VeriLoop E2 Advances Riemann Hypothesis Zero Proportion to 67.35% with Verifiable AI

VeriLoop E2: Verifiable Post-Training for Code, Mathematics, and Physics

Since summer 2026, frontier large models have entered a new phase of rapid evolution: parameter scales grow, contexts extend, test-time compute deepens, and tool-use capabilities strengthen. Yet a fundamental problem becomes more prominent: more computation does not guarantee more correctness. A code change may fix the main test but break regression tests; a mathematical derivation may reduce local error but violate another constraint; a physical approximation may improve fit but break conservation laws. The question shifts from "can the model compute more?" to "can added computation be proven to constitute genuine progress?"

VeriLoop-Governed Recurrence (VGR) Mechanism

To address this structural gap, a team from Tsinghua University Shenzhen International Graduate School, led by Professor Liu Houdé and Chairman Zhou Chaonan, with postdoctoral researcher Wang Libo and master's students Li Haiqi and Xiang Anlin, developed VeriLoop E2. Based on Qwen 3.8-27B, VeriLoop E2 targets complex reasoning tasks with externally verifiable constraints in code, mathematics, and physics. The core innovation is VeriLoop-Governed Recurrence (VGR), which separates the right to propose candidate states from the right to persist them. When the model generates a new candidate hidden state, it must first be externalized as a checkable artifact via a fixed decoder, then judged by an independent Verifier against a fixed set of protected obligations. Only if all protected evidences are non-regressive and at least one strictly improves does the candidate earn commit rights; otherwise the current certified state is retained.

Permission Separation: Propose, Decode, Verify, Commit

VGR redefines runtime permissions without rewriting Attention or FFN. At run start, task identity, model contract, Verifier identity, Verifier contract, and evidence compiler contract are fixed. A baseline latent state is obtained via standard forward pass, immediately decoded to artifact, and verified to produce initial evidence rank. The first recursion begins from this externally checked baseline, not from model self-evaluation. At each attempt t, a learned proposer reads the current retained latent state, source anchor, last rejected exploration memory, and evidence features, outputting a bounded displacement. The controller checks shape, dtype, device, finiteness, and per-token radius before constructing the candidate state. The candidate then undergoes fixed decoding to produce output, witness, and metadata, bound together as a candidate artifact. The Verifier receives only task and artifact, not model internal scores. The permission chain: standard forward → external verification → bounded candidate proposal → controller constructs candidate state → fixed decode → re-verification → protected evidence comparison → atomic commit/reject.

Submission Criterion: Strict Protected Dominance

Let m protected obligations be fixed at run start. Current retained evidence rank is vector r_t, candidate evidence rank is r_c. Candidate may commit iff ∀i: r_c[i] ≤ r_t[i] (no regression) and ∃j: r_c[j] < r_t[j] (strict improvement). For example, if current rank is (0,1,1), candidate (0,0,1) may commit; candidate (1,0,0) must be rejected even though total error drops from 2 to 1, because the first already-satisfied obligation regresses. This avoids "total score improvement" rules that can mask regression on protected coordinates. The persisted object is a CertifiedBundle containing artifact, latent state, verification result, and binding digest. Rejection preserves the entire prior certified bundle; rejected displacement is stored in exploration memory to influence next proposal but gains no persistence rights. A potential function Φ mapping evidence rank to non-negative integer ensures finite strict commits: each commit reduces Φ by at least 1, so total commits are bounded.

Post-Training: Converting Verifiable Progress into Learning Signals

VGR solves runtime "what results qualify to stay"; post-training asks "how to make qualifying candidates more likely to be proposed". Training samples are constructed around the same incumbent: multiple candidates generated from one incumbent, decoded with fixed decoder, verified by fixed External Verifier. Only candidates satisfying strict protected dominance become positive samples; others (no-op, regression, Pareto-incomparable, total-score traps) become negative. The preference loss uses teacher-forced sequence log-probabilities of chosen vs rejected candidates with a softplus margin. Crucially, intermediate progress (non-zero rank but strict improvement) supervises action SFT/NLL (teaching "how to continue fixing"), while only zero-rank accepted final artifacts enter certified generation NLL (teaching "what constitutes a terminal output"). This avoids conflating valuable intermediate actions with completed answers. Across domains, verification objects change but logical structure stays consistent: code learns modifications that fix target errors while preserving passed tests and build/interface constraints; mathematics learns derivations that don't break already-closed conditions; physics learns to improve one fitting target while maintaining other physical constraints. The Verifier remains a discrete external fact producer, not a learnable scorer; optimizer learns to increase probability of committable candidates, while deployment-time commit decisions remain with external Verifier.

Technical Contributions: From State Evolution to Protected State Transitions

VGR addresses six specific gaps converging to three research contributions: (1) introducing protected partial order into state submission — progress defined as all protected coordinates non-regressive with at least one strict improvement; (2) binding state, artifact, and verification result into an atomic persistent object — Verifier checks actual checkable consequences via fixed decoder, rejection preserves entire predecessor certified bundle; (3) extending the same submission predicate to post-training supervision interface — runtime commit qualification and training positive-sample definition share identical evidence semantics, eliminating semantic misalignment between training reward and deployment gating.

Riemann ζ Critical-Line Zero Proportion: 67.25% → 67.35% Strict Finite-Dimensional Certificate

Positioning: From Public Baseline into Verifiable Research Loop

Beyond static benchmarks (9 benchmarks: SWE-bench Pro 76.2%, Terminal-Bench 2.1 88.8%, Terminal-Bench 3.0 29.7%, Terminal-Bench 4.0 37.9%, DeepSWE v1.1 64.6%, SWE-Marathon v1.1 45.0%, AIME 2026 98.3%, GPQA Diamond 93.94%, Apex 2025 89.6%), the team tested dynamic reasoning on an open research task: improving the lower bound κ of the proportion of non-trivial Riemann ζ zeros on the critical line Re(s)=1/2. The baseline was Anthropic's ~67.25% result and proof framework. VeriLoop E2 with a non-public Harness formed an evidence-driven loop: model proposes candidate structures, reconstructs problem representations, selects research lines, adjusts strategies from failure evidence; Harness fixes baseline and result boundaries, converts unknowns into evidence obligations, organizes counterexample search, deterministic computation, interval verification. Only observations that decisively support or refute current judgment enter next research state. Model retains exploration freedom but cannot self-decide what is established.

Reasoning Pivot: From Window Optimization to Matrix/Pressure Certificate

After the 67.25% baseline, E2 separated window-only optimization from matrix-term contributions. The upstream relation remains κ = H(v) + Δ(M). New candidate's window witness H(v) > 0.672167187145431; raising H(v) alone insufficient. E2 reconstructed the research object from "keep polishing numbers" to "establish independently verifiable finite-dimensional positive lower bound for Δ(M) and reassemble into κ". The new line was compressed into a finite certificate chain: matrix structure localization → explicit energy lower bound → strict local inequalities → pressure global envelope → exact rational assembly. Harness verified each obligation; any unclosed level kept candidate in exploration; only full evidence chain reassembling to same κ lower bound earned freeze rights.

Counterexample-Driven: From Higher Candidate Rejection to 67.35% Closure

Counterexamples drove recursion. E2 produced two higher candidates (67.350352375073% and 67.35006335392536%), but Harness's strengthened counterexample search found early seed families covered only ~1.04 and ~1.975, missing a large gap near ~2.915. E2 concluded the issue was incomplete search representation, not insufficient compute, and expanded seed family to {1.04, 1.975, 2.915}^q; local εs re-bounded, original candidate lost strict acceptance. Higher numbers were actively abandoned; counterexamples became input for next search representation. At freeze diagnosis (q=12 peak, thin-sample inflation at higher dimensions), E2 elevated "three-level seed complete enumeration, deterministic re-test" to protected condition. A low-cost short hierarchy at 67.275055959117140% verified a key strategy: derive required certificate margin from target threshold, then decide verification precision and compute budget, making proof cost driven by conclusion need rather than "compute a bit more". After low-cost loop verification, same certificate structure pushed to 67.35% target, compressed to three local inequalities:

s = 1/2, ε = 526/78125 = 0.0067328

s = 19/20, ε = 9879/1250000 = 0.0079032

s = 1, ε = 20033/2500000 = 0.0080132

Strict path completed 327/327 hard wells, 3/3 local certificates, 190,375,830 branch-and-bound nodes failure-closed; verify_exact.py reassembled with exact rationals reproducing 67.350003708785593%. Final verification expanded float bounds toward safety, used π intervals and Taylor remainder envelopes for trigonometric functions, upgrading "no counterexample found" to failure-closed coverage over specified finite domain. Conclusion remains strictly a finite-dimensional computer-assisted certificate; upstream Zeta23 formal bridge and Lean/nanoda end-to-end replay are ongoing.

Quantization Releases

Official GGUF quantizations released alongside standard weights: BF16, Q8_0, Q6_K, Q5_K_M, Q4_K_M, Q3_K_M, IQ2_S, IQ1_M. Quality judged against canonical BF16 GGUF under frozen paired protocol measuring PPL, KL divergence, token probability drift, Same top-p. Distinction made between quantization fidelity and downstream capability loss; full 9-benchmark re-run not performed for each quantization, so PPL drift (e.g., +0.3191% for IQ1_M, +0.4450% for Q5_K_M) cannot be interpreted as proportional task score drops. Developer selection by deployment target: Q6_K (20.566 GiB, -58.96% vs BF16, PPL -0.0395%, Mean KLD 0.004409) quality/efficiency sweet spot; Q5_K_M (18.965 GiB, -62.16%, PPL +0.4450%) memory/quality sweet spot; IQ1_M (16.790 GiB, -66.50%, PPL +0.3191%, Mean KLD 0.014357, Same top-p 95.870%) minimal footprint sweet spot; IQ2_S (16.799 GiB, Mean KLD 0.014023) lower-footprint alternative with better distribution fidelity. Note: IQ1_M and IQ2_S are mixed-precision, not uniform 1/2-bit; model file size ≠ actual VRAM usage (KV cache, compute buffers, optional MTP need headroom).

Philosophical Stance: The Future Should Not Be Finished

Contrasting Dario Amodei's call to slow frontier capabilities for alignment to catch up, VeriLoop pursues advancing frontier capabilities while simultaneously growing verification, safety, and governance capabilities. The divergence is not on safety's importance but on its nature: Anthropic seeks enforced risk control chasing model capability; VeriLoop builds verification, safety, and responsibility boundaries as capabilities that co-expand with the frontier. Strong intelligence tempts compression of world uncertainty; but if future is pre-computed by superior intelligence, human choice degrades from creating future to confirming answers. A fully optimized world may not be a truly free world. Unknown is not a bug but the space where civilization continues to happen. Science's greatness lies not in eliminating all problems but in turning unknowable into askable, problems into testable propositions, opened by new evidence. VeriLoop aims not to give AI more answers for humans, but to give humans ability to reach deeper unknowns. What should expand is not "answer inventory" but "cognitive frontiers humans can personally reach". Exploration can be aggressive; commit must be strict; capability can grow fast; evidence standards must rise in lockstep. Safety need not be a brake ahead of capability; it can be a structure co-evolving with capability. Team plans to release programming agent Voder based on VeriLoop Harness, making "model proposes — external verifies — rollback/commit — evidence freeze" chain independently auditable and extensible. The deeper fear: humanity's longing for omniscient, uncertainty-eliminating entity. If we cast that longing into a machine and surrender judgment, choice, meaning to it, technology hasn't helped transcend myth — it has engineered myth for the first time. From oracle to algorithm, priest to model, if the end is still humans ceding agency to higher authority, the vessel changed but spiritual structure didn't. VeriLoop needs no such god. Humans need no savior to complete their future. The boundary to guard is human subjectivity: machines can help understand world, not prescribe its meaning; compute possibilities, not monopolize what's worth choosing; see farther, not thereby own the future. The stronger the intelligence, the more critical this boundary. AGI's endpoint is not completing the future for humans, but expanding the unknown humans can still personally reach. If future is fully predictable, failure fully eliminable, death fully excludable, humans may gain absolute safety but lose the very act of transcending fear. No unknown → no genuine choice; no unfinished future → humans no longer authors of future. Civilization's greatness lies not in how many fears it ultimately erased, nor in finding a guarantor of endings, but in that, knowing experiments may fail, theories may be wrong, models may regress, unknowns may persist lifetimes unanswered, we still choose to enter. No one can guarantee the endpoint; no one should complete it for humanity. Hope comes not from a guaranteed future, but from the future being unfinished. And humans, in an unfinished, unguaranteed, savior-less world, still choosing to explore, bear responsibility, correct, and advance — that will has a name far older than AI: courage. The future should not be finished; it should be continually opened.