Where Does Jagged Competence Come From?

Ioannis Tsiokos

cs.LGOct 2, 2026

Where Does Jagged Competence Come From?

Authors: Ioannis Tsiokos

Organizations: Automorph Inc., Wilmington, DE, USA

Abstract

Capable systems often show jagged competence: low average error alongside failures on particular inputs. We ask where it comes from in a task built from two known layers. A lower layer A computes five per-slot sums from records; an upper layer B uses the slot-1 sum and a mode carried over from earlier boards to predict the next board's category, so B cannot be computed from the current A alone: B is a strict extension of A. We train small recurrent networks on B and measure, against exact ground truth, what they acquire of A. The central finding is that learning B gives the network a jagged version of A, measured through probe readability and supervised outputs. It is readable where B needs it (slot 1, 97-98% by a linear probe) and becomes less readable where B does not (the other slots, 5-10%); category misreads concentrate near the slot-1 cutoffs; and when A is trained explicitly it comes out only approximately right. The networks show jagged competence in B: natural KL below $3\times10^{-4}$ bits coexists with a maximum law TV of about 0.25 on constructed histories. Matched experiments show that even a good A is not enough: connecting a learned A to B cuts misreads 1.3 to 17-fold, the same exact A gives fewer misreads as one-hot inputs (0-2) than as numerical inputs (10-317), one B update makes an exact A inexact unless the B gradient is blocked, and exact A still leaves some B failures. In this task, uneven acquisition, access and preservation of the lower theory explain part of the jagged competence of a network that learns the theory built on it; some failures remain unexplained. Whether the same holds in larger systems is a test to run.

Figures & tables

Figure 1: Two task layers and their history witness. (a) A maps four records to five per-slot sums. B uses their slot-1 category and the preceding mode to update the mode and determine the next-category law. The annotations identify training conditions, not evidence that a network learned the depicted decomposition. (b) The histories end on the same neutral sum board but retain different modes and require different laws. Values are integer twelfths; law entries are ordered L/N/H. The task has 24,435 completed sum boards and 1,555 local histories. Everything shown is exact task ground truth, not model output.

Figure 2: Recoverability, computation and access are distinct measurements, not interchangeable routes. (a) The sums output head and its loss attach to the shared carrier; a dashed optional connection feeds sums output to prediction in the connected variants. (b) The matrix identifies the routes and losses in the three versions and r13 variants; probability targets retain their base version’s routes and replace sampled targets with the exact law. The probe fits a reader on frozen states; the swap replaces an aligned donor block, with selectivity checked separately. No measurement, seed average or control result is implied by the icons; procedures and their calibration are in Appendix C.

condition	recoverability	computation	access	preservation	reliable prediction
r8: records $\to$ prediction	$\checkmark$ 1/5 $\cdot$ 1/5 $\cdot$ 1/5	–	–	–	$\times$ 0/3
r10: records $\to$ prediction	$\checkmark$ 5/5 $\cdot$ 5/5 $\cdot$ 5/5	–	–	–	$\times$ 0/3
r10: records $\to$ prediction+sums	$\checkmark$ 5/5 $\cdot$ 5/5 $\cdot$ 5/5	–	–	–	$\times$ 0/3
r11: records $\to$ prediction (Erosion continuation)	$\checkmark$ 1/5 $\cdot$ 1/5 $\cdot$ 1/5	$\times$ 0/3	–	$\times$	$\times$ 0/3
r11: records $\to$ prediction+sums (Erosion continuation)	$\checkmark$ 3/5 $\cdot$ 1/5 $\cdot$ 1/5	$\checkmark$ 1/3	–	$\times$	$\checkmark$ 1/3
r12: records $\to$ prediction+sums	$\checkmark$ 5/5 $\cdot$ 5/5 $\cdot$ 5/5	$\times$ 0/3	–	–	$\times$ 0/3

Table 1: Five achievements, original law.

\checkmark

supported;

\times

not supported; – not established: available evidence does not establish this achievement. Fractions are seeds passing out of three, except recoverability: supported slots out of five in seed order 0/1/2. Access:

D

by design (exact sums supplied);

I

demonstrated by a matched connection intervention. Complete scope, counts and controls: Supplement Table S1.

Figure 3: Recoverable sums and low average error coexist with specific failures. (a,b) r8 linear/MLP; (c) r10 linear: slots 1–5, 512 held-out boards/slot, majority ticks, shuffled floors and oracle ceiling. Open/filled: update 0/20k (a–c), r10/r8 (d,e); shapes: seeds 0/1/2. (d) Natural KL: 60,000 positions, board-only/mode-blind floors; the log axis omits the oracle’s 0 bits. (e) Law TV: 576 r8 histories/180 r10 rows, zero oracle, r8 bar 0.02 and r10 original-law 0.05 bar (its equal-law bar is 0.02). (f) r12 uniform categorical misreads: 24,435 boards, rendering 0, untrained floor, shaded cutoff bands. Compare seeds within panels; r8/r10 comparisons are historical and tasks differ. Paired probe intervals: Appendix E.

Figure 4: Target noise, access and encoding affect measured errors without guaranteeing prediction closure. Matched r13 comparisons: targets within versions (a,b), numbered access variants (c–e), encodings of exact sums (f). Shapes: seeds 0/1/2; grey crosses: matched update 0; dashed line: zero-error oracle except in (d), whose 0-bit oracle lies outside the logarithmic axis. Categorical misreads: 24,435 boards, rendering 0; natural KL: 60,000 positions; maximum long-panel TV: 384 N3–N8 endpoints, bar 0.02. No pooled means or bands. Controls are matched within panels.

Figure 5: Prediction gradients erode exact sums at update 1, not uniquely under AdamW. Columns: seeds 0–2. (a–c) Compare r11 losses: exact starts on 24,435 boards; updates 1/5/10/50/100 use a fixed 128-board subset. Symlog separates 0/1. (d–f) r13, all 1,555 local histories: every tested active optimizer loses exactness at update 1. Update-1 correct counts, seeds 0/1/2, each out of 1,555: AdamW: 393 / 5 / 518; reduced-lr AdamW: 1,554 / 1,468 / 1,553; SGD: 1,440 / 89 / 1,495; blocked prediction gradient: 1,555 / 1,555 / 1,555. Blocked prediction gradient is a separate weight-decay control; dashed line: exact start/oracle. Movement-matched points: fig. D.2 . r11/r13 is historical; no pooling or bands.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

study	recorded compute	version/review records
r8	CPU-era; processor model not recorded in saved per-run summaries	11
r9	CPU-era; processor model not recorded in saved per-run summaries	1
r10	GPU 3 / CUDA fp32 execution recorded in summaries and host verification; GPU model and driver not recorded in the public study artifacts	20
r11	GPU 3 / CUDA fp32 execution recorded in summaries and host verification; GPU model and driver not recorded in the public study artifacts	13
r12	GPU 3 / CUDA fp32 execution recorded in summaries and host verification; GPU model and driver not recorded in the public study artifacts	14
r13	GPU 3 / CUDA fp32 execution recorded in summaries and host verification; GPU model and driver not recorded in the public study artifacts	44

Appendix

Table A.1: Analytical appendix A. Exact recorded counts or decisions; no pooled reader accuracies. Complete support, settings, controls and per-seed records are in the numbered supplementary tables and released data.

study	registered passes	original-law runs	law TV bar / support
r8	5	9	0.02 (576 history endpoints)
r10	0	9	0.05 (180 law rows)
r11	1	24	0.02 (576 history endpoints)
r12	2	15	0.02 (576 history endpoints)
r13	1	27	0.02 (576 history endpoints)

Appendix

Table B.1: Analytical appendix B. Exact recorded counts or decisions; no pooled reader accuracies. Complete support, settings, controls and per-seed records are in the numbered supplementary tables and released data.

condition	natural KL (bits)	law TV (maximum)	B passes
r8: records $\to$ prediction	0.00029829 7.047e-05 0.00021024	0.25 0.2548 0.25649	0/3
r8: records+sums $\to$ prediction (sums route only)	0.00033865 6.7874e-05 6.5306e-05	0.017721 0.011403 0.020775	2/3
r8: records+sums $\to$ prediction	0.00025483 6.8052e-05 8.0944e-05	0.018665 0.0051108 0.0059933	3/3
r10: records $\to$ prediction	0.016034 0.019138 0.016491	0.3219 0.28963 0.30706	0/3
r10: records $\to$ prediction+sums	0.032325 0.033959 0.035928	0.27737 0.25519 0.23295	0/3
r10: records+sums $\to$ prediction (sums route only)	0.0070603 0.0059933 0.0057117	0.12242 0.079414 0.10517	0/3

Appendix

Table B.2: Access does not guarantee registered reliability. Each vertically ordered triple is seeds 0/1/2, never a pooled mean. Compare within study; cross-study comparisons are historical. r8 full history maximum, r12/r13 long N3–N8 maximum: bar 0.02. r10 is a different task: 180 law rows, original bar 0.05 (equal bar 0.02). Natural panels: 5,000 episodes x 12 rounds. Complete per-seed failed bars, census support and labels: Supplement Table S2.

Figure C.1: Separate probe (a) and swap-test (b) procedures. Readers and alignment fit on training identities only; scoring identities are held out. Positive, no-op and shuffled controls calibrate the procedure, while donor-follow and selectivity concern the evaluated carrier. This schematic has no empirical seed average or accuracy; full controls and support are in Appendix C.

study	law	positive / null	deciding quantity	pass	support	measured value
r8	original	current_board_null	B deciding criteria	no	576	0.1875
r8	original	forced_A	B deciding criteria	yes	576	0
r8	original	forced_raw	B deciding criteria	yes	576	0
r10	original	exact_oracle	B deciding criteria	yes	180 law rows	4.4703e-08
r10	original	mode_blind	B deciding criteria	no	180 law rows	0.48635
r10	original	untrained	B deciding criteria	no	180 law rows	0.42896

Appendix

Table C.1: Analytical appendix C. Exact recorded counts or decisions; no pooled reader accuracies. Complete support, settings, controls and per-seed records are in the numbered supplementary tables and released data.

update-1 condition	correct counts (seeds 0/1/2)	histories per seed
AdamW	393 / 5 / 518	1,555
reduced-lr AdamW	1554 / 1468 / 1553	1,555
SGD	1440 / 89 / 1495	1,555
blocked prediction gradient	1555 / 1555 / 1555	1,555

Appendix

Table D.1: Analytical appendix D. Exact recorded counts or decisions; no pooled reader accuracies. Complete support, settings, controls and per-seed records are in the numbered supplementary tables and released data.

Figure D.1: Recorded r12 learning curves show changes in average error, not sufficient evidence of reliable competence. Every seed and condition is retained; (a–c) original and (d–f) equal laws are separate rows, in seed order 0–2. Open points mark each condition’s own update-0 control; dashed zero is the exact oracle. The quantity is natural KL in bits on 5,000 natural episodes of 12 rounds (60,000 scored positions), not a loss or a census frequency. No smoothing, selected checkpoint or across-seed band is used. Compare conditions within the same seed and law only; cross-study comparisons are historical. Natural KL alone does not establish reliability.

Figure D.2: Exact sums erode across active optimizers at common recorded movement. (a–c) Seeds 0–2, all 1,555 local histories. Compare AdamW, reduced-rate AdamW and SGD within each seed’s three-active-arm common range; points are observed, not interpolated. Blocked prediction gradient is excluded from that range and remains a separate control in fig. 5 . Dashed line: exact start/oracle. No pooled means or confidence bands.

Figure D.3: Recorded r13 noise learning curves show changes in average error, not sufficient evidence of reliable competence. Every seed and condition is retained; (a–c) original; no equal-law arms were run, in seed order 0–2. Open points mark each condition’s own update-0 control; dashed zero is the exact oracle. The quantity is natural KL in bits on 5,000 natural episodes of 12 rounds (60,000 scored positions), not a loss or a census frequency. No smoothing, selected checkpoint or across-seed band is used. Compare conditions within the same seed and law only; cross-study comparisons are historical. Natural KL alone does not establish reliability.

raw linear slot-1 sums reader	seed	correct	boards	gain CI95	converged
r11 records $\to$ prediction+sums (sums dose 100%)	0	484	512	0.5332/0.62695	yes
r11 records $\to$ prediction+sums (sums dose 100%)	1	491	512	0.60938/0.69336	yes
r11 records $\to$ prediction+sums (sums dose 100%)	2	500	512	0.68555/0.76172	yes
r12 records $\to$ prediction+sums	0	483	512	0.53315/0.62305	yes
r12 records $\to$ prediction+sums	1	509	512	0.64648/0.72852	yes
r12 records $\to$ prediction+sums	2	505	512	0.69722/0.77148	yes

Appendix

Table E.1: Analytical appendix E. Exact recorded counts or decisions; no pooled reader accuracies. Complete support, settings, controls and per-seed records are in the numbered supplementary tables and released data.

Figure E.1: Cutoff discrimination can succeed while prediction still fails. The supplementary panel was bound before states were read, with disjoint fit/evaluation identities and fixed renderings. Three seed columns show saved 0/5k/20k sampled- and probability-target prediction-loss checkpoints only, with no network updates: (a–c) accuracy and (d–f) cross-entropy, in seed order 0–2. Compare sampled- and probability-target readers for the same seed and pair; circles/solid lines are 12/13 (135 fitting pairs, 150 evaluation pairs; 300 endpoint labels), squares/dashed lines are 23/24 (12/13 pairs; 26 labels). Grey crosses are the saved shuffled-label floors; balanced constant floors are 0.5 accuracy and ln 2 loss, and the exact ceiling is 1 accuracy / 0 loss. No confidence bands are implied. Distances and rendering variation remain descriptive in Appendix E’s support tables; successful discrimination is not proof of predictive use or of a precision ceiling.

study	condition (all seeds)	seconds (0/1/2)	execution scope	aggregation
r13	AdamW	21.962/19.645/20.531	200-update erosion; recorded execution	examples, not totals
r13	SGD	22.219/18.44/18.835	200-update erosion; recorded execution	examples, not totals
r13	blocked prediction gradient	19.329/17.844/17.231	200-update erosion; recorded execution	examples, not totals
r13	reduced-lr AdamW	19.241/19.387/20.737	200-update erosion; recorded execution	examples, not totals
r8	records $\to$ prediction	1855.1/1783.3/1895.1	CPU endpoint	examples, not totals
r8	records+sums $\to$ prediction	1765.2/1810.8/1837.4	CPU endpoint	examples, not totals

Appendix

Table F.1: Analytical appendix F. Exact recorded counts or decisions; no pooled reader accuracies. Complete support, settings, controls and per-seed records are in the numbered supplementary tables and released data.

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