CAILculator MCP Server

High-dimensional mathematical structure analysis for autonomous AI systems

Applied Pathological Mathematics™ was born from this hypothesis:

Higher-dimensional algebras following the Cayley-Dickson sequence—often dismissed as "pathological"—can be interpreted and exploited for computational advantage.

CAILculator puts that hypothesis to work — a Model Context Protocol (MCP) server that empowers AI agents to analyze and compute within high-dimensional algebraic spaces (16D sedenions to 256D), exposing the Chavez Transform and the Zero Divisor Transmission Protocol (ZDTP) as experimental measurements, with Lean 4 proofs of the gateway zero-divisor identities and of the Chavez Transform's stability bound.

Formal Verification

Parts of CAILculator's mathematical foundation are proved in Lean 4. The proofs cover the statements listed below, not the tools' numerical outputs, which are computed in floating point. Two files check finite computations with native_decide, which trusts compiled code instead of the Lean kernel; lean/README.md explains what that means and lists each file's axiom footprint.

• [BilateralCollapse.lean](./lean/BilateralCollapse.lean): Proves the bilateral zero divisor identity (PQ = 0 ∧ QP = 0) for gateway S1, and the collapse formula, without native_decide. The identities for all six gateways are proved in RHForcingArgument.lean, by native_decide. At runtime, transmissions are gated by a numerical check, not by a Lean call.
• [ChavezTransform_genuine.lean](./lean/ChavezTransform_genuine.lean): Establishes the stability constant M, guaranteeing transform outputs never exceed theoretical bounds (|C[f]| ≤ M·‖f‖₁).
• [e8_weyl_orbit_unification.lean](./lean/e8_weyl_orbit_unification.lean): An exploratory result connecting the Canonical Six to E8 lattice structure. The Lean 4 file proves that the five distinct gateway P-vectors have norm² = 2 (consistent with the E8 first shell) and that an antipodal pair is related by a Weyl reflection, both by native_decide, and gives evidence (not a proof) that the vectors lie in a single Weyl orbit. This is the most preliminary of CAILculator's formal components and an active area of work; the associated tooling (map_e8_orbit) is correspondingly experimental.

The Chavez Transform

The Chavez Transform is an integral transform whose kernel is built from sedenion zero divisor pairs: elements whose product is zero although neither factor is. To our knowledge, no earlier integral transform uses zero divisors in its kernel. That novelty of construction is the claim; no property comparable to those of the classical transforms has been shown.

What the shipped tools compute (2.2.0). The 1D transform behind the chavez_transform tool and analyze_dataset has an exact closed form. The series is embedded on a single basis element, so the kernel is the same for every gateway pair of the shipped shape, zero divisor or not, and the output is a fixed, center-weighted sum of the series: T = Σᵢ dataᵢ·wᵢ, with weights that depend only on the series length, α and d. The formally verified property is the stability bound |C[f]| ≤ M·‖f‖₁. No noise-filtering or structure-detecting property has been verified or validated, and the 1D value carries no zero-divisor information. Both values are reported as experimental measurements. The n-dimensional transform, where zero divisors can annihilate part of the kernel, is not exposed by any tool and has not been tested. Details: docs/chavez_transform.md.

The Zero Divisor Transmission Protocol (ZDTP)

ZDTP lifts a 16D sedenion state into 256D space through each of the six Canonical gateways and compares the resulting magnitudes. The six gateways have three independent K_Z kernel components — S3A and S3B share a Fano intersection origin, as do S4 and S5 — but all six are run independently in transmission, and the convergence score reflects the full spread of their output magnitudes.

Closed form (2.2.0). The outputs reduce exactly to magnitude_256d(k) = √(‖x‖² + 64x₀² + 16sₖ²), where sₖ is a signed sum of four input components, and the convergence score is a function of those six numbers. The zero-divisor property of the gateways does not enter the result: pairs of the same shape that are not zero divisors give the same values. ZDTP outputs are reported as experimental measurements, not structural findings. Derivation and checks: docs/ZDTP_CLOSED_FORM.md.

Transmission Mechanics

Each of the six Canonical Gateway Pairs is a verified bilateral zero divisor in the Cayley-Dickson algebra: two sedenion elements P and Q satisfying both PQ = 0 and QP = 0. Computed (not formally proved): S2 is also bilateral in the Clifford framework at 16D–256D, the only gateway that is. Before any transmission begins, the oracle reconfirms this property numerically at 10⁻¹⁵ precision.

The transmission step is the four-factor interaction sum:

interaction = Px + xQ + Qx + xP

where x is the 16D input. Because Cayley-Dickson algebras are quadratic (uv + vu = −2⟨u, v⟩ for pure imaginary u, v), this sum simplifies to 2x₀(P+Q) − 2⟨P+Q, x⟩e₀: only the input's real part and one signed four-term sum survive. The result is appended to, not substituted for, the original input. The 16D input occupies the first 16 components of the output state unchanged.

This append-and-expand pattern repeats recursively: 16D → 32D → 64D → 128D → 256D. At each stage the original 16D gateway pair is zero-padded into the current dimension and the interaction is appended (the upper-half interaction term is identically zero). The original 16D state is always recoverable as the first 16 components of any higher-dimensional output.

Convergence Scoring

A single transmission through one gateway produces a 256D state. The full cascade runs the same 16D input through all six gateways and compares the resulting 256D magnitudes. The convergence score is:

score = 1 − std / mean   (over the six gateway output magnitudes)

Since 2.2.0 the tools no longer report these levels as verdicts. stability_level and regime.classification are "not_assessed". On 200 seeded null series the score fell in 0.54–0.65, the same range as every structured series tested (including a 50% step break), so the bands above did not separate structure from noise. The score itself is still reported. See docs/EVALUATION_CONTRACT.md.

By the closed form, the six magnitudes differ only through the six signed four-term sums sₖ. Similar magnitudes mean those sums are similar in size; different magnitudes mean they are not. That is a statement about four-component sums of the input, not about structure in the data.

Formal Verification Basis

The six gateway pairs are proved bilateral zero divisors in Lean 4: S1 in BilateralCollapse.lean (kernel-checked), and all six in RHForcingArgument.lean (by native_decide). Those proofs were run once, offline. The verified coordinates are hardcoded as constants in the engine with the Lean file as their attribution source. At runtime, the oracle independently reconfirms the bilateral property numerically before each transmission — not as a Lean call, but as a 10⁻¹⁵-precision arithmetic gate.

For S1, bilateral_collapse additionally proves the exact formula (aP + bQ)(bP + cQ) = −2b(a+c), a real scalar, for products of two elements of the pair's span with matched cross-coefficient b; scalar_channel is its existence form. It does not cover arbitrary pairs of linear combinations.

The Lean proof is the mathematical guarantee that these six pairs are valid gateways; the runtime oracle is the numerical lock that enforces it.

Why "Pathological" Means "Powerful"

Beyond the 8D Octonions, algebras following the Cayley-Dickson construction lose traditional properties like associativity and division algebra structure. The working hypothesis of Applied Pathological Mathematics is that these "pathologies" could be useful features for AI research:

• Non-associativity: could encode order-dependence and context-sensitivity directly into the algebraic operation.
• Zero Divisors: could mark branching structures and bifurcation points in high-dimensional representations.
• Structural invariants: might expose structure in data that simpler methods miss.

These are hypotheses, not results. So far, the tested cases have not supported them: the shipped Chavez and ZDTP values reduce to closed forms that do not use the zero-divisor property (above), and our own pre-registered tests on real campaign-finance data (internal, 2026) found no information in zero-divisor quantities beyond plain statistics. CAILculator is a tool for testing such hypotheses, with controls.

Deep-Dive Documentation

Extended theoretical frameworks and protocol specifications for researchers:

• [Chavez Transform Explainer](./docs/chavez_transform.md): The historical context, zero divisor kernel, the 1D closed form, stability bound, scalar channel theorem, formal verification chain, and research applications across RHI, finance, and journalism.
• [ZDTP Protocol Specification](./docs/zdtp_protocol.md): The Zero Divisor Transmission Protocol in full — Cayley-Dickson tower mechanics, four-factor interaction, recursive 16D→256D expansion, convergence scoring, and the Lean-to-runtime verification pipeline. Read with [ZDTP_CLOSED_FORM.md](./docs/ZDTP_CLOSED_FORM.md), which derives what the outputs reduce to.
• [Project Glossary](./GLOSSARY.md): Definitive terminology for high-dimensional algebraic structures (sedenions, pathions, and higher-dimensional algebras) and their domain projections in journalism and quantitative finance.

Python Environment

• Required: Python 3.10, 3.11, 3.12, or 3.13 (64-bit).
• Incompatible: Python 3.14+ (pending numba support) and all 32-bit versions.

Supported Operating Systems

• Windows 10/11
• macOS 10.15+
• Linux (Ubuntu 20.04+, Debian 10+)

API Key Acquisition

CAILculator requires a valid API key for tool execution.

1. Visit the Portal: Access the CAILculator API Portal to review subscription tiers (Individual, Journalist, Academic, Commercial, and Quantitative Finance).
2. Request Access: Email [[email protected]](mailto:[email protected]). Keys are typically issued within 24 hours.
3. Enterprise/Research: For custom profile development or large-scale research collaborations, include project details in your request.

Installation & Setup

1. Install CAILculator

pip install cailculator-mcp

This will download several hundred MB of scientific computing dependencies (`numpy`, `scipy`, `numba`).

2. Configure Your MCP Client

#### Claude Desktop Add the following to your configuration file. The surest way to open the right file is Settings → Developer → Edit Config in Claude Desktop. Typical locations:

• Windows: %APPDATA%\Claude\claude_desktop_config.json (the Microsoft Store version keeps it under %LOCALAPPDATA%\Packages\Claude_…\LocalCache\Roaming\Claude\)
• macOS: ~/Library/Application Support/Claude/claude_desktop_config.json

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

#### Any HTTP-mode MCP Client (including Gemini CLI) To leverage larger context windows, run the server locally over HTTP:

1. Install with HTTP support:

   pip install "cailculator-mcp[http]"

1. Start the local server:

   cailculator-mcp --transport http --port 8080

1. Register in your client's settings:

   {
     "mcpServers": {
       "cailculator": {
         "manifestUrl": "http://localhost:8080/mcp/manifest"
       }
     }
   }

Specialized Profiles

The Profile Manager projects universal algebraic patterns into domain-specific intelligence:

• Journalism Profile: Reports measured structural quantities for public-records data. Since 3.0.0 it gives no verdicts (measured_only).
• Quant Equity Profile: Reports measured structural quantities and standard technical indicators for OHLCV price data. Since 2.2.0 the regime verdict is withheld (not_assessed).

The RHI (Riemann Hypothesis Investigation) profile was removed in 2.2.0. The RH research described in the docs continues, using the general tools.

Available Tools

CAILculator computes natively across two fundamentally different algebraic frameworks — non-associative Cayley-Dickson and associative Clifford (Geometric) — at dimensions from 16D to 256D. Most mathematical software commits to one framework. CAILculator runs the same computation in both and shows where they disagree. For example, computed (not formally proved): S2 is the only Canonical Six gateway that is bilateral in both frameworks. With verify_bilateral_oracle's framework argument you can reproduce this directly.

High-Precision Research

• `chavez_transform` (experimental): 1D Chavez Transform of a series. Returns a fixed center-weighted sum of the series (closed form above) and the verified stability bound. Not a structure detector.
• `detect_patterns`: Multi-stage pipeline identifying linear, geometric, Fibonacci, and complex symmetry patterns.
• `verify_bilateral_oracle`: Exact bilateral zero-divisor check (PQ = 0 ∧ QP = 0) at 10⁻¹⁵ precision, runnable in either the non-associative Cayley-Dickson or associative Clifford framework via the framework argument. Running both shows where the frameworks differ: computed, S2 is the only Canonical Six gateway bilateral in both, while the others (e.g. S1) are bilateral in Cayley-Dickson but one-sided in Clifford (‖QP‖ = 2√2). Returns the residual ‖PQ‖ and ‖QP‖ norms.
• `map_e8_orbit` (experimental): Projects a vector onto E8 first-shell Weyl orbits. Currently maps the first 8 coordinates (the octonion half) only — 16D inputs with support in e₈–e₁₅ have that half dropped, so a full-16D vector can report as off-shell. Under active development.
• `compute_high_dimensional`: Direct sedenion algebra operations (multiply, add, conjugate, norm, zero divisor classification) extended into 32D–256D spaces.

Analysis & Visualization

• `analyze_dataset`: Pattern detection, the 1D Chavez value and the ZDTP full cascade in a single call. Returns top patterns, the convergence score and per-gateway magnitudes; the Chavez and ZDTP values are experimental measurements with closed forms (above). The regime classification is withheld (not_assessed) until a detector passes a validation gate. Accepts a close-price list or OHLCV dict. Minimum 16 data points.
• `zdtp_transmit` (experimental): Lifts a 16D input through one or all six Canonical gateways (S1, S2, S3A, S3B, S4, S5) to 256D. Outputs have the closed form above.
• `list_domain_profiles`: Lists the available domain profiles (Quant Equity, Journalism, General Data, Developer) and how to request a custom one.
• `illustrate`: Generate mathematical visualizations (bar charts, heatmaps, multi-panel plots) saved as high-fidelity PNG files.
• `get_version`: Verify engine status and formal verification metadata.

Additional financial tools (regime_detection, load_market_data, batch_analyze_market) are implemented as standalone modules and will be wired into a future release.

Technical Specifications

• Arithmetic: 64-bit floating point (numpy). Rounding error grows with dimension; products agree with exact values to about 10⁻¹⁵ at 16D and about 10⁻¹⁴ at 64D.
• Zero divisor thresholds: verify_bilateral_oracle, compute_high_dimensional's is_zero_divisor and the transmission gate all require ‖PQ‖ and ‖QP‖ below 10⁻¹⁵. This is a classification threshold, not a precision guarantee (see docs/EVALUATION_CONTRACT.md).
• Research Citation: Grounded in systematic computational enumeration published at Zenodo: 10.5281/zenodo.17402495.
• Core Libraries: numpy and scipy.

Glossary & Terminology

To support rigorous cross-disciplinary collaboration, we maintain a definitive [Project Glossary](./GLOSSARY.md) establishing terminology for high-dimensional algebraic structures (sedenions, pathions, and higher-dimensional algebras) and their domain projections in journalism and quantitative finance.

Contact & Collaboration

Research & Engineering: [email protected] GitHub: ChavezAILabs/cailculator-mcp

Chavez AI Labs "Better math, less suffering"