Microcanonical algorithms

TL;DR

All microcanonical routes share the two-stage philosophy documented in Microcanonical sampling — construct one feasible state, then sample the target measure on the constraint fiber. The concrete constructors and kernels differ per constraint; this page maps each route to its algorithm and proof.

Constraint	Constructor	Sampling kernel	Exactness	Detailed page
(E,T)	feasible support/occupations	direct conditional sampler (uniform support + fixed-total Gibbs for ME)	exact direct	—
(k,T)	exact-degree support + occupations	degree-support MCMC + occupation allocator, fixed total	exact stationary MCMC	—
s	compressed feasible table	4-cycle occupation MCMC on the residual state	exact stationary MCMC	—
(s,E)	exact feasible state (biased initialization-only repair for residual E)	local exact-\(E\) 4-cycle kernel + censored bridge mixture	exact stationary MCMC	fixed (s,E)
(s,k)	extras-first combinatorial exact constructor	capped first-return degree trace of the fixed-\((s,E)\) kernel \(K_E\)	exact stationary MCMC	fixed (s,k)
s + cost	feasible strength state	cost-biased chain with fitted gamma; strengths exact, cost expected	hybrid	[fixed strength+cost] — see Spatial costs

Shared philosophy → concrete algorithms

The common view is:

constraints
   |
   v
construct one feasible state
   |
   v
sample the target ensemble around/on the corresponding constraint fiber
   |
   v
diagnostics / burn-in / thinning / independent draws as appropriate

Every route instantiates it differently:

• (E,T) constructs a uniform support/occupation allocation and draws directly from the conditional law;
• (k,T) constructs a support with exact degrees and allocates occupations under a fixed total;
• s compresses the fixed-strength state and runs the occupied-cell (4-cycle) chain;
• (s,E) builds an exact-\(E\) state and mixes the local kernel with the censored bridge;
• (s,k) constructs extras-first (combinatorially, no MCMC) and traces the degree-distance auxiliary chain onto the degree fiber;
• s + cost fits \(\gamma\) so expected cost matches, then runs the cost-biased chain with strengths exact.

Construction is not required to be reversible; sampling exactness is a property of the sampling law/kernel, not of how the initial feasible state was found (Validation).

Where the code lives

• shared capability registry: menobis.capabilities (backend column);
• routing: menobis/routing.py (sample_model_detailed exactness assignment);
• Rust kernels: crates/menobis-core (constructors, chains, trace kernels, gamma fitting);
• pyo3 bindings: crates/menobis-python.

Extending the routes

When adding or changing a microcanonical route, follow the contributor policy: document exactness classification, feasibility, diagnostics, and tested scales (Contributor documentation policy), update the capability registry, and regenerate the support tables (Supported models).