# Finite State-Space Tables and Position-Specific Audit (B001-B004) ## Status This is a finite-observation working note. It records what was built and seen inside the observed local-state-word tables for B001-B004. It is not a theory paper. There is no proof, no general result, and no claim beyond the finite CSV sample. The miss/control split is used only as a bookkeeping label, never as a direction of cause. Frequent bundle labels are kept separate from stable ones, and bundle labels are not promoted to state variables here. ## Purpose Two passes are consolidated: 1. A finite state-space table built from observed local state words, with transitions, branching, and change-feature bundles attached. 2. A position-specific audit that re-ran the transition view with the position pair kept as an explicit condition, focused on the `-4 -> -3` step. ## Input Artifacts The row-level sources were the finite local-state-automaton files (`local_state_dictionary.csv`, `local_state_words_by_pair.csv`, and `collapse_edges_minus4_to_minus3.csv`), plus the basin material table for continuity. Broader Collatz notes were treated as background, not as row sources. Derived tables produced and consolidated here: `state_table.csv`, `transition_table.csv`, `branching_table.csv`, `bundle_check.csv`, `bundle_frequency.csv`, `state_equation_candidate.md`; and for the audit, `position_transition_table.csv`, `minus4_to_minus3_transition_table.csv`, `position_branching_table.csv`, `minus4_to_minus3_branching_table.csv`, `position_bundle_frequency.csv`, `minus4_to_minus3_bundle_frequency.csv`, `minus4_to_minus3_64_127_to_32_63.csv`; and for the conditioned resolution audit, `conditioned_level_summary.csv` and `conditioned_branching_by_source.csv`. ## 1. State Definition (observation cards) A state is a composite observation card built from the observed coordinates: `state_id = band | R_before -> R_after | front | chain_status | transition_k | residue32` Each card is a finite observation point, nothing more. The exported table holds 33 such states (one, `S000/MISSING`, is an observed placeholder and is dropped from the transition and bundle tables). ## 2. Transition Construction Consecutive state changes inside each local word (over positions -5, -4, -3, -2, -1, 0) were counted as observed transitions. `probability_from_source` is the outgoing count divided by the total outgoing count for that source state. The all-position table holds 37 distinct transitions from 30 source states. ## 3. Bundle Definition `B(x -> y) = { feature_i : feature_i(x) != feature_i(y) }` A bundle string is a changed-feature set attached to a transition. It is a transition label, not a state coordinate. Seven bundle strings appear in the finite state table. ## 4. All-Position View: Branching Appeared Read across all positions at once, most source states were function-like but some split. The all-position `branching_table.csv` labels 24 sources deterministic_like, 2 weak_branching, and 4 branching. The visible splits concentrate in the `64-127` band, for example the source at `R=67->62` (out_degree 3, dominant share 0.5), `R=67->63` (out_degree 2, dominant share 0.5), `R=70->63` (out_degree 2, dominant share 0.5), `R=67->60` (dominant share ~0.67), and `R=64->63` (dominant share ~0.77). At this point the transition view looked partly like a branching map: same card, more than one observed next card. ## 5. Position-Specific Audit: F_p(x) The audit re-ran the transition view with the position pair kept explicit: - All-position finite transition: `F(x) = { y | x -> y is observed }` - Position-conditioned: `F_p(x) = { y | x -> y is observed at position pair p }` - Focused slice: `F_{-4->-3}(x) = { y | x -> y is observed from -4 to -3 }` - Position-conditioned bundle: `B_p(x,y) = changed features of x -> y at position pair p` Each of the five position pairs carries 100 observations. ## 6. Result at -4 -> -3 The function_like result holds at the basin-conditioned level, not for bare source ids. 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 (dominant share 1.0, entropy 0), across all 22 rows. The observation-weighted side split of the slice is control_only 24, miss_only 36, mixed 40. But this is conditioned on `basin_id`. If `basin_id` is removed and only `source_state_id` is used, several source states still map to more than one target across B001-B004, because each basin has its own `-4 -> -3` target (B001 -> S012, B002/B003 -> S013, B004 -> S014). Observed examples: - S021 -> S012 | S013 | S014 - S017 -> S012 | S013 - S023 -> S012 | S014 - S029 -> S012 | S014 - S019 -> S013 | S014 So conditioning on the `-4 -> -3` position separated transitions that the all-position table had merged, but it did not remove all branching. Some remains once basin context is dropped. The fuller conditioned audit in Section 8 traces which combination of conditions completes the resolution over the whole table, and shows that position by itself does not. ## 7. The -4 -> -3 Slice Is Entirely 64-127 -> 32-63 In this B001-B004 sample, all 22 rows of the `-4 -> -3` slice have source band `64-127` and target band `32-63`. The row-level side split there is mixed 1, miss_only 18, control_only 3. The single mixed row is the B001 `S017 -> S012` entrance (40 observations, 14 miss / 26 control). So the `-4 -> -3` step, in this finite sample, coincides with the `64-127 -> 32-63` entrance face. ## 8. Conditioned Resolution: position -> side -> basin A separate conditioned audit steps through the conditions one at a time, to see which one actually completes the resolution of the visible branching: - `F(x) = { y | x -> y is observed }` - `F_p(x) = { y | x -> y is observed at position pair p }` - `F_{p,side}(x) = { y | x -> y is observed at position pair p and side }` - `F_{p,side,basin}(x) = { y | x -> y is observed at position pair p, side, and basin }` Level summary (source rows are the conditioned source rows at each level): | level | source rows | function_like | weak | strong | function_like share | | --- | ---: | ---: | ---: | ---: | ---: | | F(x) | 30 | 24 | 2 | 4 | 0.800000 | | F_p(x) | 30 | 24 | 2 | 4 | 0.800000 | | F_{p,side}(x) | 38 | 31 | 2 | 5 | 0.815789 | | F_{p,side,basin}(x) | 56 | 56 | 0 | 0 | 1.000000 | At `F(x)`, six source states are not function_like: S011 and S017 (weak_branching), and S019, S021, S023, S029 (strong_branching). Following those six down the ladder: - At `F_p(x)`, 0 of the six become fully resolved. The source-row counts are identical to `F(x)`: 24 function_like, 2 weak, 4 strong. - At `F_{p,side}(x)`, 0 of the six become fully resolved. Adding side creates more conditioned rows but still leaves weak and strong branching (31 / 2 / 5). - At `F_{p,side,basin}(x)`, all six become fully resolved. All 56 conditioned source rows are function_like. - Remaining unresolved after all three conditions: 0. So the apparent branching in `F(x)` is not fully resolved by adding position alone, and not fully resolved by adding side after position either. In this finite B001-B004 table, it is resolved only when `(position_pair, side, basin_id)` are all conditioned. Phrased carefully: position is important, but not sufficient; side is also not sufficient; in this finite table, basin/suffix context is the condition that completes the observed resolution of the branching. The branching therefore looks, in this sample, like a mixture of basin/suffix contexts rather than a split that persists after all currently tracked conditions. This is a finite-map observation, not a formula and not a general result. Side and basin are conditioning labels here; the audit does not claim that side or basin causes or explains the branching, only that conditioning on them removes it in this table. `-4 -> -3` is a focused audit slice, not a privileged final answer, and basin/suffix context is treated as a finite-table context label, not a promoted state variable. ## 9. Bundle Labels at -4 -> -3 The `-4 -> -3` bundle frequencies keep the frequent/stable distinction visible: - `band+R_before+R_after+transition_k+residue16+residue32`: rows 3, observations 52, delta patterns 2 — the exported table tags this as a *stable_candidate*. That is a tentative candidate label (few delta patterns), not a confirmed stable unit; it is not being declared stable here. - `all_except_chain_status`: rows 9, observations 18, delta patterns 8 — frequent_but_diverse. - `all_features`: rows 9, observations 18, delta patterns 9 — frequent_but_diverse. - `band+R_before+R_after`: rows 1, observations 12, delta pattern 1 — sparse. `all_except_front` did not appear in the `-4 -> -3` slice. Bundles remain transition labels; none is promoted to a state coordinate. ## 10. What Looks Function-Like vs What Still Branches Within this finite table, under the current threshold, full resolution appears only at the end of the conditioning ladder. Branching is still present at `F(x)` (six non-function-like sources), unchanged at `F_p(x)` (position alone resolves none of the six), and still present at `F_{p,side}(x)` (position plus side resolves none of the six). Only at `F_{p,side,basin}(x)` are all 56 conditioned source rows function_like. The persistent splits sit mostly in the `64-127` band and in the cross-basin sources (S011, S017, S019, S021, S023, S029), and they close once basin/suffix context is added — position removes one layer of mixing, side another sliver, but basin/suffix context is what completes the resolution here. The open question is which structural coordinate that basin/suffix context stands for, and whether the pattern survives a larger sample. ## 11. Unresolved - Whether `R_after` and `transition_k` are state coordinates or edge attributes of a transition. They currently sit inside the observation card but read partly as step-level quantities. - Whether `front` is state-like or transition-like. - Whether `residue16` adds resolution beyond `residue32`, or is redundant here. - Whether the bundle labels are only coarse transition labels, or whether any is compact enough (few delta patterns) to eventually stand as a coordinate. Not decided. - Whether the `(position_pair, side, basin_id)` resolution holds outside B001-B004. The full-resolution result rests on this finite table only (56 conditioned source rows). - What basin/suffix context represents structurally. It is the condition that completes the observed resolution here, but the audit does not say what it stands for. - Whether a coordinate can later replace `basin_id`. For now `basin_id` is kept as a finite-table context label and is not promoted to a mathematical state variable; if a measurable coordinate reproduces the same resolution, it could stand in later. - Whether `R_after` and `transition_k` are state coordinates or edge attributes, whether `front` is state-like or transition-like, and whether `residue16` adds resolution beyond `residue32` — all still open (Sections above). - Whether side-skewed rows stay side-skewed once more coordinates are added, and whether the miss/control split persists after conditioning on state. miss/control stays a bookkeeping split, not a driver. - Source, target, and delta rows are kept visible beside the conditioned summaries; the ladder counts are search handles, not replacements for the row-level evidence. ## 12. Suggested Next Inspection Keep arrow/state/delta rows beside the bundle labels; do not collapse to bundle strings. The most useful next checks are: test whether the `(position_pair, side, basin_id)` resolution holds outside B001-B004; inspect what basin/suffix context represents structurally and whether a measurable coordinate could stand in for `basin_id`; and try adding or removing `residue16` and `exit_distance` to see whether the branching profile changes before basin is conditioned. This stays a finite map of the observed tables, not a general formula.