The Letter Tax: How Much Shorter Is a Letter-Free Reasoning Trace Than Length-Optimized Language?

Abstract

PRE-REGISTRATION — The Letter Tax: how much shorter is a letter-free reasoning trace than length-optimized language? Registered 2026-09-19, before any GPU-hour. Code, data, and protocol frozen at commit d207c263df938f6003e51ff3da90f13367b40d99 ("v8 pre-registration freeze (r4)") in https://github.com/ZechenWei7/letter-tax ; follow-up commit 471f0561 records the hash in the README. Full protocol: README §2 of that commit. This page is the binding summary. QUESTION. On a synthetic deductive task certified to require an intermediate reasoning trace, how much shorter is an RL-length-optimized trace that may not use any letter-containing token than one that may, at matched accuracy — and does the letter-free trace carry the computation? Prior reports of 3–16× token savings from symbolic/abstract chain-of-thought compare against verbal chains never length-optimized, on tasks their models largely solve with no chain, without a filler control. We measure the missing quantity. No claim about a general non-human code, frontier models, non-scorable content, or the cited papers' numbers in their settings; the estimand is a "letter tax", not a language tax. TASK. Unique linear extension with disjunctive precedence: n events, h hard constraints i≺j, d disjunctions (i≺j)|(k≺l); exactly one order satisfies all; answer = the order as n digits. Symbolic prompt for all arms. Cells tried in order (n,h,d) = (8,4,5), (8,4,6), (9,5,6), (9,5,7); first cell passing admission is used; backup task unique list-coloring under the same tests. Guided generation (consistent disjuncts from uncovered covering edges; inconsistent sides uniform), uniqueness by brute force, decision depth S ≥ 1 under closure-only failed-disjunct probing. Disclosed: every covering edge of the answer appears in the prompt; decision content is d bits; one level of assume-propagate-refute solves every instance; minimal-trace floors are ≈ 30 tokens (Kahn) and ≈ 35 tokens (decision) at 2.5 tokens per element, and converged lengths are reported in absolute tokens beside them. ADMISSION (frozen Qwen3-4B, n = 500 unless stated; all required): direct-answer exact ≤ 2 × 2^−d on the 2,000-item direct split (1/#LE(hard) reported); native-trace accuracy 60–80%; forced-budget curve rising (smallest budget within 5 pp of native ≥ 0.5 M); whitelist-unigram filler never within 10 pp of native at ≥ 0.5 M; a native trace from a different instance (random derangement) lowers accuracy at least halfway to direct; frozen letter-banned accuracy ≤ direct + 10 pp; shortcut detectors fire on fewer than 5% of native traces; median native length ≥ 5 × Kahn floor. M = median of the 500 admission traces. Cell rejection statistics per README §2.1 (all four cells pass). Post-training, per arm, every 50 steps: once with-trace accuracy has reached frozen direct + 10 pp, the run stops and is marked uninterpretable if direct ≥ with-trace − (with-trace − frozen direct)/2 on two consecutive checks; C_rand exempt. ARMS (same checkpoint, reward, hyperparameters, stopping rule; LoRA on attention, MLP, embeddings): A (unconstrained, 3 seeds); A″ (single-letter tokens reachable, multi-letter tokens banned; whitelist 8,455; 2 seeds); B (no letter-containing tokens; audited whitelist 8,351; 4 seeds); C_rand (letter ban; context trace tokens drawn from B seed i's converged unigram, length drawn once per prompt group from B's distribution; loss on answer only; 2 seeds); B_warm-SFT (A's converged traces under a fixed word-to-symbol map, SFT, no RL; 1 per A seed, evaluated under B's mask); B_warm-RL (2 seeds, only if B seed 1's best accuracy by step 200 is more than 15 pp below A seed 1's best and the cost projection fits under $300; else recorded as a search-budget result). A and A″ must pass transplant/resample collapse to serve as comparators. REWARD/OPTIMIZATION. r = 10[c(1 − min(λL/M, 0.9)) − 0.5v(1−c) − 0.05vc]; hard violations (non-whitespace text after the first closing think tag, any extra think tag) force c = 0; λ = 0.5; GRPO, Dr. GRPO loss, β = 0, 2 prompts × 16 generations, cap 5,120, forced close = closing think tag + two newlines + opening answer tag, with 40 reserved tokens; the closing think tag is on every whitelist. Stop when the last three 50-step evals show |ΔL|/L below 5% and |Δacc| below 4 pp for both consecutive pairs, else the step-400 eval is the designated checkpoint. Manipulation gate: A seed 1 must cut ≥ 30% tokens at ≤ 5 pp with L_median ≥ 2 × Kahn floor; else λ = 0.3, then λ = 1.0, then G = 32, applied to all arms, pre-tune curve reported. Kill: A seed 1's converged L_median ≤ 1.5 × decision floor (no headroom). Order: A1 (20-step cost check first) → B_warm-SFT(A1) → B1 → warm decision → B2–4 → A2–3 (+ their B_warm-SFT) → A″ × 2 → C_rand × 2 → conditional B_warm-RL. ENDPOINTS. E1 letter tax = (L_A − L_B)/L_A at matched total accuracy: per converged policy, forced-budget curve on a held-out reporting split (isotonic per seed); matched y = min over arms of the seed-mean accuracy at converged budget, minus 5 pp; E1 not computed if any arm's curve does not cross it (then the registered no-matched-accuracy paper — admission tables, frozen-to-A reduction, E2, C_rand/transplant, instrument control — is the result). Item-level paired bootstrap with equal-weight seed blocks; tokens primary; code points and zstd bits must fall in the same tier, tiers assigned from interval endpoints: [67,∞) / [25,67) / [10,25) / (−10,10) / ≤ −10. E2: B_warm-SFT valid if within 5 pp of its source A seed; its length ratio is the encoding-only tax. E3 (all required for B): B above C_rand at matched length; transplant and unigram-resample each lower B's accuracy at least halfway to its direct accuracy; prompt-copy rate below 30%; a prefix-only decoder with target-instance labels applies if B's parse rate ≥ 50%. Strategy classes, soundness/completeness, and a cipher statistic are descriptive. Expectation stated now: the tax will most likely land in [10,25) or (−10,10). READINGS (fixed): tax ≥ 25% with E3 → RL under a letter ban produces a shorter load-bearing trace from the supplied primitives. A″ ≈ B, both shorter than A → constraint, not letters. tax below 10% with A's letter fraction below 30% → length optimization sheds letters on its own. tax below 10% with letters retained → words cost nothing here. tax ≤ −10% → the ban costs length. Cold B fails, B_warm-RL succeeds → usable but not findable from cold start. Both fail → search-budget result. B above C_rand but no collapse under transplant/resample → structured filler. E1 not computable → the no-matched-accuracy paper. BUDGET. One A100, ceiling $300 (coded), 11 RL runs + B_warm-SFT, up to 2 conditional. Cost measured on A seed 1 before the matrix; over budget → stop and decide, no automatic seed cuts. Deviations logged with commit hashes; full rollout archive released.