CDT++ samples a triangulation T with target weight proportional to exp(-S(T)). For a proposed triangulation T', the implemented acceptance probability is

min(1, exp(S(T) - S(T')) * q(T | T') / q(T' | T)).

This is the Metropolis-Hastings rule of Hastings Hastings1970Hastings1970. The three-dimensional causal triangulations, Regge action, and local move set follow Ambjørn, Jurkiewicz, and Loll AmbjornJurkiewiczLoll2001-3DAmbjornJurkiewiczLoll2001-3D.

Proposal kernel

Each transition first chooses one of the five 3D move types uniformly. It then chooses one raw site uniformly from the move-specific domain below. The move is attempted only at that site. An inapplicable site, failed construction, or invalid geometry delta is an explicit rejected self-transition.

Move	Raw proposal sites in T	Reverse move
(2,3)	N3_22(T) two-two cells	(3,2)
(3,2)	N1_TL(T) timelike edges	(2,3)
(2,6)	N3_13(T) one-three cells	(6,2)
(6,2)	N0(T) vertices	(2,6)
(4,4)	N1_SL(T) spacelike edges	(4,4)

For a raw-site count C_m(T), a particular proposal has probability

q(T' | T) = 1 / (5 * C_m(T)).

The factor 1/5 cancels between a move and its reverse, giving

q(T | T') / q(T' | T) = C_m(T) / C_reverse(m)(T').

The site definitions give a unique inverse site for each successful local retriangulation: a (2,3) face becomes the timelike edge used by (3,2); a successful (2,6) move creates the vertex used by (6,2); and a (4,4) pivot edge becomes the opposite pivot edge. Each (2,6) proposal examines exactly one uniformly selected (1,3) cell. Failed raw sites are not silently skipped, because conditioning on only the movable subset would change q and invalidate the count ratio above.

Move-type selection, raw-site selection, internal candidate-construction ordering, and the acceptance draw all consume the same run-owned random engine. Given the same starting manifold, an explicit seed therefore replays the complete transition sequence, including the technical edge ordering used to construct a (6,2) candidate. Initialization uses a separate named stream derived from the same root seed, so changes in point-generation draw counts do not shift the transition sequence. The CLI, checkpoint metadata, stream ownership, and future parallel policy are documented in Reproducible random runs.

State and geometry deltas

Candidates are constructed off to the side and checked against the exact topology delta before the acceptance calculation. Only an accepted candidate replaces the canonical manifold. Thus every decision uses the state committed by the immediately preceding transition.

Move	ΔN3	ΔN3_31_13	ΔN3_22	ΔN1_TL
(2,3)	+1	0	+1	+1
(3,2)	-1	0	-1	-1
(2,6)	+4	+4	0	+2
(6,2)	-4	-4	0	-2
(4,4)	0	0	0	0

In particular, (6,2) removes two (3,1) and two (1,3) simplices while leaving N3_22 unchanged. The runtime invariant check also covers N2, total and classified edges, spacelike edges, vertices, validity, and foliation bounds.

Counters

The counters obey two identities:

proposed = accepted + rejected

attempted = succeeded + failed = proposed

succeeded means that candidate construction produced a valid manifold. Such a candidate can still be rejected by Metropolis-Hastings. failed means the raw site was inapplicable or candidate construction violated an invariant; every failure is therefore also a rejection.

Numerical policy

The action, action difference, exponential, proposal ratio, and acceptance probability remain MPFR values at 256-bit precision. Every MPFR operation uses round-to-nearest with ties to an even significand (MPFR_RNDN). Conversion to long double occurs only for diagnostic output; an acceptance draw is compared directly with the MPFR probability.

References

Bibliographic metadata for Hastings1970Hastings1970 and AmbjornJurkiewiczLoll2001-3DAmbjornJurkiewiczLoll2001-3D is maintained in the repository-wide REFERENCES.md.