Consistency Math Kernel™
Consistency Math Kernel™
Consistency Math Kernel™
Deterministic Verification for mathematical correctness
Deterministic certification for mathematical correctness
Also see the Consistency Logic Kernel™
Deterministic certification for logic.
CONSISTENCY MATH KERNEL™ (CMK)
Deterministic certification for mathematical correctness
Public Core Package — Non-Confidential
Overview: what CMK is
The Consistency Math Kernel™ (CMK) is a deterministic certification engine for mathematical correctness.
CMK runs at inference time as a certification layer and enforces a strict production contract:
CERTIFIED
CMK deterministically validates correctness within its supported scope.
NOT CERTIFIED
When deterministic certification cannot be established under explicit constraints, CMK returns NOT CERTIFIED by design.
Where CMK fits in real systems
CMK is built for workflows where math must be repeatable, auditable, and safe to automate:
- Certify math inside LLM outputs before downstream execution (pricing, eligibility, disclosures, risk logic).
- Prevent incorrect calculations from entering regulated records, customer communications, or audit trails.
- Harden deterministic decisioning (underwriting thresholds, scoring rules, limits, compliance gates).
- Guardrail agentic/tooling systems by blocking actions triggered by incorrect arithmetic, algebra, probability, or unit conversion.
- Enable safe automation at scale: certified outputs proceed; not certified outcomes route into retry/reformulate/escalation policies.
How CMK differs from common approaches
CMK is certification-first. It is not a generator, and it is not a “best-effort” math tool. It enforces a correct-or-not certified contract suitable for production systems.
| Approach (examples) | Correctness guarantee | Failure mode | Integration posture | Best fit |
|---|---|---|---|---|
| LLMs (e.g., GPT/Claude/Gemini) | No deterministic guarantee | Can be confident but wrong on math | Probabilistic generation | Drafting, exploration, candidate generation |
| SMT/SAT solvers (Z3, cvc5, Yices, Boolector) | Strong within formal encodings/theories | Requires precise encoding; scope depends on theory/support | Solver backend / formal methods tool | Constraints, satisfiability, formal verification tasks |
| CAS (Mathematica, Maple, SymPy, SageMath) | High within supported symbolic domains | Rigid input formats; may fail outside scope or on ambiguous NL | Interactive / programmatic math tooling | Symbolic manipulation, exact algebra/calculus workflows |
| Numeric computing (NumPy, SciPy) | High for implemented numeric methods | No semantic verification; requires correct formulation/code | Engineering library | Numerical analysis, optimization, scientific computing |
| CMK™ | Deterministic within certified scope | NOT CERTIFIED when not certifiable (coverage/policy/bounds) | Inference-time certification layer | Correct-or-not certified pipelines for safe automation |
Benchmarks and evaluation results
DeepMind Mathematics
Evaluated on arithmetic, algebra, calculus, probability, comparison, and structured reasoning tasks.
- 560,000 problems evaluated
- 322,694 scored.
- 322,694 certified correct(57.624% coverage)
- Incorrect: 0
- Coverage: 57.6239%
- Runtime: 148.24 seconds
- Average time per problem: 0.2647 ms
AsyMOB Mathematics
Structurally adversarial symbolic suite: altered structure with preserved semantics.
- 17,000+ problems evaluated
- Coverage expansion and validation in progress
- 0 incorrect results among answered outputs
- Remaining cases NOT CERTIFIED by design
Hendrycks MATH (Geometry Subset)
Evaluated on the geometry subset of the Hendrycks MATH benchmark to measure deterministic, fail-closed solving under formal certification constraints.
- 479 problems evaluated
- 257 answered
- 257 certified correct
- Incorrect: 0
- Not certified (abstain): 222
- Coverage: 53.65%
- Runtime: 0.684 seconds
- Average time per problem: 1.428 ms
Updated 3/1/2026
Consistency Math Kernel™ Capabilities
Arithmetic and expression evaluation
- Integer/fraction arithmetic & expression evaluation
- Radical (square-root) simplification
- Percentages & proportional reasoning
- Percent of / percent change / discounts
- Rates & ratios
- Unit rate & cost/rate problems
- Measurement & unit conversion
- Unit conversion & time measurement
- Comparison & ordering
- Numeric comparison / min-max selection
Exponentials and logarithms
- Exact exponentiation and logarithms (supported identities only)
Algebra
Equations
- Linear equations & linear function evaluation
Systems
- Systems of linear equations (2×2 / 3×3)
Polynomials
- Binomial expansion
- Polynomial interpolation (Lagrange)
- Quadratics (roots, factoring, Vieta)
Rational expressions
- Partial fractions
- Rational simplification / cancellation
Inequalities
- Intervals & absolute value
- Linear/quadratic/rational inequalities & inequality tools
Calculus
- Differentiation & polynomial calculus
Trigonometry
- Trig evaluation / identities within supported scope
Complex numbers
- Complex arithmetic (basic)
Linear algebra
- Determinants (up to 4×4)
- Diagonalization (2×2 rational)
- Eigenvalues/eigenvectors (2×2)
- Matrix inverse (3×3)
- Matrix powers
- Vector algebra & inequalities
Geometry
Measurement
- Area/length on piecewise-linear geometry (polylines)
Euclidean
- Circles & tangency
- Polygons (including regular polygons) & quadrilaterals
- Triangles, similarity, Pythagorean relationships
Analytic
- Conic sections (including parabolas)
Coordinate
- Lines/segments, point–line relations, coordinate transforms
Discrete / coordinate
- Lattice geometry (Pick’s theorem, boundary points)
3D
- Planes, spheres, quadrics, and 3D incidence
Probability
- Basic probability, conditional probability & Bayes’ rule
- Discrete distributions (binomial, hypergeometric)
- Expectation / expected value
- Markov chains (basic)
Statistics
- Descriptive statistics (discrete)
Number theory
Core
- Diophantine & continued fractions
- Integers: gcd/lcm/division with remainder
- Primes, factorization & squarefree structure
- Wilson-type prime properties
Modular arithmetic
- Congruences, modular arithmetic & CRT
- Euler φ, Carmichael λ, multiplicative order
- Quadratic residues: Legendre/Jacobi, Hensel lifting, square roots mod n
Algorithms
- Discrete logarithm (BSGS / Pohlig–Hellman)
Arithmetic functions
- Arithmetic functions & transforms (Möbius/φ/σ/τ)
Combinatorics
- Counting & inclusion–exclusion
- Generating functions (basic)
Discrete math
- Bitset convolutions (AND/OR/XOR)
- Sequences & linear recurrences
Also see the Consistency Logic Kernel™
Deterministic certification for logic.
CONSISTENCY MATH KERNEL™ (CMK)
Deterministic certification for mathematical correctness
Public Core Package — Non-Confidential
Overview: what CMK is
The Consistency Math Kernel™ (CMK) is a deterministic certification engine for mathematical correctness.
CMK runs at inference time as a certification layer and enforces a strict production contract:
CERTIFIED
CMK deterministically validates correctness within its supported scope.
NOT CERTIFIED
When deterministic certification cannot be established under explicit constraints, CMK returns NOT CERTIFIED by design.
Where CMK fits in real systems
CMK is built for workflows where math must be repeatable, auditable, and safe to automate:
- Certify math inside LLM outputs before downstream execution (pricing, eligibility, disclosures, risk logic).
- Prevent incorrect calculations from entering regulated records, customer communications, or audit trails.
- Harden deterministic decisioning (underwriting thresholds, scoring rules, limits, compliance gates).
- Guardrail agentic/tooling systems by blocking actions triggered by incorrect arithmetic, algebra, probability, or unit conversion.
- Enable safe automation at scale: certified outputs proceed; not certified outcomes route into retry/reformulate/escalation policies.
How CMK differs from common approaches
CMK is certification-first. It is not a generator, and it is not a “best-effort” math tool. It enforces a correct-or-not certified contract suitable for production systems.
| Approach (examples) | Correctness guarantee | Failure mode | Integration posture | Best fit |
|---|---|---|---|---|
| LLMs (e.g., GPT/Claude/Gemini) | No deterministic guarantee | Can be confident but wrong on math | Probabilistic generation | Drafting, exploration, candidate generation |
| SMT/SAT solvers (Z3, cvc5, Yices, Boolector) | Strong within formal encodings/theories | Requires precise encoding; scope depends on theory/support | Solver backend / formal methods tool | Constraints, satisfiability, formal verification tasks |
| CAS (Mathematica, Maple, SymPy, SageMath) | High within supported symbolic domains | Rigid input formats; may fail outside scope or on ambiguous NL | Interactive / programmatic math tooling | Symbolic manipulation, exact algebra/calculus workflows |
| Numeric computing (NumPy, SciPy) | High for implemented numeric methods | No semantic verification; requires correct formulation/code | Engineering library | Numerical analysis, optimization, scientific computing |
| CMK™ | Deterministic within certified scope | NOT CERTIFIED when not certifiable (coverage/policy/bounds) | Inference-time certification layer | Correct-or-not certified pipelines for safe automation |
Benchmarks and evaluation results
DeepMind Mathematics
Evaluated on arithmetic, algebra, calculus, probability, comparison, and structured reasoning tasks.
- 560,000 problems evaluated
- 322,694 scored.
- 322,694 certified correct(57.624% coverage)
- Incorrect: 0
- Coverage: 57.6239%
- Runtime: 148.24 seconds
- Average time per problem: 0.2647 ms
AsyMOB Mathematics
Structurally adversarial symbolic suite: altered structure with preserved semantics.
- 17,000+ problems evaluated
- Coverage expansion and validation in progress
- 0 incorrect results among answered outputs
- Remaining cases NOT CERTIFIED by design
Hendrycks MATH (Geometry Subset)
Evaluated on the geometry subset of the Hendrycks MATH benchmark to measure deterministic, fail-closed solving under formal certification constraints.
- 479 problems evaluated
- 257 answered
- 257 certified correct
- Incorrect: 0
- Not certified (abstain): 222
- Coverage: 53.65%
- Runtime: 0.684 seconds
- Average time per problem: 1.428 ms
Updated 3/1/2026
Consistency Math Kernel™ Capabilities
Arithmetic and expression evaluation
- Integer/fraction arithmetic & expression evaluation
- Radical (square-root) simplification
- Percentages & proportional reasoning
- Percent of / percent change / discounts
- Rates & ratios
- Unit rate & cost/rate problems
- Measurement & unit conversion
- Unit conversion & time measurement
- Comparison & ordering
- Numeric comparison / min-max selection
Exponentials and logarithms
- Exact exponentiation and logarithms (supported identities only)
Algebra
Equations
- Linear equations & linear function evaluation
Systems
- Systems of linear equations (2×2 / 3×3)
Polynomials
- Binomial expansion
- Polynomial interpolation (Lagrange)
- Quadratics (roots, factoring, Vieta)
Rational expressions
- Partial fractions
- Rational simplification / cancellation
Inequalities
- Intervals & absolute value
- Linear/quadratic/rational inequalities & inequality tools
Calculus
- Differentiation & polynomial calculus
Trigonometry
- Trig evaluation / identities within supported scope
Complex numbers
- Complex arithmetic (basic)
Linear algebra
- Determinants (up to 4×4)
- Diagonalization (2×2 rational)
- Eigenvalues/eigenvectors (2×2)
- Matrix inverse (3×3)
- Matrix powers
- Vector algebra & inequalities
Geometry
Measurement
- Area/length on piecewise-linear geometry (polylines)
Euclidean
- Circles & tangency
- Polygons (including regular polygons) & quadrilaterals
- Triangles, similarity, Pythagorean relationships
Analytic
- Conic sections (including parabolas)
Coordinate
- Lines/segments, point–line relations, coordinate transforms
Discrete / coordinate
- Lattice geometry (Pick’s theorem, boundary points)
3D
- Planes, spheres, quadrics, and 3D incidence
Probability
- Basic probability, conditional probability & Bayes’ rule
- Discrete distributions (binomial, hypergeometric)
- Expectation / expected value
- Markov chains (basic)
Statistics
- Descriptive statistics (discrete)
Number theory
Core
- Diophantine & continued fractions
- Integers: gcd/lcm/division with remainder
- Primes, factorization & squarefree structure
- Wilson-type prime properties
Modular arithmetic
- Congruences, modular arithmetic & CRT
- Euler φ, Carmichael λ, multiplicative order
- Quadratic residues: Legendre/Jacobi, Hensel lifting, square roots mod n
Algorithms
- Discrete logarithm (BSGS / Pohlig–Hellman)
Arithmetic functions
- Arithmetic functions & transforms (Möbius/φ/σ/τ)
Combinatorics
- Counting & inclusion–exclusion
- Generating functions (basic)
Discrete math
- Bitset convolutions (AND/OR/XOR)
- Sequences & linear recurrences
