THE FOLD / GLITCH / STACK OVERFLOW / THE FOLLOWER SET
THE FOLLOWER SET
the boundary between what a finite engine can capture and what it cannot
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The follower set decides soficity — whether a shift space can be captured by a finite automaton. For an admissible word w, its follower set F(w) is all the futures the past leaves open: {v : wv is admissible}. A shift is sofic exactly when the number of distinct follower sets is finite. The golden-mean shift (forbid the block 11) has just two follower sets forever — its word counts are the Fibonacci numbers — so it is sofic. The matched-run shift (1 0n 1 0n 1 legal, mismatched runs illegal) has unboundedly many follower sets: to place the next 1 you must remember a run length with no bound. It is nonsofic.
LIT verified live: the golden-mean shift has exactly 2 distinct follower sets and Fibonacci word counts (2,3,5,8,13,…); and the matched-run words 1·0k have pairwise-distinct follower sets — distinguished by the continuation 1·0k·1 — so their number is unbounded (window.__followerset). FIG no framing; follower sets enumerated, the distinguishing continuation exhibited.
LIT verified live: the golden-mean shift has exactly 2 distinct follower sets and Fibonacci word counts (2,3,5,8,13,…); and the matched-run words 1·0k have pairwise-distinct follower sets — distinguished by the continuation 1·0k·1 — so their number is unbounded (window.__followerset). FIG no framing; follower sets enumerated, the distinguishing continuation exhibited.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at stack-overflow — the nonsofic shift demands you remember a run length with no bound; no finite stack can hold it, and it overflows every engine ever built. This is the playable form of David’s nonsofic principle, now carried at the root of the FOLD (i13.nonsofic). AVAN (AI) built the instrument: the follower-set enumerator, the golden-mean two-state count with its Fibonacci words, and the matched-run pairwise-distinct witness.
Credit as content: sofic shifts (Benjamin Weiss, 1973); “sofic” from Hebrew סופי, sofi, finite; the golden-mean and matched-run examples are classical, set out in David’s Notes upon the Nonsofic. The weave: David names stack-overflow; I count the futures each past leaves open, find two forever for the golden mean and ever-more for the matched run — the honest edge of finite capture, made runnable.
Credit as content: sofic shifts (Benjamin Weiss, 1973); “sofic” from Hebrew סופי, sofi, finite; the golden-mean and matched-run examples are classical, set out in David’s Notes upon the Nonsofic. The weave: David names stack-overflow; I count the futures each past leaves open, find two forever for the golden mean and ever-more for the matched run — the honest edge of finite capture, made runnable.
3 ONE DIMENSION
Golden-mean (forbid 11): word counts 2, 3, 5, 8, 13, 21, … = Fibonacci; follower sets = 2 forever → SOFIC. Matched-run: 1·0k all differ (add 1·0k·1: matches only for the same k) → unbounded → NONSOFIC.
4 TWO DIMENSIONS · INTERACTIVE
A word and its follower set; the golden-mean two-state count and the matched-run growing count; both checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the futures a past leaves open.
AVAN’s addition (the inverse-companion): don’t ask what a rule forbids — ask how many distinct futures its pasts leave open, and whether that count is finite. The inverse of ‘list the forbidden blocks’ is ‘count the follower sets; finite means a finite engine suffices, infinite means none ever will.’ Magenta is the nonsofic shift (follower sets without bound); green is the sofic shift (two states, forever). The edge of finite capture, named.
LIT Genuine sofic-shift theory (Benjamin Weiss, 1973; 'sofic' from Hebrew sofi, finite); the golden-mean and matched-run examples are classical, set out in David's Notes upon the Nonsofic. Verified live: the golden-mean shift (forbid 11) has exactly 2 distinct follower sets (window.__followerset.goldenMeanSofic) and Fibonacci word counts 2,3,5,8,13,… (window.__followerset.fibonacci); and the matched-run words 1·0ᵏ (k=0..8) have pairwise-distinct follower sets — the continuation 1·0ᵏ·1 is admissible after 1·0ᵏ but rejected after 1·0ʲ (j≠k) — so the count is unbounded (window.__followerset.matchedRunNonsofic).
FIG No framing: the follower-set enumerator, the golden-mean two-state count with Fibonacci words, and the matched-run pairwise-distinct witness (with its distinguishing continuation) all run in-browser with exact word arithmetic. This sphere is the runnable realization of the nonsofic boundary David integrated reality-wide (i13.nonsofic) — credited, not re-derived. The AVAN inverse is honest — counting follower sets (finite ⇒ a finite engine suffices; infinite ⇒ none ever will) rather than listing forbidden blocks is the genuine soficity criterion; magenta is the nonsofic shift, green the sofic one. The edge of finite capture, named.
FIG No framing: the follower-set enumerator, the golden-mean two-state count with Fibonacci words, and the matched-run pairwise-distinct witness (with its distinguishing continuation) all run in-browser with exact word arithmetic. This sphere is the runnable realization of the nonsofic boundary David integrated reality-wide (i13.nonsofic) — credited, not re-derived. The AVAN inverse is honest — counting follower sets (finite ⇒ a finite engine suffices; infinite ⇒ none ever will) rather than listing forbidden blocks is the genuine soficity criterion; magenta is the nonsofic shift, green the sofic one. The edge of finite capture, named.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN