Collatz Conditioned State Audit

The same state sometimes appeared to lead to more than one next state. We therefore asked: what differences make those destinations separate in this dataset?

This does not show that the basin group causes the destination. The basin label may be standing in for some other difference that has not yet been identified.

This is not a proof of the Collatz conjecture, and it is not a general law for every integer. It is an observation in the B001–B004 data tables examined here.

That is the short version. The sections below explain how the states were built, how conditions were added, the numerical results, unresolved questions, and the source files.

Observed local state words for B001–B004 were folded into composite state descriptors, then into transition, branching, and bundle tables.

This is not a mathematically settled state definition. It is a composite state descriptor assembled from coordinates that were visible in the finite tables. The S000/MISSING placeholder is dropped from the transition and bundle tables.

Read across all positions at once, most source states appeared function-like, but some showed more than one observed next-state descriptor.

← scroll table →

This does not mean the Collatz map itself branches. In this finite table, different position pairs, sides, and basin contexts were merged into the same state descriptor, so the split may be an artifact of that merge — apparent branching on the table, nothing more.

The audit adds one condition at a time and re-reads the same transitions. The bar shows the function-like share of conditioned source rows at each level.

1. F(x) state only 0.800000

   { y | x → y is observed } · 30 source rows · 24 / 2 / 4

2. F_p(x) + position pair 0.800000

   { y | observed at position pair p } · 30 source rows · 24 / 2 / 4

3. F_{p,side}(x) + position pair, side 0.815789

   { y | observed at p and side } · 38 source rows · 31 / 2 / 5

4. F_{p,side,basin}(x) + position pair, side, basin 1.000000

   { y | observed at p, side, and basin } · 56 source rows · 56 / 0 / 0

The ladder is an audit procedure, not a formula. Each step keeps the same observed rows and only narrows the grouping key. Side and basin are conditioning labels used for bookkeeping. Here, F is not an equation for predicting unseen transitions; it is a relation that organizes transitions observed in this finite sample.

← scroll table →

At F(x), six source states were not function-like. Their class was tracked down the ladder.

At F_p(x), none of the six were fully resolved. At F_{p,side}(x), none were fully resolved either; S017 splits into conditioned rows of mixed class. Only at F_{p,side,basin}(x) are all conditioned source rows for these six function-like in this table. Class labels for S011, S017, S019, S021, S023, S029 read from conditioned_branching_by_source.csv.

The transition view was re-run with the position pair kept as an explicit condition, focused on one slice.

Inside the −4 → −3 slice, every basin-conditioned source row is function-like: for each (position_pair, basin_id, source_state_id) triple the observed target is unique in this finite table. Drop basin_id and keep only source_state_id, and several source states show more than one target across B001–B004.

Observed without basin context

• S021 → S012 | S013 | S014
• S017 → S012 | S013
• S023 → S012 | S014
• S029 → S012 | S014
• S019 → S013 | S014

Target by basin in this slice

This should be interpreted as an observation about the table, not about the map: the same source ID shows several targets once basin context is dropped, because each basin carried its own target in this sample. Conditioning on position separated transitions the all-position table had merged, but it did not remove all branching. The slice coincides with the 64–127 → 32–63 entrance face in this sample.

A bundle is the set of features that changed across a transition. It is a transition label, not a state coordinate.

stable_candidate is a tentative label given for having few delta patterns. It is not a confirmed stable unit, and it is not declared stable here. Frequent is not the same as stable: all_features and all_except_chain_status appear often and stay diverse. No bundle is promoted to a state coordinate. all_except_front did not appear in the −4 → −3 slice.

Root

• README.md Project overview.

scripts/

• local_state_automaton.py
• build_state_space_tables.py
• build_position_specific_audit.py
• build_conditioned_transition_audit.py

source/

• local_state_dictionary.csv
• local_state_words_by_pair.csv
• collapse_edges_minus4_to_minus3.csv

state-space/

• state_table.csv
• transition_table.csv
• branching_table.csv
• bundle_check.csv
• bundle_frequency.csv
• state_equation_candidate.md

position-audit/

• position_transition_table.csv
• position_branching_table.csv
• position_bundle_frequency.csv
• minus4_to_minus3_transition_table.csv
• minus4_to_minus3_branching_table.csv
• minus4_to_minus3_bundle_frequency.csv
• minus4_to_minus3_64_127_to_32_63.csv

conditioned-audit/

• conditioned_level_summary.csv
• conditioned_branching_by_source.csv