What it is. The framework's first step beyond discrete answer sets, in two tracks. Track 1 replicates the "turtle paper" (outcome-based RL to predict the future) on Polymarket binary questions: the model reads a question written before resolution and outputs a probability, rewarded by a proper scoring rule once the outcome is known. Track 2 moves to continuous targets, where the model outputs a full density — first a single Beta distribution, then a mixture of Betas. Everything is GRPO + LoRA on Qwen3-4B (scale probes at 14B/32B), trained on 4.7k synthetic "future-as-label" examples, tested on 1,265 real Polymarket questions (zero train/test contamination verified by exact-match and Jaccard checks).

Track 1 — Polymarket binary forecasting

Headline table below is the historical parse-conditioned view — each model scored only on the questions it parsed. The parse % column is the new addition that exposes the catch: the Beta-format methods abstain on ~20% of questions, so their Brier is not comparable to the ~99%-parse point methods. The set-matched comparison directly below corrects this.

Brier/ECE/Acc here are each model's own parsed subset (1,265-question turtle test set). Parse % from forecasting_matched.py; Beta-format rows abstain heavily (PP-Beta 4B parses 1014/1265). o1 / ReMax / Polymarket predictions are full-coverage references shipped with the test set.

Findings.

• Log > Brier as RL reward (0.241 vs 0.256): the quadratic Brier penalty is too gentle on confident-and-wrong predictions.
• The Beta output format is a structural regularizer — but part of the apparent edge is abstention. A 14B base model asked to emit \(\mathrm{Beta}(\alpha,\beta)\) with zero training posts 0.211 Brier with 2% extremes. For a binary outcome the Beta marginal likelihood reduces exactly to \(\log \mathbb{E}[\theta]\) — identical to a point probability — so any gain is a change in the reasoning pathway, not the scoring. But it parses only 81% of questions: on the set-matched subset (below) the 14B point-vs-Beta gap shrinks from 0.347→0.211 (=0.136) to 0.282→0.200 (=0.082), so ~40% of the "massive" advantage was the Beta format abstaining on the hard questions the point format had to answer.
• Bottleneck is training data, not algorithm: the gap to ReMax (0.022 Brier) tracks their 110k vs our 4.7k training examples.
• Entropy bonuses and probability-clamping hacks both failed to generalize.

Example: one question, three prompt formats (click to expand)

The same question ("Will Apple launch an iPhone SE on February 19?", gold = No) answered by the 14B base model under three prompt formats — full reasoning and completion each:

Set-matched comparison (added 2026-06-18)

Parse failure is informative abstention: a model that declines 20% of questions is scored only on the (easier) 80% it answered, so a parse-conditioned Brier flatters it. The project-wide fix (see Overview → Measurement standards) is to compare every method on the matched subset — the questions all methods in the group parsed — and to show parse rate beside the score. Full-coverage references (o1, ReMax, Polymarket) are scored on the same subset. Tooling: 4-regression-forecasting/forecasting_matched.py.

The 4B reward comparison (log vs Brier vs Beta-format RL), matched n=1005:

On the matched subset Beta-format RL still leads the 4B field (0.213 vs log 0.232) — that win is real, not an artifact. But log/Brier improve when restricted to the questions Beta answered (0.241→0.232, 0.256→0.236), confirming those 1005 questions are easier; and Beta carries a 20-point lower parse rate. The honest reading: Beta is better per answered question while answering fewer of them.

The base-model format/scale comparison (14B/32B point vs Beta), matched n=964:

The point format's Brier falls hard on the matched subset (0.348→0.282): the questions Beta abstains on are exactly where the point format flails. The Beta win survives matching (0.200 vs 0.282) but is ~40% smaller than the parse-conditioned gap implied.

Example rollouts — per method (click to expand)

For each method, a confident-correct rollout and a confident-wrong one (the overconfident miss is the failure a proper score punishes). Each shows the full prompt, reasoning, gold resolution, and completion.

Track 2 — continuous targets and branching uncertainty

For continuous outcomes the model emits an XML mixture of Betas (K ≤ 4 components, normalized weights), inducing \( p(y) = \sum_k w_k\, \mathrm{Beta}(y;\alpha_k,\beta_k) \) on the range-normalized target, rewarded with the mixture log-likelihood. Unlike the binary case, here the full density is scored — mixtures can genuinely help.

Synthetic branching benchmark — the architectural test

Oracle ceilings: true mixture +0.377 nats mean LL; best possible single Beta (moment-matched) +0.106 nats. The mixture's theoretical advantage is therefore +0.271 nats.

Set-matched check: unlike Track 1, the Track 2 headline rows do not need a matched-subset correction — at s800 both the mixture and single-Beta parse 300/300 test cases (parse rate 1.00), so every continuous metric is already computed over the same complete set. Parse rates only dip at s200 (single-Beta 0.89, mixture 0.98); those are not the reported architectural-comparison rows.

Metaculus continuous questions (real data)

Mixture beats single Beta by −26% MSE and +0.27 nats at matched steps; s800 keeps improving LL but MSE worsens slightly — log-likelihood mildly incentivizes oversharpening. A per-component sharpness bonus (λ=0.1) was ablated and does not help (−0.12 nats on branching, flat on Metaculus).

Open items → and the bridge to project 5

• Can RL on the discrete-quantized format match trained Beta? (Would isolate format-vs-training gains.)
• Does the mixture advantage persist on genuinely branching real tasks? Mean K is suspiciously similar on branched (1.8–2.0) and unbranched (1.8–1.95) data — general regularizer or task-responsive?
• Close the 0.022 Brier gap to ReMax with a 110k-scale training set.

Status: both tracks complete. Last activity: prompt-ablation study, 2026-05-28.