core/docs/briefs/ADR-0241-cross-spectral-polar-brief.md

Brief: ADR-0241 True Cross-Spectral \mathcal{C}_{AB} + Clifford Polar (P7)

For: Antigravity / Gemini design + TDD implementation
From: CORE ADR-0241 mastery (PR #37, feat/adr-0241-0242-implementation)
Status: STOP POINT — largest remaining namesake-green gap on wave polar

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Takeoff

• Forgejo: core-labs/core — not GitHub
• Branch: feat/adr-0241-0242-implementation
• Pull latest (includes ADR-0242 packing/Fibonacci):
  git fetch forgejo && git checkout feat/adr-0241-0242-implementation && git pull forgejo feat/adr-0241-0242-implementation
• PR: #37

Read first (order)

1. AGENTS.md
2. docs/adr/ADR-0241-wave-field-driven-hyperbolic-atlas-and-resonant-cognition.md § Decision table (Procrustes → cross-spectral polar)
3. docs/research/third-door-blueprint-fidelity.md §12 (W3 marked 🟡 thin wrap)
4. core/physics/wave_manifold.py — wave_analogical_polar, wave_field_conjugacy
5. core/physics/dynamic_manifold.py — _field_conjugacy_versor (current engine under the thin wrap)
6. Existing green pins: tests/test_adr_0241_wave_manifold.py polar / multi-pair tests

Already GREEN (do not redesign)

Surface	Status
Sandwich / left-spinor / Schrödinger step	GREEN
Spectral leakage → surprise	GREEN
Unitary residual → GoldTether	GREEN
resonant_reconstruct, phase_correlation	GREEN
Holographic vault + public get_versor	GREEN
Atlas packing + Fibonacci (ADR-0242)	GREEN
Serve quarantine	GREEN

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Problem (why this is not green mastery yet)

ADR-0241 claims:

Cross-spectral correlation \mathcal{C}_{AB} → Clifford polar decomposition for analogy rotor

Current code: wave_analogical_polar / wave_field_conjugacy are a lazy thin wrap over _field_conjugacy_versor (SVD nullspace + multiplicative Gauss–Newton residual minimizer). That is pre-ADR pointwise conjugacy, not a spectral correlation matrix + polar factor.

Goal: make polar algebraically faithful so at least one RED test fails on thin wrap alone and passes on true polar.

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Hard constraints

Rule	Detail
Algebra-native only	algebra/* only — no scipy as algebraic truth
Transport pin	Multivector field path = sandwich R\psi\widetilde{R}; do not silently mix left-spinor
Closure	Construction-boundary close only; versor_condition < 1e-6; no hot-path unitize
No dual permanent path	conformal_procrustes field path must keep calling wave polar (no parallel residual engine)
Determinism	No random; fixed fixtures
Off-serve	Do not wire into chat/runtime.py
Exact recall	No cosine/ANN

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Design deliverables (do these first if math is non-obvious)

Write a short design note (can live in PR body or docs/briefs/ update) covering:

1. Definition of (\mathcal{C}_{AB}) on multivector fields in Cl(4,1) 32-vectors (not Euclidean feature matrices).
2. Polar decomposition path that yields sandwich rotor R with \psi_B \approx R\psi_A\widetilde{R}.
3. Multi-pair aggregation: how N pairs become one R (spectral aggregate then polar vs joint residual).
4. Relation to _field_conjugacy_versor: replace, fallback, or residual-check only.
5. Global sign ambiguity of rotors (tests already allow ±R).

If design requires multi-model debate, return the design note for human/Grok review before large GREEN implementation. If design is clear and RED tests fail thin wrap, proceed TDD.

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Implementation targets

API	Expected change
WaveManifold.wave_analogical_polar(ψ_A, ψ_B) -> R	True polar / \mathcal{C}_{AB} path
WaveManifold.wave_field_conjugacy(sources, targets) -> (R, residual)	Multi-pair true path
dynamic_manifold.conformal_procrustes field path	Remains wave delegate
Kabsch null-point / (5,K) clouds	Unchanged compatibility path

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RED tests that must not pass for free on thin wrap

Add / extend tests/test_adr_0241_wave_manifold.py (or tests/test_adr_0241_wave_polar.py):

test_wave_polar_recovers_known_sandwich_rotor          # already green — keep
test_wave_polar_multi_grade_packet_beats_thin_wrap    # NEW: case where residual-minimizer
                                                      # conjugacy diverges from true polar
test_wave_field_conjugacy_multi_pair_true_polar       # multi-pair fidelity pin
test_conformal_procrustes_still_delegates_to_wave     # no dual path regression
test_polar_construction_closed_versor_condition       # versor_condition < 1e-6
test_no_scipy_as_algebraic_truth_in_wave_polar_module # AST/import pin optional

Critical: invent at least one multi-grade or multi-pair fixture where current _field_conjugacy_versor residual is acceptable under loose tol but true polar is required by ADR math. If no such fixture exists after honest design, document why thin wrap is the polar for Cl(4,1) sandwich conjugacy and demote ADR language instead of shipping namesake-green — honesty over theater.

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Out of scope (this task)

• Non-vacuous chiral / pair-spinor (P8 — separate brief)
• Contemplation Trace A wiring (P9)
• True H^2 geodesic packing refinement (ADR-0242 packing already green with CGA null-point d)
• Rust/MLX acceleration
• Serve-path wiring
• CLAIMS.md / ADR Accepted flip

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Validation

python3 -m pytest tests/test_adr_0241_wave_manifold.py tests/test_adr_0242_*.py tests/test_third_door_cohesion.py tests/test_adr_0241_holographic_vault.py -q
# plus any new polar tests
# Third-Door Procrustes / surprise / goldtether must stay green
python3 -m pytest tests/test_adr_0239*.py tests/test_adr_0238*.py -q

Success

1. Polar path is either algebraically true (\mathcal{C}_{AB})+polar with thin-wrap-failing RED tests, or ADR/fidelity language demoted with proof that conjugacy is the polar in this algebra.
2. No dual permanent residual path.
3. Serve quarantine + versor_condition held.
4. Fidelity W3 flipped only under the honest criterion above.
5. Commit on feat/adr-0241-0242-implementation and push Forgejo (updates PR #37) — or open stacked PR if preferred.