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.