README.md

metadata

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 (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 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	58.32	-6.3%	1.171×
F3.1, conservative-only with the split step	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 (not conservative)	53.12	+2.8%	1.066×
the arm 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.

Trained first order, the same model is level (added 2026-10-22). FO-b-PMX is this model trained as a true first-order gradient flow (the wells and the stiff modes relaxed exactly, no velocity anywhere): it settles at 54.02 against 54.63 here, −1.1%, a tie. Without the wells the same comparison gave second order a 4.9% advantage; the wells help the first-order model twice as much as this one. Its wells are attractive by a majority where this model's are balanced, and one extra layer costs it +0.3% against +956% here. The reading proposed, the memory-substitution hypothesis (the second-order advantage is the advantage of a memory, redundant once an explicit one exists), is pre-registered in the protocol (§5.13, MS1–MS7). The first-vs-second-order collection holds the pair.

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.

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.
• 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.

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:

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

Synchronization and binding (SY programme)

Do the couplings of these models synchronize tokens, and does that group related words? The question was put to every Gen 3 model and to the matched GPT-2 with criteria fixed before any measurement. Full results and figures: Spontaneous_Synchronization_and_Semantic_Binding_Test_Programme.md, Part II; pre-registrations and scoring: Depth_Ladder_and_Matched_Baseline_Protocol.md, §5.22.

Terms.

• Coupling: any term through which another token's state moves this token: the pair potential $V_{\varphi }V\_\{\backslash phi\}$, the Poisson-mode wells, the Fock register read, the exchange field, and the context field $\xi \backslash xi$, which couples tokens because it is built from the earlier ones.
• Synchronization, contraction $SS$: how much closer the token states end up because of one coupling, measured against a replay in which that coupling pushes every token alike. $S>0S\; \backslash gt\; 0$: drawn together; $S<0S\; \backslash lt\; 0$: spread apart.
• Binding contrast: how much more alike the states of related words are than those of unrelated words at the same distance in the text, in cosine units. An effect counts only if it is at least 0.02 and its 95% interval excludes zero.
• Noun chunk: the words of one simple noun phrase, such as the old wooden table.
• Head–dependent pair: a word and the word it grammatically depends on, such as keys and are in the keys to the cabinet are.
• Reference, coreference, antecedent: what an expression stands for. Two expressions are coreferent when they pick out the same thing, as Mary and she do in Mary said she would come; the earlier one, Mary, is the antecedent. Binding reference means linking a pronoun to the expression it stands for.
• Prediction control: what remains of a grouping after removing everything that the similarity of the words' predicted next words explains. A grouping that survives it is not just a side effect of predicting the next word.
• Coreference probe: a small attention-shaped reader trained to pick each pronoun's antecedent from the states. It is compared with always picking the nearest earlier mention, which is right 54.1% of the time (chance: 13.6%).
• Extra depth, over-synchronization: the trained model run for more steps than it was trained for, at the trained step size. Over-synchronization is the tokens drawing together (\(S \geq 0.05\)) or losing dimensions (effective rank down by 20% or more) as it fails.

Findings across the models (each call was pre-registered):

• The couplings do draw tokens together (SY-L0: hit), the pair potential modestly and the Poisson wells strongly, though toward their fixed centres rather than toward each other. None groups related words on the routes the test was designed around (SY-L1: miss).
• The context field $\xi \backslash xi$ groups the words of noun chunks in every model, and the grouping survives the prediction control. It spreads the tokens apart overall while drawing each chunk's words together: clustered synchronization, as in oscillator networks that split into synchronized groups.
• The matched GPT-2 forms the same grouping, built by attention in its middle layers. The grouping is generic to language models at this budget; how it is made is not.
• No model binds reference. No probe beats the nearest-mention rule by a margin the test can resolve, in these models or in GPT-2.
• Extra depth collapses the tokens at the catastrophic stage of its failure (SY-G2: hit), not at its onset.

Measured on this model (PMX, Poisson-mode wells):

measurement	result
contraction by each coupling, last layer	Vφ +0.021; the wells +0.98, toward their fixed centres rather than toward each other
noun-chunk binding made by ξ, last layer	+0.038 and head–dependent separation −0.033, both passing (the replicated pair)
noun-chunk grouping after the prediction control	+0.042
coreference probe, last layer / best layer	0.532 / 0.532, at the nearest-mention baseline (0.541)
first extra step	perplexity +15%, S +0.054: over-synchronizes; one step further perplexity 205, S +0.50

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	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	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	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	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	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)
Fock_Mechanism_v2.1_Principle_of_Operation.md	how the Fock mechanism of these models works, as implemented: the exchange field, the registers and their write port, and the reverse channel that is the registers' only read port; their equations, the virtual-particle reading, the relation to attention, and measurements on the trained G3′. An exploratory tool for studying attention-like routing in a conservative model
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 → the arm attention_potential answers the same question by construction instead
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	whether the layer stack is a continuous flow or a sequence of maps; Gates 0–3 on the L=2 the arm no-exchange returned MAPS

Theoretical supplement: Poisson-mode registers

In reading order:

note	read it for
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	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	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	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	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	§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

@misc{Gueorguiev2026SemSim,
  author    = {Gueorguiev, Dimitar P.},
  title     = {Semantic Simulation: A Prescriptive Lagrangian Framework
               for Semantic Inference --- Conservative Potentials,
               Conservative and Non-Conservative Memory, and Competing Mechanisms},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.19712427},
  url       = {https://doi.org/10.5281/zenodo.19712427},
  note      = {Version v25 (Nov 9, 2026).
               Companion code repository (DOI 10.5281/zenodo.24066016):
               \url{https://github.com/dimitarpg13/semsimula-paper}}
}

Card generated from _ladder/ladder.json on 2026-11-08.