Arithmetic J-Space Training Lab

Arithmetic J-Space Training Lab v0.4.0

Event-sampler hotfix

This build replaces rare-event rejection sampling with a constructive exactly-one-carry or exactly-one-borrow generator. It fixes the deterministic carry_discovery failure at the first 10-digit OOD evaluation. An original v0.4.0 discovery run is accepted only when its resolved configuration differs solely by the allowlisted package-manifest update. The migration is recorded in package_hotfix_migration.json, and training resumes from the latest valid rolling checkpoint.

This package trains standard decoder-only transformers from scratch on answer-only addition and subtraction, then tests a candidate balanced causal arithmetic subspace. It does not insert a workspace, supervise a hidden space, use answer labels during subspace discovery, or modify a trained model to create the candidate.

The shipped task uses positive integer operands and nonnegative results; subtraction operands are ordered to avoid negative answers. It does not test signed arithmetic, zero identities, multiplication, or division.

Version 0.4 is deliberately candidate-only. It can establish behavioral competence and run target-blind geometric and causal diagnostics, but it cannot establish a J-space equivalent. Formal causal mediation, native shared multi-query access, and component-specific upstream-writer rescue are not implemented. Every profile therefore reports:

JSPACE_CLAIM=NOT_ESTABLISHED

Recommended Windows run

Use the known komorebi-full Python explicitly:

$d="$env:USERPROFILE\Downloads\Arithmetic_JSpace_Training_Lab_v0.4.0"; & "$d\Run_Arithmetic_Workspace_Experiment.ps1" -Profile carry_discovery -Python "$env:USERPROFILE\miniconda3\envs\komorebi-full\python.exe" -Device cuda -SkipInstall

The command runs all three discovery seeds. It is resumable: rerun the identical command after interruption. Progress is printed and flushed every 100 optimizer steps. Light evaluation runs every 500 steps and full evaluation every 2,000 steps in the serious profiles.

Profiles

carry_confirmatory requires the workspace_lock.json written by a completed discovery run. It is a locked candidate replication—not a profile capable of establishing a J-space claim.

On an 8 GB RTX 3070 Ti Laptop GPU, budget minutes for smoke, roughly 3–9 days for discovery, and roughly 12–35 days for the full confirmation. These are planning ranges. Use the measured ETA after the first 500 steps of each arm. Keep roughly 30 GB free for the default discovery checkpoints and 80 GB for the full locked replication. Exact use depends on checkpoint serialization and the number of completed arms; outputs are not deleted automatically.

Scratch model and objective

• fixed 260-token byte vocabulary (0..255, PAD, BOS, SEP, EOS)
• prompt-masked causal loss on answer bytes and EOS only
• pre-normalized causal decoder with rotary positions
• no pretrained weights, scratchpad, carry trace, auxiliary representation loss, bottleneck, or probe loss
• AdamW with warmup-stable-decay learning-rate schedule
• BF16 when supported, otherwise scaled FP16 on CUDA
• rolling atomic resume checkpoints and immutable research checkpoints

The primary model is not Qwen or another downloaded model.

Four arms

correct, pseudo_fst, and deranged use the same input-expression order and paired initial weights. no_carry necessarily uses a different restricted training stream; it is not globally prompt-, token-, or answer-width-matched. The exact shared comparison with it is the sealed no_transfer panel.

Each derangement window first attempts a seeded random value permutation within operator/output-width nuisance strata, never a lexical sort-and-rotate. An underdetermined stratum is deterministically coarsened to output-width-only pooling when needed. Exact empirical target-multiset preservation and zero source/assigned value fixed points remain mandatory for every nonsynthetic stratum, including a successfully coarsened stratum. Within a seeded candidate, synthetic labels are not considered until exact operator/output-width and exact output-width-only constructions are mathematically impossible. Additional seeded candidates serve the residual-correlation and string-agreement gates.

Only when an output-width-only stratum of at most 31 rows makes such a value derangement mathematically impossible does the final fallback synthesize canonical same-width decimal labels, deterministically and with no source/assigned value fixed points. Synthetic rows therefore do not preserve the empirical target multiset. Their count and total-window fraction are explicitly reported in the control audit. The 31-row limit applies to each impossible width stratum separately. The strict fraction cap is 0.001 (0.1%); the audit records the cap and pass/fail result, and the run fails closed if the cap is exceeded. A constructed window that uses the fallback emits one DERANGEMENT_SYNTHETIC_FALLBACK telemetry line. This fallback weakens the deranged control, but prevents a rare finite training window from becoming impossible to construct. The residual-correlation gate uses the absolute row-weighted Pearson correlation over predeclared eligible strata. Tiny or degenerate strata are still reported but excluded from that gate using the fixed eligibility rule in workspace_data.py.

Target-blind balanced-space discovery

Discovery receives tokenized prefixes ending at SEP. It rejects answer tokens and EOS. Upstream residual perturbations estimate empirical controllability; gradients of fixed, answer-independent output summaries estimate local observability. The summaries are exactly:

• digit log-probability mass
• EOS log probability
• minus-sign log probability
• output entropy
• top log probability
• four sealed random projections of the vocabulary log-probability vector

The leading directions of

form the candidate. Rank is selected by heldout prediction of changes in those same target-free summaries. Rank-matched controls are PCA, one seeded random subspace, and a complementary balanced-spectrum subspace. There is no bootstrap-stability assay or distribution of many random subspaces in v0.4.

Geometry runs only after target-task competence and is fitted independently in the competent correct and pseudo_fst arms. Their causal effects are compared as a limited specificity diagnostic. The pseudo-FST controls digit values but deliberately uses the true answer width to match output length. It therefore retains a coarse arithmetic-derived schema signal and is not an entirely arithmetic-free control.

Implemented causal assays

On sealed, semantically matched carry/borrow counterfactual pairs, every basis (candidate, PCA, random, complement) receives the same free-generation tests at the final prefix SEP position:

1. candidate-coordinate ablation
2. clean-to-corrupt coordinate patching with directional digit audit
3. independent transplant of a corrupt donor residual at an earlier layer
4. restoration of only the registered source-layer coordinates after that upstream transplant

Control interventions are rank-matched and scaled to the candidate intervention energy for the same state. Candidate fitting and selection never receive answers. After the candidate is frozen, sealed answers and their widths are used to construct/match counterfactual pairs, define eligibility, and score free generation.

Primary rates are computed only on qualified pairs. Ablation requires a valid, EOS-terminated natural clean answer. Patching requires valid and correct natural clean and corrupt answers with distinct targets. Rescue additionally requires the unrescued upstream corrupt-donor transplant to generate the corrupt target exactly. Discovery requires at least 128 baseline-qualified pairs and 32 such directional rescue pairs; locked confirmation requires 512 and 128. A missing denominator fails closed rather than being treated as a null effect.

The upstream transplant is a residual-state intervention, not a head/MLP writer intervention. Coordinate rescue is not a formal mediation analysis. No native shared multi-query assay or unrelated-language preservation assay is bundled.

Statistics and evidence ceiling

Confirmation retains item-level outputs. Seed is the replication unit for candidate-minus-strongest-control ablation, patch, and rescue effects. The registered analysis uses paired seed tests, Holm familywise correction, and a family-clustered hierarchical bootstrap of directional patch effects.

These statistics can support a replicated candidate effect. They cannot override the v0.4 claim ceiling. Missing formal mediation, shared multi-query access, and component-specific writer rescue keep the claim state NOT_ESTABLISHED even if every implemented comparison passes.

Output and resume

The result directory contains:

resolved_experiment.json
runtime.json
progress.jsonl
training/seed_<n>/<arm>/
behavior/seed_<n>/
behavior/seed_<n>/control_audits.json
geometry/seed_<n>/
causal/seed_<n>/
causal/confirmation_statistics.json
workspace_lock.json              # discovery only, if a selection passes
experiment_summary.json

Rerunning the same command resumes. A resolved-config mismatch fails closed; use a new -Output path for a changed run. -NoResume disables resumption.

For exact preregistered comparisons and limitations, read EXPERIMENT_DESIGN.md and WORKSPACE_PROTOCOL.md.