Deterministic chaos with an exact fossil record

Does the two-line B3/S23 rule really manufacture EXACT, reproducible constants out of deterministic transient chaos — the R-pentomino's 1103-generation lifetime, its 116-cell fossil record, and gliders at exactly c/4 — and do those constants belong to the rule rather than to luck, position, or the details of the seed?

Units: generations to stabilization of the R-pentomino (secondary knowns: population 116 = 86 ash + 6 gliders, glider c/4, LWSS c/2, diehard 130, acorn 5206/633/13)

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The finding

Deterministic chaos with an exact fossil record: Conway's B3/S23 rule burns the 5-cell R-pentomino for EXACTLY 1103 generations and stabilizes at EXACTLY 116 cells — an 86-cell ash (4 blinkers + 8 blocks + 1 boat + 4 beehives + 1 ship + 1 loaf, recovered as the exact size histogram) plus 6 gliders receding at EXACTLY c/4 (each verified by a 100-cell diagonal displacement over 400 generations, zero fit); the lifetime is recovered by two independent detectors (in-ash-box period-2 suffix scan and last population change) that agree at 1103; seed chaos vs rule invariance: remove ONE seed cell → lifetime collapses 1103 → 9 (123×), diehard vanishes at exactly 130, acorn burns for exactly 5206 generations to 633 cells with 13 gliders — every one at exact c/4; LWSS at exact c/2; rotate/mirror/translate the seed → identical fate; rivals die: add one birth condition (B36/S23) → the same seed EXTINCT at generation 9, remove one survival condition (B3/S2) → extinct at 3 — the cascade and its census are manufactured by exactly B3/S23; the module's toroidal stepper is bit-identical to the oracle's infinite plane through the R-pentomino's first margin contact (gen 176/177) and the seed-7 on-screen gen/live counters are pinned and scrape-verified 5/5

Method

Generator = the rule and nothing else: a sparse Set-of-live-cells implementation of B3/S23 on an UNBOUNDED plane (the module's verbatim neighbour-count update minus the torus wrap). The lifetime is DISCOVERED, not asserted, by two independent detectors: (1) pattern-level — decompose a late snapshot (gen 4000) into 8-connected components, classify gliders by exact ±1-diagonal translation over 4 generations, take the remaining ash's bounding box, and find the smallest generation T from which the in-box state repeats with period 2 all the way to the 4400-generation horizon (suffix scan); (2) the last generation at which total population changes. Census from component sizes; glider speeds from EXACT translation matching (each glider must reappear displaced exactly (±100,±100) at +400 generations — an exact rational, no fit). No 1103, 116, 6, 130, 5206 or 1/4 appears in generation or detection; knowns live in scripts/oracles/conway.reference.json and enter only the scorer. Gates: (A) lifetime 1103 by BOTH detectors; (B) census 116 = 86 ash cells in 19 components with size histogram {3:4, 4:8, 5:1, 6:5, 7:1}, 4 period-2 + 15 still, + 6 gliders; (C) all 6 gliders at exact c/4; (D) canonical glider (100,100)/400 = c/4, LWSS (200,0)/400 = c/2, blinker period 2, block still; (E) seed family — T-tetromino settles at 9 (<20 gate, structural), diehard population 0 at exactly 130, acorn exactly 5206/633/13 gliders all at exact c/4; (F) rotated+translated and mirrored+translated seeds → identical 1103/116/6; (G) rivals B36/S23 and B3/S2 through the identical machinery → same seed extinct at 9 and 3; (H) module mirror — ConwayModule's verbatim toroidal stepper bit-identical to the sparse plane through first margin contact (176/177) and seed-7 soup population pins at gens 0/50/100/250/500; (I) scoring self-test — tampered known_value ⇒ headline FAIL, recovery unchanged.

The law it recovers

Conway's Game of Life, B3/S23 (Gardner 1970): the R-pentomino stabilizes at generation 1103 with final population 116 — ash of still lifes and blinkers plus 6 gliders at the exact glider speed c/4; LWSS at c/2 (Berlekamp–Conway–Guy 1982); methuselah lifetimes are chaotically seed-dependent (diehard 130, acorn 5206).

Measurements, controls & cross-checks

Headline

Lifetime pattern detector

1103

Lifetime pop detector

1103

Population

116

Ash cells

86

Ash components

19

Ash histogram

3

4

4

8

5

1

6

5

7

1

Blinkers

4

Still lifes

15

Gliders

6

Gliders exact c4

6

Seed family

Ttetromino lifetime

9

Collapse factor

123

Diehard extinct at

130

Acorn lifetime

5206

Acorn population

633

Acorn gliders

13

Acorn gliders exact c4

13

Note

one seed cell changes the lifetime by two orders of magnitude while every emitted glider, from every seed, moves at exactly c/4 — trajectories belong to the seed, constants to the rule

Speed laws

Glider

0.25

Lwss

0.5

Note

recovered as exact integer displacements (100,100)/400 and (200,0)/400 — zero-parameter recoveries

Symmetry

Rot90 translate

• 1103
• 116
• 6

Mirror translate

• 1103
• 116
• 6

Rival

Highlife B36 S23 extinct at

9

Highlife peak

8

No s3 B3 S2 extinct at

3

No s3 peak

5

Note

add one birth condition or remove one survival condition and the same 5 cells die within single-digit generations — no cascade, no ash, no gliders

Module mirror

Torus plane bitexact through

176

Margin contact at

177

Seed7 pins

0

2547

50

973

100

786

250

519

500

239

Scrape

5/5 live (gen, live) pairs scraped from ?world=conway&seed=7 match the mirror exactly (gens 137–209)

Gates

14/14 pass in 2.7 s (9 physics gates first-run 2026-07-18, +5 honest-module cert gates J–N first-run 2026-07-26, prototype-first both times); tamper (known_value → 1096) ⇒ headline FAIL, exit 1, recovered lifetime byte-unchanged at 1103

Module certificate

What

gates J–N EXECUTE the shipped src/modules/ConwayModule.ts (sha-pinned, type-stripped, headless against recorder stubs) rather than a hand mirror

Rng accounting

frozen-init: ONE mulberry32 soup (7920 draws at init) + ONE Math.random site exercised ONLY on the no-seed fallback (executed proof: 0 draws seeded, exactly 1 unseeded, fallback seed === (draw·0xffffffff)>>>0); the fixedUpdate-onward slice draws NOTHING — the dynamics are a pure deterministic function of the seed

Lockstep

6720 calls at fl(1/120) bit-exact vs an independent replica in grid + generation + population + accumulator + thinInstanceCount every call (full 126720-float instance buffer on every generation advance); generation census {0:6217, 1:503}; final gen 503 pop 227; seed frozen

Accumulator class

NEW (5th) class — incommensurate-subtractive: fl(1/9) is not a float multiple of fl(1/120), so the accumulator never returns to an exact fixed point; generations fire on a 13/14-call alternation with a walking residue (max acc 0.11111111111111109 pinned); the acc > 3·GEN_DT overflow reset is DEAD by execution

Display

HUD === replica template at all 1344 %5 writes — gen/live shown as verbatim integers of the executed state, density = (live/7920·100).toFixed(1), no free constants; LIVE by distinct count 504 = generations+1 (#125); bufferUpdated/boundsRefresh/frame counts pinned

Closure

the EXECUTED shipped _step() on the centered R-pentomino is bit-identical to the oracle's sparse infinite-plane generator through generation 176 (margin contact 177), matching gate H's hand mirror exactly; the executed module's seed-7 populations at generations 0/50/100/250/500 === the reference pins — ZERO instrument gap: every shown number is an exact integer of the executed state

Tampers

surgical by design: sha ⇒ only J; rule flip B3→B4 in the executable ⇒ L (first grid diff at call 14, the first generation call) + M + N with J/K unchanged; hand-tampered known_value ⇒ only scoring gates (A/F/I), exit 1, recovery byte-unchanged

What it reduces to

The canonical Game-of-Life constants: the R-pentomino's stabilization at generation 1103 with final population 116 and 6 escaped gliders (Gardner, Sci. Am. 223(4), 120 (1970); LifeWiki 'R-pentomino'), the glider's exact speed c/4 and the LWSS's c/2 (Berlekamp–Conway–Guy, Winning Ways vol. 2, 1982), diehard's complete disappearance at 130 and the acorn's 5206-generation burn to 633 cells with 13 gliders (LifeWiki). Non-circular because the generator is the bare two-line rule on a sparse unbounded plane and every number is discovered: the lifetime by a from-scratch period-2 suffix scan over an ash box that itself comes from component decomposition of a late snapshot (cross-checked against an independent last-population-change detector — the two conventions agree exactly at 1103), the census by component size histogram, and the speeds by exact translation matching over a 400-generation baseline — 1103, 116, 6, 130, 5206, 633, 13, 1/4 and 1/2 never enter generation or detection, only the scorer (proven by the tamper self-test: wrong known_value flips the headline to FAIL with the recovered 1103 byte-unchanged). The recovery is sharper than 'Life is complex' in three ways: every scored quantity is an EXACT integer (lifetimes, censuses, glider counts, displacements — equality tolerance throughout, the same standard as ECA's dyadic fractal and ANT's period 104); the seed-family sweep separates what is chaotic (lifetimes: 9 / 130 / 1103 / 5206 from single-cell seed changes) from what is invariant (every glider from every seed at exactly c/4, every ash period ≤ 2); and the rule-genericity rival is falsified from both directions — adding one birth condition (B36/S23) or removing one survival condition (B3/S2) sends the identical seed extinct in single-digit generations, so the 1103-generation cascade and its 116-cell fossil record are manufactured by exactly B3/S23. Limits: the T-tetromino gate is structural (<20 generations, recovered 9) because its 'traffic light at ~generation 9' has no sharp canonical citation in the reference set used; Life's Turing-completeness (and hence undecidability of the general fate) is cited, not re-proven; the module's toroidal wrap makes its long-run soup fate deliberately differ from the infinite plane — disclosed as a module systematic, with the bit-exact pre-wrap window proving the local physics is identical.

Module systematics

HONEST-MODULE CERTIFIED (2026-07-26, gates J–N execute the shipped file — see result.module_certificate). The on-screen ConwayModule shows live 'gen', 'live' and 'density' counters for a random toroidal soup (110×72, mulberry32 seed, fill 0.32) — it makes no numeric physics claim beyond those counters, so honesty here means the counters must be exactly right, and they are proven so at two levels: gate H replicates the stepper verbatim as a hand mirror (bit-identical to the infinite-plane generator through generation 176, margin contact 177; seed-7 population pins 2547/973/786/519/239 at gens 0/50/100/250/500; live scrape of ?world=conway&seed=7 matched 5/5 on-screen (gen, live) pairs, gens 137–209), and gates J–N now EXECUTE the sha-pinned shipped source itself — init soup and full instance buffer bit-equal to an independent replica, a 6720-call fl(1/120) lockstep bit-exact in grid/generation/population/accumulator/thinInstanceCount every call, the HUD template byte-equal at all 1344 writes, and the executed _step() bit-identical to the oracle plane through the same 176/177 window. Disclosed systematic (unchanged): the module's torus WRAPS — gliders re-enter and collide, so the module's long-run soup fate deliberately differs from infinite-plane Life; the R-pentomino headline numbers (1103/116/6) live on the unbounded plane and are not shown on screen. The catalog entry still reads kind: 'investigating' with no numeric claim — that is conservative, not false (updating it would touch the shared worldCatalog.ts, a smoke:all-gated creator-style edit deliberately not bundled into this cert run). No module code was changed this run.

What the screen shows

toroidal soup with live counters: 'game of life · B3/S23 · gen N · live P · density d%' — counters certified by executing the shipped module (gates J–N): verbatim integers of the executed state, template byte-equal at every write, zero instrument gap

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- conway (scripts/conway-derisk.mjs)

Oracle

scripts/oracles/conway.reference.json

Sources

M. Gardner, 'Mathematical Games — The fantastic combinations of John Conway's new solitaire game "life"', Scientific American 223(4), 120–123 (October 1970); E. R. Berlekamp, J. H. Conway, R. K. Guy, 'Winning Ways for Your Mathematical Plays', vol. 2, Academic Press (1982) — glider c/4, LWSS c/2, Turing-completeness; LifeWiki: 'R-pentomino' (1103, 116, 6 gliders), 'Die hard' (130), 'Acorn' (5206, 633, 13 gliders).