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REP-002: supplied-coordinate findings
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# REP-002: adequate coordinates help, but fitted operations still matter The supplied Fourier representation improved every paired seed on the declared 16-operation evaluation. Both arms used a **learned decoder**, trained only on the same 84 training states, and identical initial decoder/head tensors. The median full-state accuracy increased from **35.24% to 94.23%**. The experiment completed ten supervised RTX 5070 trials, each with 1,000 updates. This is a post-outcome diagnostic on the REP-001 world and split. It demonstrates the benefit of a supplied mathematical prior under these conditions. It does not establish a general method for learning that prior from language. ## What was compared The [frozen protocol](REPRESENTATION_FIXED_PROTOCOL.md) and [specification](SPEC.md) define R22 and R23. The learned arm is the unchanged width-10 REP-001 M3 model. The fixed arm computes ten Fourier features arithmetically from the observation, with 14 stored FP32 cosine/sine scalars. Both use the same decoder architecture and eight affine operation heads, paired initial tensors, data and update count. The fixed arm has 1,760 trainable parameters; the learned arm has 2,634. Supplying coordinates also removes encoder optimization, and its consistency term is constant with respect to trainable parameters. This is not an equal-parameter or equal-effective-objective comparison. No evaluation score selected a head, checkpoint, seed or stopping point. The world contains 98 states, eight operators, reversible transformations and a noninvertible projection. Only 580 training-state one-step edges train the models. Evaluations reuse REP-001's operation words and support strata. ## Native trained-head outcomes Exact full-state accuracy, across all 98 initial states; values below are percentages. Each seed has 3,136 length-16 evaluation cases. | Seed | Learned coordinates | Fourier coordinates | Paired improvement | |---|---:|---:|---:| | 0 | 48.95 | 94.23 | +45.28 pp | | 1 | 46.52 | 95.85 | +49.33 pp | | 2 | 20.41 | 92.06 | +71.65 pp | | 3 | 35.24 | 54.46 | +19.23 pp | | 4 | 33.64 | 97.64 | +64.00 pp | | Median | **35.24** | **94.23** | **+49.33 pp** | The median paired difference differs from subtracting the two marginal medians. The five improvements are descriptive evidence, not a population confidence interval. The poor fixed-coordinate seed remains part of every summary. | Operations composed | Learned median | Fourier median | |---|---:|---:| | 1 | 94.39% | 100% | | 2 | 87.74% | 100% | | 4 | 83.93% | 100% | | 8 | 53.22% | 100% | | 16 | 35.24% | 94.23% | All five Fourier decoders reconstructed all 98 states exactly, including held-out states. All five also answered every one-step evaluation correctly. Yet native long compositions still failed. Held-initial-state length-16 accuracy shows the same pattern: learned median 36.83%, Fourier median 93.75%.  ## The head refit separates an important remaining failure The registered secondary diagnostic fits affine heads by float64 least squares using only the same training edges, then executes them in FP32 through each arm's **same learned decoder**. This yields 100% exact answers for every declared composition length and all five Fourier seeds. Learned-coordinate refits reach only 37.05% median at length 16. The primary native results above are not replaced by these refitted results. A separate [NumPy refit audit](evidence/representation-fixed-refit-audit.json) independently reconstructed the five Fourier least-squares fits from saved coordinates and training edges, then checked all 82,320 one-step/composition answers through the saved learned decoders. All were correct. This CPU audit uses the same NumPy solver family, so it is not an independent numerical-library replication. It uses no new training data and does not change the failed native continuous replay. This supports a narrower conclusion: in the supplied-coordinate arm, the trained decoder and representation can support the declared operations, while the native jointly trained heads are an avoidable source of errors. It does not explain why optimization reached those heads or prove that the learned encoder lost information. Fourier seed 3 is an explicit counterexample to using perfect one-step answers as an adequacy certificate. Its native H-then-H identity answers are only 69.39% correct; H-S-H-S, another exact identity in this world, reaches 20.41%. The corresponding normalized latent relation errors are about .555 and .620. Operation relations remain necessary measurements even when decoding one step looks perfect. Post-outcome [inspection](evidence/representation-fixed-head-diagnostic.json) singles out H, the context-sign flip: its normalized latent error is .123–.137 in four Fourier seeds and .461 in seed 3, while every other action is below .009. Every Fourier seed answers all five declared length-16 words without H correctly (490 cases per seed); errors occur only in words containing H. This association is not yet a causal intervention. ## Distance and numerical diagnostics Native outputs are saved for reconstruction, one-step transitions, and every declared composition: 16,562 cases per trial. Distances use each arm's final encoded 98-state reference set, including held-out states, strictly for analysis. The live decoder is never replaced with nearest-state lookup or snapping. At length 16, four Fourier seeds have the correct state as their nearest reference in every case, yet their learned decoders still make 74 to 249 errors. Seed 3 has 76.56% nearest-reference accuracy and 54.46% decoder accuracy. These counts distinguish coordinate displacement from how the decoder handles those displacements. They do not by themselves establish a causal decomposition. The learned arm actually has the smaller median mean distance to *some* valid code (.215 versus .264, normalized by sqrt(10)), while answering far fewer questions correctly. Each arm has its own geometry, so cross-arm distances are not a universal semantic scale. Even within an arm, closeness to the wrong state is insufficient; distance alone cannot certify operational meaning. Quantization also matters. At 16 operations, the Fourier arm's median decoded accuracy is 4.37% with 2 bits per coordinate, 84.53% with 4 bits, 94.80% with 8 bits and 94.23% with 16 bits, quantizing after every operation as registered. These are FP32 computations over dequantized coordinates, not integer kernels. The modest nonmonotonic difference does not select a preferred precision. ## Verification, retained failure and resource evidence - **646 tests and 17 subtests passed**, with no skips; one expected Ripser diagnostic warning. The mathematical check also passed. - Both arms passed initial CPU/CUDA forward, loss and gradient comparisons. Maximum gradient difference was 4.47e-8; maximum gradient tolerance ratio .00422. The arithmetic Fourier map differed from independent binary64 features by at most 6.16e-8, below its declared 1e-6 tolerance. - The completed receipt verifies 97 payloads, exact design/configuration, shared initialization, final checkpoints, numeric NPZ arrays, recomputed diagnostics and resource limits. This verification does not retrain models. - Independent NumPy checkpoint replay matched **all 165,620 decoded answers** and all audited reconstruction, one-step and composition aggregate metrics. All saved distance/margin diagnostics also matched independent recalculation. - **Strict continuous replay failed**: eight of 4,315,920 compared scalars exceeded atol=1e-6 + rtol=1e-4 times the saved magnitude. Six trials passed the entire replay, four failed; all failures were in learned-coordinate seeds 0,2,3,4 at length 8 or 16. The largest absolute discrepancy was 3.4571e-6; the largest tolerance ratio was 1.26336. No categorical answer changed. The failed report is retained without widening tolerances or relabeling it as passed. - The official worker ran for 75.66 seconds; the parent attempt recorded 77.11 seconds before final verification. Peak CUDA tensor allocation was 66.76 MiB, reservation 68 MiB, and process-tree host memory 1.82 GiB. Small memory use reflects this small finite experiment, not absence of CUDA. The separate 100-update probe also completed and is retained. Evidence: [summary](evidence/representation-fixed-summary.json), [strict replay](evidence/representation-fixed-replay.json), [replay witnesses](evidence/representation-fixed-replay-witnesses.json), [test receipt](evidence/representation-fixed-tests.xml), [source identity](evidence/representation-fixed-source.json). Official run: `G:/RenigmaLLM_Artifacts/representation-fixed-runs/rep-a98ae7a10fd3`. Runtime SHA-256: `81c8691236213584088caa27f593656f6be36e18786ccfb992e3a8c3540334df`. Receipt SHA-256: `23470a50894fd6ec6bb3f1daefd2becf662a8722d2fa867c8bf43dd8dc13cfc1`. All 33 historical runtime files match REP-001's recorded hashes; only new runtime modules were added. Historical archives, receipts and failed replay remain intact. ## Next experimental decision The next discriminating diagnostic is a frozen-head intervention: compare native heads, replacement of H alone by its already train-only refit, replacement of all other heads alone, and replacement of all heads. Freeze encoders, decoders, checkpoints, words and all five seeds; perform no further optimizer updates. This would test whether the H association localizes the remaining Fourier errors. It is a proposed next registered experiment, not an outcome established here. Then compare ways of fitting operations that preserve their relations. Include inverse, periodic, context-dependent and idempotent relations; retain decoded consequences as the primary criterion. Do not assume every operator is invertible or impose an inappropriate group law on projection P. Investigate continuous replay sensitivity separately, with precision controls and tolerances frozen before new results. Then test the chosen mechanism on a newly frozen world/split before integrating learned language. This study does not justify adding a missing context variable: the supplied ten coordinates already suffice for the declared finite world. The human-facing purpose remains local control, inspectable interpretations and traceable consequences. Same-host computation and agent review are not independent-host replication, publication, or general language comprehension.