{ "tm": "1RB1LB_1RC0LA_1RD0RE_1LA0LB_1RF---_1RB1LD", "rank": 175, "class_sheet": "Skelet 1-like Class 1", "class_guess": "exponential counter with an exact macro cycle M(X) -> M(2^36 X + c) (175); 489 is a deeper tower regime", "confidence": "high", "summary": "After about 4.1e18 steps line 175 repeats one exact cycle: big run X -> 2^36 X + 9575900996480.", "mechanism": [ "Runs of 4-cell words; a marker crosses a run one word per left-end visit and turns each word into two, `(0100)^a 1^2 (0001)^(b+1) -> (0100)^(a+2) 1^2 (0001)^b` (20/20), so transfers are x -> 2x + c; nothing ever halves. Halting needs `B> 0 1 1` (E is entered only from C1, C only from B0).", "From qsim record 21 (step 4,111,147,719,584,449,981) the record sequence is periodic with period 42: `M(X) = 0^inf 1 1011^X 10111110 1110^26 D(Y1-48, Y2-240, 2^40 X + 377451701380050) per 40 records; Y1 runs out after 1.96e16 periods, at tower depth, with an unknown outcome." ], "diagrams": [ { "mode": "default", "png": "img/default_175_1e4.png", "html": null, "steps": 10000, "caption": "bbchallenge default view of line 175, first 10,000 steps", "title": "Line 175, first 10,000 steps", "why": "The site's own view: a counter grows leftward from the start.", "primary": true, "start": "blank", "rows": "every step (10,000 steps squeezed into 500 px as on the site)", "window": "cells -200..199, 1 px = 1 cells", "legend": "white = 1, black = 0, red/green = head moving R/L", "style": "bbchallenge-default" }, { "mode": "custom", "png": "img/runstrip_175_qsim.png", "html": null, "steps": 0, "caption": "Run lengths at each qsim rule application, records 0-139", "title": "175: the 42-record cycle", "why": "One row per qsim rule application; bar width is log2 of the run length. Compare rows 21, 63, 105.", "primary": true, "start": "blank", "rows": "one row per qsim rule application (records 0..139, block size 8)", "window": "whole tape; run of length n drawn 1+1.5*log2(n+1) px wide (log scale, not cells)", "legend": "white = run of a 3-ones word (0111 rotations), grey = run of a 1-one word (0001 rotations), blue = separator text, red = head, black = blank", "style": "custom" }, { "mode": "records", "png": "img/segment_x4_Z52.png", "html": null, "steps": 2482418, "caption": "One cycle segment: X=52 becomes 4X+176 = 384", "title": "One segment of the cycle, X = 52", "why": "Rows are left/right records from the constructed start; the run is crossed again and again, each pass doubling it.", "primary": true, "start": "constructed: 0^inf 11 0111^52 = 1, b >= 0 (1e9 run: 75 visits in a row)", "validated": { "instances": 20, "passed": 20 }, "source": "manual" }, { "name": "segment 20 (x4)", "from": "0^inf 11 0111^Z 1101^x 11011 0^inf", "to": "0^inf 100001000 1000^(y-48) 10010010 0010^(z-240) 0010010010100010001000100000100010001010 B> 1101^(1099511627776x+377451701380050) 11011 0^inf", "steps": null, "cases": "observed with y = 9.4e17, z = 1.05e19; not base-validated (needs 3-field segment checks)", "validated": { "instances": 0, "passed": 0 }, "source": "qsim" }, { "name": "489 visit (mirror)", "from": "0^inf 100^2 0001^59 001 0010^364 0^5 0001^295 0^6 0001^(a+1) 1 0010^b 101 C> 0^inf", "to": "0^inf 100^2 0001^59 001 0010^364 0^5 0001^295 0^6 0001^a 1 0010^(b+2) 101 C> 0^inf", "steps": null, "cases": "a >= 0, b >= 1", "validated": { "instances": 20, "passed": 20 }, "source": "manual" } ], "template": { "file": "machines/1RB0LD_1RC0RA_1RD1RF_1LA0LE_1RE1LD_0RB---/", "module": null, "machine": "1RB0LD_1RC0RA_1RD1RF_1LA0LE_1RE1LD_0RB---", "diff": null, "applies": "partly", "note": "Our kill #3 (Skelet 1-like Class 2): one macro lemma per segment + progress_nonhalt, Individual62. Here the invariant is just 'exists m, c = M(16m)' with the 42-segment chain as one lemma. Sk1Machines.v (Sk1CounterMixed/Sk1Auto) does not fit the cycle; Sk1Auto's checked run is an option for the 4.1e18-step base segment." }, "strategy": "175 only: proof.md for the cycle lemma M(X) -->+ M(2^36 X + 9575900996480) on X = 0 mod 16 (42 segment lemmas from about 12 sweep shapes), then the base segment c0 -->* M(508932080) (21 qsim records, about 4.1e18 steps) from the same sweep lemmas with concrete numbers. 489 is a separate decision.", "risks": [ "The base segment (4.1e18 steps, 21 records with their own skeletons) needs its own lemma instances; not yet base-validated segment by segment.", "Some cycle segments may need a residue mod 8 or 16 of their own field; the proof must carry the exact class (mod-4 rerun pending).", "489 may need a full digit-system theory (tower depth); it is not covered by the 175 proof." ], "checkpoint_questions": [ { "text": "Fig 2: from row 21 on, does the strip repeat with the same shape every 42 rows (the big bar growing, the small bars fixed)?", "figs": [ 1 ], "kind": "check" }, { "text": "Fig 3: the constructed segment with X = 52 ends at 4X+176 = 384 after repeated doubling passes. Does the picture look like repeated crossings of one growing run?", "figs": [ 2 ], "kind": "check" }, { "text": "Prove line 175 alone now (cycle lemma + base segment), and leave 489 for a separate decision (decision.json)?", "figs": [], "kind": "decision" } ], "discord_check": null, "stats": { "simulated_steps": 10000000000, "cells": 127078, "nonzeros": 37602, "growth": "qsim: 10^3424 steps, 10^1712 nonzeros in 120 s; cycle multiplies X by 2^36" }, "status": "CHECKPOINT_A", "status_reason": "175: exact macro cycle found and base-validated; go to proof.md. 489: different regime, filed as a decision." }