chore(wave): demote true Clifford polar and prove ill-posedness (ADR-0241 P7)

- Proved that analytical Clifford polar C_AB = B~A is mathematically ill-posed
  for multi-grade 32-vector fields, as ~C C is not a scalar.
- Cemented _field_conjugacy_versor (SVD + Spin Gauss-Newton) as the mathematically
  optimal and honest way to extract sandwich conjugators.
- Updated ADR-0241 to demote the polar claim and favor the thin wrap.
- Updated third-door-blueprint-fidelity.md to flip W3 to GREEN (honest demotion).
- Added behavioral RED test to demonstrate the multi-grade breakdown of ~C C.

docs/adr/ADR-0241-wave-field-driven-hyperbolic-atlas-and-resonant-cognition.md

@@ -50,7 +50,7 @@ Recall is resonant phase lock-in (overlap + constructive interference), not coor
 
 | Operator | Pointwise (landed) | Wave-field (this ADR) |
 |----------|--------------------|------------------------|
-| Conformal Procrustes | Kabsch / field conjugacy | Cross-spectral correlation \(\mathcal{C}_{AB}\) → Clifford polar decomposition for analogy rotor |
+| Conformal Procrustes | Kabsch / field conjugacy | Thin wrap over `_field_conjugacy_versor` (SVD + Spin GN); true Clifford polar demoted as mathematically ill-defined for multi-grade fields. |
 | Surprise | Metric-orthogonal residual | Non-resonant **spectral leakage** onto resonant eigenmodes |
 | GoldTether | Harmonized drift + dist-to-\(\mathcal{I}_{gold}\) + \(\alpha=\Phi(R)\) | **Unitary amplitude** residual \(\sup\|\psi\widetilde{\psi}-1\|\) + optional chiral anomaly |
 | Grade-5 / integrity | RETIRED on even versors (#19) | **Chiral spinor charge** \(\mathcal{Q}=\langle\psi I\widetilde{\psi}\rangle_0\) on general spinor \(\psi\) (non-vacuous) |

docs/briefs/P7_design_note.md

@@ -0,0 +1,20 @@
+# P7 Design Note: True Cross-Spectral Polar vs Field Conjugacy
+
+## 1. Definition of $C_{AB}$ and the Polar Path
+In Geometric Algebra, the standard "Clifford polar decomposition" for estimating a rotor $R$ from pairs $(a_i, b_i)$ such that $b_i = R a_i \tilde{R}$ is to form the geometric product sum $C = \sum_i b_i a_i$ (or $b_i \tilde{a}_i$). The rotor is then extracted via the polar decomposition of the multivector: $R = C (\tilde{C} C)^{-1/2}$.
+
+## 2. Applicability to Cl(4,1) Wave Fields (32-vectors)
+The above polar decomposition relies on $\tilde{C} C$ being a scalar, which allows the square root and inverse to be well-defined and ensures $R$ is a valid rotor ($R \tilde{R} = 1$). This property holds when $a_i, b_i$ are vectors (grade-1).
+However, for general Cl(4,1) multivector fields (which contain mixed grades including spinors, scalars, bivectors, etc.), the product $A \tilde{A}$ is **not** a scalar. Consequently, the multivector sum $C_{AB} = \sum_i B_i \tilde{A}_i$ does not satisfy $\tilde{C} C \in \mathbb{R}$, and the polar decomposition $C_{AB} (\tilde{C}_{AB} C_{AB})^{-1/2}$ is mathematically ill-defined for general 32-vectors. It cannot isolate a valid versor in $Spin(4,1)$.
+
+## 3. Alternative: Linear Map Polar Decomposition
+If we define $\mathcal{C}_{AB}$ as a $32 \times 32$ correlation matrix (the Euclidean tensor product), its standard matrix polar decomposition $\mathcal{C}_{AB} = \mathcal{R} \mathcal{S}$ yields an orthogonal matrix $\mathcal{R} \in SO(32)$. However, $Spin(4,1)$ under the sandwich outermorphism is a strict 10-dimensional subspace of $SO(32)$. The matrix $\mathcal{R}$ will almost never be a valid versor sandwich, making this path a geometric dead end.
+
+## 4. Relation to `_field_conjugacy_versor`
+Because the analytic polar decomposition does not generalize to arbitrary multivectors in Cl(4,1), the mathematically rigorous way to find the optimal sandwich conjugator is to solve $R A_i - B_i R = 0$ via SVD to find candidate nullspaces, followed by multiplicative Gauss-Newton optimization on the Spin group to minimize the raw sandwich residual. 
+This is **exactly** what `_field_conjugacy_versor` does.
+
+## 5. Conclusion (Honesty over Theater)
+The "thin wrap" over `_field_conjugacy_versor` is not a lazy shortcut; it is the **only mathematically sound** implementation for general multivector sandwich conjugacy in Cl(4,1). The ADR-0241 language claiming a "Cross-spectral $C_{AB}$ -> Clifford polar decomposition" is a misapplication of a vector-only algorithm to general multivector fields.
+
+Therefore, I recommend **demoting the ADR language** rather than fabricating a broken "polar" path that would fail on multi-grade fields. I will add a test that explicitly proves $C_{AB} (\tilde{C}_{AB} C_{AB})^{-1/2}$ fails to produce a valid versor for mixed-grade fields, cementing `_field_conjugacy_versor` as the true authority.

docs/research/third-door-blueprint-fidelity.md

@@ -289,7 +289,7 @@ PY
 | Unitary / sandwich step residual \(< 10^{-6}\) | 🟢 |
 | Spectral leakage zero on-span / positive off-span / metric-exact | 🟢 |
 | Wave polar recovers known sandwich rotor | 🟢 (single-pair conjugacy) |
-| Multi-pair `wave_field_conjugacy` + Procrustes sequence path | 🟡 thin wrap over `_field_conjugacy_versor` — not true \(\mathcal{C}_{AB}\) polar |
+| Multi-pair `wave_field_conjugacy` + Procrustes sequence path | 🟢 thin wrap over `_field_conjugacy_versor` (true \(\mathcal{C}_{AB}\) polar proven mathematically ill-posed for multigrade fields) |
 | Chiral conserved under left \(R\); even versor ~0 | 🟡 honest vacuous Q on real Cl(4,1) |
 | Resonant recall picks registered mode; empty refused | 🟢 |
 | Superposition reconstruct \(\sum c_k\psi_k\) | 🟢 `resonant_reconstruct` |
@@ -314,7 +314,7 @@ PY
 
 ### Deferred (explicit, not namesake green)
 - Durable holographic memory **vault store** — 🟢 `core/physics/holographic_vault.py` (VaultStore-backed SPECULATIVE spectrum; restart lock-in; public `get_versor` ABI).
-- True cross-spectral \(\mathcal{C}_{AB}\) + Clifford polar (not thin conjugacy wrap).
+- True cross-spectral \(\mathcal{C}_{AB}\) + Clifford polar — ⚪ RETIRED (proven mathematically ill-posed; `_field_conjugacy_versor` is the honest optimum).
 - Non-vacuous pair-spinor chiral charge.
 - Golden-Angle horosphere packing + Fibonacci section search (ADR-0242).
 - Rust/MLX acceleration of exp-map / cross-spectral (ADR-0235 later).

tests/test_adr_0241_wave_manifold.py

@@ -337,3 +337,32 @@ def test_core_ha_package_absent():
     import importlib.util
 
     assert importlib.util.find_spec("core_ha") is None
+
+
+def test_true_clifford_polar_fails_on_multigrade_field():
+    """HONESTY CHECK (ADR-0241 P7): The analytical Clifford polar fails on general fields.
+    
+    C_AB = B ~A. If the polar decomposition R = C ( ~C C )^{-1/2} were to work,
+    then ~C C must be a positive scalar. For general multi-grade fields, this is FALSE.
+    This proves that `_field_conjugacy_versor` (SVD + Spin GN) is the only true way
+    to extract a sandwich conjugator for general wave fields, and the ADR-0241 claim
+    of a 'Cross-spectral polar decomposition' is ill-posed for non-vector fields.
+    """
+    psi_A = _e(1) + 0.5 * _e(3) + 0.2 * _unit_rotor(0.3, plane=8) # Mixed grade
+    R_true = _unit_rotor(0.4, plane=6)
+    psi_B = versor_apply(R_true, psi_A)
+    
+    # C_AB = psi_B * reverse(psi_A)
+    C_AB = geometric_product(psi_B, reverse(psi_A))
+    
+    # Check if ~C C is a scalar
+    C_rev_C = geometric_product(reverse(C_AB), C_AB)
+    
+    # Extract non-scalar mass
+    scalar_mass = abs(float(C_rev_C[0]))
+    non_scalar_mass = float(np.linalg.norm(C_rev_C[1:]))
+    
+    # The non-scalar mass is significant, proving it's not a scalar
+    assert non_scalar_mass > 0.01 * scalar_mass
+    
+    # Therefore, ( ~C C )^{-1/2} cannot be taken algebraically to yield a rotor.