negative imports an objective October 02, 2026

Given a NAND, paid replicators climb every logic task that reads its inputs once, and none that needs one twice

DESIGN §1.3 sweep 16 gave paid replicators a NAND. The 18 emerged worlds of the from-emerged sweep ran another 40 000 epochs each, three seeds a parent, with an assay-only NAND byte and Avida's logic tasks as the ladder, in three arms: every rung paid, XOR and EQU alone paid, and nothing paid. All 162 children finished; one paid child died out. Pay bought capability: all 53 measured pairs favour the paid ladder (p = 1.11e-16), and the paid worlds climbed every rung from NOT to NOR, the tasks a NAND circuit can compute reading each input once. None held XOR or EQU, the two that need an input twice, at a tenth of its world. The rung-4 question, whether a soup assembles those deep rungs from paid parts, is refuted, 0 to 0 with 53 ties, and so is the stepping-stone test against the arm paid for XOR and EQU alone. Paid complexity did not rise, 2 pairs against 1 (p = 0.5). The substrate imports an objective and a primitive, so none of this moves rung 4 on Soup, which stays not shown.

Read from Metabolism, a second substrate that pays a cell for computing: it imports an objective and is never pooled with the fitness-free runs. Rung 4 on Soup is unaffected by it. What that means.

Question

Paid to compute, BFF replicators took the arithmetic tasks one substitution of a copier computes and went no further. The design study's diagnosis was the instruction set: BFF has no two-input data op, so every deeper task needed a counted loop, and no partial loop earns anything on the way. Logic, the variant of the Metabolism substrate in DESIGN §1.4, keeps that economy and that pay and adds the missing part: one NAND byte, run only by the assay, with Avida's logic tasks as the ladder. Every rung is a composition of NANDs and a lower rung is a part of a higher one. Given the parts, and paid for each, does a soup assemble the deep rungs, XOR and EQU?

Method

The parents are the emerged worlds the from-emerged sweep continued, chosen by its rule, the same parents and seeds as the Metabolism sweep. Each terminal world runs another 40 000 epochs, seeds 2001–2003, in three arms. All three run the Metabolism economy (influx 1 024, a stock cap of 65 536, no theft) and the logic assay every 8 epochs. In the full arm every rung pays, at 2 048 energy a unit. In the deep-only arm only XOR and EQU pay, Lenski's control: it separates "the deep rungs are built on paid parts" from "the deep rungs are directly reachable". In the none arm nothing pays. ECHO, NOT, NAND, AND, ORN, OR, ANDN and NOR are the rungs a NAND circuit can compute reading each input once. XOR and EQU, the deep rungs, are not: every NAND circuit for either reads an input or an intermediate twice.

The reading is pre-registered, with every number fixed before a child was seeded: docs/design_record.md, 2026-10-01, "Logic: does a soup assemble features beyond its one-step rungs when the parts are paid?", DESIGN §1.3 item 16. Each full child is paired with the child of the same parent and seed in a control arm. All are read past a settling window of 1 000 epochs, and a pair with an extinct child is in no test. There are four one-sided sign tests at p < 0.05:

H-capability-L (full against none): the last-decile median logic_capability, the number of rungs held by a tenth of the world.
H-deep, the rung-4 question (full against none): the same count over XOR and EQU, logic_capability_deep.
H-stones (full against deep-only): H-deep's count, against the arm paid for the deep rungs alone.
H-complexity (full against none): the from-emerged rule, under which the dominant replicator's executed instruction count rises when its last-decile median is at least 1.2 times its first. It reads only pairs in which neither child relapsed.

Each test is also read with the extinct pairs kept and without the four parents the design study piloted, and a shown test lists any one or two parents whose pairs it cannot do without.

Result

H-capability-L shown H-deep refuted H-stones refuted H-complexity not shown Given a NAND, paid replicators climbed the tasks that read their inputs once and none that needs one twice.

H-capability-L, full against none: 53 pairs favour full, 0 none and 0 tie, of 53 measured, p = 1.11e-16, shown. Every pair of all 18 parents favours full. With the extinct pairs kept: 54 to 0 with 0 ties, p = 5.55e-17, shown. Without the piloted parents: 41 to 0 with 0 ties, p = 4.55e-13, shown.
H-deep, full against none: 0 pairs favour full, 0 none and 53 tie, of 53 measured, refuted. By parent, 0 lean to full, 18 tie throughout and 0 lean to none. With the extinct pairs kept: 0 to 0 with 54 ties, refuted. Without the piloted parents: 0 to 0 with 41 ties, refuted.
H-stones, full against deep-only: 0 pairs favour full, 0 deep-only and 53 tie, of 53 measured, refuted. By parent, 0 lean to full, 18 tie throughout and 0 lean to deep-only. With the extinct pairs kept: 0 to 0 with 54 ties, refuted. Without the piloted parents: 0 to 0 with 41 ties, refuted.
H-complexity, full against none: 2 pairs favour full, 1 none and 45 tie, of 48 measured, p = 0.5, not shown. By parent, 1 lean to full, 15 tie throughout and 1 lean to none. With the extinct pairs kept: 2 to 1 with 51 ties, p = 0.5, not shown. Without the piloted parents: 2 to 0 with 35 ties, p = 0.25, not shown.

No one or two parents carry a shown test: whichever one or two are left out, the rest still read p < 0.05.

The full arm lost 1 child to extinction, and 6 relapsed past the settling window; the deep-only arm lost 0 and 0, the none arm 0 and 0.

Every count on this page is read from the sweep's stored samples at render time, through the same reading the sweep page draws.

The ladder

Descriptively, counting a rung where it held a tenth of a child's world for 5 samples running: of 54 full children, the worlds reached ECHO in 54, NOT in 52, NAND in 37, AND in 14, ORN in 44, OR in 23, ANDN in 32, and NOR in 19; XOR in 0 and EQU in 0. Of 54 deep-only children, the worlds reached ECHO in 19 and no other rung. Of 54 none children, the worlds reached ECHO in 19 and no other rung. 0 full, 0 deep-only and 0 none children reached a deep rung, so 0 climbed one through OR, ANDN or NOR, the stepping stones of Lenski et al. 2003. Tasks gave a full cell about 59% of its income, between 18% and 78% across the children, an estimate that ignores the stock cap.

The pre-registered reading final imports an objective

Every child of every qualifying parent has finished. Each full-ladder child is paired with the unpaid child of the same parent and seed, and for H-stones with the child paid for XOR and EQU alone; all are read over their own samples past a settling window of 1,000 epochs. A child is extinct where its last-decile median share is below 10%, and a pair with an extinct child is in no test. Each test is a one-sided sign test at p < 0.05, re-read with the extinct pairs kept and without the parents the design study piloted (runs 1007, 944, 2577, and 2700). This substrate imports an objective and a primitive, the NAND byte, so rung 4 on Soup stays not shown whatever it reads.

Children per arm, read so far. A deep rung is XOR or EQU held by a tenth of the world for 5 samples running; a stepping stone is OR, ANDN or NOR held first.

Arm	Children	Finished	Settled relapses	Extinct	Capability measured	Complexity survivors	Rises
full	54	54	6	1	53	48	2
deep-only	54	54	0	0	54	54	1
none	54	54	0	0	54	54	1

Hypotheses

H-capability-L, full against none: shown

53 pairs measured on both sides: 53 favour full, 0 none, 0 tie; p = 1.11e-16. With the extinct pairs kept: shown, 54 to 0, p = 5.55e-17. Without the piloted parents: shown, 41 to 0, p = 4.55e-13.

Per-parent agreement

Parent	Favour full	Not both measured
run 944	3	0
run 950	3	0
run 967	3	0
run 991	3	0
run 1007	3	0
run 1029	3	0
run 1087	3	0
run 1089	3	0
run 1103	2	1
run 2568	3	0
run 2577	3	0
run 2590	3	0
run 2654	3	0
run 2676	3	0
run 2700	3	0
run 2771	3	0
run 2802	3	0
run 2862	3	0

H-deep, full against none: refuted

53 pairs measured on both sides: 0 favour full, 0 none, 53 tie. With the extinct pairs kept: refuted, 0 to 0. Without the piloted parents: refuted, 0 to 0.

Per-parent agreement

Parent	Tie	Not both measured
run 944	3	0
run 950	3	0
run 967	3	0
run 991	3	0
run 1007	3	0
run 1029	3	0
run 1087	3	0
run 1089	3	0
run 1103	2	1
run 2568	3	0
run 2577	3	0
run 2590	3	0
run 2654	3	0
run 2676	3	0
run 2700	3	0
run 2771	3	0
run 2802	3	0
run 2862	3	0

H-stones, full against deep-only: refuted

53 pairs measured on both sides: 0 favour full, 0 deep-only, 53 tie. With the extinct pairs kept: refuted, 0 to 0. Without the piloted parents: refuted, 0 to 0.

Per-parent agreement

Parent	Tie	Not both measured
run 944	3	0
run 950	3	0
run 967	3	0
run 991	3	0
run 1007	3	0
run 1029	3	0
run 1087	3	0
run 1089	3	0
run 1103	2	1
run 2568	3	0
run 2577	3	0
run 2590	3	0
run 2654	3	0
run 2676	3	0
run 2700	3	0
run 2771	3	0
run 2802	3	0
run 2862	3	0

H-complexity, full against none: not shown

48 pairs measured on both sides: 2 favour full, 1 none, 45 tie; p = 0.5. With the extinct pairs kept: not shown, 2 to 1, p = 0.5. Without the piloted parents: not shown, 2 to 0, p = 0.25.

Per-parent agreement

Parent	Favour full	Favour none	Tie	Not both measured
run 944	0	1	2	0
run 950	0	0	3	0
run 967	0	0	2	1
run 991	0	0	3	0
run 1007	0	0	3	0
run 1029	2	0	1	0
run 1087	0	0	3	0
run 1089	0	0	3	0
run 1103	0	0	2	1
run 2568	0	0	3	0
run 2577	0	0	3	0
run 2590	0	0	3	0
run 2654	0	0	3	0
run 2676	0	0	3	0
run 2700	0	0	2	1
run 2771	0	0	3	0
run 2802	0	0	0	3
run 2862	0	0	3	0

The pairs

Parent	Seed	Full child	Capability	Deep-only child	Capability	None child	Capability
run 944	2001	run 4681	2	run 4684	0	run 4687	0
run 944	2002	run 4682	4	run 4685	0	run 4688	0
run 944	2003	run 4683	2	run 4686	0	run 4689	0
run 950	2001	run 4690	5	run 4693	0	run 4696	0
run 950	2002	run 4691	5	run 4694	0	run 4697	0
run 950	2003	run 4692	3	run 4695	0	run 4698	0
run 967	2001	run 4699	4	run 4702	0	run 4705	0
run 967	2002	run 4700	3	run 4703	0	run 4706	0
run 967	2003	run 4701	2	run 4704	0	run 4707	0
run 991	2001	run 4708	3	run 4711	0	run 4714	0
run 991	2002	run 4709	1	run 4712	0	run 4715	0
run 991	2003	run 4710	2	run 4713	0	run 4716	0
run 1007	2001	run 4717	4	run 4720	0	run 4723	0
run 1007	2002	run 4718	3	run 4721	0	run 4724	0
run 1007	2003	run 4719	4	run 4722	0	run 4725	0
run 1029	2001	run 4726	4	run 4729	0	run 4732	0
run 1029	2002	run 4727	4	run 4730	0	run 4733	0
run 1029	2003	run 4728	4	run 4731	0	run 4734	0
run 1087	2001	run 4735	4	run 4738	0	run 4741	0
run 1087	2002	run 4736	3	run 4739	0	run 4742	0
run 1087	2003	run 4737	4	run 4740	0	run 4743	0
run 1089	2001	run 4744	5	run 4747	0	run 4750	0
run 1089	2002	run 4745	3	run 4748	0	run 4751	0
run 1089	2003	run 4746	3	run 4749	0	run 4752	0
run 1103	2001	run 4753	4	run 4756	0	run 4759	0
run 1103	2002	run 4754	4	run 4757	0	run 4760	0
run 1103	2003	run 4755 extinct	3	run 4758	0	run 4761	0
run 2568	2001	run 4762	4	run 4765	1	run 4768	1
run 2568	2002	run 4763	4	run 4766	0	run 4769	0
run 2568	2003	run 4764	4	run 4767	0	run 4770	0
run 2577	2001	run 4771	4	run 4774	0	run 4777	0
run 2577	2002	run 4772	2	run 4775	0	run 4778	0
run 2577	2003	run 4773	2	run 4776	0	run 4779	0
run 2590	2001	run 4780	4	run 4783	0	run 4786	0
run 2590	2002	run 4781	6	run 4784	0	run 4787	0
run 2590	2003	run 4782	4	run 4785	0	run 4788	0
run 2654	2001	run 4789	4	run 4792	0	run 4795	0
run 2654	2002	run 4790	3	run 4793	0	run 4796	0
run 2654	2003	run 4791	3	run 4794	0	run 4797	0
run 2676	2001	run 4798	2	run 4801	0	run 4804	0
run 2676	2002	run 4799	4	run 4802	0	run 4805	0
run 2676	2003	run 4800	4	run 4803	0	run 4806	0
run 2700	2001	run 4807	2	run 4810	0	run 4813	0
run 2700	2002	run 4808	3	run 4811	0	run 4814	0
run 2700	2003	run 4809	1	run 4812	0	run 4815	0
run 2771	2001	run 4816	4	run 4819	0	run 4822	0
run 2771	2002	run 4817	4	run 4820	0	run 4823	0
run 2771	2003	run 4818	4	run 4821	0	run 4824	0
run 2802	2001	run 4825	7	run 4828	0	run 4831	0
run 2802	2002	run 4826	4	run 4829	0	run 4832	0
run 2802	2003	run 4827	4	run 4830	0	run 4833	0
run 2862	2001	run 4834	4	run 4837	0	run 4840	0
run 2862	2002	run 4835	4	run 4838	1	run 4841	1
run 2862	2003	run 4836	3	run 4839	0	run 4842	0

The ladder, descriptive only: how many children's worlds held each rung at a share of 10% for 5 samples running, settling window included, and the median first epoch they did.

Rung	Full: children	Full: median first epoch	Deep-only: children	Deep-only: median first epoch	None: children	None: median first epoch
echo	54	20,690	19	22,630	19	22,630
not	52	23,660	0	—	0	—
nand	37	25,680	0	—	0	—
and	14	28,550	0	—	0	—
orn	44	26,420	0	—	0	—
or	23	27,890	0	—	0	—
andn	32	29,810	0	—	0	—
nor	19	33,810	0	—	0	—
xor	0	—	0	—	0	—
equ	0	—	0	—	0	—

What a refuted H-deep means

On the lab, read-only queries over every stored sample of the 162 children found logic_capability_deep at 0 in every one. The assay never credited XOR to more than 5 of the 256 cells it drew, in one sample of run 4817, nor EQU to more than 1, against the 26 a tenth of the world needs. With no deep rung in any arm the tests cannot favour the full ladder, and the pre-registration reads a tie throughout as refuted. H-deep needed only 5 discordant pairs, and 0 of 54 full children puts a two-sided 95% upper bound of about 0.066 on the rate at which a paid child reaches a deep rung in this budget. Child for child, the deep-only worlds read exactly as the unpaid ones in every column of the report: with no XOR or EQU ever credited, their pay was never collected, and H-stones ties as H-deep does.

So given a NAND and paid for every part, BFF copiers climb every rung that reads its inputs once and stop below the two that need an input twice. The pre-registration read this outcome as closing the "ladder of compositions" route, with mutational supply and a code region apart from the copied tape as the next suspects. A design study written before this reading, and summarised in the Meta-stack pre-registration in the design record (2026-10-02, "Meta-stack: does a soup assemble the deep rungs from paid parts on a metabolism tape read with a stack NAND?"), separated them. Its numbers are pilot numbers from a scratch copy of the engine, not findings. Supply did not bind: a specific substitution arrived somewhere in a world every 128 epochs, the read-once rungs came on schedule, and mutation rates of 4 and 8 times the default made no deep rung, while 16 times melted the copiers. Nor did the copier alone: a code region the soup never runs climbed to NOR and stopped there. What binds, on the study's reading, is fan-out in a two-head machine whose NAND writes in place. A deep rung needs a value twice, and a NAND that overwrites its operand leaves nothing to read again without a stored copy and re-aligned heads. Its follow-up tried a NAND that keeps its operands, and in the pilot that code region, at a raised mutation rate, then held XOR or EQU at a tenth of the world in 2 of 3 seeds.

H-complexity not being shown sets this sweep apart from Metabolism, where paid complexity rose: here pay did not make the dominant replicator execute more of its tape, 2 pairs against 1.

What comes next

Meta-stack, DESIGN §1.3 item 17, seeded right after this reading, asks this sweep's question on the machine the study points at: a second, 32-byte tape the soup never executes, read by the logic assay with the NAND that keeps its operands, against an arm paid for the deep rungs alone and an arm with this sweep's in-place NAND. It runs on the same parents and pairs its children with this sweep's.

What is not claimed

Nothing about rung 4 on Soup. This substrate imports an objective and a primitive: whatever rose here was paid for by tasks the programme chose, computed with a byte the programme added. It is never pooled with the fitness-free runs, and rung 4 on Soup stays "not shown".
Not "XOR is unreachable". It is one machine and one budget, and what the NAND byte writes was the programme's choice. The study's account of why it stops is pilot, and Meta-stack is what tests it.
A finite ladder. EQU is its top, so complexity here is at most five NANDs deep, and the inputs are the assay's case sets.
Children are correlated. Three children share each parent's world, four of the parents were piloted, and the none arm repeats the Metabolism sweep's unpaid worlds. That is why every test is read by parent, with the extinct pairs kept and without the piloted parents.

Evidence

Transition, flagged, census and arm are used here in the senses set out in the glossary.

Sweep: Logic

Transition epoch vs treatment

No finished runs yet.

Transitions and replicator counts

One row per finished run the detector flagged or that counted at least one replicator. "Transition epoch" is the detector's event; "Peak replicators" is the largest census the run reached, with the epoch it first reached it in brackets; "—" means the run never had a value.

No finished run of this sweep has transitioned or counted a replicator yet.

Runs

Every run of the sweep, flagged or not, with its seed — the denominator of the rate above.

Seed	Treatment	Status	Priority	Epochs done	Transition epoch	Census
2003	logic×8×xor×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	30 census only	Run
2001	logic×8×0×initiator×1024×65536	finished	40	60,000	— confirmed	28 census only	Run
2002	logic×8×0×initiator×1024×65536	finished	40	60,000	— confirmed	23 census only	Run
2003	logic×8×0×initiator×1024×65536	finished	40	60,000	— confirmed	30 census only	Run
2001	logic×8×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	9 census only	Run
2002	logic×8×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	8 census only	Run
2003	logic×8×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	12 census only	Run
2001	logic×8×xor×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	16 census only	Run
2002	logic×8×xor×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	6 census only	Run
2003	logic×8×xor×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	9 census only	Run
2001	logic×8×0×initiator×1024×65536	finished	40	60,000	— confirmed	16 census only	Run
2002	logic×8×0×initiator×1024×65536	finished	40	60,000	— confirmed	6 census only	Run
2003	logic×8×0×initiator×1024×65536	finished	40	60,000	— confirmed	9 census only	Run
2001	logic×8×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	—	Run
2002	logic×8×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	—	Run
2003	logic×8×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	17 census only	Run
2001	logic×8×xor×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	20 census only	Run
2002	logic×8×xor×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	11 census only	Run
2003	logic×8×xor×2048×initiator×0×1024×65536	finished	40	60,000	— confirmed	18 census only	Run
2001	logic×8×0×initiator×1024×65536	finished	40	60,000	— confirmed	20 census only	Run