Applied Pathological Mathematics™

Better math.
Less suffering.

CAILculator MCP Server — v2.0.3 — Formally Verified

High-dimensional mathematical analysis for AI agents. Built on 16-dimensional sedenion algebra — the structures dismissed as pathological — now formally verified in Lean 4 and deployed as a Model Context Protocol server.

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16D Sedenion Space
10⁻¹⁵ Precision Floor
0 Lean Sorries
38 Orders of Magnitude
Lean 4 Verified — BilateralCollapse.lean · ChavezTransform_genuine.lean
Zero Sorries — Machine-verified proofs, no placeholders
DOI: 10.5281/zenodo.17402495 — Published research foundation
Dual Algebra Support — Cayley-Dickson + Clifford frameworks
MCP Native — Claude Desktop · Claude Code · Gemini CLI
Mathematical Foundation

Pathological by design.
Powerful by proof.

CAILculator treats structures that conventional mathematics discards as computational resources. Every claim is machine-verified.

Zero Divisors

Non-zero elements P, Q where P × Q = 0. In 16D sedenion space there are 84 base patterns — formally enumerated and verified. CAILculator's oracle layer dispatches on (P,Q) object pairs, not labels.

✓ Lean Verified

Chavez Transform

38 orders of magnitude dynamic range. Stability constant M = 2·(‖P‖²+‖Q‖²)/(α·e) is Lean-proved — not empirically estimated. Transform outputs are checked against the verified bound at runtime.

✓ Stability Proved

Canonical Six

Six framework-independent bilateral zero-divisor patterns persisting from 16D through 256D in both Cayley-Dickson and Clifford algebras. Identified as lying on the E8 first shell, forming a single Weyl orbit.

✓ E8 Orbit Verified

Dual Algebra Frameworks

Native support for both non-associative Cayley-Dickson algebras and associative Clifford (geometric) algebras. The bilateral oracle runs parity-verified across both frameworks simultaneously.

✓ Clifford Bridge Verified

ZDTP Transmission

Zero Divisor Transmission Protocol — transmit data through verified mathematical gateways (S1–S6) to 32D and 64D spaces. Bilateral annihilation confirmed universal across all 50 Riemann zeros tested.

✓ Bilateral Collapse Proved

Domain Profile System

Universal algebraic layer stays Lean-anchored. Domain projections are pluggable profiles: Journalism, Quant Equity, and RHI ship as reference implementations. Custom profiles available on request.

✓ v2.0 Architecture
MCP Tools

8 tools. One verified core.

Every tool dispatches on Lean-verified (P,Q) object pairs. No label-based heuristics. Stability bounds checked at output time.

chavez_transform
Apply the formally verified Chavez Transform to find hidden structure in data. Stability bound M checked at every output. 38 orders of magnitude dynamic range.
verify_bilateral_oracle
High-precision check (10⁻¹⁵) for any zero divisor pair P·Q=0 ∧ Q·P=0. Supports both Cayley-Dickson and Clifford frameworks with full parity verification.
detect_patterns
Algebraic detection of Tipping Points and Pattern Consistency via oracle-backed dispatch. All pattern_id routing is on verified (P,Q) objects, not semantic labels.
map_e8_orbit
Project 16D/32D vectors onto verified E8 Weyl orbits. The Canonical Six are confirmed members of the E8 first shell, forming a single 24-element Weyl orbit family.
illustrate
Generate high-precision visualizations of algebraic structures — zero divisor lattices, Weyl orbit projections, transform convergence plots, and gateway interaction diagrams.
get_version
Returns current server version, active profile, Lean proof inventory, and axiom footprint. Use to verify your deployment matches the expected verified core.
list_domain_profiles
Enumerate available domain profiles (Journalism, Quant Equity, RHI). Each profile declares its coefficient mappings, gateway labels, used theorems, and empirical dependencies separately.
zdtp_transmit
Transmit data through verified mathematical gateways S1–S6 to 32D and 64D spaces. Gateway semantic labels are profile-specific; the underlying (P,Q) pairs are Lean-proved universal.
regime_detection
Dual-method market analysis combining Hidden Markov Model baseline with Chavez Transform structural analysis. Bridges statistical and algebraic approaches for regime identification.
batch_analyze_market
Smart sampling strategy for large-scale datasets. Handles up to 5M records (Professional) or unlimited GB-scale (Elite). Preserves OHLCV coefficient mapping through profile invariants.
Domain Profiles

Universal algebra.
Domain intelligence.

The verified core stays universal. Domain projections are declared profile objects with versioned manifests, explicit evidence references, and auditable empirical dependencies.

Journalism

Investigative
Analysis

Algebraic pattern detection optimized for data reporting. Treat structural collapses in budgets, consensus, and policy the way zero divisors treat dimensionality.

  • Tipping point detection via bilateral zeros
  • Sourcing confidence from transform convergence
  • Pattern consistency across data series
  • Three-tier terminology: technical / standard / plain
Quant Equity

Market
Regime Detection

Dual-method analysis bridging HMM statistical baselines with algebraic structural analysis. OHLCV coefficient mapping to sedenion gateways, RSI/MACD/Bollinger interpretation.

  • Regime detection: HMM + Chavez Transform
  • 10 technical indicators with algebraic anchors
  • Volatility anchors at zero-divisor loci
  • Bifurcation risk via bilateral threshold analysis
RHI — Research

Riemann Hypothesis
Investigation

Spectral research mapping and prime embedding analysis. The active profile powering an open-science Riemann Hypothesis investigation, with Phase 72 introducing the Sedenionic Hamiltonian H(s).

  • Prime embedding: log p → ROOT_16D
  • ZDTP convergence tracking across γₙ
  • S3B=S4 bilateral gateway pairing (proved)
  • Sedenionic Hamiltonian operator support
Formal Verification

Not approximated.
Proved.

Core algebraic properties are machine-verified in Lean 4 with Aristotle co-authorship. Every oracle tool has non-triviality tests confirming it produces distinct outputs on distinct inputs.

✓

BilateralCollapse.lean

Proves P·Q = 0 ∧ Q·P = 0 for all Canonical Six pairs. The oracle used to gate every ZDTP transmission.

✓

ChavezTransform_genuine.lean

Stability constant M, unconditional absolute convergence, pattern invariance across dimensional scaling.

✓

e8_weyl_orbit_unification.lean

Canonical Six confirmed on E8 first shell, forming a single Weyl orbit generating a 24-element zero divisor family.

Published research foundation
DOI: 10.5281/zenodo.17402495
Canonical Six v1.4 · Addendum D: Bilateral Collapse Theorem
-- BilateralCollapse.lean (excerpt) -- 0 errors · 0 sorries · standard axioms only theorem bilateral_collapse_universal (P Q : Sedenion) (hP : P ∈ canonical_six) (hQ : Q ∈ canonical_six) : P * Q = 0 ∧ Q * P = 0 := by exact bilateral_collapse_iff.mpr ⟨canonical_mul_zero hP hQ, canonical_mul_zero hQ hP⟩ theorem stability_bound (f : ℝ → Sedenion) (α : ℝ) (hα : 0 < α) (P Q : Sedenion) (hPQ : P * Q = 0) : ‖C[f]‖ ≤ M P Q α := by exact chavez_stability hα hPQ where M P Q α := 2 * (‖P‖² + ‖Q‖²) / (α * Real.exp 1)
Pricing

Choose your
depth of analysis.

API key delivered within 24 hours of subscription. Students save with code STUDENT2025 at checkout.

Core / Research
Quant Equity
Individual
$ 79 .99/mo
25,000 requests / month
  • Zero divisor detection
  • Pattern analysis
  • Chavez Transform
  • Advanced visualizations
  • E8 lattice analysis
  • Email support
Subscribe
Academic
$ 199 /mo
75,000 requests / month
  • Everything in Individual
  • Priority support
  • Research collaboration
  • Citation assistance
  • Classroom use
Subscribe
Commercial
$ 299 /seat/mo
250,000 requests / month
  • Everything in Academic
  • Dedicated support
  • Custom integrations
  • Usage analytics
  • Team management
Subscribe
Enterprise
Custom
Unlimited requests
  • Multiple API keys
  • SLA guarantees
  • White-label options
  • Phone & Slack support
Contact Us
Quant Explorer
$ 599 /mo
Try our unique dual-method approach
  • Full dual-method regime detection
  • All 10 technical indicators
  • All 3 terminology levels
  • Up to 50K records per analysis
  • Email support (24hr response)
Subscribe
Quant Professional
$ 1,499 /mo
Scale it up for serious analysis
  • Everything in Explorer, plus:
  • Up to 5M records (100× more)
  • Batch processing for large datasets
  • API access for automation
  • 1 custom pattern analysis / month
  • Priority support (2hr response)
Subscribe
Quant Elite
$ 3,499 /mo
Partner with us to master it
  • Everything in Professional, plus:
  • Unlimited records + GB-scale processing
  • Monthly 60min strategy session
  • 2 custom developments / month
  • Dedicated support (Slack/Discord)
  • White-label option available
Subscribe
Quant Enterprise
$10K+
Custom deployment & research partnership
  • Everything in Elite, plus:
  • On-premise deployment options
  • Team training & onboarding
  • Research partnership collaboration
  • Custom indicators & methodologies
  • SLA guarantees
Contact Us
🎓 Students: use code STUDENT2025 at checkout for savings off Individual pricing.
Installation

Up in
three steps.

Python 3.10–3.13, 64-bit. Windows 10/11, macOS 10.15+, Ubuntu 20.04+.

Windows Claude Desktop

$ pip install cailculator_mcp

Open %APPDATA%\Claude\claude_desktop_config.json and add:

{
  "mcpServers": {
    "cailculator": {
      "command": "cailculator-mcp",
      "args": ["--transport", "stdio"],
      "env": {
        "CAILCULATOR_API_KEY": "your_api_key"
      }
    }
  }
}

macOS / Linux Claude Desktop

$ pip install cailculator_mcp

Open ~/Library/Application Support/Claude/claude_desktop_config.json and add:

{
  "mcpServers": {
    "cailculator": {
      "command": "cailculator-mcp",
      "args": ["--transport", "stdio"],
      "env": {
        "CAILCULATOR_API_KEY": "your_api_key"
      }
    }
  }
}

Claude Code / Gemini CLI HTTP Transport

CLAUDE CODE

claude mcp add cailculator \
  --transport http \
  --url https://cailculator-mcp-production.up.railway.app/mcp \
  --header "X-API-Key: your_api_key"

GEMINI CLI

gemini mcp add \
  --name cailculator \
  --url https://cailculator-mcp-production.up.railway.app/mcp \
  --api-key your_api_key