Abstract

Reinforcement learning from verifiable feedback is usually studied on one skill and scored by pass rate; production needs three heterogeneous behaviours at once, including the one that matters most in deployment—declining what the facts do not support. We post-train a single 1.5B parameter code model across three verifiably distinct tasks with RLVF: proof carrying with honest abstention, intent fidelity with findings, and action routing with named refusals. Every label is computed by an exact oracle—a proof kernel, a fidelity audit, a closed policy—nothing is hand-labelled, and eight gates were registered before training. Across nine configurations—data dose, checkpoints, sequential refreshes, learning-rate schedule, weight soup—the official candidate passes seven: intent $90.9$, router $81.0$, ESCALATE recall $84.7$, decidable-suite yield $97.8$, $0$ faults, and a clean plain-prompt serving battery, with the single residual on the held-out undecidable gap family ($15.2\%$ overclaim against the registered $\le5\%$). The nine probes do more than select a candidate—they map one exchange surface at this capacity: closing the gap family to zero overclaim costs intent along a measured rate ($90.9\to65.5$ against $15.2\to0$), so the residual is a capacity-point property of the surface rather than an artifact of the training recipe, and the full per-configuration record lives in an appendix rather than in prose. A premise-fidelity probe from that appendix exposed and closed a laundering channel no artifact-level oracle can see: invented preconditions $7\to0$; undecidable-side overclaim dropped from $26.5\%$ to $0.0\%$, and recall reached $100\%$ under a pre-registered fidelity-aware reward—a sound oracle is not sufficient, the reward must see the task-level obligation. Serving-side, the candidate fails closed when deactivated ($10/10$ declines), and 8-bit quantization reproduces the full gate suite at parity ($7/8$) at $1.8\times$ decode. Our findings characterize the Pareto frontier specifically for a single 1.5B architecture under single-seed training protocols.

1. Introduction

Reinforcement learning from verifiable feedback is usually studied on one skill at a time—a math domain, a coding domain—with pass rate as the score; the RLVR/GRPO line makes that scalar work harder through denser credit, better baselines, and teacher guidance [1, 2, 3]. Production systems need three heterogeneous behaviours at once, and, more importantly, they need the negative behaviour: declining what the facts do not support. We ask a narrower and more measurable question: given three components that each have an exact oracle, does joint post-training on all three compose one fail-closed policy, or do the skills interfere?

The three components and their oracles:

component skill oracle
prover proof-carrying code or ABSTAIN proof kernel (certify)
intent BUILD/ASK/blocked + findings fidelity audit (audit)
router ROUTE/ESCALATE/ABSTAIN + reason closed policy (decision_dict)

Every training and evaluation label is oracle-derived; nothing is hand-labelled. The oracles operate offline during training: every candidate completion is disposed by its component's oracle, verdicts that fail the typed reward table $\rho$ (including the laundering detectors) are excluded, and only oracle-consistent completions become training targets; at evaluation, the same oracles dispose the model's outputs. The hypothesis: all three teach the same meta- behaviour—decide only what the facts support, otherwise decline and name what is missing—so they should compose, subject to capacity.

Contributions: (1) a harness and evaluation protocol with eight pre-registered gates spanning templated accuracy and plain-prompt serving; (2) the nine-configuration frontier: one residual, structurally mapped; (3) the reward-laundering result—a sound per-artifact oracle is not sufficient; the reward must see the task-level obligation—with a registered prediction confirmed; (4) serving and deployment findings: fail-closed deactivation, plain-prompt skill separability, and quantization parity measured by re-running the gates, not perplexity.

RLVR. Reinforcement learning with verifiable rewards directly optimizes task success using automatically checkable outcomes, with GRPO the standard critic-free approach for mathematical and code reasoning [1, 2]. Outcome rewards, however, are sparse and coarse: GRPO assigns the same trajectory-level advantage to every token and yields no gradient when all responses in a group receive the same reward [4, 5]. The literature addresses those limitations through dynamic sampling [6], correcting optimization and length biases [7], or finer-grained credit assignment [8, 9, 10]; DiffGate [3] complements outcome rewards with dense teacher guidance where the group-relative signal is weak. All of these make the outcome scalar denser. RLVR asks: how do we make a scalar success reward denser so the policy gradient flows? Our setting asks the orthogonal question: what happens when the oracle is exact but the reward can be gamed, the right answer is "I don't know," and three verifiable skills must coexist in one small model? Density of signal is their bottleneck; honesty of signal and composition are ours. Our position between the two lines is explicit: the feedback interface is typed (Definition 1), the optimization used for the reported candidates is oracle-guided maximum likelihood with reward-table filtering (§3.1), and group-relative policy optimization over typed verdict alphabets is posed as future work (§9). The distinction is not cosmetic: a denser outcome reward is blind to premise invention—the verifier certifies the artifact—and so would a teacher-guidance channel that reshapes credit inside the same reward, which is exactly the channel our premise_invention $=-1.00$ term had to open (§5).

Verifiable feedback beyond the scalar. Verifier- and process-reward training raises math and coding performance [11, 12, 13]; RLAI replaces human labels with model judgments [14]. Our feedback is neither prose-judged nor a pass rate: it is typed verdicts from pinned deterministic oracles, and honest abstention is positively scored. Learned judges are a documented failure regime—position, verbosity, and self-enhancement biases [15] and reward hacking under specification mismatch [16]; we make "exact-compute-or-nothing" a standing rule and measure one laundering incident when the rule was violated (§5).

Interference and composition. Multitask bias [17], gradient-based forgetting mitigation [18, 19], and model soups [20] frame our result. We contribute a per-class dose–response frontier at one capacity point: sequential refreshes are anti-compositional (fix shown, poison hidden), while weight averaging partially composes across trajectories—but interferes on one class (ASK), consistent with task-vector interference [21]. The locus of the abstention boundary also connects to selective prediction [22] and to the $\Omega$-certifiability framing of claim-vs-decidable behavior [23].

Efficient fine-tuning and deployment. LoRA [24] throughout; quantization judged the way it should be—by re-running the registered evaluation—rather than by perplexity, in the spirit of the quantization literature [25, 26, 27].

3. Setup

3.1 Formal setup: typed feedback against a scalar reward

Definition 1 (Components and oracle verdicts). Three components $k\in\mathcal{K}=\{P,I,R\}$ (prover, intent, router) share one policy $\pi_\theta$ over prompts $x\sim\mathcal{D}_k$. Each component has a pinned deterministic oracle $\Omega_k:\mathcal{Y}\to\mathcal{V}_k$ over a typed verdict alphabet: $$ \mathcal{V}_P=\{\textsf{cert},\textsf{ref},\textsf{h-abstain}, \textsf{abstain-on-A}\},\quad \mathcal{V}_I=\{\textsf{build},\textsf{ask},\textsf{blocked}\}, \quad \mathcal{V}_R=\{\textsf{route},\textsf{esc},\textsf{abstain}\}, $$ plus the laundering detectors $\mathcal{V}'\supseteq\{\textsf{fabricated},\textsf{premise\_inv}\}$ evaluated on certified artifacts. Feedback is the typed verdict $\Omega(x,y)\in\mathcal{V}_k\cup\mathcal{V}'$, never a scalar.

For contrast, the RLVR/GRPO scalarization of the same contact surface assigns trajectory reward $R_i$ and the group-relative advantage

$$ A_i^{\mathrm{GRPO}} =\frac{R_i-\bar{R}_G}{s_G+\varepsilon}, \qquad \nabla_\theta J_{\mathrm{GRPO}} =\frac{1}{G}\sum_{i=1}^{G}\frac{1}{|y_i|} \sum_{t=1}^{|y_i|} A_i^{\mathrm{GRPO}}\, \nabla_\theta \log \pi_\theta(y_{i,t}\mid x,y_{i,<t}). \tag{1} $$

Observation 1 (Vanishing group signal). If all rewards in a group agree ($s_G=0$), then $A_i^{\mathrm{GRPO}}=0$ for every $i$ and $\nabla_\theta J_{\mathrm{GRPO}}=\mathbf{0}$: the step contributes nothing regardless of trajectory quality. Under output-only reward every token in a trajectory also receives the identical advantage.

Proof (Derivation). $\bar R_G=R$ and $s_G=0$ give $A_i=0$ termwise in Eq. (1); with only a terminal reward, the return at every position equals $R_i$, so the advantage has no $t$-dependence [4, 5].

Remark 1 (Typed feedback and variance). Observation 1 is a standard property of zero-variance normalization, recorded here to locate the failure the typed table addresses. Under a group-relative scheme over the alphabets of Definition 1, within-group variance $s_G>0$ whenever a group contains heterogeneous verdict classes—including the positively scored honest-abstain class $\rho(\textsf{h-abstain})=+0.75$—so the typed table restores exactly the gradient that scalar pass/fail rewards lose when a group saturates. Formalizing this requires the on-policy instantiation of §9; it is a design remark here, not a proved result.

Observation 1 is the density problem the RLVR fixes target. We keep the scalar where it belongs and open two channels it cannot express:

Definition 2 (Typed reward table). For reward-shaped stages the feedback is scored by the fixed table $\rho:\mathcal{V}_k\cup\mathcal{V}'\to\mathbb{R}$, $$ \rho(v)= \begin{cases} +1.00 & v=\textsf{cert} \\ -1.00 & v=\textsf{ref} \\ +0.75 & v=\textsf{h-abstain}\ \text{(honest decline scored positive)} \\ \phantom{-}0.00 & v=\textsf{abstain-on-A}\ \text{(decline where decidable)} \\ -1.00 & v=\textsf{fabricated} \\ -1.00 & v=\textsf{premise\_inv}, \end{cases} \tag{2} $$ where $\textsf{premise\_inv}$ is the fidelity-aware term added after the laundering incident ($\lambda_p=1.00$ registered before retraining).

Definition 3 (Laundering event). Given task obligation $\mathrm{Ob}(x,y)$ (the prompt's contract) and artifact-level oracle, $$ \mathcal{L}(x,y) =\mathbb{I}\!\left[\Omega(x,y)=\textsf{cert}\ \wedge\ \neg\,\mathrm{Ob}(x,y)\right]. \tag{3} $$ The base reward $\rho\circ\Omega$ is blind to $\mathcal{L}$ by construction: the artifact is sound, so the oracle cannot see the invented precondition. The fidelity-aware reward is $r(x,y)=\rho(\Omega(x,y))-\lambda_p\,\mathbb{I}[\textsf{premise\_inv}] -\lambda_f\,\mathbb{I}[\textsf{fabricated}]$, and our claim is exactly that $\mathcal{L}$ is measurable—and reducible—only through the second channel.

Definition 4 (Composition objective and gates). Training is prompt-masked maximum likelihood on oracle-consistent targets with contradicted rows dropped, never relabelled: $$ \min_\theta\; \mathbb{E}_{k\sim\mathcal{K}}\; \mathbb{E}_{(x,y^*)\sim\mathcal{D}_k^{\mathrm{filt}}} \big[-\log \pi_\theta(y^*\mid x)\big], \qquad y^*=\Omega_k\text{-consistent target}, \tag{4} $$ with refresh variants resuming $\theta$ on targeted slices of $\mathcal{D}_k^{\mathrm{filt}}$. Evaluation is eight inequalities registered before v2, each a boolean gate $g_j(v)\ge\tau_j$ (direction-aware; thresholds: intent $\tau_1{=}87$, router $\tau_2{=}78$, ESC-recall $\tau_3{=}66$, $C$-overclaim $\tau_4{=}5$ (upper), $C$-recall $\tau_5{=}90$, $A$-yield $\tau_6{=}92$, faults $\tau_7{=}0$, battery $\tau_8{=}(6,5,\mathrm{all})$): $$ \Gamma(v)=\sum_{j=1}^{8}\mathbb{I}\!\left[g_j(v)\ \mathrm{satisfies}\ \tau_j\right] \in\{0,\ldots,8\}, \qquad v^\star=\arg\max_{v\in\text{configs}}\Gamma(v). \tag{5} $$

Remark 2 (How Eq. (2) and Eq. (4) connect). The reported candidates—including $v_4$—are trained by oracle-guided maximum likelihood (Eq. (4)); no policy-gradient update (PPO, REINFORCE, or GRPO) is used anywhere in this paper. The typed table $\rho$ enters the pipeline in exactly two places: (i) as an acceptance filter defining $\mathcal{D}_k^{\mathrm{filt}}$—rows whose oracle verdict scores negative, or whose laundering detectors ($\textsf{fabricated}$, $\textsf{premise\_inv}$) fire, are excluded (dropped, never relabelled); and (ii) inside reward-shaped secondary stages, where $\rho$ scores rollouts directly. RLVF names this feedback interface—typed verdicts from exact oracles flowing into data selection and objectives —not a specific reinforcement-learning algorithm; the group-relative RL instantiation over typed alphabets is posed in §9.

On the prover axis, the $2\times2$ matrix specializes the $\Omega$-certifiability pair [23]:

$$ \mathrm{yield}_A=\frac{\#\{\text{claim}\wedge\text{cert on }A\}}{|A|}, \qquad \mathrm{overclaim}_C=\frac{\#\{\text{claim on }C\}}{|C|}, \qquad \mathrm{recall}_C=\frac{\#\{\text{decline on }C\}}{|C|}, \tag{6} $$

with faults counted separately, never averaged in. The empirical frontier is characterized by the intent budget function

$$ a^\star(c)\;=\;\sup\{\,a(v)\;:\;v\ \text{configuration},\;\, c(v)\le c\,\}, \tag{7} $$

the best intent accuracy attainable under an overclaim budget $c$; $a^\star$ is weakly increasing by construction, so its slope is the exchange rate between the axes. The controlled dose lineage of §4 measures that rate directly: driving Suite-$C$ overclaim $15.2\to0$ moves intent $90.9\to65.5$, i.e. $\Delta a/\Delta c=(90.9-65.5)/(15.2-0)\approx+1.67$ intent points per overclaim point removed—within the lineage neither endpoint dominates (lower overclaim is purchased with lower intent). Across all nine configurations the budget rises from $a^\star(0)=87.3$ to $a^\star(15.2)=90.9$: at this capacity the two axes trade, and the residual of §4 is the cost of moving along $a^\star$.

3.2 Model, data, and protocol

Model and training. Qwen2.5-Coder-1.5B-Instruct, LoRA rank 8, lr $10^{-5}$, batch 1, ${\sim}600$ iterations (${\sim}1.6$ epochs), seed 0, prompt-masked SFT on oracle-derived completions. Refresh variants resume adapters with targeted slices at $3\times10^{-6}$.

Data. Intent population: parameterized families (unit swaps/drops, entity loss, negation violations, vague asks, $\ldots$) re-disposed through the audit oracle; cases whose oracle behaviour contradicts their intended family are dropped, not relabelled (v4: $171$ population $\to$ $9$ dropped $\to$ $106$ train / $55$ eval). Router population: closed-vocabulary synthetic cells plus $740$ declared benchmark rows, labelled by the policy (v4: $853$ population; ROUTE $517$ / ABSTAIN $194$ / ESCALATE $142$; $106$ train / $295$ eval). Prover: held-out task suite, $K=4$ rollouts. Splits are content-hashed and seed-stable; per-component caps, minority oversampling (ESCALATE$\times5$), plain-prompt curriculum rows, and system-prompt randomization are the recipe knobs. The serving battery's $26$ prompts per cell are excluded from training at both the templated and plain levels.

The eight gates (registered before v2). Templated axis: intent accuracy $\ge87\%$; router accuracy $\ge78\%$; ESCALATE recall $\ge66\%$; undecidable-suite overclaim $\le5\%$ with recall $\ge90\%$; decidable-suite yield $\ge92\%$; plus $0$ faults. Serving axis (a four-cell battery of $26$ plain prompts per cell—no task templates): trained+active intent $\ge6/8$, router $\ge5/8$, and every out-of-fragment prover ask declined. Controls: the base model with only a system prompt must still show its raw prior ($10/10$ fenced code), and the bare base must show degenerate routing.

Evaluation instrument. The prover axis is scored on a frozen suite of $184$ tasks ($0$ dropped, $0$ oracle mismatches) in the standard $2\times2$ matrix: Suite $A$ ($n=120$: straight-line code, loop summaries, premises, bounds, linear systems) and Suite $C$ ($n=64$: mutation, subscript, method, membership, identity, type, unbounded loops, premise gaps, non-polynomial expressions—the family that makes overclaim measurable). Labels are machine-validated through the authority at build time, and the manifest pins split hashes and the authority fingerprint (re-pin = re-dispose; all recorded runs re-disposed byte-identical across one soundness fix). Suite $A$ is authority-shaped by construction—we use it to rank configurations against each other, never as an open-world capability claim. The battery's four cells (trained+active, trained+inactive, base+prompt, base) run sequentially with one process each at temperature $0$; the activation system prompt is the original contract prompt verbatim, one prompt serving all three components (the prompt-design test that fixed this is in Appendix A).

Reward design (for reward-shaped stages). The scoring is exactly the typed table $\rho$ of Eq. (2) (certified $+1.0$, refuted $-1.0$, honest abstain $+0.75$, abstain-on-decidable $0$), with the two exact-penalty channels added after the laundering incident: fabricated (claim names a quantity the source lacks) and premise_invention at $-1.00$ each, i.e. $\lambda_f=\lambda_p=1.00$ in the fidelity-aware reward of §3.1. A non-vacuity gate counts a certification only with at least one non-premise discharged obligation. Verdict text never enters training as repair instructions.

4. Results

version intent router ESC C ovr/rec A yield batt gates
base 54.5 21.7 0.0 79.5/4.5 42.2 --- 0
v2 87.3 74.5 66.1 0/100 92.2 3/2/0 4
v3 89.1 76.6 0.0 23.5/76.5 98.9 6/4/5 4
v4 90.9 81.0 84.7 15.2/84.8 97.8 6/6/6 7
v5 58.2 85.4 67.9 0/100 93.3 6/5/6 6
v6 65.5 85.4 67.9 0/100 93.3 6/5/6 6
v6ck900 85.5 78.6 57.6 12.9/87.1 96.1 6/3/6 4
v7 83.6 84.1 56.9 1.5/98.5 92.2 8/5/7 6
v8 87.3 58.0 94.9 68.9/6.1 100 8/5/0 5
soup 80.0 87.1 84.7 9.8/90.2 94.4 6/4/7 5
cosine 87.3 70.8 0.0 26.5/65.9 93.9 6/2/5 4

Table 1. Nine configurations against the eight gates (best in column bold). C: undecidable-suite overclaim/recall; batt: battery intent/router/prover-declines of 8/8/10.

Composition happens, for free only in the large. Every trained version beats base by a wide margin on all three axes (Table 1): the skills do compose into one prior. What does not come for free is calibration sub-classes: the primary optimization trade-offs emerge between overclaim, abstention, ESCALATE, and ASK behaviors, rather than across component domains.

The official candidate. $v_4$ attains $\Gamma(v_4)=7$ in Eq. (5) (no configuration reaches $8$): intent $90.9$ ($n{=}55$, Wilson $[80.4,96.1]$), router $81.0$ ($n{=}295$), ESCALATE $84.7$ ($n{=}59$), $A$-yield $97.8$ (false-abstain $1.1$), $0$ faults, battery $6/6/6$ with every out-of-fragment ask declined. Sole residual: the held-out C-gap family ($\mathrm{overclaim}_C=15.2\%>\tau_4$, recall $84.8$—still $19\times$ better than base on C). Alternate: v7 if C-calibration outranks two sub-bar templated numbers ($C$ $1.5/98.5$, battery intent $8/8$).

Plain-prompt serving is a separable skill. Plain-ask inference is trained explicitly from v3 on—curriculum rows plus system-prompt randomization—and every later version holds it (the pre-v3 record is part of the Appendix battery). Deactivation fails closed: the trained+inactive cell declines $10/10$ with zero fences, while the base cells show the raw prior ($10/10$ fenced, $5/5$ fenced+prose). Contract form never leaked: $0$ fenced, $0$ prose outputs in every trained active cell.

version prover (ab/ba/fe/pr) intent router OOF declines
v2 0/10/0/0 3/8 2/8 ---
v3 5/5/0/0 6/8 4/8 all
v4 6/4/0/0 6/8 6/8 all
v6 7/3/0/0 6/8 5/8 all
v7 7/3/0/0 8/8 5/8 all
v8 0/10/0/0 8/8 5/8 none
soup 7/3/0/0 6/8 4/8 all
cosine 5/5/0/0 6/8 2/8 all
base+prompt 0/0/10/0 3/8 2/8 ---
base 0/0/5/5 6/8 2/8 ---

Table 2. Serving battery, trained+active cell ($26$ plain prompts: $10$ prover / $8$ intent / $8$ router). Prover shapes: abstain/bare/fenced/prose. Every trained cell keeps contract form; the per-version anomalies are catalogued in Appendix A.

Table 2 gives the battery in full. Controls held in every battery: base+prompt shows $10/10$ fenced code (the system prompt alone does not teach decline), the bare base shows $5$ fenced $+$ $5$ prose, and base router is the degenerate always-ABSTAIN ($2/8$, correct by luck on abstain cells only).

The frontier. Nine probes map the intent budget function $a^\star(c)$ of Eq. (7). The controlled dose lineage ($0\to32$ gap-decline rows) measures the exchange rate directly: Suite-$C$ overclaim $15.2\to0$ with intent $90.9\to65.5$, i.e. $\Delta a/\Delta c\approx+1.67$ intent points per overclaim point removed, so within the lineage no endpoint dominates the other—lower overclaim is purchased with lower intent. Across all nine configurations, $a^\star$ rises from $87.3$ at budget $0$ to $90.9$ at budget $15.2$. What each configuration contributed to that map—refreshes, schedule, soup—is catalogued class by class in Appendix A: every knob moves along the surface rather than off it, and a regression on any gate is a signal to stop, not to average. The structural claim: at 1.5B / rank 8 / ${\sim}1.6$ epochs the residual is a capacity-point property of the surface, not an artifact of the training recipe; the one untested combination is more epochs so the gap family converges before a gentle tail.

Protocol. Every configuration is evaluated with greedy decoding, per-class Wilson intervals, and content-hashed eval splits the model never trained on; the prover axis is $78$ held-out tasks $\times K=4$ ($312$ rollouts) per version. A secondary DPO arm on mined failures plus mechanical guards is recorded in Appendix A.

5. Reward laundering, and the fix is in the reward

A premise-fidelity probe—registered with the battery before retraining—quantified a channel that the artifact-level oracle of Eq. (3) cannot see by construction: certified outputs whose emitted assertions strengthen an undeclared precondition, converting honest failure into certified success.

Schematic example of the class. The obligation is universal—"return the sum of the positive elements, for any list $xs$." The emitted artifact computes that sum but first asserts $\texttt{assert len(xs) > 0}$ as an additional premise. The artifact-level kernel then verifies a strengthened claim: the conditional statement ($xs\neq[] \Rightarrow$ correct) under an assumption the obligation never granted. The empty-list case—part of the actual contract—was never proved, yet the artifact certifies. The prompt declares no non-emptiness parameter; the artifact invents one.

Detection. $\texttt{premise\_invention}$ is detected by an exact, parameter-only, directional check: an emitted assertion whose antecedent quantifies over a parameter the obligation never declares is flagged. No syntactic parser over arbitrary code and no auxiliary model is involved—the check reads the obligation's declared parameters and the assertion's antecedent, nothing else.

Measured on the original corrective run, the channel carried $7$ of $52$ certified outputs ($13.5\%$); the probe and its repair were registered as one prediction, and confirmed:

A yield (%) A ovr (%) C ovr (%) C recall (%) inventions
original corrective run 95.6 4.4 26.5 67.6 7
fidelity-aware ($\lambda_p{=}1.00$) 93.3 4.4 0.0 100.0 0

The gain generalized beyond the gap family to the whole undecidable side, at a cost of $2.2\%$ decidable-side false abstention. The incident record—what an invention looks like, why the honest-failure controls never produce one—is kept with the battery (Appendix A). The standing lesson: a sound oracle is not sufficient; the reward must see the task-level obligation. The reward, not the model, is the trust boundary.

6. Deployment: quantization measured, not assumed

artifact config size decode gates verdict
sft-all3v4 (fp16) reference 2.9 GB 55.5 7/8 scoring reference
fp8 8-bit float, 8.25 b/w 1.5 GB ${\sim}100$ ($1.8\times$) 7/8 parity deploy default
q4g64 4-bit group-64 839 MB 151–162 ($2.9\times$) 6/8 max-speed option
m34 mixed 3/4-bit 679 MB ${\sim}140$ ($2.5\times$) 4/8 rejected (3-bit cliff)

Table 3. Quantization parity: fp16 reference vs compressed artifacts, identical eight-gate protocol re-run against the fp16 baseline. Decode is tokens/s measured with the MLX runtime on Apple silicon, batch size $1$, on a fixed coding prompt (fp16: $55.5$); sizes are on-disk artifact sizes.

Table 3: 8-bit reproduces every router class and the battery bit-for-bit; 4-bit's single slip is the templated ROUTE class ($94\to69.5$) while Suite $C$ improves ($6/20$ gap declines vs $2/20$) and the plain-prompt battery remains identical; 3-bit loses Suite $A$-yield $-25$ points and ASK $\to0$. The 4-bit asymmetry—templated classification degrading while decline behavior survives—is reported as observed; we do not have a mechanistic attribution for it (open challenge six, §9). Reading: compression tracks the training frontier—declines survive compression, capacity does not—so deployment moves only on parity evidence: fp8 by default, int4 when throughput outranks the ROUTE gate, fp16 as the scoring reference.

7. Limitations

Training-seed variance is uncharacterized: every reported version uses a single training seed ($S=1$), and multi-seed replication ($S\ge3$, the standard publication bar) is the binding open item (§9). Uncertainty we can bound is reported as such: evaluation-split uncertainty via Wilson score intervals (intent $n=55$, ASK cell $n=9$, ESCALATE $n=59$), $K{=}4$ rollouts. One backbone, one capacity point; the budget-function claim is scoped there. The prover suite mirrors the authority's vocabulary (authority-shaped by construction, disclosed). The battery is $n{=}8$ per capability—a serving smoke test with control cells, not a statistical benchmark. Cross-backbone results elsewhere suggest capacity transfers while calibration judgment does not—unexplained at $n{=}3$.

8. Conclusion

Three oracle-derived skills compose in one 1.5B model into a fail-closed prior—measurably, against gates registered before training: 7/8 for the official candidate against the base's 0/8, intent $90.9$, router $81.0$, yield $97.8$, serving behaviour that fails closed when deactivated ($10/10$ declines), and a quantized deployment at full parity ($1.8\times$ decode). The residual is quantified by the intent budget function $a^\star(c)$: nine probes agree on one exchange rate at this capacity, and every per-configuration diagnosis is catalogued in the appendix. A pre-registered reward-honesty probe found a laundering channel an artifact-level oracle structurally cannot see, and its repair was confirmed on every axis ($7\to0$ inventions, overclaim $26.5\to0.0\%$, recall $\to100\%$)—the reward must see the task-level obligation, and now the reward does. Exact oracles everywhere, registered gates, typed verdict reports, and optimization weights treated as secondary to the measured exchange surface: this is the methodology that makes the reported numbers interpretable.

9. Future Work and Open Challenges

This work was conducted within the hybrid-neurosymbolic research program of Optimality.ai (neural proposers, symbolic authorities, fail-closed contracts). The open challenges are as follows.

First and binding: replicate the official candidate at $S\ge3$ training seeds—training-seed variance is currently uncharacterized, and this is the precondition for treating the reported frontier as more than a single-seed characterization. Second, run the one untested training knob (more epochs, followed by a gentle decay) to determine whether the Suite-$C$ residual is a capacity property or a convergence artifact. Third, instantiate the group-relative RL arm over typed verdict alphabets: GRPO with reward $\rho$ (Eq. (2)) rather than a scalar, so the group signal carries honesty (abstain vs overclaim), the variance argument of Remark 1 made empirical, and the feedback interface of Remark 2 exercised as reinforcement learning rather than filtering. Fourth, scale capacity to ask whether the budget function $a^\star(c)$ (Eq. (7)) shifts or persists. Fifth, replicate the judgment-transfer question across backbones under a controlled factorial design until the $n=3$ confound dissolves. Sixth, attribute the 4-bit templated-ROUTE degradation (§6) mechanistically—class- specific quantization sensitivity is observed, not explained.

A. Battery of Tests

The main text presents the primary empirical results; this appendix details the underlying validation and diagnostic experiments—every registered probe, control, and per-configuration diagnosis.

A.1 Per-configuration signatures

Each row was diagnosed from the class-level ledger before any further knob was turned (Table 4).

version signature
v3 router ESCALATE $\to 0\%$: minority swamped by ABSTAIN oversampling
v5 intent blocked $\to 37.5\%$ (answers ASK); $A$ false-abstain $1.1\to12.8$—one-sided decline demos taught a surface heuristic
v6 staggered convergence: intent peaks at ck800–900 ($85.5$), collapses to $65.5$ at ck1000 while gap/C and router finish
v7 intent-only refresh lifts intent $+18.1$ but erodes unshown ESCALATE $67.9\to56.9$
v8 refresh composes exactly what it shows: ESC $56.9\to94.9$, but ABSTAIN $88.7\to25.8$ and gap attempts $20/20\to0/20$; serving: $0/10$ declines (twin-heavy slice, no out-of-fragment asks in the slice)
soup composes router/ESC/C but ASK collapses to $11\%$ (subspace interference, not midpoint)
cosine intent remains stable ($87.3$), though late training iterations damage previously converged classes; gap/ESC never finish learning—the same steps carry damage and progress

Table 4. Per-configuration signatures across the nine probes.

A.2 Serving-battery and prompt-bleed tests

(i) v2 reached $C$ $0/100$ and $A$ $92.2$ yet scored $0$ prover declines and $3/8$ intent on plain prompts: templated calibration does not transfer to plain asks untrained—the test that forced the v3 curriculum. (ii) v8's twin-heavy refresh placed no out-of-fragment ask in its slice: $0/10$ declines, OOF column not exercised. (iii) A draft activation prompt that added per-component role paragraphs was measured to bleed answers across skills and rejected; the shipping prompt is the original contract prompt verbatim, one prompt serving all three. All three sit beside the control rows of Table 2.

A.3 Reward-laundering incident record

The channel of §5 in full: an invention is an emitted assertion that strengthens an undeclared precondition—e.g. promoting a contextual assumption into a checked premise—so the artifact verifies while the task obligation is violated. Rate on the original corrective run: $\widehat{\mathbb{E}}[\mathcal{L}]=7/52=13.5\%$. Controls: base and self-imitation policies never invent ($\widehat{\mathbb{E}}[\mathcal{L}]=0$); they fail honestly, which is why the channel is attributable to the reward shape rather than the data. Repair as registered: $\lambda_p=1.00$ (params-only, directional check), gap family enriched $3\to8$, retrain; outcome in the table of §5: $0$ inventions, $C$-overclaim $26.5\to0.0\%$, $C$-recall $67.6\to100.0\%$, $A$-yield cost $2.2\%$ false abstention.

A.4 Secondary arm: DPO

A DPO arm on mined v4 failures plus mechanical guards ($\beta=0.1$, ${\sim}620$ pairs) reached intent $85.5$, router $82.7$, ESCALATE $84.7$, $C$ $18.2/81.8$, $A$-yield $98.9$, $0$ faults, battery $6/6/6$—$\Gamma=7$ at best, missing the same two gates as the SFT frontier, consistent with the surface being a capacity property rather than an optimizer property.

A.5 Registered tests and outcomes

test challenge outcome
eight gates ($\tau_j$ fixed before v2) whether composition beats base under pre-commitment $\Gamma(v_4)=7$ vs base $0$ — pass
serving battery controls whether the system prompt alone teaches decline base+prompt $10/10$ fenced (control holds)
activation-prompt role paragraphs whether prompts must stay one contract bleed measured; draft rejected
reward-fidelity probe ($\lambda_p$) whether an artifact-level oracle suffices channel found ($7/52$), repair confirmed ($0$)
dose probe on gap-decline rows whether $a^\star$ is tradeable monotone: $15.2\to0$ overclaim costs $90.9\to65.5$ intent
quantization parity (re-run gates) whether compression preserves behaviour fp8 $7/8$ parity, battery bit-for-bit

Table 5. The registration ledger: every pre-registered test, the claim it challenged, and its outcome.

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