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README.md

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+---
+pipeline_tag: text-generation
+library_name: pytorch
+language:
+  - en
+license: cc-by-4.0
+tags:
+  - research
+  - semsimula
+  - conservative-language-model
+  - scalar-potential
+  - lagrangian-mechanics
+  - energy-based-model
+  - physics-informed
+  - openwebtext
+  - mechanism-ablation
+  - ladder-study
+  - pre-registered
+  - live-gradients
+  - generation-3
+  - poisson-mode-registers
+  - exact-integrator
+  - matched-token-budget
+  - fock-space
+  - parflm
+  - anisotropic-gaussian-potential
+  - cfc
+  - baoab
+  - fully-conservative
+  - riemannian-geometry
+  - riemannian-geodesics
+  - jacobi-metric
+  - non-transformer
+  - attention-free
+  - constant-memory-inference
+datasets:
+  - Skylion007/openwebtext
+metrics:
+  - perplexity
+model-index:
+  - name: semsimula-ladder-live-owt-d384-l2-none-norc-pm64-pmx16-sr2
+    results:
+      - task:
+          type: text-generation
+        dataset:
+          type: Skylion007/openwebtext
+          name: OpenWebText
+          split: validation
+        metrics:
+          - type: perplexity
+            value: 54.63
+            name: Validation perplexity at block 512, settled (mean of the last three 500-step evals at block 512)
+            verified: false
+---
+
+# Poisson-mode registers with the wells integrated exactly (PMX): the best conservative model on the ladder (Gen 3)
+
+> **Generation 3: current.** The first published model with **Poisson-mode registers**:
+> 64 shared modes that tokens populate as they pass, each mode a Gaussian well whose
+> depth grows with its occupation, so earlier context deepens the wells later tokens fall into.
+> It is [SR2](https://huggingface.co/dimitarpg13/semsimula-ladder-live-owt-d384-l2-none-norc-sr2) (the conservative-only model with exact damped stiff modes) plus the
+> registers, with the wells integrated **exactly** in the same damped flow. Every force is
+> still a gradient. It is the best conservative model on the ladder. Under its pre-registered
+> criterion it is **not** refinement-ready; SR2 is.
+> [The Gen 3 collection](https://huggingface.co/collections/dimitarpg13/semantic-simulation-gen-3-cfc-baoab-live-gradients-6abf371ea3c83f127057020c) holds the live-gradient ladder.
+
+## Result
+
+| model | settled PPL | this model against it | vs GPT-2 |
+| --- | ---: | ---: | ---: |
+| **this model (PMX)** | **54.63** | | **1.097×** |
+| [SR2, the same model without the registers](https://huggingface.co/dimitarpg13/semsimula-ladder-live-owt-d384-l2-none-norc-sr2) | 58.32 | -6.3% | 1.171× |
+| [F3.1, conservative-only with the split step](https://huggingface.co/dimitarpg13/semsimula-ladder-live-owt-d384-l2-none-norc) | 57.76 | -5.4% | 1.160× |
+| PM1, the registers with explicit wells on F3.1 (not published) | 55.17 | -1.0% | 1.108× |
+| [Fock-PARFLM, L=2, live gradients](https://huggingface.co/dimitarpg13/semsimula-ladder-live-owt-d384-l2-none) (not conservative) | 53.12 | +2.8% | 1.066× |
+| the arm [gpt2-matched](https://huggingface.co/dimitarpg13/semsimula-ladder-owt-d384-l2-gpt2-matched) | 49.81 | +9.7% | 1.000× |
+
+- Settled is the mean of the last three evals (56.07, 54.61, 53.21); best 53.15 at step 30,500.
+- **The registers are worth 6.3% on the exact-flow base,** and the exact integration of the wells keeps the gain PM1 found with explicit wells (PM1: 55.17).
+- **The cost:** 2.98 s per step against 1.72 for SR2 on an A100, from a 32-wide batched eigensolve per token and layer.
+- Run health: 0 watchdog triggers, 0 spikes; the global gradient norm exceeded 1.0 on 15 of 490 logged steps after the probe (max 1.50).
+
+**This model sees exactly 532,480,000 training tokens** — 32,500 steps × 32 × 512 = 532,480,000 tokens — the same budget, corpus, tokenizer and validation batches as every other model in the table. At 76.6M parameters that is about 7 tokens per parameter,
+a third of the compute-optimal ratio: every result below is for an undertrained model.
+
+## Poisson-mode registers
+
+Each of the 64 modes has a centre μ, a width κ, a survival probability λ per token and a
+depth per layer. Token s creates particles in mode v at its overlap with the mode,
+exp(−κ²‖h − μ‖²). The particles survive from token to token with probability λ, so the
+occupation φ that token t sees is a decaying count over the tokens before it (the strict
+past only). Token t then feels a sum of Gaussian wells with depths φ × depth, and its force
+is the gradient of that sum in its own coordinates.
+
+- **What is bosonic about it:** the occupation is exactly the mean of a Poisson
+  immigration–death process, the coherent state of the bosonic Doi–Peliti formalism.
+- **What is not:** the coupling is linear in the occupation, so no measurement on this model
+  can see bosonic fluctuations, only their mean.
+- **Why the step stays conservative:** the occupation depends only on earlier tokens, so at
+  fixed context each token's force is an exact gradient. Verified on the trained weights below.
+
+## The wells integrated exactly
+
+With explicit wells (PM1) the wells were the weak point: they sat at the explicit stability
+wall (ω·Δt about 2.1); at layer 1 the explicit kick made up 94% of the step (its kick share,
+0.94); and they caused the refinement failure. PMX freezes the wells' quadratic at the start of each step and adds it to
+the exact damped flow SR2 built for Vθ's stiff modes:
+
+- every well's force and isotropic curvature α, plus the rank part of the token's 16
+  strongest wells, form a spring on all of the 384 directions;
+- that spring is indefinite (an attractive well's outer slope is unstable radially), so it
+  is diagonalised in a 32-wide eigensolve that handles negative curvature, and each mode is
+  integrated as an exact damped oscillator or exact hyperbolic motion, friction included;
+- the explicit kick carries only the remainder.
+
+The explicit part of the layer-1 step falls from 0.94 (PM1) to 0.288. The derivations, with
+figures, are in the theoretical supplement below.
+
+## Refinement: the per-layer flow
+
+The inference mode `substeps_per_layer = k` runs each layer as k substeps with the context
+and both quadratics frozen at the layer's entry and one LayerNorm projection per layer.
+k = 1 is the trained model, bit for bit.
+
+```python
+model.eval()
+model.cfg.substeps_per_layer = 4      # each layer as 4 substeps of dt/4
+with torch.enable_grad():             # the forces use autograd, also at inference
+    logits, _ = model(idx)
+model.cfg.substeps_per_layer = 1
+```
+
+Validation perplexity against k (FLOW-R's 12 × 4 × 512 fresh tokens; k = 1 is the trained model on the same tokens):
+
+| substeps per layer k | 1 | 2 | 3 | 4 | 6 | 8 |
+| --- | ---: | ---: | ---: | ---: | ---: | ---: |
+| **this model, per-layer flow** | 52.26 | 121.19 | 69.96 | 79.16 | 76.90 | 76.95 |
+| SR2, per-layer flow | 55.04 | 67.64 | 59.83 | 61.00 | 61.08 | 61.09 |
+
+- **The flow reaches a limit, but slowly and far from the trained step.** Between k = 6 and k = 8 it moves by 0.001 nats, but from k = 4 to 6 by 0.029, and its limit sits 0.387 nats above the trained model (SR2: 0.104).
+- **Pre-registered, and missed:** CX3 asked for the last two differences to be at most 0.01 and 0.02 nats; the second is 0.029. CX3′ asked for a limit within 0.25 nats. Both are recorded as misses. **SR2 is the refinement-ready model of the ladder; this one is not, at this budget.**
+- **It still removes most of the standard refinement's damage:** at k = 4 the standard refinement (everything recomputed every substep) gives 614, the per-layer flow 79.16.
+
+## The standard gates
+
+| reading (Cell 6b-7) | this model | SR2 |
+| --- | ---: | ---: |
+| velocity reset (gate 1) | +91% | +3.9% |
+| one extra step at the trained Δt (gate 2) | +956% | +35% |
+| eight steps at the trained Δt (gate 2) | +23,884% | +567% |
+| standard refinement, 3 steps at fixed T (gate 3), nats | 1.097 | 0.405 |
+| standard refinement, 4 steps at fixed T (gate 3), nats | 1.172 | 0.986 |
+| standard refinement, 8 steps at fixed T (gate 3), nats | 2.579 | 2.110 |
+
+- **The momentum lock is gone at the end of training.** Gate 1 was +30,600% at the 8,000-step probe and is +91% now (PM1 ended at +10,600%).
+- **Extra depth is catastrophic** (gate 2), far more than in SR2. Whether that is the tokens over-synchronizing in the deep wells is an open, pre-registered question (protocol §5.22, SY5).
+
+## Which coupling carries the model
+
+Each coupling removed at inference from the trained weights (12 × 4 × 512 tokens; full model 53.24), cost in nats:
+
+| removed | cost | PPL change |
+| --- | ---: | ---: |
+| Vφ, the pair potential | 0.0145 ± 0.0012 | +1.5% |
+| the Poisson-mode wells | 1.063 ± 0.056 | +190% |
+| both | 1.088 ± 0.054 | +197% |
+
+- **The wells carry the model; Vφ is marginal,** as it already is in SR2 (0.0135 nats). With explicit wells (PM1) both were load-bearing (0.77 and 0.32): the exact flow, not the wells, is what sidelines Vφ.
+- **A removal cost measures how much this trained network relies on a term, not what a model trained without it would lose.** Trained without the registers (SR2) the model is 6.3% worse, against a removal cost of +190%.
+
+## Geometry
+
+| reading | this model | SR2 |
+| --- | ---: | ---: |
+| distance from the Vθ-only geodesic, R(geo) (6b-9) | 1.020 | 0.620 |
+| the wells' share of the step (6b-9) | -0.518 | — |
+| Vφ's share of the step (6b-9) | -0.0025 | -0.002 |
+| ω·dt median at the endpoint, all layers (6b-13) | 2.65 | 7.55 |
+| ω·dt median, layer 0 (6b-13) | 2.19 | 8.93 |
+
+- **The spring the model integrates** (one forward, 256 tokens): at layer 0 the wells are weak and repulsive (α < 0 for every token, most modes hyperbolic); at layer 1 they are stiff and attractive (α > 0 for every token, |ω·Δt| median 1.7, at most one unstable radial direction per token). Both regimes are integrated exactly.
+- **The wells force every token** (6b-12): 97.7% of tokens at layer 1 are moved substantially by them. About half the learned well depths are positive and half negative: the wells attract toward some centres and repel from others.
+
+## Conservativity and causality
+
+- **Conservativity on the trained weights:** every force term passes, per token at fixed context: Vθ, Vφ, the Poisson-mode wells and the total are conservative at both layers (worst Jacobian asymmetry 2.4e-4, for Vφ at layer 1; the wells at most 2.2e-7; worst closed-loop work 3.3e-4). Vφ's straight-through router term is at most 2.2% of the total force (p95, layer 0, t ≥ 16) and about 0 at layer 1. Output: [`results/conservativity_test_output.txt`](results/conservativity_test_output.txt).
+- **Causality:** SCAF leak monitor CLEAN at every audit (5k in the probe, then 10k to 30k; future perturbation exactly 0, leak tax at most 1.8e-4 nats; the 32.5k audit printed nan for its PPLs, as every arm's final audit does). Independent check on the best weights: future perturbation at five cut points exactly 0, batch independence exactly 0, prefix-only scoring leak tax −1.0e-5 nats (max |Δlogit| 6.0e-3, the straight-through router's known short-prefix effect). Output: [`results/causality_check_output.txt`](results/causality_check_output.txt).
+
+## Pre-registration
+
+Protocol §5.15 (PMX), written before the run: CX0 (step 8,000 at or below 84.0): HIT, 79.43. CX1 (settled 1% better than SR2): HIT, 54.63. CX1′ (at or below PM1's 55.17, called 30%): HIT. CX2 (standard Gate 3 at N = 3 at or below 0.405 nats): MISS, 1.097. CX3 (the per-layer flow converges by FLOW-R's criterion): MISS on the letter, |pen12 − pen8| 0.029 against 0.02 while pen16 − pen12 is 0.001. CX3′ (limit within 0.25 nats): MISS, 0.387. CX4 (Gate 1 at most +100%): HIT, +91%. CX5 (conservativity): HIT. CX6 (layer-1 kick share at most 0.3): HIT, 0.288.
+
+## Reproduction
+
+Set in Cell 0 of `colab_fock_cfc_baoab_lowrank_depth_ladder_openwebtext_d384.ipynb`
+and run Cell 0 → 1 → 1b → 1c → 1d → 2 → 3 → 4 → 5 → 5b → 6:
+
+```python
+LADDER_MECHANISM='none', REVERSE_CHANNEL=False, LADDER_L=2, LADDER_LR=1.2e-3,
+VPHI_GRAD_PATH='live', XI_GRAD_PATH='live',
+LOWRANK_DAMPED_FLOW=True,
+POISSON_MODES=64, POISSON_MODE_CLIP=0.3,
+POISSON_WELLS_EXACT=True, POISSON_WELLS_EXACT_MODES=16
+# tag components 'norc', 'vplive', 'xilive', 'pm64', 'pmx16', 'sr2'
+```
+
+Cell 5b prints the POISSON-MODE REGISTERS and EXACT DAMPED STIFF-MODE FLOW banners, and
+states that there is no O-step under PMX. The model refuses to build if V_θ's retained modes
+plus the wells' rank part exceed 32. The probe to step 8,000 ran the original basis routine;
+the continuation ran the mathematically identical faster routines (split basis, Cholesky QR,
+short series), verified to float32 rounding. The shipped code adds one robustness fix: the
+eigensolve falls back to `eigh` if the SVD fails to converge on repeated eigenvalues, and is
+bit-identical whenever it converges.
+
+Training artefacts were written to `semsimula_fock_cfc_baoab_owt_xi5long_topk16_dt32da16_mh4_aniso_dcvt5x8_vtjoint_cgqk_norc_vplive_xilive_pm64_pmx16_sr2_L2probe_ob_untied_wsd_e5c_plgate_rep0.05_fockreg0.005_g0.1_baoab_cfc_lowrank_idt4_lr0p0012_noattn/`.
+
+## Glossary
+
+| term | meaning |
+| --- | --- |
+| settled PPL | validation perplexity at block 512, the mean of the last three evals (steps 31,500–32,500). Best is the lowest single eval |
+| Gen 1 / Gen 2 / Gen 3 | the model generations. Gen 1: explicit integrators. Gen 2: the CfC/BAOAB ladder with the sources of Vφ and ξ (and of an exchange field) *detached*, which keeps every force causal, but no loss gradient ever reaches an earlier token through them. Gen 3: the same forward pass with those gradients *live* |
+| live / detached gradients | whether the loss gradient flows into the earlier tokens that act as sources of a force. Detaching changes no forward value; it only cuts that backward path |
+| Vθ | the learned scalar potential on a token's state h, given its context ξ: a bank of anisotropic Gaussian wells. Its negative gradient is the conservative force on the token |
+| Vφ | a pair potential between a token and a few selected earlier tokens (sparse top-k) |
+| ξ | the causal context channels: exponential moving averages of earlier states at several horizons, which condition Vθ |
+| Fock mechanism, registers, reverse channel | a pool of register particles, created and destroyed from the token context. The reverse channel is the register-to-token force; it is not the gradient of any potential on the tokens, so it is the model's non-conservative part |
+| exchange field | an attention-like token-to-token interaction added to the conservative core. Each token is pushed by earlier tokens through learned query, key and value projections, as in attention. The ladder tests it in two forms, `attention` and `attention_potential` |
+| `attention` (non-conservative exchange field) | the exchange term enters as a force directly. It is not the gradient of anything, so the step stops being a geodesic step |
+| `attention_potential` (attention as a potential) | the same interaction written as a scalar potential whose gradient is the force. The routing weights are computed from the hidden states and held fixed with respect to the state being moved, so the force stays an exact gradient and the model stays conservative |
+| no-exchange (`LADDER_MECHANISM = 'none'`) | the arm with no exchange field at all: Vθ, Vφ and ξ, plus the Fock register mechanism unless the arm is conservative-only |
+| live exchange field (`RELAX_GRAD_PATH = 'live'`) | `attention_potential` with the identical forward force written out over live tensors, so the loss gradient reaches the field's sources and routing. In Gen 2 these were detached, and the field sent no gradient into any hidden state |
+| Gen 2 probe | a Gen 2 arm retrained with one gradient path changed, to test a hypothesis before the full Gen 3 ladder. The live-exchange-field probe (`attention_potential` with `RELAX_GRAD_PATH = 'live'`, the full 32,500 steps) is the one that found the gradient starvation |
+| gpt2-matched | the reference transformer: GPT-2 architecture, d = 384, L = 8, tied embeddings (33.7M parameters), trained on the same tokenizer, corpus, batches and token budget. Shared by Gen 2 and Gen 3 |
+| tag components (`vplive`, `xilive`, `rglive`, `norc`, `L4probe`, `idt…`) | pieces of the run's variant tag, so that every configuration gets its own Drive folder: live Vφ sources, live ξ, live exchange field, no reverse channel, the depth L, and the integrator step Δt |
+| conservative-only | a model in which every force is a gradient: no reverse channel and no non-conservative exchange field |
+| CfC/BAOAB low-rank propagator | the integrator. Each layer step is split into position (A), force kick (B) and friction (O) substeps, and the stiff harmonic modes of Vθ are integrated exactly in closed form |
+| ω·Δt, θ | a stiff mode's frequency times the step size. Past 2, an explicit integrator would be unstable (the *wall*); the closed-form propagator is not. θ/sin θ is the factor by which a split step's dissipation depends on the mode's phase, largest near θ = π |
+| geodesic step, R(geo), share | the step a damped particle would take under Vθ alone; R(geo) is the distance of the real step from it, relative to the step's length. A component's *share* is how much it moves that distance (6b-9 attribution) |
+| Gates 1–3, FLOW / MAPS | see **Terms** in the geometry section |
+| CG1–CG7, 6b-N | the CfC-geometry experiments, and the ladder-notebook cells that run them |
+| SCAF, Tier A, Tier B | the in-flight causal-leak monitor. Its core test changes future tokens and requires the logits at earlier positions to stay bit-identical. Tier A asks the same of the hidden states, as 1 − cos between the clean and perturbed state at each earlier position and layer. Tier B asks whether the perturbation moves any earlier state into a different dominant Vθ well (the basin-crossing rate). All three read 0 for a causal model |
+| leak tax, prefix-only | the loss difference between scoring a position with only its prefix and with the full sequence; about 0 for a causal model |
+| clip-hit | the share of logged steps whose gradient norm exceeded the clipping threshold (1.0) |
+| pre-registered | a prediction recorded, with its threshold, before the run |
+
+## Theoretical foundations of the ladder
+
+The design and theory behind every arm of the ladder: the integrator they share, the
+potential they share, and the mechanisms the ladder compares. The same list appears on
+every ladder card.
+
+| note | what it establishes |
+| --- | --- |
+| [`Closed_Form_and_Hybrid_Integration_Strategies_for_Fock-PARFLM.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Closed_Form_and_Hybrid_Integration_Strategies_for_Fock-PARFLM.md) | why the integrator is closed-form: which parts of the Vθ dynamics admit exact propagation, and three strategies in decreasing order of exactness, following the CfC lesson from liquid neural networks |
+| [`Blended_CfC_BAOAB_Deep_Dive.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Blended_CfC_BAOAB_Deep_Dive.md) | the full construction of the CfC-BAOAB step every arm integrates with: the Gaussian Vθ's exact harmonic propagator placed inside the BAOAB Langevin splitting, every matrix written out |
+| [`Modified_BAOAB_with_STP_identity_Detailed_Analysis.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Modified_BAOAB_with_STP_identity_Detailed_Analysis.md) | the BAOAB splitting with the STP identity as the simulator's integrator backbone: component-by-component analysis, coupling to the equations of motion, theoretical properties and trade-offs |
+| [`Analytic_Multi_Channel_Integration_in_Structured_Vtheta.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Analytic_Multi_Channel_Integration_in_Structured_Vtheta.md) | the joint Vθ (`vtjoint` in the run tags): the five ξ context channels fused into one closed-form well bank instead of summed as separable ones, and why CfC-BAOAB stays valid |
+| [`Conservative_Obstruction_and_Virtual_Particle_Necessity.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Conservative_Obstruction_and_Virtual_Particle_Necessity.md) | the theorem the ladder tests: no scalar potential on the tokens alone reproduces attention's structure, and the Fock registers are the minimal extension. Gen 2 read the +31.3% Fock gap as that obstruction's price; the reading is withdrawn (detached sources, see the banner) |
+| [`Improving_the_Fock_Mechanism_to_match_Attention.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Improving_the_Fock_Mechanism_to_match_Attention.md) | how the Fock mechanism was redesigned toward attention (Q/K/V-structured creation, the reverse channel). Its headline PPL figures predate the second causal-leak fix and are flagged as uncertified in the note itself |
+| [`Measuring_the_Price_of_Conservativity.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Measuring_the_Price_of_Conservativity.md) | the λ-gated relaxation protocol first proposed to price conservativity, and how the ladder's pair the arm [attention](https://huggingface.co/dimitarpg13/semsimula-ladder-owt-d384-l2-attention) → the arm [attention_potential](https://huggingface.co/dimitarpg13/semsimula-ladder-owt-d384-l2-attention-potential) answers the same question by construction instead |
+| [`Geodesic_Preservation_Experiment.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Geodesic_Preservation_Experiment.md) | why measuring the geometry of the model's own dynamics is the capability to demonstrate, and the design and first analyses of the geodesic measurements (d=384 and d=1024) |
+| [`Composing_Single_Layer_Inferences_Flow_or_Maps.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Composing_Single_Layer_Inferences_Flow_or_Maps.md) | whether the layer stack is a continuous flow or a sequence of maps; Gates 0–3 on the L=2 the arm [no-exchange](https://huggingface.co/dimitarpg13/semsimula-ladder-owt-d384-l2-none) returned MAPS |
+
+## Theoretical supplement: Poisson-mode registers
+
+In reading order:
+
+| note | read it for |
+| --- | --- |
+| [`Poisson_Mode_Registers_PM1.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Poisson_Mode_Registers_PM1.md) | **start here**: the mechanism, a worked example, the theorem that the occupation is exactly a Poisson mean, why the step stays conservative, how it differs from the Fock slot registers, and PM1's results |
+| [`Exact_Integration_of_Poisson_Mode_Wells_PMX.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Exact_Integration_of_Poisson_Mode_Wells_PMX.md) | how this model integrates the wells: their indefinite Hessian, the 32-wide hybrid, the diagonalisation, the closed-form damped flow for curvature of either sign, the splitting and its energy identity, the per-layer flow, and what the trained model integrates, with figures |
+| [`Single_Particle_Hilbert_Space_in_Semantic_Simulation.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Single_Particle_Hilbert_Space_in_Semantic_Simulation.md) | what a mode is: single-particle states as Gaussian root densities, repeated occupation, the vacuum, and the bridge to Doi–Peliti, where coherent states are Poisson distributions |
+| [`Doi_Peliti_Dynamics_of_Semantic_Particles_and_Registers.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Doi_Peliti_Dynamics_of_Semantic_Particles_and_Registers.md) | why the bosonic formalism fits these registers and not the Fock slot registers: the master equation, exclusion against bosons, the coherent-state path integral, and the measurements |
+| [`Two_Clustering_Mechanisms_Saturation_and_Synchronization.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Two_Clustering_Mechanisms_Saturation_and_Synchronization.md) | the registers as synchronization through shared geometry, and the conservative and non-conservative routes for carrying context |
+
+## Files
+
+| path | what |
+| --- | --- |
+| `README.md` | this card |
+| `config.json` | architecture, training, integrator, the registers, gradient convention, results |
+| `checkpoints/*_best.pt` | weights at the best eval (step 30,500), optimizer state stripped |
+| `checkpoints/*_step500_best.pt` | weights at step 500, an initialisation reference |
+| `results/L2_PMX_32500steps_output.txt` | the run log from the probe stop to 32,500, with per-step diagnostics and the SCAF lines |
+| `results/L2probe_PMX16_on_SR2_8000steps_output.txt` | the probe to step 8,000 |
+| `results/training_log.jsonl` | per-eval record; the settled PPL is the mean of its last three rows |
+| `results/scaf_leak_monitor.jsonl` | the in-flight causal-leak audits |
+| `results/causality_check_output.txt` | the independent causality check on the best weights |
+| `results/conservativity_test_output.txt` | Jacobian symmetry and closed-loop work per force term on the best weights, the wells included |
+| `results/conservativity_router_term_by_position_output.txt` | Vφ's straight-through router term, exact, by position band |
+| `results/refinement_flow_confirmation_pmx_seed20261009_output.txt` | FLOW-R: the per-layer flow, the kick share and the standard refinement (CX3, CX3′, CX6) |
+| `results/coupling_dominance_ablation_PMX_output.txt` | removal costs: Vφ, the wells and both, removed at inference |
+| `results/pmx_spring_spectrum_output.txt` | the spring each token carries in the exact flow, per layer |
+| `results/Cell-6b-7_flow_or_maps_32500steps_output.txt` | flow or maps: inertia and the standard refinement gates on the best weights |
+| `results/Cell-6b-7_flow_or_maps_step8000_output.txt` | the same at the probe stop |
+| `results/Cell-6b-9_layer_step_damped_V_theta_geodesic_32500steps_output.txt` | deflection from the damped Vθ geodesic step; the Vφ and wells attributions |
+| `results/Cell-6b-12_sparse_forcing_32500steps_output.txt` | per-token deflection: the wells' forcing is uniform across tokens |
+| `results/Cell-6b-13_omega_dt_at_endpoint_cross-arm_comp_32500steps_output.txt` | ω·dt at the endpoint, on the cross-arm token set |
+| `results/Cell-6b-15_PMX_tuning_what_limits_Poisson_modes_clip=0.3_32500steps_output.txt` | what limits the modes: step size, weight decay, occupation scale, placement |
+| `results/Cell-6b-15_PMX16_tuning_what_limits_Poisson_modes_step8000_output.txt` | the same at the probe stop |
+| `results/Cell-6b-16_PMX_GPU_step_profile_output.txt` | where the GPU time of a PMX step goes |
+| `results/omega_dt_endpoint.jsonl` | the 6b-13 record |
+| `results/pm_tuning_diagnostics.jsonl` | the 6b-15 record |
+| `results/verify_pm_wells_exact_output.txt` | verification of the exact well integration (unit tests against RK4, consistency, causality, guards) |
+| `results/verify_pmx_split_output.txt` | verification of the split basis (identical operator) |
+| `results/verify_pmx_fast_output.txt` | verification of the Cholesky basis and the short series (identical operator) |
+| `results/logfreq_surprisal_openwebtext.npy` | the log-frequency table the mass model reads |
+| `code/` | the notebook and the model code, including the exact well integration and `substeps_per_layer` |
+
+## Companion notes
+
+| note | what it covers |
+| --- | --- |
+| [`Depth_Ladder_and_Matched_Baseline_Protocol.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Depth_Ladder_and_Matched_Baseline_Protocol.md) | §5.15: PM1, PM1-cap and PMX, their pre-registrations and scoring; §5.19: the per-layer flow; §5.20: the removal costs; §5.22: the synchronization test programme |
+
+## Caveats
+
+- **One seed.**
+- **Not refinement-ready under its pre-registered criterion** (CX3, CX3′). SR2 is the refinement-ready arm.
+- **Undertrained:** 532.5M tokens, about 7 per parameter, a third of compute-optimal. A refinement failure at this budget does not show that a fully trained model of this kind cannot refine.
+- **The per-layer-flow and removal-cost evaluations use 12 × 4 × 512 tokens,** not the full validation eval behind the settled PPL.
+- **L = 2 only.** Two layer steps.
+- **Its explicit-wells sibling PM1 is not published;** its numbers are quoted from the protocol.
+- **The matched GPT-2 is a reference, not a tuned transformer,** and its depth is not matched (L = 8 against L = 2).
+- **OpenWebText only.**
+
+## Citation
+
+```bibtex
+@misc{Gueorguiev2026SemSim,
+  author    = {Gueorguiev, Dimitar P.},
+  title     = {Semantic Simulation: A Prescriptive Lagrangian Framework
+               for Semantic Inference --- Conservative Potentials,
+               Non-Conservative Memory, and the Shared-Potential Separator},
+  year      = {2026},
+  publisher = {Zenodo},
+  doi       = {10.5281/zenodo.19712427},
+  url       = {https://doi.org/10.5281/zenodo.19712427},
+  note      = {Version v22 (Oct 5, 2026).
+               Companion code repository (DOI 10.5281/zenodo.23152203):
+               \url{https://github.com/dimitarpg13/semsimula-paper}}
+}
+```
+
+*Card generated from `_ladder/ladder.json` on 2026-10-11.*