Add the tent map's four generating partitions as one family - #4
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The existing tent-map examples read the Misiurewicz-point dynamics through
the two-letter kneading partition, split only at the critical point.
Refining by both order-1 preimages of the critical point gives a four-letter
generating partition and a five-state ε-machine for the same dynamics, which
is the natural companion for comparing how the anatomy split depends on the
partition: both presentations share the entropy rate log2(a), but the refined
one has r_mu = 0.
Probabilities are derived in Q(a) from the exact interval Markov chain on the
forward-orbit closure of {c, L, R}. Reducing by the minimal polynomial
a**3 = 2a + 2 leaves every transition a quadratic in a with rational
coefficients, so unlike the kneading presentation none carries an a-dependent
denominator, and the rows normalize identically rather than only at the root.
The 2013 supplement figures cover only the kneading partition, so the
docstrings cite James et al. for the map, parameter and anatomy framework
while stating that this presentation is derived rather than reproduced.
Co-authored-by: Cursor <cursoragent@cursor.com>
Refining the kneading partition by the critical point's order-1 preimages
gives four generating partitions of the same dynamics, but only {c} and
{L, c, R} were represented, the latter as a bolted-on "preimage" special
case. Define all four together in one table with uniform accessors keyed by
the cuts they make, so each is reachable the same way and their states are
comparable -- named by decreasing stationary weight throughout, which for
{c} relabels the published figure.
Every machine was re-derived exactly over Q(a) with a**3 = 2a + 2, turning
each transition probability and ephemeral rate into a quadratic in a with
rational coefficients. Two identities fall out and are now tested: the
published kneading rate reduces to (59 + 7a - 11a**2)/57, and the ephemeral
rate is exactly modular over the two cuts, with the L cut's share equal to
the invariant measure of the two cells it separates.
Co-authored-by: Cursor <cursoragent@cursor.com>
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Summary
At the Misiurewicz point the tent map's kneading partition (cut at the critical point
c = 1/2) can be refined by either or both ofc's order-1 preimages,L = 1/(2a)andR = 1 - 1/(2a). That gives four generating partitions of the same dynamics. This PR defines all four together rather than treating the fully-refined one as a special case.tent_map_misiurewicz_partition_{cuts,forward,symbol_matrices,information_expected}plusTENT_MAP_MISIUREWICZ_PARTITIONS. States are named by decreasing stationary weight in every partition so they are comparable across the family; for"c"that relabels the published figure, andtent_map_misiurewicz_fig7_symbol_matricesstill carries the figure's own labels.Q(a)witha**3 = 2a + 2, so each transition probability and each ephemeral rate is a quadratic inawith rational coefficients — noa-dependent denominators.All four are generating, hence share
h_mu = log2(a), and all four are unifilar and strictly sofic with infinite Markov and cryptic order. Only the anatomy split moves:r_mur_mu≈"c"c(59 + 7a - 11a**2)/57"Lc"L, c(56 + 25a - 23a**2)/57"cR"c, R(1 - 6a + 4a**2)/19"LcR"L, c, R0Two exact identities came out of the derivation and are now tested:
(1/4)(3 - 2/(a+1) - 4/(a+2) + 9/(2a+3)), reduces modulo the minimal polynomial to(59 + 7a - 11a**2)/57— an independent check of the whole derivation against the paper.r_mu("c") - r_mu("Lc") - r_mu("cR") + r_mu("LcR") = 0identically ina. Each cut is worth a fixed number of bits whether or not the other has been made, and theLcut's share is exactly the invariant measure of the two interval-Markov cells it separates. Excess entropy is modular too; statistical complexity is not.Test plan
pytest— 927 passedruff check .andruff format --check .ty check— no new diagnosticssphinx-build -W --keep-going -b html docs— no content warningschaos_anatomyfloat pipeline (edges, causal-state weights,h_mu/b_mu/r_mu, excess entropy) to ~1e-12a, and that the modularity identity and the reduction of the published formula hold symbolically