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REP-001: representation comparison
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# REP-001: what the commuting-diagram experiment established The proposed criterion is now executable, with exact controls and 45 completed GPU trials. Adding an operation-consistency objective helped the five-coordinate models across all five paired seeds on the registered 16-operation test. It helped four of five ten-coordinate models and only one of five two-coordinate models. Long compositions remain unreliable. This is conditional evidence for the training objective, not a proof that learned compression preserves meaning. ## The mathematical core Use two requirements, each relative to a declared purpose: \[ F_a E \approx E T_a, \qquad D E \approx o. \] The first preserves an operation; the second preserves its required observation. A constant encoder can satisfy the first exactly while failing the second. Operators that change type need separate representation maps on their inputs and outputs. Coordinate covariance requires transforming the relevant metric and tensor structures consistently; it does not establish an arbitrary compression's adequacy. Topology, geometry and dynamics are separate contracts. In a finite deterministic world, a precise task-relative equivalence is \[ x\sim y \iff \forall w,\quad o(T_w x)=o(T_w y), \] where w ranges over permitted finite action words, including the empty word. Iterative partition refinement computes these distinctions in this experiment. This makes missing context an explicit pair of indistinguishable inputs with different required consequences. A neural fitting failure alone is weaker evidence: it can reflect optimization, coverage or restrictions on the operators. The exact world has 98 states `(x,y,h)`, coordinates modulo seven, and eight operations including rotations, translations, a context flip and a noninvertible projection. With observations `(x,y)`, dropping h preserves seven operations. The conditional translation `S(x,y,h)=(x+h,y,h)` breaks the quotient: - Best squared Frobenius transition residual: **49**. - Mean squared row residual: **1/2**; maximum row total variation: **1/2**. - Restoring h makes every operation exact. Refinement splits 49 blocks into 98. - A balanced context-free deterministic predictor cannot exceed **50%** on S. The result is independent of optimizer performance. Both changing a valid coarse transition and genuinely breaking its partition are tested, with Q refitted after the change. The user's previously reported Markov residuals were not independently replicated because their matrices were not supplied. Exact rational controls also verified the induced-metric coordinate calculation, the bulk Euler-Lagrange cancellation for a total derivative with fixed endpoints, and derivative/context and encoder-degeneracy counterexamples. These are mathematical controls, not a GR simulation or borrowed evidence for an ML theory. ## Learned comparison All models receive 16 categorical input coordinates, including h. Thus the learned experiment does not claim to discover a hidden context bit. The exact context-omission test and the learned representation comparison are distinct. There are 84 training states, 14 held-out coordinate combinations and 580 training edges with both endpoints inside training support. Training contains only individual operations. At each width, five seeds share identical initial tensors across the three objectives, architecture, optimizer and 1,000 updates. M1 trains reconstruction features; M2 adds decoded prediction; M3 also adds 0.1 times latent consistency. All three train their operation heads. M2 includes the shared reconstruction anchor, rather than being a pure prediction model. The following are medians across five seeds, using the actual trained heads. The 16-operation evaluation contains 32 fixed words over all 98 starting states (3,136 predictions per seed). It mixes the three separately recorded support strata; the plot below shows within-support and held-out-start results separately. | Latent coordinates | M1 reconstruction | M2 + prediction | M3 + consistency | M3 beats M2, paired seeds | |---|---:|---:|---:|---:| | 2 | 2.77% | 4.85% | 3.19% | 1/5 | | 5 | 2.46% | 11.77% | 37.85% | 5/5 | | 10 | 10.46% | 25.70% | 35.24% | 4/5 | The median **paired** M3-minus-M2 differences are -1.66, +13.20 and +7.94 percentage points at widths 2, 5 and 10. These are not the differences between the two marginal medians. At width five the paired differences range from +1.37 to +37.28 points; at width ten, from -11.54 to +23.28. Five initializations on one fixed synthetic world provide descriptive evidence, not a significance claim or a basis for calling five coordinates universally optimal.  At width five, median held-out-start accuracy for M3 declines from 92.86% after one operation to 37.50% after sixteen. At width ten it declines from 94.64% to 36.83%. Preserving approximate one-step effects has not produced reliable long-horizon execution. The two-coordinate result is a particularly useful warning: M3 reduces median one-step normalized latent error from 0.403 to 0.043 compared with M2, while median full-state reconstruction falls from 11.22% to 4.08% and 16-operation accuracy falls from 4.85% to 3.19%. Low latent error can accompany failure to recover useful distinctions even with fixed encoder norm. Latent errors refer to each model's own coordinates and do not by themselves rank retained information. All five two-coordinate M3 models retain 98 unique FP32 codes, so this is not proof that their float representations irretrievably erased x/y. However, their two-bit-per-coordinate codebooks contain exactly two codes, corresponding only to h. Mean effective rank is 1.024. The context bit alone is a valid closed two-state quotient: H flips it and every other action preserves it. The learner has approached a simpler operational system while failing the full observation contract. This is a concrete instance of the distinction the exact controls were designed to catch. At width ten, S remains a specific bottleneck. Mean exact S accuracy is 62.86% for M2 and 62.04% for M3, with y and h correct throughout; the errors concern x. M3 lowers S latent error in every seed but improves decoded S accuracy in only two. Erasing h reduces mean S x accuracy to approximately 49%, respecting the 50% information ceiling. These per-action results are not visible in an aggregate commutator alone. The analytic ten-coordinate Fourier reference gives 100% on all registered sequence tests, with maximum latent-coordinate error 4.02e-14. Its operators are fitted only on training edges, but its decoder compares against **all 98 known exact codewords**. This is a supplied mathematical capacity control with an oracle decoder, not learned held-out generalization. Supplemental refitted heads for learned encoders are reported separately and never substituted for the primary trained-head score. Full per-action, per-field, relation, code-rank, pair-separation, quantization and clipping measurements remain in each trial's `result.json`. A compact aggregate is in [representation-summary.json](evidence/representation-summary.json). ## Cost and execution The exact world needs seven bits for a unique state identifier; a simple implementation uses one byte plus a transition algorithm or table. FP32 latent payloads use 64, 160 and 320 bits at widths 2, 5 and 10. Model parameter payloads are 5,128, 6,676 and 10,536 bytes respectively, before optimizer state, activations, library workspaces and JSON serialization. Fewer coordinates are not automatically fewer bits than a purpose-built discrete representation. Two-, four-, eight- and sixteen-bit coordinate experiments serialize actual packed state codebooks and quantize at initialization and after every operation. Execution dequantizes to FP32 and retains FP32 heads/decoder; this does not measure low-bit compute acceleration. Collision counts and clipping are retained. At width two and two bits per coordinate, at most 16 codes are possible, giving an exact uniform reconstruction ceiling of 16/98 before any decoder limitations. All 45 trials completed on the local RTX 5070 in a 340.62-second worker run; the recorded parent invocation took 342.17 seconds before final verification. Peak recorded GPU allocation was 66.77 MiB, peak reservation 68 MiB, and peak parent/child host memory 1.82 GiB. The workload is deliberately small; these figures do not establish a GPU speed advantage. A live NVIDIA sample showed 8% utilization on the selected RTX 5070 during the run. CPU/CUDA forward/loss/gradient controls passed at rtol=1e-4, atol=1e-6; the maximum recorded gradient difference was 7.45e-8 and the largest elementwise error/tolerance ratio was 0.121. The full project suite passed **596 tests plus 17 subtests**, with one existing Ripser calibration warning and no skipped tests. A separate NumPy FP32 implementation replayed all 45 checkpoints against an independent transition oracle: 463,050 state/word evaluations and 14,400 metric comparisons. **44 models passed strict decoded-accuracy agreement; one failed.** Width two, seed two, M2 differed by a net one correct x/full-state prediction among 6,272 length-two cases, also affecting the corresponding support stratum. All continuous-error comparisons passed, with maximum numeric absolute difference 2.38e-7. The failed audit is preserved in [representation-replay.json](evidence/representation-replay.json). This must not be described as perfect cross-backend categorical reproducibility. It does not change any 16-operation score in the table above. A separately supervised, inference-only diagnostic reproduced the original CUDA aggregate exactly and located the discrepancy: applying Y then P to `(3,6,+1)` should yield `(3,0,+1)`. CUDA and PyTorch CPU choose x=3; NumPy chooses x=5. The leading CUDA x logits differ by **4.47e-8**; NumPy reverses those two candidates with a **1.49e-8** margin. The largest cross-backend x-logit difference there is 7.45e-8. This establishes FP32 near-tie sensitivity in this case. It does not make the failed strict replay pass, and no tolerance or weight was changed. The [diagnostic record](evidence/representation-replay-diagnostic.json) links the frozen checkpoint and supervised measurement. The first development probe completed its GPU work but failed receipt validation because the verifier used an obsolete metric field name. It remains labelled failed. The corrected probe passed; the official study used the subsequently frozen source and protocol. No model/data/loss choice was changed using probe outcomes. Initial weights, the latest periodic recovery checkpoint/final weights, and loss curves are retained; older periodic weight snapshots were overwritten by design and are not an archived training trajectory. ## Evidence and reproduction Official folder: `G:/RenigmaLLM_Artifacts/representation-runs/rep-6aa4d565b0f9`. It contains 224 hashed payload files plus the receipt, including the source snapshot, fixed inputs, all 45 initializations/checkpoints/results, numerical controls, environment, logs and supervision. Each checkpoint is readable JSON. - Runtime source SHA-256: `79bb68dfc63ff997d38cf6a1f81668f83583c6de94f10bcf59966e23aa8aa1cd`. - Official receipt SHA-256: `93b45c937adca237f02e9f875a0e1663fd10d6f04aa70c22b45fe65a9fb80465`. - Canonical design SHA-256: `159b68733c8aaf14242fb9349c192cd53038842c63d5c775150639c9e2a70613`. - Design file SHA-256 (used by the independent replay): `25a3e7b8505d21d53b45d24c8a7f938f878e68d884fb1cbe6db9b900c8c902df`. - Protocol file SHA-256: `f5c707efa2d26db24ea30f97f165bda024647decf9e82afeb83a181704825b73`. Read-only verification: ```powershell & 'G:/RenigmaLLM_Artifacts/environment-gpu/Scripts/python.exe' scripts/run_representation.py --verify 'G:/RenigmaLLM_Artifacts/representation-runs/rep-6aa4d565b0f9' ``` The [frozen protocol](REPRESENTATION_PROTOCOL.md) contains full methods and primary references. Related losses have prior art, including [Lusch, Kutz and Brunton](https://www.nature.com/articles/s41467-018-07210-0). The purpose here is a local, falsifiable implementation with inspectable consequences. Same-host checks and agent reviews are not independent-host or Claude Code replication; neither remote publication nor general language understanding has been established. ## Projected next leg: REP-002, not yet executed The most useful next distinction is **finding adequate coordinates versus learning and executing operations in them**. The exact reference proves that closed coordinates exist for this world; the learned models do not reliably find and use them across long sequences. 1. Compare learned coordinates with the supplied Fourier coordinates using the same training-only **learned decoder and trained heads**, removing the analytic control's all-state oracle decoder. Register this as a diagnostic follow-up rather than fresh confirmation on an untouched benchmark. 2. Measure accumulated distance from valid encoded states and violations of inverse/idempotent operation laws. Test a separately registered relation or multi-step objective only after fixing its controls; do not infer that another context variable is needed from this optimization failure. Record decoded decision margins and precision sensitivity as well: small continuous discrepancies can change exact categorical answers near a tie. 3. Replicate the chosen mechanism on a newly frozen world/split before any learned-language integration. Keep direct decoded accuracy and explicit missing-context counterexamples as acceptance requirements. The long-term human-facing contract remains inspectable interpretation, traceable consequences, visible uncertainty and local control. This study advances a testable mechanism toward that purpose; it does not equate numerical geometry, a GPU architecture or an abstract invariant with human meaning.