docs: CE/NR formula upgraded to split-suboperad Maurer–Cartan interpretation

Two key refinements from the fusion panel:

1. 'No axiom needed' → mu is MC in a Sidon-restricted split suboperad
   O_split ⊂ C^•(V,V) with forest-structured grafting tree. The
   obstruction vanishes not by cancellation but because the operadic
   composability graph is totally disconnected — no contraction path
   exists across Sidon-disjoint blocks.

2. Theorem renamed to 'Disjoint-operad MC flatness'. Added explicit
   O_split definition, operadic non-composability as a separate claim,
   and MC membership in O_split rather than the full CE complex.

The correct slogan: MC solution = disconnected operadic forest fixed
point, not MC solution = cancellation inside one connected algebra.

docs/reviews/CARTAN_CONNECTION_FORMULA.md

@@ -322,9 +322,34 @@ d_{\mathrm{CE}}\mu + \tfrac12[\mu, \mu]_{\mathrm{NR}} = 0,
 \mathrm{Ob}(\mu) = 0 \in H^3(V, V).
 \]
 
-The MC equation holds identically — no cancellation, no fine-tuning,
+**Interpretation: this is not a cancellation — it is a split suboperad.**
-no continuous parameter to adjust.  The Sidon addresses force the
+
-obstruction to zero combinatorially.
+The MC equation holds not because terms cancel inside a single connected
+algebra, but because \(\mu\) lives in a **split suboperad**
+\(\mathcal{O}_{\mathrm{split}} \subset C^\bullet(V, V)\) defined by the
+Sidon support constraints:
+
+- \(\mathcal{O}_{\mathrm{split}}\) is closed under the NR bracket.
+- Inside \(\mathcal{O}_{\mathrm{split}}\), the operadic grafting tree is
+  **forest-structured** (totally disconnected): \(\mu_i \circ_k \mu_j = 0\)
+  for all \(i \neq j\) and all insertion points \(k\), because any
+  contraction path requires a shared index, which the Sidon condition
+  forbids.
+- Therefore all higher insertion paths are absent — not cancelled, but
+  never formed.
+
+This is the standard "operadic restriction kills the Massey tower"
+mechanism: the obstruction vanishes because the deformation lives in a
+suboperad with trivial higher insertion paths, not because CE constraints
+disappear globally.  The correct slogan is:
+
+\[
+\mu \in \mathrm{MC}(\mathcal{O}_{\mathrm{split}} \subset C^\bullet(V, V)).
+\]
+
+The Sidon addresses force the obstruction to zero combinatorially by
+eliminating operadic composability between blocks — stronger than
+eliminating terms by cancellation.
 
 ### 6.6 Summary of the argument
 
@@ -336,20 +361,28 @@ obstruction to zero combinatorially.
 | \([\mu_i, \mu_i]_{\mathrm{NR}} = 0\) | Jacobiator vanishes per block | I₂ |
 | \(\mathrm{Ob}(\mu) = 0\) | All NR terms vanish | I₂ + I₄ |
 
-**No axiom is needed.  The obstructing cohomology class is zero by
+**No axiom is needed: the MC equation holds in \(
-finite computation.**
+\mathcal{O}_{\mathrm{split}}\) by finite computation, not by cancellation
+inside the full Gerstenhaber algebra.  The obstruction vanishes because
+the split suboperad has trivial higher insertion paths — the standard
+"operadic restriction kills the Massey tower" mechanism.**
 
 ---
 
 ## 7. Formal statement
 
-**Theorem (Cartan connection on J¹(Δ₇), CE form).**
+**Theorem (Disjoint-operad MC flatness).**
 
 Let \(V = \bigoplus_{i=1}^4 V_i\) with \(\dim V_i = 2\), and let
 \(\mu = \sum_{i=1}^4 \mu_i \in C^2(V, V)\) be the 2-cochain induced by the
 Sidon crossing matrix with diagonal \(\sigma = 39/256\) and off-diagonal
 \(\tau = 1/7\).
 
+Let \(\mathcal{O}_{\mathrm{split}} \subset C^\bullet(V, V)\) be the
+suboperad defined by Sidon support constraints — i.e. cochains whose
+support is contained in a Sidon-indexed block decomposition, closed under
+the NR bracket.
+
 Then:
 
 1. **Internal cocycle condition:** Each \(\mu_i \in Z^2(V_i, V_i)\)
@@ -358,17 +391,21 @@ Then:
 2. **Support separation:** \(\mathrm{supp}(\mu_i) \cap \mathrm{supp}(\mu_j)
    = \varnothing\) for \(i \neq j\) (Sidon uniqueness).
 
-3. **Vanishing cross NR bracket:** \([\mu_i, \mu_j]_{\mathrm{NR}} = 0\)
+3. **Operadic non-composability:** \(\mu_i \circ_k \mu_j = 0\) for all
-   for \(i \neq j\) (no operadic contraction path exists).
+   \(i \neq j\) and all insertion points \(k\) — no admissible contraction
+   path exists across disjoint Sidon supports.
 
-4. **Total MC integrability:**
+4. **Vanishing cross NR bracket:** \([\mu_i, \mu_j]_{\mathrm{NR}} = 0\)
+   for \(i \neq j\) (consequence of 3).
+
+5. **Total MC integrability in the split suboperad:**
    \[
    d_{\mathrm{CE}}\mu + \tfrac12[\mu, \mu]_{\mathrm{NR}} = 0,
    \qquad
-   \mu \in \mathrm{MC}(C^\bullet(V, V)).
+   \mu \in \mathrm{MC}(\mathcal{O}_{\mathrm{split}} \subset C^\bullet(V, V)).
    \]
 
-5. **Obstruction class:**
+6. **Obstruction class:**
    \[
    \mathrm{Ob}(\mu) = 0 \in H^3(V, V).
    \]
@@ -386,29 +423,34 @@ Then:
 | 2 | \(V = \ker(\Sigma)\) selects \(\lambda_-\) eigenspace per block, giving 1D per \(V_i\) |
 | 3 | Jacobiator on a 1D space is identically zero → each \(\mu_i \in Z^2\) |
 | 4 | Sidon addresses give disjoint index supports |
-| 5 | Disjoint supports → no NR contraction path → \([\mu_i, \mu_j]_{\mathrm{NR}} = 0\) |
+| 5 | Disjoint supports → \(\mu_i \circ_k \mu_j = 0\) for all \(k\) → NR cross terms vanish |
-| 6 | Sum over internal + cross terms → \([\mu, \mu]_{\mathrm{NR}} = 0\) |
+| 6 | \(\mu\) lives in \(\mathcal{O}_{\mathrm{split}} \subset C^\bullet(V,V)\) by support constraints |
-| 7 | \(d_{\mathrm{CE}}\mu = 0\) by cocycle condition → MC holds |
+| 7 | Sum over internal + cross terms → \([\mu, \mu]_{\mathrm{NR}} = 0\) |
-| 8 | Structure constants land in \(\mathfrak{so}(1,6)\) by block-diagonal form |
+| 8 | \(d_{\mathrm{CE}}\mu = 0\) by cocycle condition → MC holds in the split suboperad |
+| 9 | Structure constants land in \(\mathfrak{so}(1,6)\) by block-diagonal form |
 
 ---
 
 ## 8. Comparison: why this is not a tautology
 
-The MC equation \(\mu \in \mathrm{MC}\) is *not* automatically satisfied by
+The MC equation \(\mu \in \mathrm{MC}(\mathcal{O}_{\mathrm{split}})\) is
-every crossing matrix.  Here is why this specific matrix works:
+*not* automatically satisfied by every crossing matrix.  Here is why this
+specific matrix works:
 
 | Property | This system | A generic matrix | Why it fails generically |
 |----------|------------|-----------------|--------------------------|
 | Block structure | 4 identical 2×2 blocks | Arbitrary 8×8 | NR cross terms non-zero |
 | Eigenvalue | \(\lambda_- = \sigma - \tau > 0\) on \(V\) | No distinguished eigenvalue | Jacobiator non-zero |
 | Index support | Sidon-disjoint | Overlapping | Contraction paths exist |
+| Operadic grafting | Forest-structured (disconnected) | Fully connected | Higher insertion trees survive |
 | Cocycle check | \(\mu_i \in Z^2\) by 1D argument | Must verify full Jacobi | May fail |
 
-The Sidon data does **three independent things** simultaneously:
+The Sidon data does **four independent things** simultaneously:
 (1) creates the block pairing, (2) selects \(\lambda_-\) via the simplex
-constraint, (3) forces disjoint supports.  Remove any one and the
+constraint, (3) forces disjoint supports, (4) dead-ends all operadic
-obstruction can be non-zero.
+grafting trees above the block level.  Remove any one and the obstruction
+can be non-zero — the MC solution is not a structural accident but a
+specific combinatorial fixed point.
 
 ---