Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
QMA Verification and Local Hamiltonian — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Circuit Syntax, Denotational Semantics, and Measurement
- Clifford+T Exact and Approximate Synthesis
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Continuous-Gate Universality and Exact Synthesis
- Controlled Gates, Phase, and Entanglement
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite-Dimensional Complex Inner-Product Spaces and Matrix Interfaces
- Formal Languages, Encodings, and Decision Problems
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Fundamental Trigonometric Identities
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Recurrences and Rational Generating Functions
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurements, POVMs, and Instruments
- Metric Spaces
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Order, Zorn's Lemma, and the Axiom of Choice
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Properties of the Integral and the Working FTC
- Pure and Mixed States, Ensembles, and Schmidt Decomposition
- QMA Verification and Local Hamiltonian
- Qubits, Bloch Sphere, Pauli Algebra, and One-Qubit Gates
- Registers, Tensor Products, and Partial Trace
- Relations, Functions, and Quotients
- Resource Bounds and Machine Invariance
- Reversible Computation, Ancillas, and Coherent Oracles
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Splitting Fields
- Strongly Continuous Semigroups and Hille Yosida
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tensor Products of Modules
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Polynomial Hierarchy and Relativization
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The ZFC Axioms and the Basic Set Constructions
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Turing Machines, Configurations, and Computation
- Uniform Circuit Families, Promise Problems, and Complexity
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
For one Hadamard gate on input , the example writes the two-clock history state, evaluates the propagation quadratic form, and checks that the scaled output penalty equals the rejection probability .
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
One-gate history Hamiltonian energy check
Example
Take one work qubit in , one gate , and declare outcome accepting. Then the history is Verify that propagation energy is zero and that the scaled output penalty has energy , equal to the verifier's rejection probability.
Facts & Assumptions
Given: The history Hamiltonian and the stated one-qubit circuit.
Feynman–Kitaev history Hamiltonian: at , propagation is the half-weight edge penalty and output penalty coefficient is .
Propagation kernel and path-Laplacian gap: the edge penalty vanishes exactly when the second computation component is applied to the first.
Input and output penalties on history states: the scaled output penalty equals the terminal rejection probability.
Proof
The propagation quadratic form on an arbitrary vector is . Here , so the norm is zero and propagation energy vanishes.
The terminal state is , whose probability of rejecting the accept-on- predicate is . Directly, the clock- history component has squared norm , its reject projection has probability within that component, and the coefficient 2 cancels the clock factor ; the total output energy is . This matches [F3].