# 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.

**Reorganized 2026-07-02 (covariant semi-symmetry test).** §§2–4 were
ordered by "blocked on Mathlib." They are re-subordinated to a new
**§0 active milestone**: a direct curvature computation that *names* the
covariant semi-symmetry hypothesis on the classical rung (symmetric /
semisymmetric / pseudosymmetric) using an explicit metric + connection —
no ℂℙⁿ, jet bundles, or Berger classification required. Resolve §0 first;
§§2–4 are then corollaries or get re-scoped by its result.

---

## 0. ACTIVE MILESTONE — Covariant Semi-Symmetry Discriminator

**Goal.** Place the geometric object on the classical ladder —
**symmetric** (∇R = 0, Cartan) / **semisymmetric** (R·R = 0, Szabó) /
**pseudosymmetric** (R·R = f·Q(g,R), Deszcz) — by finite tensor
computation on a written-down metric + connection. Unlike §§2–4 this needs
no missing Mathlib infrastructure; it is computable now (by hand / CAS /
the 12-language rig, then Lean once the tensors are pinned).

**Load-bearing correction this milestone must resolve.** The *static*
Fisher–Rao metric on Δ₇, `g_ij = δ_ij / p_i`, is **positive-definite**
(all `p_i > 0`) — signature **(7,0)**, Riemannian. Under `p ↦ 2√p` it is
isometric to an orthant of the round sphere S⁷: **constant curvature**,
hence **fully symmetric** (∇R = 0), holonomy **SO(7)**. This contradicts
§4's claimed signature (1,6) / SO⁰(1,6): the bare Fisher metric is at the
**top** of the ladder, not "semi," and (1,6) cannot come from it.

**Where "semi" and (1,6) actually come from — the Kelvin/Rossby upgrade.**
`rossbyDriftFromChirality` (`BraidStateN.lean`) supplies a signed,
directional β-term (left = +1, right = −1, scarred = ±½, achiral = 0),
explicitly "analogous to the planetary vorticity gradient β." Kelvin/Rossby
waves solve a **hyperbolic** operator whose signature in 7D is
**(1, n−1) = (1, 6)**: the one time-like direction is the drift/propagation
direction the chirality selects; the six space-like directions are the
simplex. So SO⁰(1,6) is a property of the **drift-perturbed wave
operator**, not the static metric — and the directional (chiral) drift is
precisely what breaks ∇R = 0, pushing the object off "symmetric" onto the
"semi" rung. The Kelvin/Rossby directionality upgrade is therefore not
supporting evidence; it is the **load-bearing mechanism** of the hypothesis.

**Test sequence.**

- **0a — Baseline (decisive, essentially done).** Static φ-scaled
  Fisher–Rao on Δ₇ is positive-definite (7,0) and symmetric (∇R = 0),
  holonomy SO(7). Establishes that any "semi" / (1,6) structure must be
  drift-induced. 🟢
- **0b — Drift-perturbed connection.** Define the connection modified by
  `rossbyDriftFromChirality` (preferred direction / torsion / Randers–
  Finsler directional term). Show its signature is (1,6) — deriving
  SO⁰(1,6) from the wave operator, replacing §4's static-metric
  justification. 🟡
- **0c — Ladder placement.** Compute ∇R and R·R of the drift-perturbed
  structure. ∇R = 0 → symmetric; R·R = 0 with ∇R ≠ 0 → **semisymmetric**
  (Szabó) = the hypothesis; R·R = f·Q(g,R) → pseudosymmetric (Deszcz). 🟡
- **0d — Physics↔geometry edge.** Verify the ∇R obstruction direction
  equals the Rossby β / drift direction (chiral → Rossby/dispersive;
  achiral → Kelvin/eigensolid-trapped). A match is a verified edge from the
  formalism to named geophysical directionality (Rossby westward, Kelvin
  unidirectional from Coriolis) → populates `ene.relations` with
  provenance = the computation. 🟡

**Outcome.** The hypothesis is either *named* (a rung + a proof) or
*refuted* (∇R = 0 even after drift → symmetric all along). Both are
verified edges, not mirages.

---

## 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:**

```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 in Mathlib.

**Workaround (built 2026-06-26, `docs/reviews/CARTAN_CONNECTION_FORMULA.md`):**
reduce the jet-bundle Cartan connection to a **finite Chevalley–Eilenberg
Maurer–Cartan check** — the 2-cochain μ from the Sidon crossing matrix
satisfies `d_CE μ + ½[μ,μ]_NR = 0` because Sidon support-disjointness makes
the deformation operad forest-structured (`μ_i ∘ₖ μ_j = 0` across disjoint
supports), so no jet-bundle formalization is needed. Gates A (arithmetic)
and B (structural review) passed; **Gate C (build) is NOT done** — needs the
Nijenhuis–Richardson bracket defined in Lean (~30 lines) + the 1015-equation
system discharged by `dec_trivial`. ⚠️ That doc *asserts* signature (1,6) /
SO⁰(1,6) but justifies it from the Fisher–Rao metric — which is
positive-definite **(7,0)**. The (1,6) must come from the Kelvin/Rossby
drift (§0), not the static metric; §0 resolves this before §3's holonomy
containment can stand.

### 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:** 🟡 Medium **via the 2026-06-26 workaround** — the
remaining step is Gate C (define the NR bracket + `dec_trivial` on the
1015-equation system). The abstract jet-bundle route stays 🔴, but it is no
longer on the critical path.

---

## 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 |
|-----------|-------------------|------|
| **§0 Covariant semi-symmetry discriminator** | 🟡 **ACTIVE** | ∇R / R·R on the drift-perturbed metric — no Mathlib blocker |
| Eigensolid convergence | ✅ **DONE** | Sidon-orthogonality bypass (row-sum bound, dec_trivial) |
| Golden ℂℙ⁷ Kähler | 🔴 Hard | Depends on ℂℙⁿ formalization in Mathlib |
| Cartan connection | 🟡 workaround exists | Jet-bundle Cartan connection reduced to a finite Sidon-support MC/NR check (`CARTAN_CONNECTION_FORMULA.md`, 2026-06-26); needs NR bracket in Lean + Gate C |
| Holonomy SO⁰(1,6) | 🔴 Very hard → re-scoped by §0 | (1,6) is the Kelvin/Rossby **wave-operator** signature, not the static Fisher metric ((7,0), symmetric) |

**§0 is the active milestone: it names the covariant semi-symmetry
hypothesis by direct computation and unblocks §4 by relocating the (1,6)
signature to the drift-perturbed wave operator. §1 resolved; §§2–3 remain
pending Mathlib infrastructure but are downstream of §0.**
