API Reference

q6j(j1, j2, j3, j4, j5, j6; k=nothing, q=nothing, exact::Bool=false, eager::Bool=false, T=Float64)

Returns the Quantum 6j-symbol.

• If k and q are omitted, returns the abstract DCR object.
• If k is provided, projects to the root of unity (Float64 by default, Cyclotomic if exact=true).
• If q=1, computes the exact classical Ponzano-Regge limit.
• If eager=true, bypasses DCR construction for raw speed (only valid for root of unity k).

q3j(j1, j2, j3, m1, m2, m3; k=nothing, q=nothing, exact::Bool=false, eager::Bool=false, T=Float64)

Returns the quantum Wigner 3j-symbol.

qdim(j; k=nothing, q=nothing, exact::Bool=false, T=Float64)

Quantum dimension [2j+1]_q.

rmatrix(j1, j2, j3; k=nothing, q=nothing, exact::Bool=false, T=ComplexF64)

Returns the R-matrix phase (braiding eigenvalue) for j1, j2 crossing into j3.

fsymbol(j1, j2, j3, j4, j5, j6; k=nothing, q=nothing, exact::Bool=false, T=Float64)

Unitary crossing matrix element: √([d3][d6]) * {6j}.

gsymbol(j1, j2, j3, j4, j5, j6; k=nothing, q=nothing, exact::Bool=false, T=Float64)

Tetrahedrally symmetric invariant: √(Π[di]) * {6j}.

qint(n::Int, p::Int=1)

Returns the algebraic representation of ([n]_q)^p as a CyclotomicMonomial.

qfact(n::Int, p::Int=1)

Returns the algebraic representation of ([n]_q!)^p as a CyclotomicMonomial.

qbinomial(n::Int, k::Int)

Returns the cyclotomic monomial for the quantum binomial coefficient [n]_q! / ([k]_q! [n-k]_q!).

qseries(summand::Function, z_range::UnitRange{Int}; 
        prefactor::CyclotomicMonomial = ONE_MONOMIAL, 
        extract_radical::Bool = false)

A lightweight, user-friendly API for compiling a finite q-hypergeometric series into a Deferred Cyclotomic Representation (DCR).

Example

my_series = qseries(3:10) do z
    return (-1)^z * (qfact(z) / qfact(z - 3))
end

build_series(summand::Function, z_range::UnitRange; prefactor::CyclotomicMonomial=ONE_MONOMIAL, extract_radical::Bool=false)

Constructs a DCR from a user-provided summand function. summand(z) must return a CyclotomicMonomial representing the z-th term in the series.

build_dcr!(pref_func, base_func, ratio_func, z_min, z_max; kwargs...)

The master DCR compiler. Generates the full combinatorial skeleton of an arbitrary sequence by orchestrating closures over a highly optimized CycloBuffer.

Keyword Arguments:

• extract_radical::Bool (default false): If true, searches the prefactor for perfect squares and separates them into the root and radical fields.
• alternating_sign::Bool (default false): If true, automatically injects a (-1)^z term into the sequence (standard for recoupling invariants).

qeval(m::CyclotomicMonomial; k=nothing, q=nothing, exact::Bool=false, T::Type=Float64)

Universal evaluation API for a single CyclotomicMonomial.

• Classical Limit: Pass q = 1.
• Root of Unity (TQFT): Pass k (integer level).
• Complex Analytic: Pass q (complex or real parameter).
• Precision is controlled by exact (Float64 vs Rational/Cyclotomic).

qeval(dcr::DCR; k=nothing, q=nothing, exact::Bool=false, T::Type=Float64)

Universal evaluation API for a DCR.

• Classical Limit: Pass q = 1.
• Root of Unity (TQFT): Pass k (integer level).
• Complex Analytic: Pass q (complex or real parameter).
• Precision is controlled by exact (Float64 vs Rational/Cyclotomic).

CyclotomicMonomial

An unspecialized algebraic term: M(q) = σ * qᴾ * Π Φd(q)ᵉᵈ. Stored as a sparse, sorted vector of (d => exponent) pairs for efficiency. Φd(q) are cyclotomic polynomials.

DCR

The Deferred Cyclotomic Representation: A combinatorial skeleton of a q-hypergeometric series. Separates combinatorial summation from numerical/geometrical projection.

add_qint!(buf::CycloBuffer, n::Int, p::Int=1)

Algebraically multiplies the buffer by ([n]_q)^p.

add_qfact!(buf::CycloBuffer, n::Int, p::Int=1)

Algebraically multiplies the buffer by ([n]_q!)^p.

project_discrete(m::CyclotomicMonomial, k::Int, ::Type{T}=Float64)

Evaluates [n]q or dimensions at SU(2)k. Returns the real value (signs are tracked, complex phases are ignored)

project_discrete(res::DCR, k::Int, ::Type{T}=Float64)

Thread-safe, Zero-Allocation evaluator for full DCR symbols (e.g., 6j, 3j).

project_exact(m::CyclotomicMonomial, k::Int)

Evaluates a single monomial exactly. Returns a Nemo nf_elem.

project_exact(dcr::DCR, k::Int)

Evaluates a DCR series exactly for discrete level k. Returns a CompositeExactResult.

project_classical(m::CyclotomicMonomial, ::Type{T}=Float64) -> T

Evaluate monomial m at q -> 1. Returns a real value of type T.

project_classical(dcr::DCR, ::Type{T}=Float64) -> T

Evaluate the full DCR series at q -> 1 using a single-pass Streaming Log-Sum-Exp algorithm to prevent overflow/underflow.

project_classical_exact(m::CyclotomicMonomial) -> ClassicalResult

Evaluate monomial m exactly at q → 1. Returns a ClassicalResult encodes σ · √(val) where val is a Rational{BigInt}.

project_classical_exact(dcr::DCR) -> ClassicalResult

Evaluate the full DCR series exactly at q -> 1 as a Rational{BigInt}.

Algorithm (four passes, no GCD on every ratio):

1. Find the global denominator exponents
2. Denominator + N_0 reconstruction
3. Sum hot loop — accumulate numerators over D_glob
4. Fuse prefactor — multiply root and radical

Function is fully thread-safe.

project_analytic(m::CyclotomicMonomial, q::Number)

Evaluates a single quantum monomial

project_analytic(dcr::DCR, q::Number)

Fast, thread-safe for evaluating TQFT symbols at any generic q.

empty_caches!()

Useful for freeing RAM during long interactive sessions or resetting state for benchmarking.