dimitarpg13/semsimula-ladder-live-owt-d384-l2-none-norc-sr2-fob

SR2 trained as a true first-order gradient flow (FO-b): 4.9% behind its second-order twin on OpenWebText (Gen 3)

The question of this collection: does the conservative core of Semantic Simulation need second-order dynamics (a velocity and inertia), or does a first-order model trained the same way do as well? Each pair trains one model both ways, with everything else the same.

This model: SR2 trained as a true first-order gradient flow (FO-b) on OpenWebText. Verdict: second order wins by 4.9% (61.16 against 58.32). The first-order model led for the first third of training, then fell behind and stayed behind. The first-vs-second-order collection holds the pairs; the Gen 3 collection holds the OpenWebText ladder.

Result

pair	corpus, tokens	second order	first order	first order against second
OpenWebText	532.5M	SR2 58.32	FO-b-OWT 61.16	+4.9%
TinyStories, first order between layers (FO-a)	315M	SO-TS 5.16	FO-TS 5.13	−0.6%
OpenWebText, with 64 Poisson-mode wells	532.5M	PMX 54.63	FO-b-PMX 54.02	−1.1%
OpenWebText, with 16 Poisson-mode wells	532.5M	PMX-K16 54.24	FO-b-PMX-K16 53.73	−0.9%
OpenWebText, with 2 Poisson-mode wells	532.5M	PM2-PMX16 54.71	FO-b-PM2-PMX16 53.81	−1.6%
TinyStories, true first order (FO-b), no wells	315M	SO-TS 5.16	FO-b-TS 5.28	+2.3%
TinyStories, with 2 Poisson-mode wells	315M	PM2-TS 4.99	FO-b-PM2-TS 4.96	−0.5%
Python code, true first order (FO-b), no wells	315M	SO-CODE 6.00	FO-b-CODE 6.55	+9.2%
Python code, with 2 Poisson-mode wells	315M	PM2-CODE 5.71	FO-b-PM2-CODE 5.62	−1.6%

• This model: settled 61.16 (mean of the last three evals, 61.06, 61.15, 61.28); best 59.51 at step 31,000. Validation perplexity at block 512, on OpenWebText's own validation tokens.
• Trained on exactly 532,480,000 tokens (32,500 steps × 32 × 512), the Gen 3 ladder's budget and schedule.
• Pre-registered: FB2, second order earns at least 3% on OpenWebText (FO-b-OWT at or above 60.07), called 60%: HIT, +4.9%. FB4, a trained gradient flow refines (Gate 3 at N = 3 at most +10%), called 50%: MISS, +19.5%, and the per-layer flow does not converge.
• Run health: 0 watchdog triggers, 0 spike captures; SCAF leak monitor CLEAN at every audit (5k to 30k and 32.5k; the 32.5k audit printed nan for its PPLs, as other arms' final audits did).

What first order means here

SR2's layer integrates the damped Newton equation, mass × acceleration = force − mass × damping × velocity, with the stiff modes of the well potential integrated exactly as damped oscillators and the velocity handed from one layer to the next. FO-b replaces it with the gradient flow of the same potential: the hidden state moves with velocity = mobility × force, where the mobility is the layer step divided by twice the learned mass. The stiff modes relax exactly (exponentially, instead of oscillating), everything else takes one gradient step, and no velocity exists anywhere. The step keeps SR2's splitting and its single force evaluation, and adds no parameters.

How the gap formed

step	phase	SR2	FO-b (this model)	FO-b against SR2
500	warmup	461.81	451.17	−2.3%
5,000	constant	108.37	101.13	−6.7%
10,500	constant	83.52	82.77	−0.9%
11,000	constant: crossover	84.76	87.27	+3.0%
15,000	constant	76.83	79.50	+3.5%
25,000	decay	65.92	67.46	+2.3%
30,000	decay	58.07	61.01	+5.1%
32,500	decay, end	58.32	61.28	+5.1%

• First order learns faster at first (ahead from step 500, by 6.7% at step 5,000): a gradient flow's learning problem is better conditioned than an oscillator's.
• It stalls and falls behind at step 11,000, still at constant learning rate, with no gradient spike or instability, and stays about 3% behind through the constant phase.
• The decay widens the gap: from step 21,000 to the end SR2 gains 0.196 nats, this model 0.175.

Where the two models differ

reading	SR2	FO-b (this model)
context channels' decay rates at the end of training	0.386, 0.521, 0.887, 0.981, 0.995	0.357, 0.511, 0.888, 0.992, 0.982
stiffness of the wells, median ω·Δt at the endpoint	7.55	0.74
share of stiff-mode readings past the explicit-integrator wall (2.0)	99.7%	18.4%
share of the layer step carried by the kick rather than the exact quadratic	0.37, 0.51	0.995, 0.967

Same corpus, same schedule, and the same context horizons to within a few hundredths. The difference is in the wells: trained first order, the model builds them about ten times softer. A second-order layer can use a stiff well as a rotor, turning a mode through a phase and keeping the distinctions between tokens; a gradient flow collapses every token onto a stiff well's minimum within the step, so a first-order model cannot afford stiff wells and gives up the expressivity they provide. The argument is developed in Future_of_Semantic_Simulation_When_Second_Order_Earns_Its_Keep.md (§3).

Off the trained schedule (Cell 6b-7) and the per-layer flow (FLOW-R)

reading	this model	its pair partner
Gate 1: perplexity cost of resetting the velocity at inference	0 (no velocity carried)	+3.9%
Gate 2: one extra layer at the trained step	+0.6%	+35%
Gate 2: eight layers	+138%	+567%
Gate 3: three steps over the trained time (standard refinement), nats	0.178	0.405
per-layer flow: penalty at N = 16, nats	0.904	0.104

Per-layer flow of this model (FLOW-R, 12 × 4 × 512 fresh tokens; the context and the stiff modes' quadratic held at each layer's entry, one LayerNorm projection per layer), perplexity by the number of steps N over the trained time T = 8; N = 2 is the trained model, and N = 4, 6, 8 split each of its two layers into 2, 3, 4 substeps:

steps N	2 (trained)	3	4	6	8	12	16
perplexity	57.19	98.91	163.29	157.57	137.48	134.57	141.16
penalty, nats	0	0.548	1.049	1.013	0.877	0.856	0.904

• Extra layers become harmless: one more layer costs +0.6% against SR2's +35%. Without inertia nothing overshoots.
• But the stack is not refinement-ready: the per-layer flow does not converge (it moves 0.048 nats between N = 12 and 16) and sits about 0.9 nats above the trained step, where SR2's converges to 0.104. One FO-b step is first order in the step size, so at the trained step it is far from the continuous flow, and the model learned to use the coarse map. Of the models measured, only SR2's second-order step is a close sample of a convergent per-layer flow.

Reproduction

colab_fock_cfc_baoab_lowrank_depth_ladder_openwebtext_d384.ipynb, Cell 0 changed from the defaults:

REVERSE_CHANNEL=False, VPHI_GRAD_PATH='live', XI_GRAD_PATH='live',
LOWRANK_DAMPED_FLOW=True,
FIRST_ORDER_OVERDAMPED=True,              # tag 'fob'
VPHI_ANALYTIC_FORCE=True, LAYER_CHECKPOINT=False   # speed switches, same arithmetic

Run Cells 0, 1, 1b, 1c, 1d, 2, 3, 4, 5, 5b and 6, then 8, 7 and 6b-7. Cell 5b prints an FO-b block (TRUE FIRST ORDER); the diagnostics Cells 6b-9 and 6b-13 have FO-b branches (no velocity exists, and the resonance monitor does not hook the first-order routine). The per-layer flow is debug/refinement_flow_confirmation.py (CPU).

Training artefacts were written to semsimula_fock_cfc_baoab_owt_xi5long_topk16_dt32da16_mh4_aniso_dcvt5x8_vtjoint_cgqk_norc_vplive_xilive_sr2_fob_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

Files

path	what
README.md	this card
config.json	architecture, training, the first-vs-second-order switch, result and diagnostics
checkpoints/fock_cfc_*noattn_best.pt	weights at the best eval (step 31,000), optimizer state stripped
results/	the training log and full run outputs, Cell 6b-7, the per-layer flow and the endpoint stiffness
code/	the notebook and the model code the switches belong to

Companion notes

note	what it covers
Depth_Ladder_and_Matched_Baseline_Protocol.md	§5.13: the first-vs-second-order series, its pre-registrations, amendments and scored results
Future_of_Semantic_Simulation_When_Second_Order_Earns_Its_Keep.md	what the equations say about when second order earns its keep, revised after this series' OpenWebText result

Caveats

• One seed per model. The yardstick for small gaps, the seed spread, has not been measured.
• The hyperparameters were tuned for the second-order models (learning rate 1.2e-3, the WSD fractions, the clip groups); part of the OpenWebText gap may be a mismatch, and a run at a lower learning rate is queued as a check.
• L = 2. One layer boundary is the weakest possible test of a velocity carried between layers.
• OpenWebText only; the two pairs differ in corpus and token budget, so perplexities compare within a pair, not across pairs.

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 2026-11-10.