SilverSight/docs/reviews/CONJECTURE_UPGRADE_ROADMAP.md

Conjecture Upgrade Roadmap

How to turn each sorry into a theorem

Four conjectures in UnifiedCovariant.lean are currently tagged sorry. Each has a precise upgrade path from informal conjecture to formal theorem. Two can be completed now (Q16_16 arithmetic); two require Mathlib infrastructure that does not yet exist.

---

1. Eigensolid Convergence

File location: UnifiedCovariant.lean:146 Status: ✅ RESOLVED (2026-06-26, Sidon-orthogonality bypass). Location: formal/SilverSight/PIST/UnifiedCovariant.lean — Layer 2. Resolution: Replaced spectral operator norm with computable L∞ row-sum bound.

What was done

1. crossingMatrix (Matrix (Fin 8) (Fin 8) ℚ) defined with explicit Sidon entries: diagonal σ = 39/256, paired off-diagonal τ = 1/7.

2. maxRowSum — L∞ row-sum norm, computed by dec_trivial over Fin 8.

3. crossing_matrix_norm_bound proved: maxRowSum crossingMatrix ≤ 1775/1792.

4. braid_operator_contractive — for any state vector s ∈ ℚ^8, |(C·s)_i| ≤ r · ‖s‖_∞ where r = 1775/1792.

5. EigensolidConvergenceHypothesis (deprecated) removed.

6. Build: lake build SilverSight — 3307 jobs, 0 errors.

Key insight (Sidon-orthogonality bypass)

The Sidon uniqueness property (I₄) guarantees at most 2 non-zero entries per row of C. Each row sum is then a concrete rational — evaluating all 8 rows and comparing to 1775/1792 is a finite computation (dec_trivial), not a spectral analysis. No NormedSpace topology, no eigenvalues, no continuous analysis.

Documentation

• Formula doc: docs/reviews/SIDON_ORTHOGONALITY_BYPASS_FORMULA.md
• Breakglass log: BREAKGLASS_LOG.md (entry 2)

---

2. Golden ℂℙ⁷ is Kähler

File location: UnifiedCovariant.lean:217 Current status: def goldenCP7 : Type := sorry Blocking issue: ℂℙ⁷ as a complex manifold is not in Mathlib.

Upgrade to theorem

Standard fact. The complex projective space \mathbb{CP}^n with the Fubini–Study metric g_{FS} and the standard complex structure J_0 (satisfying J_0^2 = -I) is a Kähler manifold. Scaling the metric by any positive constant preserves the Kähler condition.

Theorem statement:

Let \mathbb{CP}^7 be complex projective space with the standard complex structure J_0 and the (\phi)-scaled Fubini–Study metric g = \phi \cdot g_{FS}. Then (\mathbb{CP}^7, J_0, g) is a Kähler manifold with Kähler form \omega = \phi \cdot \omega_{FS}.

Formal statement in Lean:

theorem goldenCP7_is_Kaehler : KaehlerManifold goldenCP7 := ...

where KaehlerManifold is defined by the triple (M, J, \omega) with J^2 = -I, d\omega = 0, and \omega(JX, JY) = \omega(X, Y).

The role of \phi. The golden ratio scales the metric but does not appear in the complex structure. The cohomology class of the Kähler form is [\omega] = \phi \cdot [\omega_{FS}] \in H^{1,1}(\mathbb{CP}^7). The conjecture from the unified model is that this particular scaling factor \phi is forced by the spectral gap \sigma - \tau, i.e.,

[ \phi = \frac{[\omega]}{[\omega_{FS}]} ]

relates the geometric structure to the discrete Layer-1 invariants.

Prerequisites:

• Formal definition of \mathbb{CP}^n as a complex manifold (does not exist in Mathlib as of 2026-06)
• Definition of the Fubini–Study metric and Kähler form
• Proof that d\omega_{FS} = 0 (standard)

Upgrade difficulty: 🔴 Hard — blocked by missing Mathlib infrastructure.

---

3. Cartan Connection on J^1(\Delta_7)

File location: UnifiedCovariant.lean:224 Current status: theorem Cartan_connection_on_J1_exists : admits_Cartan_connection openSimplex7 := by sorry Blocking issue: No formal model of jet bundles or Cartan connections.

Upgrade to theorem

Definition. Let M be an (m)-dimensional manifold. The first jet bundle J^1(M) is the vector bundle whose fibre at p \in M consists of 1-jets of smooth functions:

[ J^1_p(M) = { j^1_p f \mid f \in C^\infty(M) }. ]

A Cartan connection on J^1(M) is a principal bundle connection on the (GL(m,\mathbb{R}))-bundle of 1-jets satisfying the Cartan structure equations.

Theorem statement:

Let \Delta_7 be the open 7-simplex with the Fisher–Rao metric. Then J^1(\Delta_7) admits a Cartan connection whose curvature is determined by the golden-ratio spectral gap \sigma - \tau.

Prerequisites:

• Formal definition of jet bundles (not in Mathlib)
• Formal definition of Cartan connections (not in Mathlib)
• Formal definition of the Fisher–Rao metric on \Delta_7
• Construction of the specific connection

Upgrade difficulty: 🔴 Very hard — requires substantial differential geometry formalization.

---

4. Holonomy \mathrm{SO}^0(1,6)

File location: UnifiedCovariant.lean:227 Current status: theorem holonomy_is_SO_1_6 : has_SO_1_6_holonomy openSimplex7 := by sorry Blocking issue: Requires curvature computation and Berger's classification.

Upgrade to theorem

Berger's theorem. The holonomy group of a non-symmetric irreducible Riemannian manifold is one of: \mathrm{SO}(n), \mathrm{U}(n), \mathrm{SU}(n), \mathrm{Sp}(n), \mathrm{Sp}(n)\mathrm{Sp}(1), \mathrm{G}_2, or \mathrm{Spin}(7).

Theorem statement:

The holonomy group of the (\phi)-scaled Fisher–Rao metric on \Delta_7 is the identity component of the indefinite orthogonal group \mathrm{SO}^0(1,6).

Evidence. The tangent space T_p\Delta_7 \cong \mathbb{R}^7. The Fisher–Rao metric at a point p is g_{ij} = \delta_{ij}/p_i. The signature is (1,6) (one positive, six negative — the metric on the simplex is not positive-definite in the standard basis; the positive direction corresponds to the barycentric direction). The holonomy containment \mathrm{Hol}(g) \subseteq \mathrm{SO}(1,6) follows from metric compatibility. The full \mathrm{SO}^0(1,6) claim requires computing the curvature and showing the holonomy is irreducible and not a proper subgroup.

Prerequisites:

• Riemannian holonomy in Mathlib (partial — HolonomyGroup exists for Riemannian manifolds but not pseudo-Riemannian)
• Curvature computation for the Fisher–Rao metric on \Delta_7
• Berger's classification (not in Mathlib)

Upgrade difficulty: 🔴 Very hard — requires curvature computation and classification theorem.

---

Summary

Conjecture	Upgrade difficulty	Path
Eigensolid convergence	✅ DONE	Sidon-orthogonality bypass (row-sum bound, dec_trivial)
Golden ℂℙ⁷ Kähler	🔴 Hard	Depends on ℂℙⁿ formalization in Mathlib
Cartan connection	🔴 Very hard	Jet bundles not in Mathlib
Holonomy SO⁰(1,6)	🔴 Very hard	Curvature + Berger not in Mathlib

All four conjectures documented. One resolved, three pending Mathlib infrastructure.