# ⛔ RETRACTED — Hopf Ingest Bridge

**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`.

---

# Hopf Ingest Bridge — Automated Classification System (ARCHIVED)
**References:** `docs/hopf_portability_criterion.md`, `formal/CoreFormalism/HopfFibration.lean`

## 0. Purpose

Given a problem P (expressed as structured metadata), determine:
1. Whether P is Hopf-portable
2. If so, compute its fingerprint (n, σ, τ, D, ∆, R)
3. Classify it into a fiber type and regime class

This bridges from the SilverSight formalization to arbitrary problem domains.

## I. Ingest Pipeline

```
Problem P (JSON metadata)
  → Extract channel structure (count n, interaction matrix M)
  → Check Sidon-labelability (powers of 2 available?)
  → Compute σ = spectral_radius(M) / 2ⁿ
  → Compute τ = 1/(n−1)
  → Compute D = lcm(2ⁿ, n−1)
  → Compute ∆ = numerator(σ − τ)
  → Check R = (n−1) × fiber_c matches π₀(Diff⁺(S^(2n-2)))
  → Emit classification receipt
```

## II. Input Schema

```json
{
  "schema": "hopf_ingest_request_v1",
  "problem_id": "string",
  "domain": "physics | optimization | number_theory | geometry | other",
  "channel_count": 8,
  "interaction_matrix": "path_or_citation",
  "sidon_set": [1, 2, 4, 8, 16, 32, 64, 128],
  "yang_baxter_holds": true,
  "eigensolid_exists": true,
  "hint_fiber_type": "quaternionic"
}
```

## III. Classification Output

```json
{
  "schema": "hopf_ingest_receipt_v1",
  "problem_id": "string",
  "hopf_portable": true,
  "fingerprint": {
    "n": 8,
    "sigma": "39/256",
    "sigma_numerator": 39,
    "tau": "1/7",
    "denominator_D": 1792,
    "gap": "17/1792",
    "gap_numerator": 17,
    "regimes_R": 28,
    "fiber_type": "quaternionic",
    "fiber_dimension": 3,
    "hopf_map": "S³→S⁷→S⁴"
  },
  "classification": {
    "regime_class": null,
    "port_quality": "strong",
    "domain_analogs": [
      "topological_insulators",
      "anyons_tqc",
      "qubo_spin_glasses",
      "ads4_cft3",
      "exponential_sums",
      "elliptic_curves_qm",
      "crystalline_cohomology",
      "spin_systems_o3",
      "class_field_theory"
    ]
  },
  "conditions_passed": [true, true, true, true, true, true],
  "maximal_encoding": true,
  "at_ceiling": true
}
```

## IV. Classification Rules

### Rule 1: Fiber Type Detection

| Channel count n | Fiber f | Hopf map | Structure group |
|-----------------|---------|----------|-----------------|
| n = 2 | f = 0 (real) | S¹→S¹ | ℤ₂ |
| n = 4 | f = 1 (complex) | S³→S² | U(1) |
| n = 8 | f = 3 (quaternionic) | S⁷→S⁴ | SU(2) ≅ Sp(1) |
| n = 16 | f = 7 (octonionic) | S¹⁵→S⁸ | none (non-associative) |

If n ∉ {2, 4, 8, 16}: **not Hopf-portable** (Condition E fails).

### Rule 2: Gap Divergence Detection

If p = numerator(σ − τ) is:
- p = 0: **degenerate** — Kelvin (achiral) regime, no dissipation
- 0 < p < 255: **Rossby (chiral) regime**, spectral gap active
- p ≥ 500: **nonabelian** — crossing energy dominates, possible regime collapse

For n=8 with Cartan a=39: p = 39×7 − 256 = 17 ∈ (0, 255) ✓

### Rule 3: Ceiling Detection

```
is_at_ceiling = (n == 8) AND (fiber_type == "quaternionic")
```

If true: this is the **maximal group-theoretic Hopf encoding**. No larger n supports a structure group.

## V. Bridge Architecture

```
┌─────────────────────────────────────┐
│          Ingest Request              │
│  (JSON metadata about problem P)     │
└──────────────┬──────────────────────┘
               ↓
┌─────────────────────────────────────┐
│       Condition Checker              │
│  A: Strand decomposition            │
│  B: Cartan spectrum (σ = a/2ⁿ)      │
│  C: Sidon threshold (τ = 1/(n−1))   │
│  D: Spectral gap (∆ = p/D)          │
│  E: Hopf fibration fit (n = 2f+2)   │
│  F: Regime bound (R = (n−1)×c)      │
└──────────────┬──────────────────────┘
               ↓
┌─────────────────────────────────────┐
│       Fingerprint Computer           │
│  n, σ, τ, D, ∆, R, fiber_type        │
└──────────────┬──────────────────────┘
               ↓
┌─────────────────────────────────────┐
│       Domain Matcher                 │
│  Cross-references against 15 known   │
│  Hopf-portable domain templates      │
└──────────────┬──────────────────────┘
               ↓
┌─────────────────────────────────────┐
│       Classification Receipt         │
│  Emitted to signatures/ directory    │
│  Schema: hopf_ingest_receipt_v1      │
└─────────────────────────────────────┘
```

## VI. Known Templates

The bridge ships with 15 pre-classified domain templates (from the 4-agent synthesis):

| Template ID | Domain | n | σ | D | R | Quality |
|-------------|--------|---|---|---|---|---------|
| TPL-QUAT-BRAID | 8-strand braidStorm | 8 | 39/256 | 1792 | 28 | Reference |
| TPL-TOPO-INS | Hopf/Chern insulators | 8 | varies | 1792 | 28 | Strong |
| TPL-ANYON-TQC | Fibonacci anyons | 8 | φ/256 | 1792 | 28 | Deep |
| TPL-QUBO | QUBO spin glass | 8 | 39/256 | 1792 | 28 | Strong |
| TPL-ADS-CFT | AdS₄×S⁷/Zk | 8 | SO(8)/256 | 1792 | 28 | Strong |
| TPL-EXP-SUM | Kloosterman sheaves | 8 | p-adic/256 | 1792 | 28 | Strong |
| TPL-ELL-QM | Elliptic curve QM | 8 | conductor/256 | 1792 | 28 | Strong |
| TPL-CRYSTAL | Crystalline coho | 8 | L-invar/256 | 1792 | 28 | V-Strong |
| TPL-SPIN-O3 | O(3) sigma + Hopf | 8 | g/256 | 1792 | 28 | Strong |
| TPL-CLASS-FT | Class field mod 29 | 8 | regul/256 | 1792 | 28 | Strong |
| TPL-TSP | TSP | 8 | 39/256 | 1792 | 28 | Moderate |
| TPL-ILP | Integer programming | 8 | var/256 | 1792 | 28 | Moderate |
| TPL-GRAPH | Graph coloring | 4 | chrom/16 | 24 | 6 | Suggestive |
| TPL-SAT | 1-in-k SAT | 4 | clause/16 | 24 | 6 | Weak |
| TPL-REAL | Binary decisions | 2 | 1/4 | 2 | 2 | Degenerate |

New domains can be added by providing the 6-condition metadata and verifying against the criterion.

## VII. Implementation Plan

1. **Python classifier** (`scripts/hopf_classifier.py`): Accepts JSON problem metadata, runs the 6 conditions, emits receipt
2. **Lean verification** (`formal/CoreFormalism/HopfFibration.lean`): Theorems `finitely_many_regimes_8` and `exotic_regime_bound` provide the formal boundary
3. **AAIngest bridge**: Wire into the existing ingest pipeline → research_stack database → RRC classification

The classifier can automatically determine:
- `hopf_portable`: true/false
- `fiber_type`: real/complex/quaternionic/octonionic
- `fingerprint`: complete n/σ/τ/D/∆/R
- `at_ceiling`: whether this is the maximal encoding