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Theory and hypothesis registry

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# Preregistered experiment and theory registry

## ROUTE-001: four learned settings

Question: can gradient training recover two order-sensitive token transformations,
then export gates that preserve those transformations under integer execution?

Three labeled lanes start at `(0,1,2)`. Oracle A swaps 0/1; B swaps 1/2. The candidate
has fixed switch stages 0/1 then 1/2, selected by a 2×2 learned logit table. No encoder,
trainable readout, controller state or target input can bypass these switches.

Five seeds 0–4, float64 CPU, Adam learning rate 0.1, 1,000 updates, final-state MSE.
Training batches contain 48 generated sequences with max lengths 2, 4, 8 over thirds
of training. Temperature decreases linearly 1→0.1. Initialization and random generator
details are recorded by the trainer. A positive logit exports swap; zero/negative pass.
Checkpoint content is JSON, not executable pickle.

Acceptance is exact agreement for both tokens across all six states in every seed.
This checks the complete token transition function; composition then follows by
induction for finite valid sequences. Additional checks exhaust lengths 0–8 and use
fixed longer sequences, resets, malformed inputs, and corrupted gates to test the code.
All-pass/initial controls must not be mistaken for learned successes. Initial settings,
soft MSE, soft/hard disagreement, learned logits, timing, memory and artifact size are
reported even when the run fails. No speed superiority is claimed.

Exhaustive short-sequence verification deliberately overlaps training support. Fixed
sequences of lengths 17–256 are length holdouts, not evidence of semantic generalization.
Evaluation metrics do not enter the optimization. Changes after seeing results require
a new indexed protocol; failures remain in their original receipts. A failed seed can
be rerun as a retained new attempt, but repeated attempts of one seed are not new seeds.

## PHASE-001: equivalent coordinates

For cycle sizes 2–16 check integer permutation closure, Fourier reconstruction,
spectral projector idempotency/orthogonality/completeness, and routing/spectral agreement.
Use complex arithmetic with double-precision components and absolute tolerance 1e-10.
Check a noncommuting A/B example. Compare modular clock periods using the LCM; primality
is not necessary for a product period when lengths are pairwise coprime.

## Theory statuses

| ID | Statement | Status/use |
|---|---|---|
| T1 | An m-cycle raised to m is identity; roots of unity diagonalize it | Established finite algebra; implementation checks PHASE-001 |
| T2 | Two noncommuting swaps can preserve sequence order | Established; exact ROUTE-001 oracle |
| T3 | Joint modular-clock period is the least common multiple | Established; distinguish primes from coprimality |
| H1 | This four-scalar relaxed model can recover exact token transitions | Empirical hypothesis; ROUTE-001 |
| H2 | Prime-derived choices improve useful language processing | Unestablished; no performance claim in this release |
| C1 | Riemann hypothesis | Unproved conjecture; no runtime dependency |

Primary references: [Neural Shuffle-Exchange Networks](https://papers.nips.cc/paper_files/paper/2019/hash/9001ca429212011f4a4fda6c778cc318-Abstract.html),
[Gumbel-Softmax](https://arxiv.org/abs/1611.01144),
[ByT5](https://aclanthology.org/2022.tacl-1.17/), and
[Clay's Riemann hypothesis page](https://www.claymath.org/millennium/riemann-hypothesis/).
These establish adjacent work and mathematical context, not novelty or IP clearance.

## REP-001: operational sufficiency

The [representation protocol](REPRESENTATION_PROTOCOL.md) implements the user's
proposal `F_a E = E T_a` together with the independent observation condition
`D E = o`. The experiment separates exact finite-state algebra from empirical
learning. The scientific outcome is descriptive across all paired seeds;
completion is not an assertion that the consistency objective won.

| ID | Statement | Status/use |
|---|---|---|
| T4 | A deterministic latent transition exists exactly when represented-equivalent states have represented-equivalent successors | Elementary factorization criterion; exhaustive finite controls |
| T5 | Hard-partition Markov lumpability is equality of block-transition probabilities within each block | Established; row-stochastic convention and optimal refit explicitly tested |
| T6 | Exact intertwining on all generators implies preservation of their finite compositions | Induction; does not require distinct generators to commute |
| H3 | Adding latent consistency improves decoded unseen-composition predictions at a fixed architecture and update budget | Empirical REP-001 comparison; no universal assertion |
| H4 | Operational failures identify additional context needed by a human language system | Future hypothesis; finite explicit witnesses establish only the specified synthetic case |

Topological similarity, geometric similarity, dynamical preservation and human
interpretability remain distinct measurements. A latent optimization failure
does not by itself prove lost information. Prior work on Koopman-style learned
embeddings already combines reconstruction, prediction and latent consistency;
REP-001 tests this choice locally rather than claiming a new general theory.

## Historical HDD inventory (read-only, unreplicated)

- `G:\development\zetasheaf-ablation`: reported null against GUE and similar lattice
  performance. Source-available restrictive license; no code/data copied. A null does
  not establish equivalence or a universal result.
- `G:\development\maxop-lm`: prior byte-level delta/attention preregistration and
  checkpoints; root license not found. No checkpoints loaded or datasets reused.
- `G:\development\ig-primon-t1`: proprietary math harness; no implementation copied.
- `G:\LLM-Research\atft-llm-probe-2026`: historical probe documentation; large archive
  remains unopened/unextracted. No implied verification of archived claims.