Arithmetic J-Space Training Lab

Preregistered candidate-space experiment v0.4

Status and estimand

The estimand is whether answer-only arithmetic training produces a small residual-stream subspace whose target-blind local geometry predicts downstream output-summary changes and whose coordinates have selective effects on free generation under matched interventions.

The task domain is positive-operand addition/subtraction with nonnegative results; it makes no inference about signed arithmetic, zero identities, multiplication, or division.

The allowed conclusion is a replicated candidate balanced causal arithmetic subspace. Version 0.4 cannot establish a J-space equivalent. Its machine- readable claim ceiling is fixed to JSPACE_CLAIM=NOT_ESTABLISHED because formal causal mediation, native shared multi-query access, and component-specific upstream-writer rescue are not implemented.

Model and supervision

The serious profiles use the same conventional decoder architecture:

• 12 transformer blocks
• residual width 512
• 8 attention heads
• feed-forward width 2048
• approximately 38 million parameters
• fixed 260-token byte vocabulary
• maximum context 64

Weights start from scratch. Training minimizes ordinary causal next-token loss on the answer and EOS; prompt tokens are masked. No intermediate arithmetic trace, carry/borrow label, hidden-state target, auxiliary probe loss, bottleneck, or workspace module is present.

Arms and pairing

For each seed, correct, pseudo_fst, and deranged have identical initial model hashes and audited input-expression order. Targets differ. no_carry uses an intentionally restricted stream and is not claimed to match the full stream's prompts, token count, or answer-width distribution. The sealed no_transfer panel is the exact shared task comparison between correct and no_carry.

The pseudo_fst must pass its own target-task competence gate before its geometry is used as an accuracy-matched specificity control. Comparing a competent arithmetic model with an unlearned pseudo task is invalid. Because its target width is taken from the true answer width, this arm is not fully arithmetic-free and can support only a limited specificity contrast.

Derangement control

Every derangement window first attempts a random value derangement within operator/output-width nuisance strata. If a stratum is underdetermined, the construction deterministically coarsens it to output-width-only pooling. Exact empirical target-multiset preservation remains required after either form of pooling, and every assigned value must differ from its source value.

Only if an output-width-only stratum of at most 31 rows makes an empirical-multiset value derangement mathematically impossible may the construction deterministically synthesize canonical same-width decimal labels. The limit of 31 applies separately to each impossible width stratum, not to the window as a whole. Those labels still forbid all source/assigned value fixed points, but the synthetic rows do not preserve the empirical target multiset. The audit records the number and total-window fraction of synthetic rows, rather than treating them as an exact derangement. That fraction has a strict 0.001 (0.1%) cap. The audit records the configured cap and pass/fail result, and the run fails closed if the cap is exceeded. Each constructed window that uses synthetic rows emits exactly one DERANGEMENT_SYNTHETIC_FALLBACK telemetry line. This is an explicit weakening of the random-label control, accepted to prevent rare finite-window impossibility.

The construction therefore:

1. requires exact target-multiset preservation for every nonsynthetic stratum
2. forbids source/assigned value equality for both empirical and synthetic rows
3. never sorts answers by lexical or numeric value
4. records the target assignment, active nuisance strata, coarsening, synthetic-row count and total-window fraction, its 0.001 cap, and cap pass/fail
5. considers the synthetic fallback within a seeded candidate only after exact operator/output-width and exact output-width-only constructions are mathematically impossible; the up-to-eight seeded candidates test the residual-correlation and string-agreement gates, not fallback eligibility
6. audits the initial window before training and every later window at construction time; any configured correlation/string-agreement failure or synthetic-fraction-cap failure stops the run

derangement_max_abs_pearson applies to the absolute row-weighted Pearson correlation across eligible within-stratum correlations. Eligibility is frozen: at least 32 rows, at least 10 distinct values, and output width of at least two. All tiny and degenerate strata, including their correlations, remain in the audit but do not enter that rejection gate. Prefix and suffix agreement are tested as excess above the stratum chance baseline.

Counter-addressed data and splits

Training data are generated deterministically from (seed, namespace, split, counter, stage). Major train, validation, discovery, causal, IID, OOD, and final classes use disjoint relation-family partitions. Addition commutation and addition/subtraction inverse relations share a canonical family ID.

The counter stream is reproducible but does not guarantee global within-training uniqueness; finite arithmetic collisions can occur. The pipeline does not perform a global within-training collision audit, and no report may describe all training rows as unique.

The fixed structural schedule is:

1. no carry/borrow
2. a single transfer event
3. contiguous transfer chains
4. a balanced mixture

Stages use fixed step boundaries, not observed-accuracy advancement, preserving comparable schedules across the paired label arms. Structural evaluation panels include IID trained widths, OOD longer widths, carry chains, borrow chains, and the shared no-transfer panel.

Optimization and longitudinal records

Discovery uses 60,000 optimizer steps per arm and seed; locked replication uses 80,000. AdamW follows a warmup-stable-decay schedule. BF16 is used when the installed CUDA stack supports it, otherwise scaled FP16 is used. Progress is flushed every 100 optimizer steps, including stale-evaluation age. Light free- generation evaluation occurs every 500 steps and full evaluation every 2,000.

Research checkpoints are specified in each JSON profile. The rolling resume state includes model, optimizer, scaler, RNG, stream counter, stage, and runtime compatibility information. Resume refuses incompatible resolved configuration or runtime fingerprints.

Behavioral gates

Free generation is scored against each arm's trained completion, with the ordinary arithmetic oracle also stored for controls. Reports retain exact match, malformed rate, operation, digit width, and carry/borrow structure.

Candidate discovery is forbidden until the correct arm passes its probe_requires IID, carry, borrow, and malformed thresholds. The stronger claim_requires report additionally includes OOD exact match and the minimum correct-minus-deranged arithmetic-oracle margin. The stronger report is retained for scientific interpretation even though v0.4 cannot emit an established claim.

no_carry and pseudo_fst have separate target-task competence gates. Transfer panels for the no_carry model are scored on full arithmetic inputs and record evaluation_target_arm=correct; its training restriction is not silently applied to those evaluation rows.

Target-blind candidate discovery

Discovery accepts prefix token IDs only. Every prefix must end at ByteTokenizer SEP, and EOS is forbidden, preventing teacher-forced answer leakage.

At upstream layer \(u\), sealed random residual perturbations are propagated to candidate source layer \(\ell\). Their source-layer effects estimate

Gradients at \(h_\ell\) estimate observability for exactly nine fixed, answer-independent output consumers:

1. digit log-probability mass
2. EOS log probability
3. minus-sign log probability
4. output entropy
5. top log probability
6. four sealed random vocabulary log-probability projections

Thus

The candidate basis is the leading mapped eigenspace of \(B\). Candidate rank is the smallest registered rank reaching the heldout fraction of target-free consumer-response variance; if none reaches it, the best registered rank is retained with threshold_not_met. Discovery prefixes, rank-selection prefixes, and sealed causal pairs are disjoint.

Layer/rank selection is exploratory in carry_discovery. A modal selection across passing seeds is written to workspace_lock.json. Confirmation accepts only that source layer, upstream layer, and rank.

The fitted rank-matched controls are:

• PCA of discovery source states
• one seeded random orthonormal subspace
• a complementary balanced-spectrum subspace

Version 0.4 does not implement a bootstrap stability analysis or a distribution of many random subspace draws. The empirical upstream perturbation is a local residual intervention, not an exhaustive head/MLP controllability estimate.

Implemented sealed causal assays

The provider constructs semantically matched clean/corrupt pairs differing at a registered carry or borrow event column, with expected affected and unaffected answer-digit positions. Answers are unavailable during candidate fitting and selection. After the candidate is frozen, sealed answer values and widths are used for pair construction, baseline eligibility, directional evaluation, and free-generation scoring.

At the final prefix SEP position, the candidate, PCA, random, and complement bases receive:

1. Coordinate ablation: remove the registered projection around the discovery mean from a clean residual state.
2. Clean-to-corrupt patch: transplant clean donor coordinates into the corrupt recipient state; audit predicted affected digits and preservation of unaffected digits.
3. Independent upstream residual transplant: replace the clean upstream residual with its matched corrupt donor residual and propagate to the source layer.
4. Coordinate rescue: after that upstream transplant, restore only the registered source-layer coordinates from the clean source state.

The causal assay regenerates autoregressively after intervention. Rank-matched controls are energy-scaled to the candidate intervention norm for the same state. Individual outputs and intervention norms are retained.

All primary causal estimands have registered denominators. Ablation counts only pairs whose natural clean generation is a valid EOS-terminated signed decimal and exactly correct. Patch requires both natural clean and corrupt generations to be valid and correct with distinct targets. Rescue further requires the unrescued upstream corrupt-donor transplant to generate the corrupt target exactly. Discovery requires at least 128 baseline-qualified pairs and 32 directional rescue pairs; confirmation requires 512 and 128. Insufficient qualified data fails the corresponding gate.

Coordinate rescue after an upstream residual transplant is an implemented causal diagnostic. It is not formal mediation and does not identify which attention head or MLP wrote the rescued state.

Arithmetic specificity

If both correct and pseudo_fst pass target-task competence and candidate selection, the same geometry and causal battery runs in both models. The arithmetic-specificity diagnostic requires:

• IID target-task accuracy matched within the registered tolerance
• correct-arm candidate-minus-strongest-control effect exceeding the analogous pseudo-FST effect by the registered margin

This is a limited candidate specificity diagnostic. The answer-width nuisance is shared deliberately, so this contrast does not isolate every arithmetic cue and does not replace native shared access or mediation.

Locked replication statistics

Confirmation retains item-level predictions and directional patch records. Seed is the replication unit. For ablation, patch, and rescue, the registered effect is candidate minus the strongest available PCA/random/complement control. The analysis uses:

• paired seed-level effect tests
• Holm correction across the three registered causal contrasts
• family-clustered hierarchical bootstrap for item-level directional patch superiority

The configured confirmation has eight seeds and requires the registered seed count. Passing statistics means that the implemented candidate effects replicated. It cannot lift the release claim ceiling.

Mandatory assays not implemented in v0.4

The following are explicit fail-closed gates:

• formal causal mediation with a registered estimand and intervention chain
• native shared multi-query access from one held state
• component-specific upstream-writer perturbation and downstream coordinate rescue

An unrelated-language preservation panel is also not bundled because the model is arithmetic-only and has no meaningful language competence baseline.

No combination of loss, exact accuracy, balanced-space score, ablation, patch, rescue, p-value, or bootstrap interval in this release can change JSPACE_CLAIM from NOT_ESTABLISHED.