The Σ₀ LLM — design of record for the owned local model

1. The claim (falsifiable, Pareto form — imported from #2877 verbatim)

Refused: raw leaderboard SOTA at this size (open 32B-class ≈ closed frontier on SWE-bench; parameters are not our axis).

Made: on the operating point ⟨local ≤8GB box, verified answers, deterministic serving, bounded cost⟩, the best measured point on the cost–reliability frontier: more verified pass@1 than any single ≤8GB model call at ≤K× its cost, with precision-of-claimed-solve ≈ 1.0 (when it says "solved," tests pass). Registered against BENCHMARKS.md; the missing headline run is named in §8.

2. Hard constraints (operator envelope, 2026-07-23 — non-negotiable)

3. Architecture — one small looped core, verified outside

user task (verifiable: code/math/structured)

   ↓

[CHEAP TIER — pure looped recursive core, ≤3B, quantized→ternary]

   N samples, adaptive loop depth (Q-exit), JSRR ρ<1 accept-gate per §5

   ↓  real exec verifier (Fix-Rate) picks the first verified candidate

[verified? → done at cost ≈ N·c₀]

   ↓  stall (verifier-confirmed, not self-assessed)

[ESCALATION TIER — 7B-class local (capable boxes) or cloud frontier]

   inherits accumulated progress (ADR-0030 spiral memory); rescue logged as VTD fuel

   ↓  still unverified → HONEST HALT ("cannot verify"), never a fabricated pass

• The mechanism is verifier amplification, not model size (#2877 §2.1, core math IMPORTED

from pass@k, Chen et al. 2021): P(verified solve) = τ·(1−(1−p)^N) + (1−τ)·FA(N). At the ≤3B tier's smaller p, N cheap CPU samples × a real exec verifier is the product, not an optimization. MEASURED anchors: coding-golden exec pass@1 0.96 (#2173); 7B Q4 HumanEval-164 0.829 single-shot (now re-scoped to escalation-tier reference); cascade 8.3× cheaper at e≈0 with a strong cheap tier, rescue 88.4% > 84.8% with a weak one (#2798/#2800); fully-local 0.5B→7B cascade 18/18 MBPP-basic at e=6% (ADR-0030 Phase 0).

• Cascade policy is FIXED (propose → verify → escalate), not learned. The learned

value-of-information router is a late-stage, falsifier-gated bet (Grok red-team rank #2; both reviews concur): it is built only after the fixed policy is measured, and dies unless it beats the fixed policy on verified-skill/sample/watt.

• Held-out gate (NEW, forced by measurement). experiments/spiral_arc_smoketest.js

(2026-07-23, real harness + real python exec) reproduced the transduction trap: a memorizing program passes all visible tests (Fix-Rate 1.0 → "solved") yet fails held-out. Therefore visible-test pass is necessary, not sufficient: whenever tests can be split, verification holds out a subset; τ (test adequacy) is carried as a real term, not a footnote (hidden-test false-accepts 5–15% in literature; unmeasured in-repo — NOT-VALIDATED).

3.1 The core: pure looped recursion now — MoE deferred behind certification, NOT rejected

The v1 cheap tier is a weight-tied looped transformer with adaptive exit (Ouro-style; the paper's third scaling axis — loop depth, not width), because that is the only architecture our own stability machinery covers today: certificate §1.2.2 states a routed MoE-recurrent loop is a switched system that Part I does not certify (per-route contraction + dwell-time conditions are needed), and a repo-wide grep (2026-07-23, recorded in data/oracle/active-loop-runs.jsonl) confirms no switched-system / dwell-time tooling exists yet.

MoE is a live future direction, not a rejection (operator, 2026-07-23). A future fused design (anticipated, not yet specified) is expected to bring expert width into the looped core. The certificate itself names the on-ramp; this design pre-commits to it as the MoE admission gate — the future design enters v-next when these bricks exist:

1. Switched-system certification tooling — per-route contraction + average-dwell-time gate

(multiple Lyapunov functions; the certificate's own cites: arXiv:2405.03560, 2303.17858, 2008.06546) built and validated the way JSRR was: synthetic known-ρ cases first, then the real loop.

1. Route-switch dynamics measured as a stability quantity — frozen-route contraction + the

dwell-time statistics the model actually induces (certificate §1.2.2 consequence 1).

1. Expert-choice routing preferred (arXiv:2202.09368 — experts pick tokens; fixed capacity,

balance by design) as the lower-discontinuity comparator to token-choice top-k.

1. Candidate architectures held warm, not discarded: tied/universal experts shared across

recursion steps (MoUE; Tied-Expert-MoE arXiv:2606.16825; MoR arXiv:2507.10524) — width folded into the loop rather than bolted beside it, which also minimizes the switched-system surface the gate must certify.

Until the gate exists, quoting §1 stability numbers for a routed loop is forbidden (certificate §1.2.2 consequence 3) — that is the only thing being enforced here.

3.2 Anytime budget mode (operator scope enhancement, 2026-07-23)

The cascade's cost posture is a per-task budget dial, not a global minimization objective. The spiral is an anytime algorithm: it can keep proposing → verifying → refining on one problem for as long as the budget allows, and its answer only improves (the ratchet commits only verified advances). Four levers open when the dial opens, ranked by expected impact:

1. Raise N under verifier amplification — more cheap samples before escalating (pass@k

compounding; cost ~linear in N).

1. Structured search over candidate programs, not raw sampling — beam/evolutionary/inductive

refinement; the known high-value path on ARC-style work. "More spiral helps; smarter spiral helps more."

1. Escalate earlier on hard signals — probe/uncertainty streaks + failed-verify runs route

budget away from tasks the small core cannot solve (uncertainty-aware budget allocation is an activelane: arXiv:2604.14853, 2605.26849, 2606.04402 — IMPORTED, not yet run in-repo).

1. Teacher-as-repair, not replacement — escalation sends the *best local candidate + the

failing tests + partial progress*, never a blank prompt. Caution carried honestly: self-repair feedback in frozen small models is partly placebo (arXiv:2606.31511), which is why repair here means real execution feedback + a stronger repairer — a configuration we've already measured working (rescue 88.4% > 84.8%) — never tiny-model self-repair (measured harmful, 6/6→2/6).

Scope honesty: this raises score within the efficiency band; it does not chase the $10–$200/task frontier cluster. Diminishing returns on raw N are real — the budget buys structured search and repair, not indefinite blind sampling.

Base-model choice is a measured decision, not an assumption: best of {Qwen2.5-Coder-1.5B/3B-class, Ouro-1.4B native loop} by verified-pass@cost + probe signal on the reference box, with retrofit recurrence (arXiv:2511.07384) available to add looping to a non-looped winner. LoopFormer-style variable-unroll training with short/long-trajectory consistency (arXiv:2602.11451) is the candidate recipe for making one checkpoint serve both shallow-cheap and deep-hard. (Both IMPORTED; neither run in-repo — NOT-VALIDATED.)

4. Honesty stack (what "feels like Claude" without the cloud)

5. The certificate as an enforced contract (the Grok-flagged gap, closed by rule)

Rule: every Σ₀ serving or training path MUST run the certificate's runtime gates, or carry an explicit written justification for skipping. Today's reality, grounded by grep 2026-07-23:

Honest caveats carried verbatim: Part-I theorems are in-regime (local linear Jacobian), so grounding remains the actual safety mechanism; JSRR is validated as the right stability object, not a quality win (CART's null result); the empirical-Jacobian proxy, not the true JVP, feeds the gate today (#2029 deferred to a GPU run).

6. Training path (owned weights, staged, gated)

Sequential, never joint (Grok rank #3: joint ternary multi-component training is the brittle bet): (a) pick the base by measurement (§3.1) → (b) VTD on the owned escalation corpus + exec-verified open sets (TACO Apache-2.0 primary; KodCode NC-tagged) — dose-response already MEASURED: −6 attraces, ±0 atgentle; crossover needs order-of-magnitude more traces → (c) RLVR with Fix-Rate reward (ADR-0025, double-gated) → (d) ternary BitDistill (ADR-0026) with #2873 probe-survival as the critical-path acceptance test. Every promotion passes Σ_θ on fresh held-out — which also finally gives Part II its first real-run evidence. GPU spend is real money and sits with the mookman handoff (#2850); nothing here presumes it.

Explicitly deferred for v1 (with the reason on record): MoE core (deferred behind the §3.1 admission gate — a future fused design is anticipated; certification bricks unlock it, and building them is a valid parallel research lane) · learned VoI router (falsifier-gated, late) · verifier internalization (Grok rank #1 most-dangerous bet — the external verifier stays the ground truth and held-out anchor; an internal PRM head may only ever pre-rank candidates) · hypervisor / hot-swap contracts and live session migration (systems elegance, zero capability; revisit only when shipping multiple backbones) · from-scratch pretraining (7.7T-token territory).

7. Product layer (latency/cost levers + the IP reality)

• Speculative decoding (draft-model) is the biggest cheap latency win for CPU serving — and

an actively patented area: WO2024205726A1, US12229192B2 (granted 2025), US20250245430A1. Use upstream implementations (llama.cpp/EAGLE-family); do not claim novelty; FTO check before any commercial packaging.

• Edge quantization: GB2641319A (joint weight-equalization + activation-range for

edge-destined LLMs) is the closest patent to our lane; Q4_K_M measured free on-box (0.829 = fp16 baseline, quant-cliff run) — MEASURED.

• Aggressive caching/prompt reuse; deterministic serving for reproducible receipts.

8. Falsifiers & registered benchmarks (what would prove — or kill — this design)

1. The missing headline number (queued first): full HumanEval-164 verified-cascade run at

the ≤3B product tier (N samples + exec verify + escalation), cost-instrumented → one point on the §1 frontier vs. the 0.829 single-shot 7B reference. (Named missing in #2877 §4.)

1. Verifier-amplification check: does measured verified-pass at N samples track

1−(1−p)^N within the FA(N) band? (First in-repo measurement of τ.)

1. ARC-AGI-2 budgeted track (ADR-0031): land on the cost-efficiency Pareto frontier

(NVARC 24% @ ~$0.20/task band) via inductive program synthesis with the held-out gate.

1. Stability under depth: ρ-trajectory + peak-then-collapse profile of the chosen core on

real tasks (the #2029 JVP run) — the JSRR gate's first live-workload validation.

1. Router bet (late): learned router must beat the fixed policy on verified-skill/sample/watt

on the reference box, or it is dropped without sunk cost.

1. Ternary survival: #2873 — probe AUROC and verified-pass must survive W1.58A8, else the

capability path falls back to 4-bit and the §2 envelope is re-negotiated.

1. The two-budget experiment (§3.2's own falsifier): same ARC-AGI-2 split, two runs —

low budget (minimal-escalation cascade) vs high budget (anytime spiral: higher N, structured search, early escalation, repair prompts). Report score, $/task, score-per-dollar, and the local-vs-escalated solve split. This measures exactly what the budget dial buys.

9. Build order (each step measured before the next)

P0 wire the gates (JSRR on default path; arm Σ₀⁻¹ + canary, bounded) — no new weights → P1 base-model bake-off at ≤3B on the reference box (§3.1) → P2 run falsifiers 1–2 (the headline cascade number) → P3 VTD→RLVR→ternary behind Σ_θ (§6) → P4 falsifier 3–4 → P5 router bet, only if P0–P4 hold.

R (parallel, non-blocking): the MoE admission gate — build the §3.1 switched-system certification bricks (dwell-time gate first, synthetic-validated like JSRR). This lane never blocks P0–P5; when it lands, the operator's future fused MoE design has a certified doorway instead of an exception.

10. References (verified this session)

#2877 / docs/research/2026-07-23-small-model-sota-theory.md · Collapse Certificate (incl. §1.2.2, §1.2.3, §3, §8) · STARS arXiv:2605.26733 · CART arXiv:2606.01495 · Ouro arXiv:2510.25741 · Retrofitted Recurrence arXiv:2511.07384 · LoopFormer arXiv:2602.11451 · MoR arXiv:2507.10524 (rejected-for-v1 context) · Snell arXiv:2408.03314 · BitDistill arXiv:2510.13998 · TernaryLM arXiv:2602.07374 · pass@k Chen et al. arXiv:2107.03374 · FrugalGPT arXiv:2305.05176 · patents WO2024205726A1, US12229192B2, US20250245430A1, GB2641319A · Grok red-team transcript (2026-07-23, operator-relayed) · grounded pulls in data/oracle/active-loop-runs.jsonl.