SilverSight/docs/hopf_ingest_bridge.md

Hopf Ingest Bridge — Automated Classification System

Status: Specification, June 30, 2026 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

{
  "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

{
  "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