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# ⛔ RETRACTED — Hopf Ingest Bridge
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**Retraction date:** June 30, 2026. Depends on retracted Hopf Portability Criterion (`docs/hopf_portability_criterion.md`). Replaced by `scripts/cartan_fingerprint.py` and `docs/cartan_fingerprint.md`.
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---
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# Hopf Ingest Bridge — Automated Classification System (ARCHIVED)
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**References:** `docs/hopf_portability_criterion.md`, `formal/CoreFormalism/HopfFibration.lean`
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## 0. Purpose
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Given a problem P (expressed as structured metadata), determine:
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1. Whether P is Hopf-portable
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2. If so, compute its fingerprint (n, σ, τ, D, ∆, R)
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3. Classify it into a fiber type and regime class
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This bridges from the SilverSight formalization to arbitrary problem domains.
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## I. Ingest Pipeline
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```
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Problem P (JSON metadata)
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→ Extract channel structure (count n, interaction matrix M)
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→ Check Sidon-labelability (powers of 2 available?)
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→ Compute σ = spectral_radius(M) / 2ⁿ
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→ Compute τ = 1/(n−1)
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→ Compute D = lcm(2ⁿ, n−1)
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→ Compute ∆ = numerator(σ − τ)
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→ Check R = (n−1) × fiber_c matches π₀(Diff⁺(S^(2n-2)))
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→ Emit classification receipt
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```
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## II. Input Schema
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```json
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{
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"schema": "hopf_ingest_request_v1",
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"problem_id": "string",
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"domain": "physics | optimization | number_theory | geometry | other",
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"channel_count": 8,
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"interaction_matrix": "path_or_citation",
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"sidon_set": [1, 2, 4, 8, 16, 32, 64, 128],
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"yang_baxter_holds": true,
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"eigensolid_exists": true,
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"hint_fiber_type": "quaternionic"
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}
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```
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## III. Classification Output
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```json
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{
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"schema": "hopf_ingest_receipt_v1",
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"problem_id": "string",
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"hopf_portable": true,
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"fingerprint": {
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"n": 8,
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"sigma": "39/256",
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"sigma_numerator": 39,
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"tau": "1/7",
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"denominator_D": 1792,
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"gap": "17/1792",
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"gap_numerator": 17,
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"regimes_R": 28,
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"fiber_type": "quaternionic",
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"fiber_dimension": 3,
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"hopf_map": "S³→S⁷→S⁴"
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},
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"classification": {
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"regime_class": null,
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"port_quality": "strong",
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"domain_analogs": [
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"topological_insulators",
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"anyons_tqc",
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"qubo_spin_glasses",
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"ads4_cft3",
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"exponential_sums",
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"elliptic_curves_qm",
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"crystalline_cohomology",
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"spin_systems_o3",
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"class_field_theory"
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]
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},
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"conditions_passed": [true, true, true, true, true, true],
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"maximal_encoding": true,
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"at_ceiling": true
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}
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```
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## IV. Classification Rules
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### Rule 1: Fiber Type Detection
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| Channel count n | Fiber f | Hopf map | Structure group |
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|-----------------|---------|----------|-----------------|
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| n = 2 | f = 0 (real) | S¹→S¹ | ℤ₂ |
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| n = 4 | f = 1 (complex) | S³→S² | U(1) |
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| n = 8 | f = 3 (quaternionic) | S⁷→S⁴ | SU(2) ≅ Sp(1) |
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| n = 16 | f = 7 (octonionic) | S¹⁵→S⁸ | none (non-associative) |
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If n ∉ {2, 4, 8, 16}: **not Hopf-portable** (Condition E fails).
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### Rule 2: Gap Divergence Detection
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If p = numerator(σ − τ) is:
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- p = 0: **degenerate** — Kelvin (achiral) regime, no dissipation
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- 0 < p < 255: **Rossby (chiral) regime**, spectral gap active
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- p ≥ 500: **nonabelian** — crossing energy dominates, possible regime collapse
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For n=8 with Cartan a=39: p = 39×7 − 256 = 17 ∈ (0, 255) ✓
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### Rule 3: Ceiling Detection
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```
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is_at_ceiling = (n == 8) AND (fiber_type == "quaternionic")
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```
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If true: this is the **maximal group-theoretic Hopf encoding**. No larger n supports a structure group.
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## V. Bridge Architecture
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```
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┌─────────────────────────────────────┐
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│ Ingest Request │
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│ (JSON metadata about problem P) │
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└──────────────┬──────────────────────┘
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↓
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┌─────────────────────────────────────┐
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│ Condition Checker │
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│ A: Strand decomposition │
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│ B: Cartan spectrum (σ = a/2ⁿ) │
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│ C: Sidon threshold (τ = 1/(n−1)) │
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│ D: Spectral gap (∆ = p/D) │
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│ E: Hopf fibration fit (n = 2f+2) │
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│ F: Regime bound (R = (n−1)×c) │
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└──────────────┬──────────────────────┘
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↓
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┌─────────────────────────────────────┐
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│ Fingerprint Computer │
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│ n, σ, τ, D, ∆, R, fiber_type │
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└──────────────┬──────────────────────┘
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↓
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┌─────────────────────────────────────┐
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│ Domain Matcher │
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│ Cross-references against 15 known │
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│ Hopf-portable domain templates │
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└──────────────┬──────────────────────┘
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↓
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┌─────────────────────────────────────┐
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│ Classification Receipt │
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│ Emitted to signatures/ directory │
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│ Schema: hopf_ingest_receipt_v1 │
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└─────────────────────────────────────┘
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```
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## VI. Known Templates
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The bridge ships with 15 pre-classified domain templates (from the 4-agent synthesis):
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| Template ID | Domain | n | σ | D | R | Quality |
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|-------------|--------|---|---|---|---|---------|
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| TPL-QUAT-BRAID | 8-strand braidStorm | 8 | 39/256 | 1792 | 28 | Reference |
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| TPL-TOPO-INS | Hopf/Chern insulators | 8 | varies | 1792 | 28 | Strong |
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| TPL-ANYON-TQC | Fibonacci anyons | 8 | φ/256 | 1792 | 28 | Deep |
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| TPL-QUBO | QUBO spin glass | 8 | 39/256 | 1792 | 28 | Strong |
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| TPL-ADS-CFT | AdS₄×S⁷/Zk | 8 | SO(8)/256 | 1792 | 28 | Strong |
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| TPL-EXP-SUM | Kloosterman sheaves | 8 | p-adic/256 | 1792 | 28 | Strong |
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| TPL-ELL-QM | Elliptic curve QM | 8 | conductor/256 | 1792 | 28 | Strong |
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| TPL-CRYSTAL | Crystalline coho | 8 | L-invar/256 | 1792 | 28 | V-Strong |
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| TPL-SPIN-O3 | O(3) sigma + Hopf | 8 | g/256 | 1792 | 28 | Strong |
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| TPL-CLASS-FT | Class field mod 29 | 8 | regul/256 | 1792 | 28 | Strong |
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| TPL-TSP | TSP | 8 | 39/256 | 1792 | 28 | Moderate |
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| TPL-ILP | Integer programming | 8 | var/256 | 1792 | 28 | Moderate |
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| TPL-GRAPH | Graph coloring | 4 | chrom/16 | 24 | 6 | Suggestive |
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| TPL-SAT | 1-in-k SAT | 4 | clause/16 | 24 | 6 | Weak |
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| TPL-REAL | Binary decisions | 2 | 1/4 | 2 | 2 | Degenerate |
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New domains can be added by providing the 6-condition metadata and verifying against the criterion.
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## VII. Implementation Plan
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1. **Python classifier** (`scripts/hopf_classifier.py`): Accepts JSON problem metadata, runs the 6 conditions, emits receipt
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2. **Lean verification** (`formal/CoreFormalism/HopfFibration.lean`): Theorems `finitely_many_regimes_8` and `exotic_regime_bound` provide the formal boundary
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3. **AAIngest bridge**: Wire into the existing ingest pipeline → research_stack database → RRC classification
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The classifier can automatically determine:
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- `hopf_portable`: true/false
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- `fiber_type`: real/complex/quaternionic/octonionic
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- `fingerprint`: complete n/σ/τ/D/∆/R
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- `at_ceiling`: whether this is the maximal encoding
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