Prompt-Selected Presented Algebra

Status: repaired mechanics pilot passes; learned compiler not yet implemented; no novelty or general reasoning claim.

Thesis

Shohin's recent mechanisms can fit source consequences, preserve invariants, maintain particles, lower contradiction, and solve episode coefficients. None forces the latent representation to obey an algebra that determines behavior outside the observed witnesses. The one positive precedent is S7: once a learned Cayley generator was the only route to the answer, unseen-law recurrent execution reached 100% while an exact-fit ordinary Transformer failed.

Prompt-Selected Presented Algebra (PSPA) generalizes that principle. A source does not compile an opaque vector or an arbitrary feature basis. It selects and fills a finite presentation:

P(X) = (carrier slots C, generator actions {A_g}, relations R, query word q)

Each A_g is a complete action over anonymous carrier slots. A tied runtime composes generator words. Prompt relations and source witnesses constrain the same actions that execute the late query:

relation_energy = sum_(u=v in R) distance(A_u, A_v)
witness_energy  = sum_(x,w,y) CE(A_w one_hot(x), y)
answer          = A_q one_hot(query_start)

Whole action tables persist across recurrent completion. Coordinates from different presentations are never averaged. The query is absent from action completion and can only execute the completed presentation.

Architectural separator

PSPA is not a standard Transformer, MoE, LoRA, recurrent-depth block, latent scratchpad, particle average, kernel regression, or external symbolic tool. Its defining restriction is that prompt-conditioned reasoning state is a finite algebraic presentation and every update must preserve the semantics of generator composition.

Finite groups, semigroups, equivariant networks, differentiable interpreters, neural theorem proving, and learned Cayley tables are prior art. S7 is direct internal prior art. Any novelty claim is therefore provisional. The candidate separator is language-conditioned selection and completion of a presented algebra with one tied model-owned executor, rather than hard-coding one cyclic topology or emitting operation embeddings.

Minimal discriminating gate

The first mechanics board uses anonymous, freshly permuted carriers and three action families:

1. cyclic/affine actions;
2. noncommuting dihedral or permutation-generator actions; and
3. random multi-generator permutation actions with repeated binding.

Training uses words of length 1--4. Frozen development uses lengths 8 and 12, new carrier renamings and unseen generator words.

The mechanics harness evaluates:

• PSPA: source-challenge-constrained whole-generator completion and exact tied word composition;
• RECURRENT: an unconstrained tied dense state updater;
• TRANSFORMER: an ordinary two-layer sequence model;
• SHUFFLED-CHALLENGE: PSPA with source challenge outcomes exchanged across episodes; and
• LINEAGE-SWAP: PSPA with selected whole presentations exchanged across episodes before query execution.

The combined harness has 180,054 trainable neural-control parameters. The structured PSPA runtime itself is deterministic and receives strictly less trainable capacity than both controls. This first result is therefore a favorable-control mechanics test, not the final parameter/FLOP-matched learning claim. A pass authorizes a learned language-to-presentation compiler and its matched gate; it does not by itself establish a superior learned architecture.

The first seed is a fail-fast mechanics pilot. PSPA advances to a learned compiler only if it beats the best non-intervention control by at least five macro points and gains on every family, reconstructs at least 95% of complete generator tables, and loses at least five points under both causal interventions. The later learned compiler's five-seed gate requires:

• at least +10 absolute OOD macro points over the best matched control;
• positive gain on every family in at least four of five seeds;
• at least five points lost under exchanged relations;
• at least five points lost under generator-lineage swaps;
• at least 95% primitive generator fit but no answer supervision leakage; and
• one unchanged-weight confirmation pass after all development gates pass.

Failure of the first seed closes this exact structured compiler/runtime without width, duration, seed, loss, carrier, or challenge-set tuning.

Repaired mechanics result

The first immutable runtime exposed three ties among 6,144 development episodes because six source challenges did not always identify a unique complete multi-generator presentation. This was an oracle-generation defect, not a model error. The repaired generator constructs up to eight whole-candidate challenges and proves that every wrong complete presentation is eliminated before evaluation. Exhaustive local replay then recovered all 6,144 presentations and answers exactly.

The repaired seed-43 H100 pilot used 1,000 updates, 128 examples per update, 180,054 combined trainable control parameters, and 128,000 charged examples. It completed in 80.512 seconds at 1,589.829 examples/s.

PSPA is 100% exact on cyclic, dihedral, and random-permutation families at both length 8 and length 12. It clears the fixed mechanics requirements by 89.160 points over the best neural control, with large causal losses under both interventions. The result file is artifacts/reasoning/pspa_presented_pilot_071229a/seed43.json, SHA-256 b88aabc9d09ec4dce2790efd7a12814722c132e8d104d580f96468b966842539.

This is a mechanics result under favorable controls. The structured compiler still receives anonymous tables and exact challenge tuples rather than language. The next gate must learn the language-to-presentation map, retain the tied model-owned executor, remove deterministic candidate enumeration at inference, and compare against parameter- and total-FLOP-matched neural controls. A failure there closes PSPA despite this perfect mechanics score.

Resource envelope

Board generation and oracle/identifiability checks are CPU-only. Eight focused tests and an end-to-end CPU smoke already pass. At lengths 8/12, all three families have 100% exact generator reconstruction and answers; shuffled source challenges score roughly 44--67% and lineage swaps roughly 5--27% in the smoke.

The first full control pilot is one single-H100 job because one checkpoint trains both favorable controls on identical examples and reports PSPA plus both interventions. It has a one-hour Slurm ceiling, 1,000 updates, 128 examples/update, and an expected runtime under 15 minutes. Hard allocation ceiling is one H100-hour.

Only a passing mechanics pilot authorizes implementation of the learned compiler and the originally specified matched four-arm, five-seed matrix. That later matrix remains hard-capped at 20 H100-hours. No long pretraining run is authorized by mechanics or training loss alone.

Immediate implementation order

1. implement exact anonymous-carrier whole-generator composition;
2. prove query-late ownership and no fieldwise action mixing;
3. generate and audit the three-family depth-shift board;
4. implement recurrent and Transformer favorable controls;
5. pass focused gradients and CPU smoke;
6. run one combined seed-43 H100 pilot; and
7. implement the learned language-to-presentation compiler and freeze its parameter/FLOP-matched gate before scoring it.