator: 3472 = 2⁴ × 7 × 31 = dim(K₇) × dim(E₈×E₈)
- Denominator: 891 = 3⁴ × 11 = N\_gen⁴ × D\_bulk

\textbf{Alternative form}: τ\_num = 7 × 496 = dim(K₇) × dim(E₈×E₈) = 3472

This anchors τ to topological and algebraic invariants, establishing it as a geometric constant rather than a free parameter.

\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{tau\_structural\_certificate}

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\section{Part II: G₂ Holonomy Manifolds}\label{part-ii-gux2082-holonomy-manifolds}

\subsection{6. Definition and Properties}\label{definition-and-properties}

\subsubsection{6.1 G₂ as Exceptional Holonomy}\label{gux2082-as-exceptional-holonomy}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Property & Value & GIFT Role \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
dim(G₂) & 14 & Q\_Koide numerator \\
rank(G₂) & 2 & Lie rank \\
Definition & Aut(O) & Octonion automorphisms \\
\end{longtable}
\textbf{Lean Status}: G₂ Cross Product \textbf{9/11} proven:
- \texttt{epsilon\_antisymm}, \texttt{epsilon\_diag}, \texttt{cross\_apply} ✓
- \texttt{G2\_cross\_bilinear}, \texttt{G2\_cross\_antisymm}, \texttt{cross\_self} ✓
- \texttt{G2\_cross\_norm} (Lagrange identity ‖u×v‖² = ‖u‖²‖v‖² − ⟨u,v⟩²) ✓
- \texttt{reflect\_preserves\_lattice} (Weyl reflection) ✓
- Remaining: \texttt{cross\_is\_octonion\_structure} (343-case timeout), \texttt{G2\_equiv\_characterizations}

\subsubsection{6.2 G₂ as Kernel of the Lie Derivative}\label{gux2082-as-kernel-of-the-lie-derivative}

The G₂ subalgebra of so(7) admits a precise characterization as the stabilizer of the associative 3-form phi₀. For any antisymmetric matrix A in so(7), the Lie derivative of phi₀ is:

\[L_A(\varphi_0)_{ijk} = A_{ia}\varphi_{ajk} + A_{ja}\varphi_{iak} + A_{ka}\varphi_{ija}\]

The g₂ subalgebra consists of all A for which L\_A(phi₀) = 0:

\[\mathfrak{g}_2 = \ker(L) = \{A \in \mathfrak{so}(7) : L_A(\varphi_0) = 0\}\]

This yields the decomposition so(7) = g₂ + V₇, where dim(g₂) = 14 and dim(V₇) = 7. The complement V₇ carries the standard 7-dimensional representation of G₂.

In practice, the kernel is computed via singular value decomposition (SVD) of the linear map L: so(7) --\textgreater{} Lambda³(R⁷). The 14 singular vectors with eigenvalue zero span g₂; the 7 singular vectors with nonzero eigenvalue span V₇.

\textbf{Note}: A heuristic construction based on Fano-plane indices does not produce correct g₂ generators (each such generator is approximately 67\% in g₂ and 33\% in V₇). The kernel-based construction is the correct definition and must be used in all numerical computations involving g₂/V₇ decomposition.

\subsubsection{6.3 Holonomy Classification (Berger)}\label{holonomy-classification-berger}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Dimension & Holonomy & Geometry \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\textbf{7} & \textbf{G₂} & \textbf{Exceptional} \\
8 & Spin(7) & Exceptional \\
\end{longtable}
\subsubsection{6.4 Torsion: Definition and GIFT Interpretation}\label{torsion-definition-and-gift-interpretation}

\textbf{Mathematical definition}: Torsion measures failure of G₂ structure to be parallel:
\[T = \nabla\phi \neq 0\]

For a G₂ structure φ, the intrinsic torsion decomposes into four irreducible G₂-modules:

\[T \in W_1 \oplus W_7 \oplus W_{14} \oplus W_{27}\]

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Class & Dimension & Characterization \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
W₁ & 1 & Scalar: dφ = τ₀ ⋆φ \\
W₇ & 7 & Vector: dφ = 3τ₁ ∧ φ \\
W₁₄ & 14 & Co-closed part of d⋆φ \\
W₂₇ & 27 & Traceless symmetric \\
\end{longtable}
\textbf{Total dimension}: 1 + 7 + 14 + 27 = 49 = 7² = dim(K₇)²

The torsion-free condition requires all four classes to vanish simultaneously, a highly constrained state with 49 conditions.

\textbf{Torsion-free condition}:
\[\nabla\phi = 0 \Leftrightarrow d\phi = 0 \text{ and } d*\phi = 0\]

\textbf{GIFT interpretation}:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3846}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3462}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2692}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Quantity
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Meaning
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Value
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
κ\_T = 1/61 & Topological \emph{capacity} for torsion & Fixed by K₇ \\
φ\_ref & Algebraic reference form & c × φ₀ \\
T\_realized & Actual torsion for global solution & Constrained by Joyce \\
\end{longtable}
\textbf{Key insight}: The 33 dimensionless predictions use only topological invariants (b₂, b₃, dim(G₂)) and are independent of the specific torsion realization. The value κ\_T = 1/61 defines the geometric bound on deviations from φ\_ref.

\textbf{Physical interactions}: Emerge from the geometry of K₇, with deviations delta(phi) from the reference form bounded by topological constraints. This mechanism is theoretical and its detailed treatment lies beyond the scope of this supplement.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{7. Topological Invariants}\label{topological-invariants}

\subsubsection{7.1 Derived Constants}\label{derived-constants}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Constant & Formula & Value \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
det(g) & p₂ + 1/(b₂ + dim(G₂) - N\_gen) & 65/32 \\
κ\_T & 1/(b₃ - dim(G₂) - p₂) & 1/61 \\
sin²θ\_W & b₂/(b₃ + dim(G₂)) & 3/13 \\
\end{longtable}
\subsubsection{7.2 The 61 Decomposition}\label{the-61-decomposition}

\[\kappa_T^{-1} = 61 = \dim(F_4) + N_{gen}^2 = 52 + 9\]

Alternative:
\[61 = \Pi(\alpha^2_B) + 1 = 2 \times 5 \times 6 + 1\]

\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{kappa\_T\_inv\_decomposition}

\subsubsection{7.3 Spectral Geometry}\label{spectral-geometry}

The Laplace-Beltrami operator on K₇ admits a discrete spectrum with eigenvalues 0 = λ₀ \textless{} λ₁ ≤ λ₂ ≤ \ldots{} The first non-zero eigenvalue λ₁ (spectral gap) characterizes the geometry's rigidity.

\textbf{Bare topological ratio}: The ratio dim(G₂)/H* provides a topological reference scale:

\[\frac{\dim(G_2)}{H^*} = \frac{14}{b_2 + b_3 + 1} = \frac{14}{99} = 0.1414...\]

This is NOT the spectral gap itself, but a topological bound, see below.

\textbf{Analytical spectral gap}: If the TCS decomposition holds with b₂(M₁) = 11, the first eigenvalue of the scalar Laplacian admits a closed-form expression from the 1D Sturm-Liouville reduction on the TCS neck:

\[\boxed{\lambda_1 = \frac{\pi^2}{L^2 \cdot g_{ss}} = \frac{6\pi^2}{25 \cdot (b_2(M_1) + \mathrm{rank}(E_8))} = \frac{6\pi^2}{475} = 0.12467...}\]

where L = 5 is the effective neck domain length (including ACyl tails), g\_ss = (b₂(M₁) + rank(E₈))/(3·rank(G₂)) = 19/6 is the seam metric component (topologically determined by torsion minimization), and b₂(M₁) = 11 is the Picard rank of the first building block in the conjectured TCS realization. Numerical verification (Richardson extrapolation, N=800→1600): λ₁ = 0.12461, matching the formula to \textbf{0.08\%}.

The 169 metric parameters collapse to a single topological integer b₂(M₁), plus two group-theoretic constants (rank(E₈) = 8, rank(G₂) = 2). This provides a closed-form expression for a KK mass gap on a compact G₂ manifold.

\textbf{Comparison with topological ratios:}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2444}}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.1556}}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.4222}}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.1778}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Expression
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Value
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Deviation from λ₁
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Status
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
6π²/475 (analytical) & 0.12467 & 0.00\% (definition) & \textbf{EXACT} \\
NK Richardson & 0.12461 & 0.05\% & Numerical verification \\
14/99 (bare topological) & 0.14141 & +13.4\% & Topological bound only \\
13/99 (with spinor correction) & 0.13131 & +5.3\% & Approximate (see remark) \\
\end{longtable}
\emph{Remark:} The exact value λ₁ = 6π²/475 involves π (from the Sturm-Liouville eigenvalue structure), not a rational number. The product λ₁×H* = 6π²×99/475 = 12.3364, the confirmed universal invariant (see universality below).

\textbf{Lean status}: \texttt{Spectral.PhysicalSpectralGap} (28 theorems, zero axioms). \texttt{Spectral.SelbergBridge} connects the spectral gap to the mollified Dirichlet polynomial S\_w(T) via the Selberg trace formula.

\textbf{Connection to π²}: The spectral gap formula λ₁ = 6π²/475 provides a direct link between the transcendental number π² and topological integers. The near-identity dim(G₂)/√2 ≈ π² (0.30\%) finds its resolution: the eigenvalue explicitly involves π² through the Sturm-Liouville structure, while dim(G₂) enters through g\_ss.

\textbf{Universality}: The confirmed universal invariant is \textbf{λ₁ × H* = 12.3364}, holding for all 66 known G₂ manifolds (including those beyond the CHNP 2015 tabulated range, plus prior TCS literature). The universal law is empirical across all known examples; the analytical mechanism connecting the torsion-free condition to universality remains to be derived. See \texttt{canonical/scripts/construction\_classification.py} for the full scan.

\subsubsection{7.4 Continued Fraction Structure}\label{continued-fraction-structure}

The topological ratio dim(G₂)/H* admits a notable continued fraction representation:

\[\frac{14}{99} = [0; 7, 14] = \cfrac{1}{7 + \cfrac{1}{14}}\]

The only integers appearing are \textbf{7 = dim(K₇)} and \textbf{14 = dim(G₂)}, the two fundamental dimensions of GIFT geometry. Note: this is a property of the topological ratio, not of the spectral gap λ₁ = 6π²/475 (which is irrational).

\subsubsection{7.5 Pell Equation Structure}\label{pell-equation-structure}

The spectral gap parameters satisfy a Pell equation:

\[\boxed{H^{*2} - 50 \times \dim(G_2)^2 = 1}\]

Explicitly:
\[99^2 - 50 \times 14^2 = 9801 - 9800 = 1\]

where \(50 = \dim(K_7)^2 + 1 = 49 + 1\).

\textbf{Fundamental unit}: The Pell equation \(x^2 - 50y^2 = 1\) has fundamental solution \((x_0, y_0) = (99, 14)\), giving:

\[\varepsilon = 7 + \sqrt{50}, \quad \varepsilon^2 = 99 + 14\sqrt{50}\]

\textbf{Continued fraction bridge}: The discriminant \(\sqrt{50}\) has periodic continued fraction \(\sqrt{50} = [7; \overline{14}]\) with period 1, where the partial quotients are exactly dim(K₇) = 7 and dim(G₂) = 14. Combined with the selection principle κ = π²/14 (formalized in \texttt{Spectral.SelectionPrinciple}), this provides an arithmetic link between the Pell structure and the spectral gap.

\textbf{Status}: TOPOLOGICAL (algebraic identity verified in Lean)

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\section{Part III: K₇ Manifold Construction}\label{part-iii-kux2087-manifold-construction}

\subsection{8. Twisted Connected Sum Framework}\label{twisted-connected-sum-framework}

\subsubsection{8.1 TCS Construction}\label{tcs-construction}

The twisted connected sum (TCS) construction provides the primary method for constructing compact G₂ manifolds from asymptotically cylindrical building blocks.

\textbf{Key insight}: G₂ manifolds can be built by gluing two asymptotically cylindrical (ACyl) G₂ manifolds along their cylindrical ends, with the topology controlled by a twist diffeomorphism φ.

\subsubsection{8.2 Asymptotically Cylindrical G₂ Manifolds}\label{asymptotically-cylindrical-gux2082-manifolds}

\textbf{Definition}: A complete Riemannian 7-manifold (M, g) with G₂ holonomy is asymptotically cylindrical (ACyl) if there exists a compact subset K ⊂ M such that M ~K is diffeomorphic to (T₀, ∞) × N for some compact 6-manifold N.

\subsubsection{8.3 Topological Classification}\label{topological-classification}

The GIFT framework constructs an explicit G₂ metric on a compact 7-manifold K₇ with Betti numbers
(b₂, b₃) = (21, 77), certified by Newton-Kantorovich theorem ([A]). The classification of this topological type within known construction methods remains open.

\textbf{Topological status (TCS-specific).} The pair (b₂, b₃) = (21, 77) does not appear among the catalogued TCS-constructed compact G₂ manifolds. Orthogonal TCS is excluded by parity (b₂+b₃=98 is even; CHNP Lemma 6.7). A building block scan (2026-04-14) shows that b₂=21 \textgreater{} 20=max ρ(K3) forces at least one semi-Fano (non-Fano) building block, and all generic K3 lattice embeddings are excluded by the parity theorem. Non-orthogonal TCS and extra-twisted connected sums remain open paths within TCS itself. \textbf{The JK orbifold route, however, is constructive (§8.4) and realizes (21, 77) outside TCS via T³ × K3 / Z₂³ resolution.}

\textbf{CHNP 2015 range.} The CHNP 2015 explicit tabulation covered building blocks with ρ ≤ 9, giving b₂ ≤ 18. Reaching b₂=21 within TCS would require at least one semi-Fano block with ρ ≥ 20 (Shioda-Inose), outside the CHNP tabulated range. The Mori-Mukai classification allows semi-Fano threefolds with ρ up to \textasciitilde20; a TCS construction with such high-ρ blocks has not been explicitly carried out and is superseded for our purposes by the JK route below.

\textbf{Lattice-theoretic analysis.} The K3 lattice Λ\_\{K3\} = U³ ⊕ E₈(-1)² admits an
orthogonal decomposition N₁ ⊕ N₂ ⊕ ⟨2⟩ with rk(N₁) = 11, rk(N₂) = 10, satisfying
the proposed Nikulin embedding conditions (coprime discriminants, signature, discriminant form
matching). This is consistent with a TCS construction but does not constitute a proof; the Nikulin embedding is conjectured, and full verification requires an explicit construction of the matching data.

\textbf{Global properties}:
- Compact 7-manifold (no boundary)
- Simply connected: π₁(K₇) = \{1\}, b₁ = 0
- G₂ holonomy: Hol(g*) = G₂ {[}Joyce, Prop. 11.2.3{]}
- Ricci-flat: Ric(g) = 0
- Euler characteristic: χ(K₇) = 0

\textbf{Combinatorial connections}:
- b₂ = 21 = C(7,2) = edges in complete graph K₇
- b₃ = 77 = C(7,3) + 2 × b₂ = 35 + 42

\textbf{Status}: The Betti numbers (b₂, b₃) = (21, 77) are certified by the NK metric (Paper I). The spectral analysis ([B]) independently confirms 21 + 77 near-zero eigenvalues consistent with these Betti numbers. Within the TCS catalogue, the pair (21, 77) does not appear. Outside TCS, the JK Z₂³ orbifold route (§8.4) realizes (21, 77) at the topological/lattice level. An explicit closed-form positive G₂-structure ansatz at the neck level, with determinant constraint, hyperkähler rotation, and five-layer Picard-Lefschetz Wirtinger certificate, is established in [D]. A complete \emph{smooth analytic} construction (explicit polynomial Z₂³ realization on a Picard-rank-1, η²=8 K3, plus JK 2017 analytic gluing) remains open.

\subsubsection{8.4 Joyce-Karigiannis Z₂³ Construction, realizes (b₂, b₃) = (21, 77)}\label{joyce-karigiannis-zux2082uxb3-construction-realizes-bux2082-bux2083-21-77}

The Joyce-Karigiannis (JK) framework [43] resolves orbifolds T³ × K3 / G with G ⊂ Aut(T³) × Aut(K3) acting with A₁-type singularities via Eguchi-Hanson gluing, yielding compact torsion-free G₂ manifolds. For G = Z₂³ with a specific mixed-parity action, the resolved 7-manifold N has Betti numbers given by

\[b_2(N) = b_2(T^3 \times K3 / G) + b_0(\text{fixed loci}), \qquad b_3(N) = b_3(T^3 \times K3 / G) + b_1(\text{fixed loci}).\]

A four-phase computer-assisted audit (\texttt{canonical/scripts/jk\_*.py}, results in \texttt{canonical/results/jk\_*.json}, 2026-05-04) closes the count for the GIFT signature:

\textbf{Phase 1: V4 symplectic screen on CI(2,2,2).} The diagonal V4 = ⟨s₁, s₂⟩ ⊂ Z₂³ action on the CI(2,2,2) ⊂ ℂP⁵ K3 fiber has 24 raw fixed points, decomposing into \textbf{12 V4-orbits → 12 T³ components} of the fixed locus. Generators commute on the V4-invariant 9-dimensional quadric subspace.

\textbf{Phase 2, anti-symplectic obstruction.} For any diagonal τ on CI(2,2,2) ⊂ ℂP⁵ and the canonical quadric net R, the determinant ratio satisfies det(τ) / det(R) ≡ 1, so no diagonal P⁵-linear map is anti-symplectic. The full Z₂³ realization must use \emph{intrinsic K3 lattice automorphisms}, not P⁵-linear actions.

\textbf{Phase 2b: K3 lattice abstract existence.} On the K3 lattice Λ = U³ ⊕ E₈(-1)², the Nikulin involution σ₁ = E₈-swap is an order-2 isometry with trace 6 and eigenspaces (14, 8). The coinvariant T\_\{σ₁\} ≅ E₈(-2) has discriminant 2⁸. \textbf{Mukai 1988} (V4 = Z₂² ⊂ M₂₃) gives a symplectic K3 action; \textbf{Garbagnati-Sarti 2009} verifies a \textgreater{} 16 - r for the order-two non-symplectic involutions on (r, a, δ) = (11, 7, 1) and (11, 9, 1), corresponding to fixed-locus shapes (g, k) = (2, 2) and (1, 1) respectively. η² = 8 is representable in each invariant sublattice.

\textbf{Phase 4: Betti formula.} The fixed locus consists of:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lllll@{}}
\toprule\noalign{}
Component & Count & b₀ & b₁ & Source \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
T³ & 12 & 12 & 36 & V4-fixed K3 points (Phase 1) \\
S¹×Σ₂ & 1 & 1 & 5 & τ : (g,k)=(2,2) \\
S¹×ℂP¹ & 2 & 2 & 2 & τ : k=2 rational curves \\
S¹×T² & 3 & 3 & 9 & s₁τ, s₂τ, s₁s₂τ : (g,k)=(1,1) \\
S¹×ℂP¹ & 3 & 3 & 3 & s₁τ, s₂τ, s₁s₂τ : k=1 rational \\
\textbf{Total} & \textbf{21} & \textbf{21} & \textbf{55} & \\
\end{longtable}
Combined with the quotient cohomology b₂(T³ × K3 / Z₂³) = 0 and b₃(T³ × K3 / Z₂³) = 22, the JK formula gives

\[b_2(N) = 0 + 21 = 21, \qquad b_3(N) = 22 + 55 = 77, \qquad \chi(N) = 0,\]

matching the GIFT topological signature exactly.

\textbf{Lean formalization}: the four-phase audit is encoded in \texttt{GIFT/Foundations/JoyceKarigiannisConstruction.lean} with master theorem \texttt{jk\_z23\_construction\_realizes\_gift\_betti} proving \texttt{phase4.b2N\ =\ GIFT.Core.b2\ ∧\ phase4.b3N\ =\ GIFT.Core.b3} by \texttt{native\_decide}. The module introduces 0 axioms and 0 sorry; literature dependencies (Mukai, Garbagnati-Sarti) are encoded as Bool flags in a \texttt{JKPhase2bLatticeScreen} structure with explicit citation comments.

\textbf{Honest scope.} This route certifies the \emph{topological/lattice gate} only:
- ✓ Betti pair (b₂, b₃) = (21, 77) closed by exact integer arithmetic
- ✓ K3 surface with required Z₂³ action exists abstractly (Mukai/G-S)
- ⏳ Explicit polynomial coordinate model on a Picard-rank-1, η² = 8 K3, deferred (existence guaranteed but moduli construction not done here)
- ⏳ Smooth analytic torsion-free G₂ certificate, deferred to JK 2017 gluing theorem

The full four-phase topological route (lattice screens, explicit (r, a, δ) bookkeeping, and Betti formula closure) follows the Joyce-Karigiannis Z₂³ framework [43]. An explicit closed-form positive G₂-structure ansatz at the neck level, determinant-preserving radial family, hyperkähler rotation, base-coframe absorption, and five-layer Picard-Lefschetz Wirtinger certificate, is established in [D].

\textbf{Comparison to TCS within the GIFT framework.} The NK-certified metric ([A]) and the Donaldson analytic note [D] operate at the \emph{neck level} of a putative compact extension and do not depend on whether the global construction is TCS or JK. The JK route provides a constructive existence proof for the topological pair; the NK metric and [D] provide complementary analytic evidence at the neck: the former via Newton-Kantorovich certificate, the latter via explicit closed-form positive G₂-structure ansatz with Wirtinger certificate; together they argue that the GIFT signature (b₂, b₃) = (21, 77) is realizable, with the JK route being the leading candidate for the global compact construction.

\textbf{Compatibility with intermediate-curvature splitting (Chen-Hong 2026).} Recent work of Chen and Hong [46] establishes a sharp dichotomy for noncompact manifolds with nonnegative m-intermediate curvature admitting a smooth proper degree-nonzero map to M\^{}\{n−m\} × T\^{}\{m−1\} × ℝ. For 3 ≤ n ≤ 5 with 1 ≤ m ≤ n−1, or for 6 ≤ n ≤ 7 with m ∈ \{1, n−2, n−1\}, the manifold is forced to split isometrically as the Riemannian product E × T\^{}\{m−1\} × ℝ. By contrast, for 6 ≤ n ≤ 7 and 2 ≤ m ≤ n−3, they construct explicit metrics on S\^{}\{n−m\} × T\^{}\{m−1\} × ℝ with uniformly positive m-intermediate curvature, showing the algebraic gate m² − mn + m + n \textgreater{} 0 is sharp. The GIFT coassociative neck has topology K3 × T² × ℝ (n = 7, m = 3, giving m² − mn + m + n = −2 \textless{} 0), placing it within the Chen-Hong ``non-rigid'' window: metrics on this topology with nonnegative 3-intermediate curvature need not split. The k ≥ 1 Chebyshev corrections that lift the holonomy from SU(2) × U(1) to full G₂ are therefore consistent with the Chen-Hong existence regime, rather than obstructed by a forced splitting.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{9. Cohomological Structure}\label{cohomological-structure}

\subsubsection{9.1 Mayer-Vietoris Analysis}\label{mayer-vietoris-analysis}

The Mayer-Vietoris sequence provides the primary tool for computing cohomology:

\[\cdots \to H^{k-1}(N) \xrightarrow{\delta} H^k(K_7) \xrightarrow{i^*} H^k(M_1) \oplus H^k(M_2) \xrightarrow{j^*} H^k(N) \to \cdots\]

\subsubsection{9.2 Betti Number Derivation}\label{betti-number-derivation}

\textbf{Result for b₂}: b₂(K₇) = 21 is certified by the NK metric. The spectral analysis ([B]) independently confirms 21 near-zero eigenvalues of Δ₂ with gap ratio 14,635. A TCS realization would require building blocks with Picard ranks summing to 21, but the specific identification of those blocks is an open problem.

\textbf{Result for b₃}: b₃(K₇) = 77 is certified by the NK metric. The harmonic decomposition b₃ = 35 + 42 = (1+7+27) + 2×21 is confirmed by the certified metric with spectral gap 10522× between zero and non-zero modes. A splitting b₃ = b₃(M₁) + b₃(M₂) via Mayer-Vietoris is conditional on the building block identification, which remains open.

\textbf{Status}: VERIFIED (NK metric, Paper I); candidate TCS decomposition is conjectural

\subsubsection{9.3 Complete Betti Spectrum and Poincaré Duality}\label{complete-betti-spectrum-and-poincaruxe9-duality}

For a compact G₂-holonomy 7-manifold K₇, Poincaré duality gives b\_k = b\_\{7-k\}:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.1250}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3750}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.5000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
k
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
b\_k(K₇)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Derivation
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
0 & 1 & Connected \\
1 & 0 & Simply connected (G₂ holonomy) \\
2 & 21 & NK-certified; spectrally confirmed ([B]); building block decomposition open \\
3 & 77 & NK-certified; harmonic decomposition 35+42; building block decomposition open \\
4 & 77 & Poincaré duality: b₄ = b₃ \\
5 & 21 & Poincaré duality: b₅ = b₂ \\
6 & 0 & Poincaré duality: b₆ = b₁ \\
7 & 1 & Poincaré duality: b₇ = b₀ \\
\end{longtable}
\textbf{Euler characteristic}: For any compact oriented odd-dimensional manifold, χ = 0:
\[\chi(K_7) = \sum_{k=0}^{7} (-1)^k b_k = 1 - 0 + 21 - 77 + 77 - 21 + 0 - 1 = 0\]

\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{euler\_char\_K7\_is\_zero}, \texttt{poincare\_duality\_K7}

\textbf{Effective cohomological dimension}:
\[H^* = b_2 + b_3 + 1 = 21 + 77 + 1 = 99\]

\subsubsection{9.4 The Structural Constant 42}\label{the-structural-constant-42}

The number 42 appears throughout GIFT as a derived topological invariant:

\[42 = 2 \times 3 \times 7 = p_2 \times N_{gen} \times \dim(K_7)\]

\textbf{Multiple derivations}:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Formula & Value & Interpretation \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
p₂ × N\_gen × dim(K₇) & 2 × 3 × 7 = 42 & Binary × generations × fiber \\
2 × b₂ & 2 × 21 = 42 & Twice the gauge moduli \\
b₃ - C(7,3) & 77 - 35 = 42 & Global vs local 3-forms \\
\end{longtable}
\textbf{Connection to b₃ decomposition}:
\[b_3 = 77 = C(7,3) + 42 = 35 + 2 \times b_2\]

The 35 local modes correspond to Λ³(ℝ⁷) fiber forms; the 42 global modes arise from the TCS structure.

\textbf{Status}: \textbf{VERIFIED (Lean 4)}: \texttt{structural\_42\_gift\_form}, \texttt{structural\_42\_from\_b2}

\subsubsection{9.5 Third Betti Number Decomposition}\label{third-betti-number-decomposition}

The b₃ = 77 harmonic 3-forms decompose as:

\[H^3(K_7) = H^3_{\text{local}} \oplus H^3_{\text{global}}\]

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Component & Dimension & Origin \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
H³\_local & 35 = C(7,3) & Λ³(ℝ⁷) fiber forms \\
H³\_global & 42 = 2 × 21 & TCS global modes \\
\end{longtable}
\textbf{Verification}: 35 + 42 = 77

\textbf{Status}: TOPOLOGICAL

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\section{Part IV: Metric Structure and Verification}\label{part-iv-metric-structure-and-verification}

\subsection{10. Structural Metric Invariants}\label{structural-metric-invariants}

\subsubsection{10.1 Structural Metric Invariants and Normalizations}\label{structural-metric-invariants-and-normalizations}

The GIFT framework explores the hypothesis that metric invariants derive from fixed mathematical structure. The topological constraints serve as inputs; the specific geometry is then determined.

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.3143}}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2571}}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 6\tabcolsep) * \real{0.2286}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Invariant
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Formula
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Value
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Status
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
κ\_T & 1/(b₃ - dim(G₂) - p₂) & 1/61 & TOPOLOGICAL \\
det(g) & (Weyl × (rank(E₈) + Weyl))/2⁵ & 65/32 & STRUCTURAL (metric normalization, see §10.3) \\
\end{longtable}
\subsubsection{10.2 Torsion Magnitude κ\_T = 1/61}\label{torsion-magnitude-ux3ba_t-161}

\textbf{Derivation}:
\[\kappa_T = \frac{1}{b_3 - \dim(G_2) - p_2} = \frac{1}{77 - 14 - 2} = \frac{1}{61}\]

\textbf{Interpretation}:
- 61 = effective matter degrees of freedom
- b₃ = 77 total fermion modes
- dim(G₂) = 14 gauge symmetry constraints
- p₂ = 2 binary duality factor

\textbf{Status}: TOPOLOGICAL

\subsubsection{10.3 Metric Determinant det(g) = 65/32}\label{metric-determinant-detg-6532}

The metric determinant admits three independent derivations from topological invariants, providing strong evidence for its structural necessity.

\textbf{Path 1} (Weyl formula):
\[\det(g) = \frac{\text{Weyl} \times (\text{rank}(E_8) + \text{Weyl})}{2^{\text{Weyl}}} = \frac{5 \times 13}{32} = \frac{65}{32}\]

\textbf{Path 2} (Cohomological):
\[\det(g) = p_2 + \frac{1}{b_2 + \dim(G_2) - N_{\text{gen}}} = 2 + \frac{1}{21+14-3} = 2 + \frac{1}{32} = \frac{65}{32}\]

\textbf{Path 3} (H* formula):
\[\det(g) = \frac{H^* - b_2 - 13}{32} = \frac{99 - 21 - 13}{32} = \frac{65}{32}\]

The convergence of three independent algebraic paths to the same rational value is
suggestive but does not constitute a derivation from topology. The actual chain is:
the metric optimization was constrained to det(g) = 65/32 (a normalization target
chosen to be compatible with the known structural parameters g\_ss = 19/6, g\_T² = 7/6),
and the above formulas were identified post-hoc. Any rational number with small
numerator/denominator can be expressed as combinations of a few integers.

\textbf{Numerical value}: 65/32 = 2.03125 (exact rational)

\textbf{Status}: STRUCTURAL (metric normalization, algebraically exact in our metric, with three suggestive integer formulas, but not derived from topology). The 6 observables using det\_num or det\_den (six Type I observables) depend on this normalization choice.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{11. Formal Certification}\label{formal-certification}

\subsubsection{11.1 The Algebraic Reference Form}\label{the-algebraic-reference-form}

The algebraic reference form in a local G₂-adapted orthonormal coframe:

\[\varphi_{\text{ref}} = c \cdot \varphi_0, \quad c = \left(\frac{65}{32}\right)^{1/14}\]
\[g_{\text{ref}} = c^2 \cdot I_7 = \left(\frac{65}{32}\right)^{1/7} \cdot I_7\]

\textbf{Important clarification}: This representation holds in a local orthonormal frame. The manifold K₇ is curved and compact; ``I₇'' reflects the frame choice, not global flatness. The reference form φ\_ref yields det(g) = 65/32. The NK-certified model provides computational/conditional evidence for a nearby torsion-free G₂ structure within the certified analytic setup of Paper A; this does not by itself close the independent smooth compact construction problem for K₇, whose full building-block / gluing realization remains open.

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\begin{minipage}[b]{\linewidth}\raggedright
Property
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Value
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Status
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
det(g) & 65/32 & exact by imposed normalization, not fitted to experiment (§10.3) \\
φ\_ref components & 7/35 & 20\% sparsity \\
Joyce threshold & ‖T‖ \textless{} ε₀ = 0.1 & Satisfied (224× margin) \\
\end{longtable}
\subsubsection{11.2 Joyce Existence Theorem and Global Solutions}\label{joyce-existence-theorem-and-global-solutions}

\textbf{Important clarification}: The reference form φ\_ref = c·φ₀ is the canonical G₂ structure in a local orthonormal coframe, not a globally constant form on K₇. On a compact G₂ manifold, the coframe 1-forms \{eⁱ\} satisfy deⁱ ≠ 0 in general, so ``constant components'' does not imply dφ = 0 globally.

\textbf{Actual solution structure}: The topology and geometry of K₇ impose a deformation:
\[\varphi = \varphi_{\text{ref}} + \delta\varphi\]

The torsion-free condition (dφ = 0, d*φ = 0) is a \textbf{global constraint}. Joyce's perturbation theorem provides the existence criterion for a nearby torsion-free G₂ metric, conditional on the initial torsion satisfying ‖T‖ \textless{} ε₀ = 0.1. PINN validation (N=1000) confirms ‖T‖\_max = 4.46 × 10⁻⁴, providing a 224× safety margin.

\textbf{Why GIFT satisfies Joyce's criterion}: The topological bound κ\_T = 1/61 constrains ‖δφ‖, placing the manifold within Joyce's perturbative regime where a torsion-free solution exists.

\subsubsection{11.3 Independent Numerical Validation (PINN)}\label{independent-numerical-validation-pinn}

A companion numerical program constructs explicit G₂ metrics on K₇ via physics-informed neural networks (PINNs). The three-chart atlas (neck + two Calabi-Yau bulk regions) uses approximately 10⁶ trainable parameters in float64 precision.

\textbf{Initial validation} (Phase 2):

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Metric & Value & Significance \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
‖T‖\_max & 4.46 x 10⁻⁴ & 224x below Joyce epsilon₀ \\
‖T‖\_mean & 9.8 x 10⁻⁵ & T --\textgreater{} 0 confirmed \\
det(g) error & \textless{} 10⁻⁶ & Confirms 65/32 \\
\end{longtable}
\textbf{G₂ holonomy training} (Phase 3, 13 versions, v2-v13):

Over successive training protocol refinements, the holonomy quality has improved:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}llll@{}}
\toprule\noalign{}
Metric & Initial (v5) & Current best (v11) & Improvement \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
g2\_self (honest holonomy) & 3.86 & 3.25 & -16\% \\
V₇ projection score & 0.51 & 0.014 & -97\% \\
det(g) at neck & 4.69 & 2.031 & locked at target \\
phi drift & 13.4\% & 0\% & controlled \\
\end{longtable}
The g2\_score measures the normalized projection of Riemann curvature onto the complement of g₂ in so(7). A score of 0 corresponds to exact G₂ holonomy; the flat metric scores approximately 3.5. The V₇ projection score measures the fraction of curvature outside the g₂ subalgebra (using the correct kernel-based g₂ decomposition, see Section 6.2).

A critical bug in the g₂ basis construction was discovered and corrected between versions 9 and 10: the Fano-plane heuristic does not produce correct g₂ generators. The correct g₂ subalgebra is the kernel of the Lie derivative map (Section 6.2). This correction led to significant improvement in all subsequent versions.

\textbf{Robust statistical validation}: The det(g) = 65/32 prediction passes 8/8 independent tests (permutation, bootstrap, Bayesian posterior 76.3\%, joint constraint p \textless{} 6 x 10⁻⁶).

Full details of the PINN architecture, training protocol, and version-by-version results are presented in a companion paper.

\subsubsection{11.4 Lean 4 Formalization}\label{lean-4-formalization}

\textbf{Scope of verification}: The Lean formalization (134 files, 4 axioms, zero \texttt{sorry}) verifies:
1. Arithmetic identities and algebraic relations between GIFT constants
2. Numerical bounds (e.g., torsion threshold)
3. G₂ differential geometry: exterior algebra Λ*(ℝ⁷), Hodge star, ψ = ⋆φ (axiom-free \texttt{Geometry} module)
4. Spectral gap bounds and topological ratios (\texttt{Spectral.PhysicalSpectralGap}, 28 theorems, zero axioms). Analytical value: λ₁ = 6π²/475 = 0.12467 (see §7.3)
5. Selberg bridge: trace formula connecting S\_w(T) to spectral gap (\texttt{Spectral.SelbergBridge})
6. Selection principle κ = π²/14 (\texttt{Spectral.SelectionPrinciple})

\textbf{The 4 axioms are declared in \texttt{Foundation/Bounds} and used transversally throughout the formalization} (they assert numerical bounds on torsion, the NK certificate parameter h, and the spectral gap, which cannot be verified by \texttt{decide} alone due to real-arithmetic content). All other results use zero axioms.

It does \textbf{not} formalize:
- Existence of K₇ as a smooth G₂ manifold
- Physical interpretation of topological invariants
- Uniqueness of the TCS construction

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{{-}{-} GIFT.Foundations.AnalyticalMetric}

\NormalTok{def phi0\_indices : List (Fin 7 × Fin 7 × Fin 7) :=}
\NormalTok{  [(0,1,2), (0,3,4), (0,5,6), (1,3,5), (1,4,6), (2,3,6), (2,4,5)]}

\NormalTok{def phi0\_signs : List Int := [1, 1, 1, 1, {-}1, {-}1, {-}1]}

\NormalTok{def scale\_factor\_power\_14 : Rat := 65 / 32}

\NormalTok{theorem torsion\_satisfies\_joyce :}
\NormalTok{  torsion\_norm\_constant\_form \textless{} joyce\_threshold\_num := by native\_decide}

\NormalTok{theorem det\_g\_equals\_target :}
\NormalTok{  scale\_factor\_power\_14 = det\_g\_target := rfl}
\end{Highlighting}
\end{Shaded}

\textbf{Status}: VERIFIED

\subsubsection{11.5 The Derivation Chain}\label{the-derivation-chain}

The logical structure from algebra to predictions:

\begin{verbatim}
Octonions ($\mathbb{O}$)
     │
     ▼
G₂ = Aut($\mathbb{O}$), dim = 14
     │
     ▼
Standard form φ₀ (Harvey-Lawson 1982)
     │
     ▼
Scaling c = (65/32)^{1/14}    ← GIFT constraint
     │
     ▼
Metric g = c² × I₇
     │
     ▼
det(g) = 65/32               ← exact by imposed normalization, not fitted to experiment (§10.3)
     │
     ▼
sin²θ_W = 3/13, Q = 2/3, ...  ← Predictions
\end{verbatim}

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{12. Analytical G₂ Metric Details}\label{analytical-gux2082-metric-details}

\subsubsection{12.1 The Standard Form φ₀}\label{the-standard-form-ux3c6ux2080}

The associative 3-form preserved by G₂ ⊂ SO(7), introduced by Harvey and Lawson (1982) in their foundational work on calibrated geometries:

\[\varphi_0 = \sum_{(i,j,k) \in \mathcal{I}} \sigma_{ijk} \, e^{ijk}\]

where:
- $\mathcal{I}$ = \{(0,1,2), (0,3,4), (0,5,6), (1,3,5), (1,4,6), (2,3,6), (2,4,5)\}
- σ = (+1, +1, +1, +1, -1, -1, -1)

\subsubsection{12.2 Linear Index Representation}\label{linear-index-representation}

In the C(7,3) = 35 basis:

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}llllll@{}}
\toprule\noalign{}
Index & Triple & Sign & Index & Triple & Sign \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
0 & (0,1,2) & +1 & 23 & (1,4,6) & -1 \\
9 & (0,3,4) & +1 & 27 & (2,3,6) & -1 \\
14 & (0,5,6) & +1 & 28 & (2,4,5) & -1 \\
20 & (1,3,5) & +1 & & & \\
\end{longtable}
All other 28 components are exactly 0.

\subsubsection{12.3 Metric Derivation}\label{metric-derivation}

From φ₀, the metric is computed via:
\[g_{ij} = \frac{1}{6} \sum_{k,l} \varphi_{ikl} \varphi_{jkl}\]

For standard φ₀: g = I₇ (identity), det(g) = 1.

Scaling φ → c·φ gives g → c²·g, hence det(g) → c¹⁴·det(g).

Setting c¹⁴ = 65/32 yields the GIFT metric.

\subsubsection{12.4 Comparison: Fano Plane vs G₂ Form}\label{comparison-fano-plane-vs-gux2082-form}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3929}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3929}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2143}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Structure
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
7 Triples
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Role
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\textbf{Fano lines} & (0,1,3), (1,2,4), (2,3,5), (3,4,6), (4,5,0), (5,6,1), (6,0,2) & G₂ cross-product ε\_\{ijk\} \\
\textbf{G₂ form} & (0,1,2), (0,3,4), (0,5,6), (1,3,5), (1,4,6), (2,3,6), (2,4,5) & Associative 3-form \\
\end{longtable}
Both have 7 terms but different index patterns. The Fano plane defines the octonion multiplication (cross-product), while the G₂ form is the associative calibration.

\subsubsection{12.5 Verification Summary}\label{verification-summary}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2963}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.2963}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.4074}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Method
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Result
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Reference
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Algebraic & φ = (65/32)\^{}\{1/14\} × φ₀ & This section \\
Lean 4 & \texttt{det\_g\_equals\_target\ :\ rfl} & AnalyticalMetric.lean \\
PINN & Converges to constant form & gift\_core/nn/ \\
Joyce theorem & ‖T‖ \textless{} 0.1 → exists metric (224× margin) & {[}Joyce 2000{]} \\
\end{longtable}
Cross-verification between analytical and numerical methods is consistent with the conditional NK-certified solution within its analytic setup.

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{References}\label{references}

\begin{enumerate}


\item
  Adams, J.F. \emph{Lectures on Exceptional Lie Groups}
\item
  Harvey, R., Lawson, H.B. ``Calibrated geometries.'' \emph{Acta Math.} 148, 47-157 (1982)
\item
  Bryant, R.L. ``Metrics with exceptional holonomy.'' \emph{Ann. of Math.} 126, 525-576 (1987)
\item
  Joyce, D. \emph{Compact Manifolds with Special Holonomy}, Oxford U. Press (2000)
\item
  Corti, Haskins, Nordström, Pacini. \emph{G₂-manifolds and associative submanifolds}
\item
  Kovalev, A. \emph{Twisted connected sums and special Riemannian holonomy}
\item
  Conway, J.H., Sloane, N.J.A. \emph{Sphere Packings, Lattices and Groups}
\item
  \textbf{[43]} Joyce, D., Karigiannis, S. ``A new construction of compact torsion-free G₂-manifolds by gluing families of Eguchi-Hanson spaces.'' \emph{J. Differential Geom.} (2021), arXiv:1707.09325 (2017).
\item
  \textbf{[44]} Mukai, S. ``Finite groups of automorphisms of K3 surfaces and the Mathieu group.'' \emph{Invent. Math.} \textbf{94}, 183--221 (1988).
\item
  \textbf{[45]} Garbagnati, A., Sarti, A. ``Symplectic automorphisms of prime order on K3 surfaces.'' \emph{J. Algebra} \textbf{318}, 323--350 (2008); also arXiv:0712.3055.
\item
  \textbf{[46]} Chen, J., Hong, H. ``Intermediate curvature and splitting theorem,'' arXiv:2604.26529 (2026).
\end{enumerate}

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\subsection{Author's Related Works}\label{authors-related-works}

\begin{itemize}

\item
  \textbf{[A]} B. de La Fournière, ``An Explicit Approximate G₂ Metric on a Compact 7-Manifold with Certified Torsion-Free Completion,'' Zenodo \href{https://doi.org/10.5281/zenodo.19892350}{10.5281/zenodo.19892350} (2026).
\item
  \textbf{[B]} B. de La Fournière, ``Spectral Geometry of the G₂-GIFT Manifold: Betti Numbers, KK Spectrum, and Spectral Invariants,'' Zenodo \href{https://doi.org/10.5281/zenodo.19893371}{10.5281/zenodo.19893371} (2026).
\item
  \textbf{[D]} B. de La Fournière, ``An Explicit Closed-Form G₂ Ansatz on a K3-Coassociative Neck with Hyperkähler Rotation and Picard-Lefschetz Wirtinger Certificate,'' Zenodo \href{https://doi.org/10.5281/zenodo.20039066}{10.5281/zenodo.20039066} (2026).
\end{itemize}

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\emph{GIFT Framework: Supplement S1}
\emph{Mathematical Foundations: E₈ + G₂ + K₇}


\noindent\rule{\textwidth}{0.2pt}

\begin{center}
\emph{GIFT Framework --- Supplement S1 --- Mathematical Foundations}
\end{center}

\end{document}
