Fine-Tuning Fixes Mode Collapse and Over-Dispersion in LLMs

# Fine-Tuning Fixes Mode Collapse and Over-Dispersion in LLMs
Source: [https://arxiv.org/html/2609.16454](https://arxiv.org/html/2609.16454)
###### Abstract

Recent work by[Doshi and Hauser \(2024\)](https://arxiv.org/html/2609.16454#bib.bib14),[Bisbee et al\. \(2024\)](https://arxiv.org/html/2609.16454#bib.bib8), and[Xie et al\. \(2026\)](https://arxiv.org/html/2609.16454#bib.bib43)raises concerns that outputs from large language models \(LLMs\) tend to be under\-diverse: they repeat or resemble one another more often than responses from the population they are meant to represent, a phenomenon known as mode collapse\. In this work, we show that whether mode\-collapse—or its opposite—occurs depends on the specific model and dataset used\. Further, with sufficient supervised fine\-tuning \(SFT\) data, LLM output diversity converges toward that of the target distribution from which fine\-tuning data are sampled\. To quantify this comparison, we measure the probability that two responses sampled independently conditional on the same fixed prompt coincide \(collide\)\. In one set of experiments, we use exact token sequences; in the other, we use the generalised version of that—the expected similarity between responses under a kernel\. We define miscalibration as a nonzero model\-minus\-target collision gap\. We derive a bias–variance decomposition of the expected gap between the model’s and target’s collision probabilities, showing that SFT is not inherently biased toward mode collapse or its opposite: finite\-sample SFT can leave a model either under\- or over\-dispersed, depending on the model and dataset\. Finally, we show that the absolute gap is bounded by the square root of the Kullback–Leibler \(KL\) divergence from the target distribution to the model\. Consequently, a model sufficiently close to optimal under population cross\-entropy cannot exhibit arbitrarily miscalibrated diversity\. We test the decomposition and the bound in three experiments: \(1\) we fit 100 small transformers at each of 100 log\-spaced sample sizes on each of two synthetic order\-16 languages; \(2\) we fine\-tune four LLMs using low\-rank adaptation on responses from the General Social Survey, American National Election Studies, and World Values Survey; and \(3\) we repeat Experiment \(2\) on CodeNet, a dataset of human solutions to coding tasks, measuring program similarity with normalized Zhang–Shasha edit distance between canonical abstract syntax trees\. We find substantial heterogeneity in over\- and under\-diversity across models and datasets\. More target data moves model diversity toward the human \(or synthetic target\) level in all experiments, consistent with our theoretical predictions\. These results show that diversity miscalibration can arise from finite\-sample error and shrink as SFT better approximates the target distribution\. Accordingly, as the sample size of target\-distribution data increases, model diversity moves toward the target level\.

## 1Introduction

Large language models \(LLMs\) are increasingly used as simulated survey respondents\([Argyle et al\., 2023](https://arxiv.org/html/2609.16454#bib.bib3)\), for creative writing\([Chakrabarty et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib9)\), as brainstorming partners\([Terwiesch et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib17)\), and in other domains where faithfulness of the LLM’s outputs to the diversity of the target population is important\. Empirical studies show that LLMs often exhibit a narrow range of outputs, a phenomenon known as mode collapse: the model’s outputs repeat, or resemble one another, more often than samples from the population they are meant to represent\. For instance, stories written with artificial intelligence \(AI\) assistance tend to be similar to one another\([Doshi and Hauser, 2024](https://arxiv.org/html/2609.16454#bib.bib14)\); AI\-assisted users may generate ideas that are less semantically distinct\([Anderson et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib2)\); and synthetic survey respondents can reproduce aggregate patterns while suppressing individual and group\-level variation\([Bisbee et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib8);[Wang et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib40);[Xie et al\., 2026](https://arxiv.org/html/2609.16454#bib.bib43)\)\.

These findings are often interpreted as evidence that low output diversity is an inherent limitation of LLMs\([Xie and Xie, 2025](https://arxiv.org/html/2609.16454#bib.bib42)\)\. However, when a pre\-trained or instruction\-tuned model is fine\-tuned on a new dataset, its starting distribution is being compared with a target distribution it has not yet learned\. It is unclear whether this under\-diversity persists as a model learns the target distribution more accurately\. If mode collapse is inherent, increasing the amount of target\-distribution fine\-tuning data need not eliminate it; if, instead, mode collapse and other distribution miscalibrations simply reflect finite\-sample error, model diversity should converge toward the target level with sufficient fine\-tuning\. To settle this distinction, we begin with formalizing it theoretically\.

We quantify LLM output diversity by collision probability—the probability that two responses sampled independently conditional on the same fixed prompt coincide—or, more generally, by their expected similarity under a kernel \(Section[3](https://arxiv.org/html/2609.16454#S3)\)\. For next\-token comparisons, we instead fix the full context: the prompt and token prefix\. Model and target are compared under the same conditioning and similarity measure\. Their collision gap is the model’s collision probability minus the target’s; where zero indicates calibrated diversity, a positive gap indicates mode collapse, and a negative gap indicates over\-dispersion\.

This formalization allows us to make three claims \(Figure[1](https://arxiv.org/html/2609.16454#S1.F1)\)\. \(1\) SFT does not inherently produce under\- or over\-dispersion; the direction depends on the model and dataset\. The difference in collision probabilities, averaged over independent fits, decomposes into nonnegative variance and squared\-bias terms, which promote concentration, and a sign\-indefinite target–bias alignment term that can offset them \(Section[3\.1](https://arxiv.org/html/2609.16454#S3.SS1)\)\. \(2\) Magnitude: the absolute collision gap is bounded by the square root of the Kullback–Leibler \(KL\) divergence from the target distribution to the model \(Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1)\)\. This bound implies \(3\) Calibration in the limit: as this KL divergence approaches zero, the collision gap necessarily approaches zero\. In practice, this leads to the intuitive conclusion that if more target data and loss minimization bring population cross\-entropy toward the value achieved by the target distribution itself, model diversity must approach the target level\.

Figure 1:Large Langauge Models \(LLMs\) can produce diverse or colliding outputs given the same prompt\. The extent to LLMs do so can be measured\. Higher average output similarity than the target’s is referred to as mode\-collapse\. However, contrary to wide\-spread belief, mode\-collapse is not guaranteed, and depending on model and dataset, models can instead be over\-diverse \(the opposite of mode\-collapse\)\. It can be shown that fine\-tuning calibrates the models, that is, they become not too under\- and over\-diverse\.\(a\.i\)Independent outputs from the same LLM and prompt can collide or differ\.\(a\.ii\)The notation used in this figure for collision\.\(a\.iii\)The definition of the collision ratioRR\.\(b\)Shows the decomposition of Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\) using data from Experiment 1 \(§[4\.1](https://arxiv.org/html/2609.16454#S4.SS1)\)\. The two plots correspond to two different model and data set ups\. The plots demonstrate that Alignment term of Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\) can make a model either under\- or over\-diverse\. Herebh=𝔼⁡\[qh\]−phb\_\{h\}=\\mathbb\{E\}\[q\_\{h\}\]\-p\_\{h\}is the bias at contexthh, with𝔼\\mathbb\{E\}over independent fits\. The plotted curves average contexts and start atR=1R=1, with each contribution divided byC⁡\(ph\)C\(p\_\{h\}\)\.\(c\)Plots median collision ratioRRfor early\-stopped survey and final CodeNet adapters against fine\-tuning examples from Experiments 2–3 \(§§[4\.2](https://arxiv.org/html/2609.16454#S4.SS2)–[4\.3](https://arxiv.org/html/2609.16454#S4.SS3)\)\. The experiments show that additional fine\-tuning leads to diversity calibration\.\(d\)Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1)bounds the absolute model–target collision gap by the square root of their Kullback\-Leibler divergence\. Consequently, a model sufficiently close to optimal under population cross\-entropy cannot exhibit arbitrarily miscalibrated diversity\.To support our claims empirically, we set up three experiments \(Section[4](https://arxiv.org/html/2609.16454#S4)\)\. Experiment 1 tests claim \(1\) and checks the bound of claim \(2\): in two synthetic order\-1616language settings with exactly computable targets, models with the generative pre\-trained transformer \(GPT\) architecture fitted across100100log\-spaced sample sizes produce collision distortions of both signs \(Section[4\.1](https://arxiv.org/html/2609.16454#S4.SS1), Figure[2](https://arxiv.org/html/2609.16454#S4.F2)\), and the bound holds on every evaluated context \(Appendix[G](https://arxiv.org/html/2609.16454#A7), Figure[5](https://arxiv.org/html/2609.16454#A7.F5)\)\. Experiments 2 and 3 test claim \(3\) from opposite starting points\. We use low\-rank adaptation \(LoRA\) to fine\-tune four instruction\-tuned LLMs \(gemma\-2\-2b\-it\([Gemma Team, 2024](https://arxiv.org/html/2609.16454#bib.bib44)\),gemma\-3\-4b\-it\([Gemma Team, 2025](https://arxiv.org/html/2609.16454#bib.bib45)\),Qwen3\.5\-2BandQwen3\.5\-4B\([Qwen Team, 2026](https://arxiv.org/html/2609.16454#bib.bib46)\)\) on General Social Survey \(GSS\)\([Davern et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib13)\), American National Election Studies \(ANES\)\([American National Election Studies, 2022](https://arxiv.org/html/2609.16454#bib.bib1)\), and World Values Survey \(WVS\)\([Haerpfer et al\., 2022](https://arxiv.org/html/2609.16454#bib.bib19)\)\. We find that three of the four base models are1\.61\.6to2\.92\.9times under\-dispersed relative to the human populations, and SFT on human answers restores population\-level heterogeneity for every model on every survey \(Section[4\.2](https://arxiv.org/html/2609.16454#S4.SS2), Figure[3](https://arxiv.org/html/2609.16454#S4.F3)\)\. Next, we apply the same method to CodeNet\([Puri et al\., 2021](https://arxiv.org/html/2609.16454#bib.bib30)\), a dataset of human solutions to coding tasks, where we measure distance between solutions using the normalized Zhang–Shasha similarity kernel\([Zhang and Shasha, 1989](https://arxiv.org/html/2609.16454#bib.bib39)\)\. On this same dataset, some base models are under\-diverse and others over\-diverse \(R=0\.74R=0\.74–1\.641\.64\): mode collapse depends on the model\. Supervised fine\-tuning narrows the gap from both directions, as predicted by theory\.

## 2Related work

Evidence of output mode collapse\.Generative AI can enhance individual story ratings and increase similarity between AI\-assisted stories\([Doshi and Hauser, 2024](https://arxiv.org/html/2609.16454#bib.bib14)\), while co\-writing with InstructGPT reduces content diversity in essays\([Padmakumar and He, 2024](https://arxiv.org/html/2609.16454#bib.bib27)\)\. ChatGPT users generate more numerous and detailed ideas than users of another creativity\-support tool, but these ideas exhibit less semantic distinctiveness across users\([Anderson et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib2)\)\.[Sourati et al\. \(2026\)](https://arxiv.org/html/2609.16454#bib.bib36)argue that LLMs reinforce dominant styles of language and reasoning, and that widespread reliance on the same few models amplifies this convergence\. Our results show that SFT is not inherently biased toward mode collapse or its opposite: the direction of miscalibration depends on the model and dataset, and fine\-tuning can bring diversity toward the target from either direction\. LLMs are also employed as proxies for human participants in “silicon sampling” studies\([Argyle et al\., 2023](https://arxiv.org/html/2609.16454#bib.bib3)\), but recent work shows that synthetic surveys can match aggregate statistics while failing to capture individual variation and regression structures\([Bisbee et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib8)\), and can misrepresent and flatten identity groups\([Wang et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib40)\)\. A benchmark of 15 LLMs across seven surveys documents the same compression of heterogeneity\([Xie et al\., 2026](https://arxiv.org/html/2609.16454#bib.bib43)\)\. Recursive fitting on model\-generated data also removes data points from the tails of the original distribution\([Shumailov et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib33);[Seddik et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib32)\), illustrating the ease with which generative resampling can erase rare modes\.

Preference optimization and diversity\.Preference optimization has been shown to shrink output diversity\. Reinforcement learning from human feedback \(RLHF\) with proximal policy optimization \(PPO\) reduces both within\- and across\-prompt diversity for summarization, as assessed by multiple lexical and semantic metrics, though the same study finds no meaningful diversity differences for instruction following\([Kirk et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib23)\)\. The stage at which this contraction occurs varies across model lineages, with some models showing the largest reduction during SFT and others during direct preference optimization \(DPO\)\([Karouzos et al\., 2026](https://arxiv.org/html/2609.16454#bib.bib22)\)\. These effects are linked to preference data and annotator heterogeneity\. Annotators tend to favor responses to which base models assign higher likelihood even when responses are matched on correctness; consequently, a perfectly estimated preference reward can preserve typicality bias toward conventional responses\([Zhang et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib38)\)\. Finally, aggregating annotators with heterogeneous criteria implicitly implements the Borda\-count rule rather than recovering an observer\-independent utility\([Siththaranjan et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib34)\), and a single scalar reward may not represent multimodal population preferences\([Chakraborty et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib10)\)\. For a reference policyπref\\pi\_\{\\mathrm\{ref\}\}, population rewardr⋆r^\{\\star\}, and KL penalty strengthβ\\beta, the optimal policy has the formπ⋆​\(y∣x\)∝πref​\(y∣x\)​exp⁡\(r⋆​\(x,y\)/β\)\\pi^\{\\star\}\(y\\mid x\)\\propto\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\\exp\\\!\\big\(r^\{\\star\}\(x,y\)/\\beta\\big\), and DPO directly fits a policy within this implicit\-reward class\([Rafailov et al\., 2023](https://arxiv.org/html/2609.16454#bib.bib31)\)\. Finally, common reward structures and regularization regimes can produce unimodal or concentrated optima under both forward and reverse KL divergence, indicating that the KL penalty alone does not preserve the reference distribution’s diversity\([GX\-Chen et al\., 2026](https://arxiv.org/html/2609.16454#bib.bib18)\)\. These population\-level mechanisms persist as preference data size increases\. We focus instead on the finite\-sample distortion around the population objective, adding a finite\-sample component to the objective\-induced gapC⁡\(π⋆\)−C⁡\(p\)C\(\\pi^\{\\star\}\)\-C\(p\)from existing literature \(Appendix[F](https://arxiv.org/html/2609.16454#A6)\)\. Because form\-sensitive and content\-sensitive diversity metrics capture different aspects of generator behavior\([Tevet and Berant, 2021](https://arxiv.org/html/2609.16454#bib.bib37)\), our conclusions are metric\-specific\.

Relation to prior work\.[Xie et al\. \(2026\)](https://arxiv.org/html/2609.16454#bib.bib43)find compressed heterogeneity in survey microdata generated by 15 LLMs; fine\-tuning one model on 1,000 records improves aggregate realism on held\-out surveys\. Our analysis shows that the collision gap can have either sign and must vanish as excess population cross\-entropy approaches zero \(§[3\.1](https://arxiv.org/html/2609.16454#S3.SS1), Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1)\)\. Whereas the entropy\-regularized game\-theoretic method GEM\([Li et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib25)\), Selectively Encouraging Diversity in Supervised Fine\-Tuning \(SED\-SFT\)\([Chen et al\., 2026](https://arxiv.org/html/2609.16454#bib.bib11)\), and tempered focal \(TOFU\) loss\([Klypa and Cherednichenko, 2026](https://arxiv.org/html/2609.16454#bib.bib24)\)modify the SFT objective to protect diversity, our experiments show that plain SFT moves diversity toward the target across four models, three surveys, and CodeNet, from both directions\.[Banayeeanzade et al\. \(2026\)](https://arxiv.org/html/2609.16454#bib.bib7)attribute sequence\-level diversity loss to miscalibrated token probabilities; our decomposition and bound hold for any fitting procedure\.

## 3Theory

LetVVbe the token vocabulary, andV∗V^\{\*\}the set of finite token sequences\. Let promptsxxbe drawn from a distributionν\\nu, and letp⁡\(y∣x\)p\(y\\mid x\)be the target distribution over responsesy∈V∗y\\in V^\{\*\}\. The model outputs tokenata\_\{t\}given contextht:=\(x,y<t\)h\_\{t\}:=\(x,y\_\{<t\}\), wherey<ty\_\{<t\}is the response prefix before steptt\. The true conditional distribution isph​\(a\):=p​\(a∣h\)p\_\{h\}\(a\):=p\(a\\mid h\)\. For an LLM with parametersθ\\theta,fθ​\(h\)f\_\{\\theta\}\(h\)is the vector of pre\-softmax scores \(logits\) predicted by the model given contexthh\. Applying softmax givesqθ​\(a∣h\):=softmax⁡\(fθ​\(h\)\)aq\_\{\\theta\}\(a\\mid h\):=\\softmax\(f\_\{\\theta\}\(h\)\)\_\{a\}, the model’s probability of tokenaagiven contexthh\. SFT fitsθ\\thetaby minimizing empirical cross\-entropy, the average of−log⁡qθ​\(a∣h\)\-\\log q\_\{\\theta\}\(a\\mid h\)over observed context–token pairs\(h,a\)\(h,a\)in the fine\-tuning data\. A model fit produces parametersθ^\\hat\{\\theta\}and modelqθ^q\_\{\\hat\{\\theta\}\}\.

We now define collision for an arbitrary distributionπ\\piover tokens\. Collision is the probability that two tokens drawn independently fromπ\\piare equal:

c⁡\(π\):=∑a∈Vπ​\(a\)2\.\\textstyle c\(\\pi\):=\\sum\_\{a\\in V\}\\pi\(a\)^\{2\}\.\(1\)
The inverse Simpson index is the number of equally likely choices that would give the same collision probability\. ForKKequally likely tokens, it equalsKK\. For target and model distributionsppandqq, define

N2​\(p\):=1C⁡\(p\),N2​\(q\):=1C⁡\(q\),\\textstyle N\_\{2\}\(p\):=\\frac\{1\}\{C\(p\)\},\\qquad N\_\{2\}\(q\):=\\frac\{1\}\{C\(q\)\},\(2\)whereCCdenotes collision\. At a fixed contexthh,C⁡\(p\)=c⁡\(ph\)C\(p\)=c\(p\_\{h\}\)andC\(q\)=c\(qθ\(⋅∣h\)\)C\(q\)=c\(q\_\{\\theta\}\(\\cdot\\mid h\)\)\.

We compare model and target diversity through the collision ratio

R:=C⁡\(q\)C⁡\(p\)=N2​\(p\)N2​\(q\),\\textstyle R:=\\frac\{C\(q\)\}\{C\(p\)\}=\\frac\{N\_\{2\}\(p\)\}\{N\_\{2\}\(q\)\},\(3\)whereC⁡\(p\)\>0C\(p\)\>0:R=1R=1indicates calibrated diversity,R\>1R\>1mode collapse, andR<1R<1over\-dispersion\.

For a fixed promptxx, draw complete responsesYYandY′Y^\{\\prime\}independently fromp\(⋅∣x\)p\(\\cdot\\mid x\)\. Their joint law is the product measurep\(\.\|x\)⊗p\(\.\|x\)p\(\.\|x\)\\otimes p\(\.\|x\); independence is conditional onxx, and averaging over a shared random prompt need not preserve it\. Under a similarity kernelk⁡\(y,y′\)∈\[0,1\]k\(y,y^\{\\prime\}\)\\in\[0,1\], kernel collision is

Cp,k\(x\)=𝔼Y,Y′∼p\(⋅∣x\)\[k\(Y,Y′\)\]\.C\_\{p,k\}\(x\)=\\mathbb\{E\}\_\{Y,Y^\{\\prime\}\\sim p\(\\cdot\\mid x\)\}\[k\(Y,Y^\{\\prime\}\)\]\.\(4\)
The kernel functionk⁡\(y,y′\)k\(y,y^\{\\prime\}\)must satisfyk⁡\(y,y′\)∈\[0,1\]k\(y,y^\{\\prime\}\)\\in\[0,1\],k⁡\(y,y\)=1k\(y,y\)=1, andCp,k​\(x\)\>0C\_\{p,k\}\(x\)\>0\. A kernel representing exact collision is defined ask\(y,y′\)=𝟏\{y=y′\}k\(y,y^\{\\prime\}\)=\\mathbf\{1\}\\\{y=y^\{\\prime\}\\\}\. Finally, task\-specific kernels may measure embedding cosine similarity, shared cluster membership, normalized Zhang–Shasha\([Zhang and Shasha, 1989](https://arxiv.org/html/2609.16454#bib.bib39)\)similarity between code abstract syntax trees \(ASTs\), or similarity of survey responses, with all values shifted to the range\[0,1\]\[0,1\]\.

### 3\.1Sign and magnitude of the difference in collision probabilities

The collision decomposition\.For a fixed contexthh, consider fitted conditional probabilityqθ^,h=qθ^\(⋅∣h\)q\_\{\\hat\{\\theta\},h\}=q\_\{\\hat\{\\theta\}\}\(\\cdot\\mid h\)and true conditional probabilityphp\_\{h\}\. Applying Eq\. \([4](https://arxiv.org/html/2609.16454#S3.E4)\) to next\-token outputs at contexthh, with the exact\-match kernelk\(a,b\)=𝟏\{a=b\}k\(a,b\)=\\mathbf\{1\}\\\{a=b\\\}, givesCh\(q\):=∑a∈Vq\(a∣h\)2=c\(q\(⋅∣h\)\)C\_\{h\}\(q\):=\\sum\_\{a\\in V\}q\(a\\mid h\)^\{2\}=c\(q\(\\cdot\\mid h\)\)\. ThusCh​\(qθ^\)=‖qθ^,h‖22C\_\{h\}\(q\_\{\\hat\{\\theta\}\}\)=\\\|q\_\{\\hat\{\\theta\},h\}\\\|\_\{2\}^\{2\}andCh​\(p\)=‖ph‖22C\_\{h\}\(p\)=\\\|p\_\{h\}\\\|\_\{2\}^\{2\}\. We call the difference𝔼⁡\[Ch​\(qθ^\)\]−Ch​\(p\)\\mathbb\{E\}\[C\_\{h\}\(q\_\{\\hat\{\\theta\}\}\)\]\-C\_\{h\}\(p\)the expected collision gap: it is positive when the model collides more often than the target \(mode collapse\) and negative when the model is more diverse\. For an arbitrary random vectorXX, we use the standard variance decomposition identity:𝔼​‖X‖22=‖𝔼​X‖22\+Tr⁡Cov⁡\(X\)\\mathbb\{E\}\\\|X\\\|\_\{2\}^\{2\}=\\\|\\mathbb\{E\}X\\\|\_\{2\}^\{2\}\+\\Tr\\Cov\(X\)\. HereTr⁡Cov⁡\(X\)\\Tr\\Cov\(X\)is the sum of the component variances\. TakingX=qθ^,hX=q\_\{\\hat\{\\theta\},h\},𝔼⁡\[X\]\\mathbb\{E\}\[X\]is the mean fitted distribution across independent fits\. Next, define its bias relative to the target as

bh:=𝔼⁡\[qθ^,h\]−ph\.b\_\{h\}:=\\mathbb\{E\}\[q\_\{\\hat\{\\theta\},h\}\]\-p\_\{h\}\.\(5\)Thus𝔼⁡\[qθ^,h\]=ph\+bh\\mathbb\{E\}\[q\_\{\\hat\{\\theta\},h\}\]=p\_\{h\}\+b\_\{h\}\. Substituting this into the variance decomposition identity, expanding‖ph\+bh‖22\\\|p\_\{h\}\+b\_\{h\}\\\|\_\{2\}^\{2\}, and dividing byCh​\(p\)C\_\{h\}\(p\)gives

𝔼⁡\[R\]=1\+1Ch​\(p\)​\[Tr⁡Cov⁡\(qθ^,h\)⏟variance≥0\+‖bh‖22⏟squared bias≥0\+2​ph⊤​bh⏟target–bias alignment\]\.\\boxed\{\\textstyle\\;\\mathbb\{E\}\[R\]=1\+\\frac\{1\}\{C\_\{h\}\(p\)\}\\Big\[\\underbrace\{\\Tr\\Cov\(q\_\{\\hat\{\\theta\},h\}\)\}\_\{\\text\{variance\}\\,\\geq 0\}\\;\+\\;\\underbrace\{\\\|b\_\{h\}\\\|\_\{2\}^\{2\}\}\_\{\\text\{squared bias\}\\,\\geq 0\}\\;\+\\;\\underbrace\{2\\,p\_\{h\}^\{\\top\}b\_\{h\}\}\_\{\\text\{target\-\-bias alignment\}\}\\Big\]\.\\;\}\(6\)The covariance trace and squared bias are nonnegative, and thus contribute to increasing the expected collision ratio\. The target–bias alignment term2​ph⊤​bh2p\_\{h\}^\{\\top\}b\_\{h\}\(referred to as the “cross term” in what follows\) can be either positive or negative and may dominate these terms, so finite samples alone do not imply mode collapse\. Thus the collision ratioRRcan be above11\(mode collapse\) or below11\(over\-dispersion\)\. Ifbh=0b\_\{h\}=0, the expected gap isTr⁡Cov⁡\(qθ^,h\)≥0\\Tr\\Cov\(q\_\{\\hat\{\\theta\},h\}\)\\geq 0: the model is calibrated or under\-dispersed in expectation, with under\-dispersion whenever this variance is positive\.

The sign of the cross term\.Letq0,hq\_\{0,h\}denote the starting distribution at contexthh\. For a pre\-trained LLM,q0,hq\_\{0,h\}is its output distribution given contexthhbefore any fine\-tuning\. At this starting point, using the definitions of bias \([5](https://arxiv.org/html/2609.16454#S3.E5)\) and collision \([1](https://arxiv.org/html/2609.16454#S3.E1)\) we get

2​ph⊤​bh=2​ph⊤​\(q0,h−ph\)=2​\(ph⊤​q0,h−C⁡\(ph\)\)\.2\\,p\_\{h\}^\{\\top\}b\_\{h\}=2\\,p\_\{h\}^\{\\top\}\(q\_\{0,h\}\-p\_\{h\}\)\\;=\\;2\\big\(p\_\{h\}^\{\\top\}q\_\{0,h\}\-C\(p\_\{h\}\)\\big\)\.\(7\)The alignment term \([7](https://arxiv.org/html/2609.16454#S3.E7)\) is positive when the starting distribution overlaps the target’s modes more than the target overlaps itself\. Moreover, the alignment term can determine whether the expected collision gap is positive or negative\.

Using this setup, we can prove the following result bounding the collision gap between distributionsppandqq:

###### Theorem 1\(Kernel\-collision stability\)\.

Consider the similarity kernelk:V∗×V∗→\[0,1\]k:V^\{\*\}\\times V^\{\*\}\\to\[0,1\], whereV∗V^\{\*\}is the set of finite token sequences\. Write the collision metricCk​\(π\)=∑a,bπ⁡\(a\)​π​\(b\)​k​\(a,b\)C\_\{k\}\(\\pi\)=\\sum\_\{a,b\}\\pi\(a\)\\pi\(b\)k\(a,b\),Dk​\(π\)=1/Ck​\(π\)D\_\{k\}\(\\pi\)=1/C\_\{k\}\(\\pi\), whereπ\\piis a probability distribution onV∗V^\{\*\}andDk​\(π\)D\_\{k\}\(\\pi\)is its kernel effective diversity\. WriteTV\\TVfor total variation distance,⊗\\otimesfor the product measure operator, andKL\(p∥q\)=𝔼a∼p\[−logq\(a\)\]−𝔼a∼p\[−logp\(a\)\]\\mathrm\{KL\}\(p\\\|q\)=\\mathbb\{E\}\_\{a\\sim p\}\[\-\\log q\(a\)\]\-\\mathbb\{E\}\_\{a\\sim p\}\[\-\\log p\(a\)\]for excess population cross\-entropy, using natural logarithms\. For all distributionsp,qp,qonV∗V^\{\*\},

\|Ck​\(p\)−Ck​\(q\)\|≤TV⁡\(p⊗p,q⊗q\)≤KL\(p∥q\)\.\\boxed\{\\;\|C\_\{k\}\(p\)\-C\_\{k\}\(q\)\|\\;\\leq\\;\\TV\(p\\otimes p,\\,q\\otimes q\)\\;\\leq\\;\\sqrt\{\\mathrm\{KL\}\(p\\,\\\|\\,q\)\}\\;\.\}\(8\)Consequently, ifKL\(p∥q\)≤ε\\mathrm\{KL\}\(p\\,\\\|\\,q\)\\leq\\varepsilonthenmax⁡\{0,Ck​\(p\)−ε\}≤Ck​\(q\)≤min⁡\{1,Ck​\(p\)\+ε\}\\max\\\{0,C\_\{k\}\(p\)\-\\sqrt\{\\varepsilon\}\\\}\\leq C\_\{k\}\(q\)\\leq\\min\\\{1,C\_\{k\}\(p\)\+\\sqrt\{\\varepsilon\}\\\}, and the kernel effective diversity obeys

Dk​\(q\)≥1Ck​\(p\)\+ε=Dk​\(p\)1\+Dk​\(p\)​ε\.\\textstyle D\_\{k\}\(q\)\\;\\geq\\;\\frac\{1\}\{C\_\{k\}\(p\)\+\\sqrt\{\\varepsilon\}\}\\;=\\;\\frac\{D\_\{k\}\(p\)\}\{1\+D\_\{k\}\(p\)\\sqrt\{\\varepsilon\}\}\\;\.\(9\)

The proof, via total variation on product measures and Pinsker’s inequality, is in Appendix[A](https://arxiv.org/html/2609.16454#A1), together with a sharper collision\-specific bound \(Proposition[1](https://arxiv.org/html/2609.16454#Thmproposition1)\)\.

## 4Experimental results

To test the bias\-variance decomposition and convergence results in the above theory, we now run three experiments\. The first uses synthetic languages with known ground truth for each decomposition term, allowing direct measurement of every term\. The second and third use human data—survey responses from the GSS\([Davern et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib13)\), WVS\([Haerpfer et al\., 2022](https://arxiv.org/html/2609.16454#bib.bib19)\), and ANES\([American National Election Studies, 2022](https://arxiv.org/html/2609.16454#bib.bib1)\), and human solutions to programming problems from Project CodeNet\([Puri et al\., 2021](https://arxiv.org/html/2609.16454#bib.bib30)\)\. We report the collision ratioRRdefined in Eq\. \([3](https://arxiv.org/html/2609.16454#S3.E3)\); for example,R=2R=2means the model offers half as many effective choices as the data\. Experiment 1 additionally decomposes the expected normalized collision gap𝔼⁡\[R\]−1\\mathbb\{E\}\[R\]\-1into the three terms of Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\)\.

### 4\.1Experiment 1: synthetic languages with known target distribution

We sample data from a synthetic language, fit a series of small GPT models on that data, and compare the diversity of the fitted models against the diversity of the language itself\. Unlike experiments using human\-generated data, in this setting, target probabilities are known\. That allows us to repeat model fits to estimate variance, squared\-bias, and target–bias alignment terms in Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\)\. As we have established in the previous section, the latter term determines whether LLMs mode\-collapses or is over\-diverse\. In order to control this term, we initialize the GPTs using two settings: \(a\) pre\-training from random initialization, and \(b\) fine\-tuning a model pre\-trained to favor a small set of tokens\.

In setting \(a\), the language has a vocabulary of10241024tokens, and each next token depends only on the preceding1616tokens\. The target probabilities are known exactly at every context, and a small GPT can represent the generating distribution exactly, so model capacity does not limit the fit\. In setting \(b\), the language combines fixed token frequencies with adjustments based on the preceding1616tokens\. Appendix[H\.1](https://arxiv.org/html/2609.16454#A8.SS1)gives the language construction and sampling details\. The two settings differ in the model’s starting next\-token distribution, model size, and target language\. They are not intended as a controlled ablation of the starting distribution alone\. Figure[2](https://arxiv.org/html/2609.16454#S4.F2)shows pre\-training from random initialization in panel \(a\) and fine\-tuning from the pre\-trained model in panel \(b\)\.

Decomposing the expected collision gap as in Section[3\.1](https://arxiv.org/html/2609.16454#S3.SS1)requires independent fits: a single fitted model does not show the variance across runs\. We therefore fit100100models at each of100100log\-spaced sample sizes ofNNsampled sequences, giving10,00010\{,\}000GPTs per language\. Each seed draws its own dataset, initialization, and minibatch stream, ensuring that the variance termTr⁡Cov⁡\(qθ^,h\)\\Tr\\Cov\(q\_\{\\hat\{\\theta\},h\}\)in Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\) reflects total across\-run fluctuation\. Details of model sizes, data construction, and fitting configurations are provided in Appendix[H\.1](https://arxiv.org/html/2609.16454#A8.SS1)\.

We consider two settings that sit on opposite sides of the initial\-alignment condition in \([7](https://arxiv.org/html/2609.16454#S3.E7)\)\. In setting \(a\), a GPT starts from random initialization\. Its initial next\-token distributionq0,hq\_\{0,h\}at contexthhhas high entropy; we call this thediffuse prior\. Its probabilities depend on the context, but the network is randomly initialized, so it does not favor or suppress particular tokens based on the target distributionphp\_\{h\}\. In setting \(b\), we pre\-train a GPT to concentrate its probability on a fixed set of common target\-language tokens\. We call this concentrated starting distribution themode\-aligned prior\. Appendix[H\.1](https://arxiv.org/html/2609.16454#A8.SS1)describes the construction of both priors\. This mode\-aligned prior is arguably not a realistic model of pre\-training from scratch, however, it could showcase the behavior of an existing LLM pre\-trained on data from a distribution similar but not identical to the target\.

Under the diffuse prior \(Figure[2](https://arxiv.org/html/2609.16454#S4.F2)\(a\)\), the normalized cross term2​ph⊤​bh/C⁡\(ph\)2p\_\{h\}^\{\\top\}b\_\{h\}/C\(p\_\{h\}\), averaged over contexts, is negative at every sample size, and is largest in magnitude at smallNN\. The normalized variance and squared\-bias terms stay positive\. The sign of𝔼⁡\[R\]−1\\mathbb\{E\}\[R\]\-1depends onNN: at the smallestNNthe cross term dominates and the measured𝔼⁡\[R\]−1\\mathbb\{E\}\[R\]\-1is negative \(over\-dispersion\); however, over most models, the two nonnegative terms dominate and𝔼⁡\[R\]−1\\mathbb\{E\}\[R\]\-1is positive \(mode\-collapse\)\.

Under the mode\-aligned prior, the cross term increases the expected collision ratio at intermediate sample sizes \(Figure[2](https://arxiv.org/html/2609.16454#S4.F2)\(b\)\)\. Even where positive, it accounts for at most31%31\\%of the measured𝔼⁡\[R\]−1\\mathbb\{E\}\[R\]\-1, so variance and squared bias still account for most of the expected normalized collision gap\. At the smallestNN, however, the cross term is negative and mode collapse is driven by the two nonnegative terms\.

To relate these outcomes to the starting distributions, Table[1](https://arxiv.org/html/2609.16454#A2.T1)in Appendix[B](https://arxiv.org/html/2609.16454#A2)checks the initial\-alignment condition in Eq\. \([7](https://arxiv.org/html/2609.16454#S3.E7)\) using the mean initial distributionq¯0,h:=𝔼⁡\[q0,h\]\\bar\{q\}\_\{0,h\}:=\\mathbb\{E\}\[q\_\{0,h\}\], estimated across the100100seeds\. The table reports averages over contexts after the indicated normalization\. The normalized cross term is initially negative under the diffuse prior and positive under the mode\-aligned prior, but fine\-tuning does not preserve these signs at everyNN\. In the next two experiments \(Sections[4\.2](https://arxiv.org/html/2609.16454#S4.SS2)and[4\.3](https://arxiv.org/html/2609.16454#S4.SS3)\), we use human data to test whether fine\-tuning movesRRtoward11\(as Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1)predicts when excess population cross\-entropy approaches zero\)\.

Figure 2:Decomposition of the normalized gap between model and target collision probabilities\.The two synthetic languages have known target distributions \(Appendix[H\.1](https://arxiv.org/html/2609.16454#A8.SS1)\), so repeated fits let us estimate the contributions of variance, squared bias, and target–bias alignment to𝔼⁡\[R\]−1\\mathbb\{E\}\[R\]\-1using Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\)\. Contributions are averaged over contexts and plotted against sampled sequencesNN\. \(a\) shows pre\-training from the diffuse prior \(6\.1356\.135M\-parameter GPT,NNfrom3232k to11M,20002000common contexts\); \(b\) shows fine\-tuning from the mode\-aligned prior \(27\.9727\.97M\-parameter GPT,NNfrom500500to10610^\{6\}\)\. The black line shows the estimated𝔼⁡\[R\]−1\\mathbb\{E\}\[R\]\-1, averaged over contexts: positive values indicate mode collapse and negative values over\-dispersion\. The three contributions sum to this value\. The two panels carry independent vertical and horizontal scales\.
### 4\.2Experiment 2: sociological surveys \(GSS, WVS, and ANES\)

In this experiment, we LoRA fine\-tune four LLMs on human responses to three large opinion surveys, to test whether supervised fine\-tuning moves a model’s answer diversity toward the diversity of the population it is asked to imitate through fine\-tuning\.

We consider three social surveys in this section\. First, the General Social Survey \(GSS\)\([Davern et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib13)\), a nationally representative survey of U\.S\. adults since 1972\. We use the cumulative dataset spanning 1972–2024, including data from 74,485 respondents with complete demographic information\. Second, we use a random subset of one\-third of the countries and one\-third of the questions from wave 7 of the World Values Survey \(WVS\)\([Haerpfer et al\., 2022](https://arxiv.org/html/2609.16454#bib.bib19)\), a cross\-national survey of values \(the sub\-samples are used in the interest of saving compute\)\. Third, we use data from the American National Election Studies \(ANES\)\([American National Election Studies, 2022](https://arxiv.org/html/2609.16454#bib.bib1)\), a long\-running U\.S\. political survey\.

Figure 3:Experiment 2\.Collision ratioRU=C⁡\(q\)/C^UR\_\{\\mathrm\{U\}\}=C\(q\)/\\hat\{C\}\_\{\\mathrm\{U\}\}\(11= calibrated\) against fine\-tuning examples \(2020sample sizes for the GSS,1818each for the WVS and ANES\)\.C^U\\hat\{C\}\_\{\\mathrm\{U\}\}estimates how often two distinct human respondents give the same answer;C⁡\(q\)C\(q\)is the probability that two independent model answers match, using probabilities normalized over the listed options\. Each panel shows the median collision ratio across evaluated question–group pairs for early\-stopped adapters \(blue line\), with shaded bands spanning the2525th–7575th and55th–9595th percentiles across pairs\. The marker atn=0n=0is the zero\-shot base model, with vertical whiskers repeating the two bands; the horizontal axis is symmetric\-log so thatn=0n=0is shown on the same axis as the sweep, and each panel has its own y\-axis limits\.Respondents are partitioned into demographic groups based on characteristics such as country, age, sex, race/urban\-rural status, region, and education\. For each fixed question and demographic groupXX, letYYdenote the answer and letmam\_\{a\}of themmrespondents choose answeraa\. We estimate human collision probability as the proportion of pairs of distinct respondents who give the same answer:C^U:=∑ama​\(ma−1\)/\[m⁡\(m−1\)\]\\hat\{C\}\_\{\\mathrm\{U\}\}:=\\sum\_\{a\}m\_\{a\}\(m\_\{a\}\-1\)/\[m\(m\-1\)\]\. The prompt presented to the model includes the demographic group as a persona, the survey question, and its answer options, and asks the model to select a single answer option\. Applying softmax over the answer\-option logits givesq⁡\(a∣X\)q\(a\\mid X\), the model’s probability of answeraafor the fixed question and demographic groupXX\. The model’s collision probability isC⁡\(q\):=∑aq​\(a∣X\)2C\(q\):=\\sum\_\{a\}q\(a\\mid X\)^\{2\}\. We reportRU:=C⁡\(q\)/C^UR\_\{\\mathrm\{U\}\}:=C\(q\)/\\hat\{C\}\_\{\\mathrm\{U\}\}, which estimates the collision ratio in Eq\. \([3](https://arxiv.org/html/2609.16454#S3.E3)\) usingC^U\\hat\{C\}\_\{\\mathrm\{U\}\}as the human collision estimate\.

We repeat this experiment using376376GSS questions\([Davern et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib13)\),5353WVS questions\([Haerpfer et al\., 2022](https://arxiv.org/html/2609.16454#bib.bib19)\), and4444ANES questions\([American National Election Studies, 2022](https://arxiv.org/html/2609.16454#bib.bib1)\)—and four models—gemma\-2\-2b\-it\([Gemma Team, 2024](https://arxiv.org/html/2609.16454#bib.bib44)\),Qwen3\.5\-2BandQwen3\.5\-4B\([Qwen Team, 2026](https://arxiv.org/html/2609.16454#bib.bib46)\), andgemma\-3\-4b\-it\([Gemma Team, 2025](https://arxiv.org/html/2609.16454#bib.bib45)\)—using the split defined in Table[3](https://arxiv.org/html/2609.16454#A8.T3)of Appendix[H\.2](https://arxiv.org/html/2609.16454#A8.SS2)\. For each survey, model, and fine\-tuning set ofnnrespondent–question examples, we evaluateRUR\_\{\\mathrm\{U\}\}for every question–group pair with at least6060respondents\. We include the zero\-shot model asn=0n=0\. This gives a fixed set of18,27018\{,\}270pairs on the GSS,8,0158\{,\}015on the WVS, and8,2118\{,\}211on the ANES\. Figure[3](https://arxiv.org/html/2609.16454#S4.F3)plots, at eachnn, median pair\-level ratios for early\-stopped adapters, with shaded bands for the2525–75%75\\%and55–95%95\\%spread across pairs\. Movement of the median ratio toward11means that model answer diversity is approaching the human level for a typical question–group pair\.

Three of the four base models mode\-collapse across all surveys; the fourth starts closest to median calibration\. Zero\-shot,gemma\-2\-2b\-it,Qwen3\.5\-4B, andgemma\-3\-4b\-ithave median collision ratiosRUR\_\{\\mathrm\{U\}\}between1\.611\.61and2\.922\.92, corresponding to roughly one\-third to two\-thirds of the human effective diversity\. The collapse is most pronounced on the WVS—the sparsest and only cross\-national survey\. The three collapsed models all have higher median collision ratios on the WVS than on the U\.S\. surveys \(Table[2](https://arxiv.org/html/2609.16454#A3.T2)in Appendix[C](https://arxiv.org/html/2609.16454#A3)\)\. The severity of mode collapse therefore varies with both the model and the survey population\.

At the smallestnn, the three collapsed models stay near their zero\-shot collision ratio\.Qwen3\.5\-2B, the model closest to median calibration atn=0n=0, is instead driven above its zero\-shot collision ratio during early fine\-tuning on all three surveys—to2\.222\.22on the GSS, leaving less than half its starting effective diversity—before recovering \(Table[2](https://arxiv.org/html/2609.16454#A3.T2)in Appendix[C](https://arxiv.org/html/2609.16454#A3)\)\. Beyond roughly3,0003\{,\}000fine\-tuning examples, all twelve survey–model combinations maintain median collision ratios within10%10\\%of the calibrated value of11\.Qwen3\.5\-2B’s early increase in collision ratio moves it away from11, showing that small fine\-tuning datasets can worsen diversity calibration\. The later approach to11across all models and surveys shows that larger human datasets improve calibration\.

Figure 4:Experiment 3\. CodeNet diversity under normalized Zhang–Shasha similarity\.Collision ratioCZS​\(q\)/CZS​\(p\)C\_\{\\mathrm\{ZS\}\}\(q\)/C\_\{\\mathrm\{ZS\}\}\(p\)against SFT examples \(11= calibrated\)\. Lines show median finite collision ratios across problems; shaded bands span the2525th–7575th and55th–9595th percentiles\. At the full fine\-tuning dataset, each model’s median uses all2,6472\{,\}647candidate problems, while base markers cover2,6462\{,\}646,2,5692\{,\}569,986986, and2,6462\{,\}646problems in panel order\. The text separately compares the same problems before fine\-tuning and after fine\-tuning on the full dataset\. Panels have separate y\-axes and broken horizontal axes\.
### 4\.3Experiment 3: code generation using CodeNet dataset

In contrast to the previous section, exact sequence collisions are too rare to estimate for complete programs\. So on CodeNet\([Puri et al\., 2021](https://arxiv.org/html/2609.16454#bib.bib30)\)we compare model generations with accepted Python submissions using structural similarity\. We fine\-tune the same four LLMs as in Experiment 2 on accepted Python submissions\. Before fine\-tuning and at each fine\-tuning\-set size, each model generates3232programs per problem\. We map each programyyto an ordered labeled canonical Python abstract syntax treeT⁡\(y\)T\(y\), removing comments and formatting and canonicalizing identifiers and literals\.

LetdZSd\_\{\\mathrm\{ZS\}\}be the Zhang–Shasha tree edit distance\([Zhang and Shasha, 1989](https://arxiv.org/html/2609.16454#bib.bib39)\): the minimum number of node insertions, deletions, and relabelings needed to transform one tree into the other, with each operation costing one\. We define

kZS​\(y,y′\):=1−dZS​\(T⁡\(y\),T⁡\(y′\)\)\|T⁡\(y\)\|\+\|T⁡\(y′\)\|\.\\textstyle k\_\{\\mathrm\{ZS\}\}\(y,y^\{\\prime\}\):=1\-\\frac\{d\_\{\\mathrm\{ZS\}\}\\\!\\left\(T\(y\),T\(y^\{\\prime\}\)\\right\)\}\{\|T\(y\)\|\+\|T\(y^\{\\prime\}\)\|\}\.\(10\)where\|T⁡\(y\)\|\|T\(y\)\|is the number of nodes inT⁡\(y\)T\(y\)\. This similarity kernelkZS​\(y,y′\)k\_\{\\mathrm\{ZS\}\}\(y,y^\{\\prime\}\)lies in\[0,1\]\[0,1\]and equals11for identical trees\. For each problem,ppdenotes the distribution of accepted human submissions andqqthe model’s output distribution\. The kernel collisionsCZS​\(p\)C\_\{\\mathrm\{ZS\}\}\(p\)andCZS​\(q\)C\_\{\\mathrm\{ZS\}\}\(q\)in Eq\. \([4](https://arxiv.org/html/2609.16454#S3.E4)\) measure expected similarity within each source using Eq\. \([10](https://arxiv.org/html/2609.16454#S4.E10)\), conditional on both programs parsing as Python\. For each source separately, we estimate its collision by averagingkZS​\(y,y′\)k\_\{\\mathrm\{ZS\}\}\(y,y^\{\\prime\}\)over up to3232distinct unordered pairs of parseable programs sampled without replacement\. Figure[4](https://arxiv.org/html/2609.16454#S4.F4)reports the median finite ratioCZS​\(q\)/CZS​\(p\)C\_\{\\mathrm\{ZS\}\}\(q\)/C\_\{\\mathrm\{ZS\}\}\(p\)over a fixed panel of2,6472\{,\}647candidate problems\. The base models’ median collision ratios lie on both sides of the accepted\-program target: both Qwen models are over\-diverse, while both Gemma models are under\-diverse\. After fine\-tuning on the full dataset, each model’s median uses ratios from all candidate problems\. To check whether changes in which problems enter the median explain the shift toward calibration, we compare each model before and after fine\-tuning on the full dataset using only problems included at both stages \(Appendix[H\.3](https://arxiv.org/html/2609.16454#A8.SS3)\)\. All four medians still move closer to11, so the improvement persists when the set of problems is held fixed\. Appendix[H\.3](https://arxiv.org/html/2609.16454#A8.SS3)gives the complete construction, prompt, fine\-tuning configuration, and program examples\.

## 5Conclusion

LLMs can exhibit either mode collapse or over\-dispersion relative to a target population, even though recent literature has mostly emphasized mode\-collapse\. Finite\-sample fitting has no built\-in direction: variance and squared bias contribute to increasing the expected collision ratio, while target–bias alignment can offset them\. The absolute difference between collision probabilities of the model and the target population is bounded by the square root of KL divergence, so approaching the target distribution forces diversity calibration\. Across four models, our social survey and coding experiments show that increasing fine\-tuning data moves diversity toward the target from both directions\. These results support evaluating diversity relative to a specified target alongside standard performance benchmarks, and motivate fine\-tuning methods that use the diversity\-calibrating properties of SFT\. Obtaining sufficient coverage of the target distribution remains a challenge when data are limited\.

## AI use statement

In this work, we used generative AI tools to: formulate mathematical claims, provide critical ingredients for proving mathematical claims, assist in the writing of proofs, support qualitative and thematic data analysis, implement methods, polish writing\. We take responsibility for the final content of this work, including text, claims or artifacts produced with the aid of generative AI\.

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## Appendix AProofs and alignment conditions

### A\.1Kernel\-collision stability

Kernel collision is an expectation ofkkunder product measuresP=p⊗pP=p\\otimes pandQ=q⊗qQ=q\\otimes q\. Becausek∈\[0,1\]k\\in\[0,1\], the variational characterization of total variation gives

\|Ck​\(p\)−Ck​\(q\)\|=\|𝔼P​\[k\]−𝔼Q​\[k\]\|≤TV⁡\(P,Q\)\.\|C\_\{k\}\(p\)\-C\_\{k\}\(q\)\|=\|\\mathbb\{E\}\_\{P\}\[k\]\-\\mathbb\{E\}\_\{Q\}\[k\]\|\\leq\\TV\(P,Q\)\.Pinsker’s inequality\([Pinsker, 1964](https://arxiv.org/html/2609.16454#bib.bib29)\)and the product identityKL\(p⊗p∥q⊗q\)=2KL\(p∥q\)\\mathrm\{KL\}\(p\\otimes p\\\|q\\otimes q\)=2\\mathrm\{KL\}\(p\\\|q\)yield

TV⁡\(P,Q\)≤12KL\(P∥Q\)=KL\(p∥q\)\.\\TV\(P,Q\)\\leq\\sqrt\{\\tfrac\{1\}\{2\}\\mathrm\{KL\}\(P\\\|Q\)\}=\\sqrt\{\\mathrm\{KL\}\(p\\\|q\)\}\.Clipping collision to\[0,1\]\[0,1\]and inverting positive collision gives the remaining statements\. ∎

For a random fitted modelqθ^q\_\{\\hat\{\\theta\}\}, the fixed\-model bound also controls the expected kernel\-collision gap:

\|𝔼θ^​\[Ck​\(qθ^\)\]−Ck​\(p\)\|≤𝔼θ^​\[KL\(p∥qθ^\)\]≤𝔼θ^\[KL\(p∥qθ^\)\]\.\\left\|\\mathbb\{E\}\_\{\\hat\{\\theta\}\}\[C\_\{k\}\(q\_\{\\hat\{\\theta\}\}\)\]\-C\_\{k\}\(p\)\\right\|\\leq\\mathbb\{E\}\_\{\\hat\{\\theta\}\}\\\!\\left\[\\sqrt\{\\mathrm\{KL\}\(p\\\|q\_\{\\hat\{\\theta\}\}\)\}\\right\]\\leq\\sqrt\{\\mathbb\{E\}\_\{\\hat\{\\theta\}\}\[\\mathrm\{KL\}\(p\\\|q\_\{\\hat\{\\theta\}\}\)\]\}\.\(11\)The first inequality uses the triangle inequality and Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1); the second uses Jensen’s inequality\.

### A\.2A sufficient condition for nonnegative alignment

Only the target–bias alignment in Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\) can be negative\. The following condition on the fitted model is sufficient: tokens more likely under the target have no smaller bias\. A concentrated initialization does not by itself imply this condition\.

###### Assumption 1\(Target–bias comonotonicity\)\.

For every paira,b∈Va,b\\in V,

\(ph​\(a\)−ph​\(b\)\)​\(bh​\(a\)−bh​\(b\)\)≥0,bh=𝔼⁡\[qθ^,h\]−ph\.\\big\(p\_\{h\}\(a\)\-p\_\{h\}\(b\)\\big\)\\big\(b\_\{h\}\(a\)\-b\_\{h\}\(b\)\\big\)\\geq 0,\\qquad b\_\{h\}=\\mathbb\{E\}\[q\_\{\\hat\{\\theta\},h\}\]\-p\_\{h\}\.\(12\)

Chebyshev’s sum inequality then gives nonnegative covariance under the uniform measure onVV\. Since∑abh​\(a\)=0\\sum\_\{a\}b\_\{h\}\(a\)=0,

ph⊤​bh=\|V\|​Cova∼Unif⁡\(V\)⁡\(ph​\(a\),bh​\(a\)\)≥0\.p\_\{h\}^\{\\top\}b\_\{h\}=\|V\|\\,\\Cov\_\{a\\sim\\mathrm\{Unif\}\(V\)\}\\\!\\big\(p\_\{h\}\(a\),b\_\{h\}\(a\)\\big\)\\geq 0\.\(13\)All three terms inside the brackets in Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\) are therefore nonnegative, and

𝔼⁡\[Ch​\(qθ^\)\]−Ch​\(p\)≥0\.\\mathbb\{E\}\[C\_\{h\}\(q\_\{\\hat\{\\theta\}\}\)\]\-C\_\{h\}\(p\)\\geq 0\.\(14\)For a uniform target, the alignment is exactly zero because∑abh​\(a\)=0\\sum\_\{a\}b\_\{h\}\(a\)=0\. In the synthetic experiments, alignment after fitting is negative in the diffuse setting and positive only over an intermediateNNrange in the mode\-aligned setting\. Positive aggregate alignment does not by itself establish the pairwise comonotonicity assumption, and neither observation shows that initialization determines the later sign\.

### A\.3Relative collision bounds and population cross\-entropy

Dividing Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1)by exact target collision gives a relative, exact\-collision bound:

\|C⁡\(qh\)C⁡\(ph\)−1\|=\|C⁡\(qh\)−C⁡\(ph\)\|C⁡\(ph\)​≤\([8](https://arxiv.org/html/2609.16454#S3.E8)\)​KL\(ph∥qh\)C⁡\(ph\)=N2true​\(h\)​KL\(ph∥qh\)\.\\left\|\\,\\frac\{C\(q\_\{h\}\)\}\{C\(p\_\{h\}\)\}\-1\\,\\right\|\\;=\\;\\frac\{\\big\|C\(q\_\{h\}\)\-C\(p\_\{h\}\)\\big\|\}\{C\(p\_\{h\}\)\}\\;\\overset\{\\eqref\{eq:cebound\}\}\{\\leq\}\\;\\frac\{\\sqrt\{\\mathrm\{KL\}\(p\_\{h\}\\,\\\|\\,q\_\{h\}\)\}\}\{C\(p\_\{h\}\)\}\\;=\\;N\_\{2\}^\{\\mathrm\{true\}\}\(h\)\\,\\sqrt\{\\mathrm\{KL\}\(p\_\{h\}\\,\\\|\\,q\_\{h\}\)\}\.\(15\)
###### Proposition 1\(Relative bound on the collision ratio\)\.

Fix a contexthhwith true conditionalphp\_\{h\}and model conditionalqhq\_\{h\}, writeC⁡\(r\)=‖r‖22C\(r\)=\\\|r\\\|\_\{2\}^\{2\},N2​\(r\)=1/C⁡\(r\)N\_\{2\}\(r\)=1/C\(r\),N2true​\(h\)=N2​\(ph\)N\_\{2\}^\{\\mathrm\{true\}\}\(h\)=N\_\{2\}\(p\_\{h\}\)\. The local collision ratio isR=C⁡\(qh\)/C⁡\(ph\)R=C\(q\_\{h\}\)/C\(p\_\{h\}\), which equals11when the model’s diversity is calibrated and exceeds11when the model is more repetitive than the data\. Withδ=qh−ph\\delta=q\_\{h\}\-p\_\{h\}\(so∑aδa=0\\sum\_\{a\}\\delta\_\{a\}=0\),

\|C⁡\(qh\)C⁡\(ph\)−1\|≤2​N2true​\(h\)​‖qh−ph‖2\+N2true​\(h\)​‖qh−ph‖22,\\boxed\{\\;\\left\|\\,\\frac\{C\(q\_\{h\}\)\}\{C\(p\_\{h\}\)\}\-1\\,\\right\|\\;\\leq\\;2\\sqrt\{N\_\{2\}^\{\\mathrm\{true\}\}\(h\)\}\\;\\\|q\_\{h\}\-p\_\{h\}\\\|\_\{2\}\\;\+\\;N\_\{2\}^\{\\mathrm\{true\}\}\(h\)\\,\\\|q\_\{h\}\-p\_\{h\}\\\|\_\{2\}^\{2\},\\;\}\(16\)A looser bound follows using the Pearson divergenceχ2\(qh∥ph\)=∑aδa2/ph\(a\)≥∥δ∥22\\chi^\{2\}\(q\_\{h\}\\,\\\|\\,p\_\{h\}\)=\\sum\_\{a\}\\delta\_\{a\}^\{2\}/p\_\{h\}\(a\)\\geq\\\|\\delta\\\|\_\{2\}^\{2\}\(finite whenqh≪phq\_\{h\}\\ll p\_\{h\}\):

\|C⁡\(qh\)C⁡\(ph\)−1\|≤2N2true\(h\)χ2\(qh∥ph\)\+N2true\(h\)χ2\(qh∥ph\)\.\\boxed\{\\;\\left\|\\,\\frac\{C\(q\_\{h\}\)\}\{C\(p\_\{h\}\)\}\-1\\,\\right\|\\;\\leq\\;2\\sqrt\{N\_\{2\}^\{\\mathrm\{true\}\}\(h\)\\,\\chi^\{2\}\(q\_\{h\}\\,\\\|\\,p\_\{h\}\)\}\\;\+\\;N\_\{2\}^\{\\mathrm\{true\}\}\(h\)\\,\\chi^\{2\}\(q\_\{h\}\\,\\\|\\,p\_\{h\}\)\.\\;\}\(17\)

Sinceqh=ph\+δq\_\{h\}=p\_\{h\}\+\\delta,C⁡\(qh\)−C⁡\(ph\)=2​⟨ph,δ⟩\+‖δ‖22C\(q\_\{h\}\)\-C\(p\_\{h\}\)=2\\langle p\_\{h\},\\delta\\rangle\+\\\|\\delta\\\|\_\{2\}^\{2\}, soC⁡\(qh\)/C⁡\(ph\)−1=\(2​⟨ph,δ⟩\+‖δ‖22\)/C⁡\(ph\)C\(q\_\{h\}\)/C\(p\_\{h\}\)\-1=\(2\\langle p\_\{h\},\\delta\\rangle\+\\\|\\delta\\\|\_\{2\}^\{2\}\)/C\(p\_\{h\}\)\. By Cauchy–Schwarz,\|⟨ph,δ⟩\|≤‖ph‖2​‖δ‖2=C⁡\(ph\)​‖δ‖2\|\\langle p\_\{h\},\\delta\\rangle\|\\leq\\\|p\_\{h\}\\\|\_\{2\}\\\|\\delta\\\|\_\{2\}=\\sqrt\{C\(p\_\{h\}\)\}\\\|\\delta\\\|\_\{2\}, so the cross term is at most2​‖δ‖2/C⁡\(ph\)=2​N2true​\(h\)​‖δ‖22\\\|\\delta\\\|\_\{2\}/\\sqrt\{C\(p\_\{h\}\)\}=2\\sqrt\{N\_\{2\}^\{\\mathrm\{true\}\}\(h\)\}\\\|\\delta\\\|\_\{2\}, while the quadratic term isN2true​\(h\)​‖δ‖22N\_\{2\}^\{\\mathrm\{true\}\}\(h\)\\\|\\delta\\\|\_\{2\}^\{2\}; the triangle inequality gives \([16](https://arxiv.org/html/2609.16454#A1.E16)\)\. For \([17](https://arxiv.org/html/2609.16454#A1.E17)\), absolute continuity confines all sums to the support ofphp\_\{h\}, where1/ph​\(a\)≥11/p\_\{h\}\(a\)\\geq 1gives∥δ∥22≤χ2\(qh∥ph\)\\\|\\delta\\\|\_\{2\}^\{2\}\\leq\\chi^\{2\}\(q\_\{h\}\\\|p\_\{h\}\)\. ∎

Theℓ2\\ell\_\{2\}andχ2\\chi^\{2\}forms exploit exact\-collision structure\. Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1)is more general and uses forward KL, which equals population cross\-entropy above the target entropy:

KL\(ph∥qh\)=𝔼a∼ph\[−logqh\(a\)\]−𝔼a∼ph\[−logph\(a\)\],\\mathrm\{KL\}\(p\_\{h\}\\,\\\|\\,q\_\{h\}\)=\\mathbb\{E\}\_\{a\\sim p\_\{h\}\}\\\!\\left\[\-\\log q\_\{h\}\(a\)\\right\]\-\\mathbb\{E\}\_\{a\\sim p\_\{h\}\}\\\!\\left\[\-\\log p\_\{h\}\(a\)\\right\],where the first term is the population validation objective and the second is irreducible target entropy\. An empirical validation loss estimates the first term only under the same completion objective and context weighting\. In particular, the survey fine\-tuning loss includes the end\-of\-sequence \(EOS\) token, whereas its reported diversity uses an option\-conditioned answer\-position distribution; aggregate survey validation loss is not a direct estimate of the KL governing that ratio\.

Letμ\\mube a specified probability distribution over token contexts, including its token\-position weighting, and defineKL¯μ=𝔼h∼μ\[KL\(ph∥qh\)\]\\overline\{\\mathrm\{KL\}\}\_\{\\mu\}=\\mathbb\{E\}\_\{h\\sim\\mu\}\[\\mathrm\{KL\}\(p\_\{h\}\\\|q\_\{h\}\)\]\. Cauchy–Schwarz applied to Eq\. \([15](https://arxiv.org/html/2609.16454#A1.E15)\) gives

𝔼h∼μ​\|C⁡\(qh\)C⁡\(ph\)−1\|≤𝔼h∼μ​\[N2true​\(h\)2\]​KL¯μ,\\mathbb\{E\}\_\{h\\sim\\mu\}\\left\|\\,\\frac\{C\(q\_\{h\}\)\}\{C\(p\_\{h\}\)\}\-1\\,\\right\|\\leq\\sqrt\{\\mathbb\{E\}\_\{h\\sim\\mu\}\\\!\\big\[N\_\{2\}^\{\\mathrm\{true\}\}\(h\)^\{2\}\\big\]\\;\\overline\{\\mathrm\{KL\}\}\_\{\\mu\}\}\\,,\(18\)provided the displayed moments are finite\. Thus average relative collision error is controlled by population excess cross\-entropy under the same context measure\.

## Appendix BSynthetic alignment before and after fitting

Regime \(b\): a mode\-aligned starting distribution\.The second synthetic setting first pre\-trains each random GPT by soft cross\-entropy to a Zipf\-weighted distribution over the3232highest\-probability tokens of the target’s global Zipf component\. Low entropy alone does not determine alignment: Eq\. \([7](https://arxiv.org/html/2609.16454#S3.E7)\) is positive only whenph⊤​q¯0,h\>C⁡\(ph\)p\_\{h\}^\{\\top\}\\bar\{q\}\_\{0,h\}\>C\(p\_\{h\}\)\. The construction is chosen to satisfy that initial inequality\.

After SFT, the normalized alignment term2​ph⊤​bh/C⁡\(ph\)2p\_\{h\}^\{\\top\}b\_\{h\}/C\(p\_\{h\}\)is positive only forN=2\.3N=2\.3k–21\.521\.5k and peaks at0\.1930\.193nearN=4,300N=4\{,\}300\(Figure[2](https://arxiv.org/html/2609.16454#S4.F2)\(b\)\)\. Outside that interval its sign reverses\. AtN=500N=500, the normalized varianceTr⁡Cov⁡\(qθ^,h\)/C⁡\(ph\)\\Tr\\Cov\(q\_\{\\hat\{\\theta\},h\}\)/C\(p\_\{h\}\)is approximately3\.93\.9, and the ensemble normalized collision gapGh=𝔼⁡\[Ch​\(qθ^\)\]/C⁡\(ph\)−1G\_\{h\}=\\mathbb\{E\}\[C\_\{h\}\(q\_\{\\hat\{\\theta\}\}\)\]/C\(p\_\{h\}\)\-1is3\.533\.53\. Thus the starting distribution affects the observed trajectory but does not fix the sign after fine\-tuning\.

The initial sign is observable\.Defineb0,h=q¯0,h−phb\_\{0,h\}=\\bar\{q\}\_\{0,h\}\-p\_\{h\}\. Table[1](https://arxiv.org/html/2609.16454#A2.T1)evaluates the decomposition before fitting to target data, where it is independent of sample size\. Intervals are95%95\\%nonparametric bootstrap intervals from2,0002\{,\}000resamples of the common evaluation contexts\. The decomposition identity holds to machine precision \(\|derived−measured\|≤9×10−16\|\\text\{derived\}\-\\text\{measured\}\|\\leq 9\\times 10^\{\-16\}\), but these initial terms do not identify their counterparts after fitting\.

Table 1:Initialization realizes opposite alignment signs\.The four rows from variance through normalized gap form theC⁡\(ph\)C\(p\_\{h\}\)\-normalized decomposition for each100100\-seed starting ensemble; the final two rows are unnormalized\. Each quantity is averaged over contexts after any indicated normalization\. Brackets give95%95\\%bootstrap intervals over common evaluation contexts\. The mode\-aligned start satisfiesph⊤​q¯0,h\>C⁡\(ph\)p\_\{h\}^\{\\top\}\\bar\{q\}\_\{0,h\}\>C\(p\_\{h\}\), whereas the diffuse start does not\. This is an initial\-condition comparison, not a claim that initialization fixes alignment after fitting\.
## Appendix CComplete survey results

Table 2:Final\-adapter survey medians\.Values areRU=C⁡\(qopt\)/C^UR\_\{\\mathrm\{U\}\}=C\(q\_\{\\mathrm\{opt\}\}\)/\\hat\{C\}\_\{\\mathrm\{U\}\}across evaluated question–group pairs, whereC^U\\hat\{C\}\_\{\\mathrm\{U\}\}is the distinct\-pair U\-statistic\. “Base” is the zero\-shot checkpoint; the other columns summarize final adapters over the SFT data\-size grid\. At the largest pool, all medians round to0\.990\.99–1\.021\.02\.Among base checkpoints,Qwen3\.5\-2Bis closest to median calibration \(RU=0\.99R\_\{\\mathrm\{U\}\}=0\.99–1\.101\.10\), while the other three are under\-dispersed on every survey \(1\.611\.61–2\.922\.92\)\. These medians do not establish pairwise distributional fit or reveal the target–bias alignment term, which would require replicated fine\-tuning runs\.

## Appendix DMitigation methods and finite\-sample estimation

Mitigation via decoding and fitting objectives\.Several methods improve text quality and diversity in maximum\-likelihood models by modifying decoding or fitting\. Decoding methods include Top\-kkand nucleus sampling\([Holtzman et al\., 2020](https://arxiv.org/html/2609.16454#bib.bib20)\)and truncation sampling\([Finlayson et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib15)\); the unlikelihood objective instead changes the fitting criterion\([Welleck et al\., 2020](https://arxiv.org/html/2609.16454#bib.bib41)\)\. Methods specific to SFT include GEM, using entropy\-regularized distribution matching\([Li et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib25)\); SED\-SFT, selectively regularizing entropy during exploration\([Chen et al\., 2026](https://arxiv.org/html/2609.16454#bib.bib11)\); and TOFU, employing tempered focal loss to address the neglect of low\-frequency patterns\([Klypa and Cherednichenko, 2026](https://arxiv.org/html/2609.16454#bib.bib24)\)\. Finally, fine\-tuning objectives enhance diversity in creative writing\([Chung et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib12)\)\. Further approaches preserve diversity during preference optimization by separating the entropy and reference\-cross\-entropy components of the standard KL penalty\([Slocum et al\., 2025](https://arxiv.org/html/2609.16454#bib.bib35)\), or by preserving distributions or mixtures of preferences instead of compressing feedback to a single scalar value\([Siththaranjan et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib34);[Chakraborty et al\., 2024](https://arxiv.org/html/2609.16454#bib.bib10)\)\. These methods provide benchmarks for our work, which contributes a finite\-sample law quantifying the portion of observed diversity gaps attributable to estimation error\.

Finite\-sample estimation of distributional functionals\.Plug\-in entropy estimators are susceptible to downward bias in finite samples, especially when the alphabet size is large relative to the sample size \(Miller–Madow, Basharin; see[Paninski, 2003](https://arxiv.org/html/2609.16454#bib.bib28)\)\. The inverse\-Simpson number,N2=exp⁡H2N\_\{2\}=\\exp H\_\{2\}, represents the order\-2 Hill number\. Prior work characterizes minimax estimation of discrete functionals, including entropy and power sums\([Jiao et al\., 2015](https://arxiv.org/html/2609.16454#bib.bib21)\), estimates entropy for linguistic distributions\([Arora et al\., 2022](https://arxiv.org/html/2609.16454#bib.bib6)\), and investigates optimal prediction of unseen species, a task directly relevant to identifying valid, unobserved continuations\([Orlitsky et al\., 2016](https://arxiv.org/html/2609.16454#bib.bib26)\)\. Finally, diversity diagnostics have been developed, including collision\-based birthday\-paradox tests for low effective support in generative adversarial networks \(GANs\)\([Arora and Zhang, 2017](https://arxiv.org/html/2609.16454#bib.bib4);[Arora et al\., 2018](https://arxiv.org/html/2609.16454#bib.bib5)\)and the Vendi score, which measures similarity\-aware diversity beyond exact labels via the Shannon entropy of a similarity matrix’s eigenvalues\([Friedman and Dieng, 2023](https://arxiv.org/html/2609.16454#bib.bib16)\)\. These results are adapted to the context of autoregressive LLM fine\-tuning with shared neural parameters, variable prompts and context lengths, and positive\-only supervision, to interpret findings in terms of scaling laws and prior\-relative lower bounds\.

## Appendix ESequence\-level bounds and response length

Theorem[1](https://arxiv.org/html/2609.16454#Thmtheorem1)applies directly to the countable response spaceV∗V^\{\*\}\. For a bounded sequence kernelk:V∗×V∗→\[0,1\]k:V^\{\*\}\\times V^\{\*\}\\to\[0,1\], defineCp,k\(x\)=𝔼Y,Y′∼iidp\(⋅∣x\)\[k\(Y,Y′\)\]C\_\{p,k\}\(x\)=\\mathbb\{E\}\_\{Y,Y^\{\\prime\}\\overset\{\\mathrm\{iid\}\}\{\\sim\}p\(\\cdot\\mid x\)\}\[k\(Y,Y^\{\\prime\}\)\]and defineCq,k​\(x\)C\_\{q,k\}\(x\)analogously\. Then

\|Cp,k​\(x\)−Cq,k​\(x\)\|≤KL\(p\(⋅∣x\)∥q\(⋅∣x\)\)\.\\big\|\\,C\_\{p,k\}\(x\)\-C\_\{q,k\}\(x\)\\,\\big\|\\;\\leq\\;\\sqrt\{\\mathrm\{KL\}\\big\(p\(\\cdot\\mid x\)\\,\\big\\\|\\,q\(\\cdot\\mid x\)\\big\)\}\.\(19\)
Length enters through total excess cross\-entropy\.Ifppandqqfactor over the same tokenization and termination rule, the KL chain rule gives

KL\(p\(⋅∣x\)∥q\(⋅∣x\)\)=𝔼Y∼p\(⋅∣x\)\[∑t=1T⁡\(Y\)\+1KL\(pht∥qht\)\],ht=\(x,Y<t\),\\mathrm\{KL\}\\big\(p\(\\cdot\\mid x\)\\,\\big\\\|\\,q\(\\cdot\\mid x\)\\big\)\\;=\\;\\mathbb\{E\}\_\{Y\\sim p\(\\cdot\\mid x\)\}\\\!\\left\[\\;\\sum\_\{t=1\}^\{T\(Y\)\+1\}\\mathrm\{KL\}\\big\(p\_\{h\_\{t\}\}\\,\\big\\\|\\,q\_\{h\_\{t\}\}\\big\)\\right\],\\qquad h\_\{t\}=\(x,Y\_\{<t\}\),\(20\)whereT⁡\(Y\)\+1T\(Y\)\+1includes EOS\. LetL⁡\(x\)=𝔼Y∼p​\[T⁡\(Y\)\+1\]L\(x\)=\\mathbb\{E\}\_\{Y\\sim p\}\[T\(Y\)\+1\]and letKL¯tok​\(x\)\\overline\{\\mathrm\{KL\}\}\_\{\\mathrm\{tok\}\}\(x\)be the right\-hand side divided byL⁡\(x\)L\(x\)\. Equation \([19](https://arxiv.org/html/2609.16454#A5.E19)\) becomes

\|Cp,k​\(x\)−Cq,k​\(x\)\|≤L⁡\(x\)​KL¯tok​\(x\)\.\\big\|\\,C\_\{p,k\}\(x\)\-C\_\{q,k\}\(x\)\\,\\big\|\\;\\leq\\;\\sqrt\{L\(x\)\\,\\overline\{\\mathrm\{KL\}\}\_\{\\mathrm\{tok\}\}\(x\)\}\\,\.\(21\)
At fixed mean per\-token KL, this upper bound grows asL⁡\(x\)\\sqrt\{L\(x\)\}\. It improves on the trivial maximum gap of11only when total sequence KL is below one nat\. The survey statistic is a separate one\-step, option\-conditioned categorical comparison, whereas its fine\-tuning and validation losses cover the answer encoding and EOS over the full vocabulary; those aggregate losses are not the KL in a bound forqoptq\_\{\\mathrm\{opt\}\}\. CodeNet responses can be hundreds of tokens long, and the paper does not estimate their sequence\-level KL, so Eq\. \([21](https://arxiv.org/html/2609.16454#A5.E21)\) is not quantitatively informative for the reported Zhang–Shasha collision ratios\.

The relative sequence bound is usually vacuous\.Dividing the absolute exact\-match bound by target collision incurs the factor1/Cp​\(x\)1/C\_\{p\}\(x\), the target’s effective number of complete responses\. For open\-ended generation,Cp​\(x\)C\_\{p\}\(x\)can be extremely small, so1/Cp​\(x\)1/C\_\{p\}\(x\)is large and exact collisions are difficult to estimate at feasible sample sizes\. A bounded structural kernel can yield a larger, more estimableCp,k​\(x\)C\_\{p,k\}\(x\), but the usefulness of its relative bound still depends on the target kernel collision and sequence\-level KL\.

## Appendix FKL\-regularized preference optimization

The accounting framework also distinguishes finite\-sample error from a preference objective’s population optimum\. For KL\-regularized reward optimization, that optimum is the Gibbs tiltπ⋆​\(y∣x\)∝πref​\(y∣x\)​exp⁡\(r⁡\(x,y\)/β\)\\pi^\{\\star\}\(y\\mid x\)\\propto\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\\exp\\\!\\big\(r\(x,y\)/\\beta\\big\), whererris the reward,πref\\pi\_\{\\mathrm\{ref\}\}is the reference policy, andβ\>0\\beta\>0is the KL strength\. Under its preference\-model assumptions, DPO parameterizes the same implicit optimum\([Rafailov et al\., 2023](https://arxiv.org/html/2609.16454#bib.bib31)\)\. Whether that optimum is more concentrated than a human\-response target depends on the reward, reference, and target; we do not establish such concentration here\.

C⁡\(π⋆\)=∑yπref​\(y∣x\)2​exp⁡\(2​r​\(x,y\)/β\)\[∑yπref​\(y∣x\)​exp⁡\(r⁡\(x,y\)/β\)\]2,C\(\\pi^\{\\star\}\)=\\frac\{\\sum\_\{y\}\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)^\{2\}\\exp\\\!\\big\(2r\(x,y\)/\\beta\\big\)\}\{\\Big\[\\sum\_\{y\}\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\\exp\\\!\\big\(r\(x,y\)/\\beta\\big\)\\Big\]^\{2\}\},\(22\)For a random fitted policyπ^\\hat\{\\pi\}, add and subtract the collision ofπ⋆\\pi^\{\\star\}:

𝔼⁡\[C⁡\(π^\)\]−C⁡\(p\)=\(𝔼⁡\[C⁡\(π^\)\]−C⁡\(π⋆\)\)⏟estimation, approximation, and optimization\+\(C⁡\(π⋆\)−C⁡\(p\)\)⏟objective\-induced\.\\boxed\{\\;\\mathbb\{E\}\[C\(\\hat\{\\pi\}\)\]\-C\(p\)=\\underbrace\{\\big\(\\mathbb\{E\}\[C\(\\hat\{\\pi\}\)\]\-C\(\\pi^\{\\star\}\)\\big\)\}\_\{\\text\{estimation, approximation, and optimization\}\}\+\\underbrace\{\\big\(C\(\\pi^\{\\star\}\)\-C\(p\)\\big\)\}\_\{\\text\{objective\-induced\}\}\.\\;\}\(23\)The first term can vanish with growing data only under suitable consistency, capacity, and optimization conditions\. The second can remain even then becauseπ⋆\\pi^\{\\star\}need not equalpp\. Equation \([23](https://arxiv.org/html/2609.16454#A6.E23)\) is an accounting identity, not an empirical result of this paper; residual error may reflect either term\.

## Appendix GEmpirical check of the bound

Proposition[1](https://arxiv.org/html/2609.16454#Thmproposition1)provides a relative bound for fixed fitted\-model conditionals\. Across150,000150\{,\}000held\-in contexts from100100GPTs—one per pre\-training sample size from3232k to11M for a fixed6\.146\.14M\-parameter model—the measured\|C⁡\(qh\)/C⁡\(ph\)−1\|\|C\(q\_\{h\}\)/C\(p\_\{h\}\)\-1\|never exceeds theℓ2\\ell\_\{2\}bound\. The median measured\-to\-bound ratio is0\.1400\.140and the maximum is0\.9800\.980: the bound is typically loose and occasionally near\-tight\. Theχ2\\chi^\{2\}and forward\-KL forms also hold everywhere but are looser\. The minimum measured KL is0\.0353\>00\.0353\>0\.

Figure 5:Theℓ2\\ell\_\{2\}bound holds everywhere and is occasionally near\-tight\.Across150,000150\{,\}000held\-in contexts, the median measured\-to\-bound ratio is0\.1400\.140and the maximum is0\.9800\.980\. \(a\) Median measured error against median bound for eachNN; all points lie belowy=xy=x\. \(b\) Median, interquartile band, and full range of the ratio across1,5001\{,\}500contexts perNN\.Complete per\-question performance grids for all four models are available in the code repository accompanying this submission\. These grids detail performance on every measurable item\.

## Appendix HExperimental details

### H\.1Synthetic languages

How the data are generated\.In setting \(a\), the ground\-truth language is a time\-homogeneous latent\-factorkk\-gram process with exact closed\-form conditionals, vocabulary\|V\|=1024\|V\|=1024and Markov orderk=16k=16\. The tokenxtx\_\{t\}at positionttin a sequence depends only on the trailing windowht:=xt−16:t−1h\_\{t\}:=x\_\{t\-16:t\-1\}, wherex<tx\_\{<t\}denotes the sequence prefix before steptt\. Its next\-token probabilities are given by the tempered softmax of a frozen random linear mapℓ\\ellof the positionally pooled context embeddings:

p⁡\(xt∣x<t\)=p⁡\(xt∣ht\):=\[softmax⁡\(ℓ⁡\(ht\)/\(τgen​s\)\)\]xt,p\(x\_\{t\}\\mid x\_\{<t\}\)=p\(x\_\{t\}\\mid h\_\{t\}\):=\\bigl\[\\softmax\\\!\\bigl\(\\ell\(h\_\{t\}\)/\(\\tau\_\{\\mathrm\{gen\}\}\\,s\)\\bigr\)\\bigr\]\_\{x\_\{t\}\},\(24\)ℓ⁡\(h\):=poolw​\(Eh\)​W1⊤​W2⊤​Wout⊤\+b,poolw​\(Eh\):=∑j=116wj​Ehj,\\textstyle\\ell\(h\):=\\mathrm\{pool\}\_\{w\}\(E\_\{h\}\)\\,W\_\{1\}^\{\\top\}W\_\{2\}^\{\\top\}W\_\{\\mathrm\{out\}\}^\{\\top\}\+b,\\qquad\\mathrm\{pool\}\_\{w\}\(E\_\{h\}\):=\\sum\_\{j=1\}^\{16\}w\_\{j\}\\,E\_\{h\_\{j\}\},\(25\)whereE∈ℝ1024×128E\\in\\mathbb\{R\}^\{1024\\times 128\}is the token embedding table,EhE\_\{h\}collects the context\-token embeddingsEhjE\_\{h\_\{j\}\}, andw∈ℝ16w\\in\\mathbb\{R\}^\{16\}contains positional weightswjw\_\{j\}\. The matricesW1∈ℝ256×128W\_\{1\}\\in\\mathbb\{R\}^\{256\\times 128\}andW2∈ℝ256×256W\_\{2\}\\in\\mathbb\{R\}^\{256\\times 256\}are successive linear maps,Wout∈ℝ1024×256W\_\{\\mathrm\{out\}\}\\in\\mathbb\{R\}^\{1024\\times 256\}maps to token logits, andbbis the logit\-bias vector\. The generation temperature isτgen=0\.6\\tau\_\{\\mathrm\{gen\}\}=0\.6, andssis the standard deviation of logits on a reference sample of 16\-grams, so temperature is invariant to the overall scale of the random weights\. Every ingredient is drawn at random once from the master seed4242and fixed, so the language is fully reproducible\. There is no intermediate nonlinearity\. Becauseℓ\\ellis linear of rank at most256256, a small GPT can represent the generating distributionppexactly and any gap between model and truth would not be limited by model capacity\. Sequences have length6464: the firstk=16k=16tokens are drawn uniformly at random, and each subsequent token is sampled by ancestral multinomial draws fromp\(⋅∣h\)p\(\\cdot\\mid h\)on the trailing1616\-token window; the pre\-training contexts are the1616\-token windows at every prediction position16,…,6316,\\dots,63\. The pre\-training sample sizeNNcounts sampled sequences, each contributing4848supervised next\-token targets \(the data seed is derived from4242perNN\)\. Because the conditionals are closed\-form, the exact target diversityN2​\(ph\)=1/∑ap​\(a∣h\)2N\_\{2\}\(p\_\{h\}\)=1/\\sum\_\{a\}p\(a\\mid h\)^\{2\}is known at every context; it is high and varied \(median177177, range77–398398\)\. Atk=16k=16exact contexts essentially never repeat, so the count\-based/tabular prediction of Appendix[I](https://arxiv.org/html/2609.16454#A9)is structurally undefined here\.Diffuse prior in setting \(a\)\.For each seed, we initialize the GPT using its default random initialization, without preliminary fitting\. The starting distributionq0,hq\_\{0,h\}is its softmax output at contexthhbefore pre\-training on the synthetic language\. It is generally nonuniform and varies with context and initialization seed\.

What we measure\.We report the ensemble normalized gapGh=𝔼⁡\[R\]−1=𝔼⁡\[Ch​\(qθ^\)\]/Ch​\(p\)−1G\_\{h\}=\\mathbb\{E\}\[R\]\-1=\\mathbb\{E\}\[C\_\{h\}\(q\_\{\\hat\{\\theta\}\}\)\]/C\_\{h\}\(p\)\-1and its threeC⁡\(ph\)C\(p\_\{h\}\)\-normalized terms from Eq\. \([6](https://arxiv.org/html/2609.16454#S3.E6)\)\. Isolating them requires independently pre\-trained models at eachNN\. We pre\-train100100seeds at each of100100log\-spaced sequence counts from3232k to11M \(10,00010\{,\}000GPTs;6\.1356\.135M parameters\) and evaluate them on2,0002\{,\}000common contexts\. Each seed draws its own data, initialization, and minibatch stream, so the covariance captures their total across\-run variation\. The closed\-formphp\_\{h\}permits direct estimation ofbhb\_\{h\}from the across\-seed mean\. All validation losses are below the uniform baselinelog⁡1024≈6\.93\\log 1024\\approx 6\.93nats\.Setup\.We fine\-tune a27\.9727\.97M\-parameter GPT \(dmodel=256d\_\{\\text\{model\}\}=256,44heads,33layers,K=50,000K=50\{,\}000\) on a heavy\-tailed order\-1616Zipf\-plus\-context language with exponent1\.071\.07\. Each length\-1717sequence contributes one prediction target\. Before SFT, we select the3232most probable tokens in the target language’s fixed token\-frequency distribution and rescale their probabilities to sum to one\. We then pre\-train each GPT for400400steps on random contexts, using this same desired next\-token distribution for every context\. The analyzed artifacts contain100100seeds at each of100100sequence counts from500500to10610^\{6\}; every model then undergoes4,0004\{,\}000SFT steps\.

### H\.2Surveys

Datasets\.We study three long\-running probability surveys of human attitudes\. The General Social Survey \(GSS\) is a nationally representative survey of adults in the United States, conducted since19721972; we use the19721972–20242024cumulative file \(74,48574\{,\}485respondents with complete demographics\)\. The World Values Survey \(WVS\) is a cross\-national survey of social, political, religious, and economic values; we use wave 7 \(fieldwork20172017–20232023\)\. For computational tractability we restrict the WVS to a random one\-third of its countries and a random one\-third of its measurable questions, both drawn with a fixed seed \(4242\) and frozen before any distribution is computed; the resulting subset comprises32,22732\{,\}227respondents with complete demographics across2222countries spanning every inhabited continent \(e\.g\. India, Pakistan, Kenya, Morocco, Serbia, South Korea, Canada, the Netherlands\)\. The American National Election Studies \(ANES\) is a long\-running U\.S\. political survey; we use the Time\-Series Cumulative Data File \(19481948–20202020;69,78469\{,\}784respondents with complete demographics\)\. From each survey we take every suitable categorical opinion item—substantive attitude/value/policy questions with a small fixed answer set, with value labels transcribed verbatim from each survey’s codebook \(no guessed labels\)—excluding identifiers, weights, dates, interviewer/geographic metadata, raw numerics, multi\-select “mention” batteries, and behavioral/frequency/factual items\. For the GSS, items are identified by their attitudinal value\-label sets \(favor/oppose, agree–disagree, spending too\-little/too\-much, confidence, importance, likely, should, true/false\)\. We then keep only the measurable items—those with at least one demographic group of≥60\\geq 60respondents, the same threshold used for respondent\-sample target estimation—leaving376376items for the GSS,159159for the WVS \(of which the random\-third restriction above retains5353\), and4444for the ANES; every retained item is evaluated\. The seven curated items used in an earlier version of this experiment \(capital punishmentcappun, gun permitsgunlaw, welfare spendingnatfare, happinesshappy, financial satisfactionsatfin, political viewspolviews, party identificationpartyidfor the GSS, and the analogous WVS seven\) are a strict subset of the GSS measurable set, which we exploit for the seven\-vs\-full comparison \(GSS only: the WVS random\-third restriction does not retain all seven curated WVS items\)\.

Grouping, conditioning, and respondent\-sample target\.Respondents are partitioned into demographic groupsXXdefined by a tuple of variables, brought to GSS parity across surveys:

- •GSS:X=\(decade,age bucket,sex,race,region,degree\)X=\(\\text\{decade\},\\ \\text\{age bucket\},\\ \\text\{sex\},\\ \\text\{race\},\\ \\text\{region\},\\ \\text\{degree\}\);
- •WVS:X=\(country,survey year,urban/rural,age bucket,sex,education\)X=\(\\text\{country\},\\ \\text\{survey year\},\\ \\text\{urban/rural\},\\ \\text\{age bucket\},\\ \\text\{sex\},\\ \\text\{education\}\);
- •ANES:X=\(decade,age bucket,sex,race,region,education\)X=\(\\text\{decade\},\\ \\text\{age bucket\},\\ \\text\{sex\},\\ \\text\{race\},\\ \\text\{region\},\\ \\text\{education\}\),

Age was bucketed into groups\{18​–​29,30​–​44,45​–​59,60\+\}\\\{18\\text\{\-\-\}29,30\\text\{\-\-\}44,45\\text\{\-\-\}59,60\+\\\}in the GSS and\{16​–​29,30​–​44,45​–​59,60\+\}\\\{16\\text\{\-\-\}29,30\\text\{\-\-\}44,45\\text\{\-\-\}59,60\+\\\}in the WVS and ANES\. ANES and WVS aggregate calendar years \(to decade and fieldwork year, respectively\) to avoid sparse groups\. For each fixed question and groupXX, withYYdenoting the answer, observed countsmam\_\{a\}amongmmrespondents define the unweighted plug\-in conditionalp^a=ma/m\\hat\{p\}\_\{a\}=m\_\{a\}/mand plug\-in collisionC^plug=∑ap^a2\\hat\{C\}\_\{\\mathrm\{plug\}\}=\\sum\_\{a\}\\hat\{p\}\_\{a\}^\{2\}\. We report instead the distinct\-pair U\-statisticC^U=∑ama​\(ma−1\)/\[m⁡\(m−1\)\]=\(m​C^plug−1\)/\(m−1\)\\hat\{C\}\_\{\\mathrm\{U\}\}=\\sum\_\{a\}m\_\{a\}\(m\_\{a\}\-1\)/\[m\(m\-1\)\]=\(m\\hat\{C\}\_\{\\mathrm\{plug\}\}\-1\)/\(m\-1\)\. The target uses all eligible respondents, including fine\-tuning and validation records\. It represents this unweighted sample, not a held\-out or survey\-weighted estimate of the wider population\. We compute each pair’s collision ratioRU=C⁡\(qopt\)/C^UR\_\{\\mathrm\{U\}\}=C\(q\_\{\\mathrm\{opt\}\}\)/\\hat\{C\}\_\{\\mathrm\{U\}\}before taking medians across pairs\. Evaluation requires at least6060respondents for a question–group pair\. Table[3](https://arxiv.org/html/2609.16454#A8.T3)gives the sample sizes\.

Table 3:Experiment 2 datasets \(measurable\-only\)\.Each survey is restricted to its measurable opinion questions \(a question with≥1\\geq 1group of≥60\\geq 60respondents\)\. “Groups” is the number of demographic tuplesXX; “eval groups” counts groups holding≥60\\geq 60respondents; “eval pairs” counts evaluated\(question,X\)\(\\text\{question\},X\)pairs\. “Examples” are pooled \(respondent, question\) pairs, split at the respondent level; thenn\-grid is anchored to each survey’s full poolNNand halves down to the smallestn≥8n\\geq 8, so each survey has its own grid\.Prompt and measurement\.The prompt presents the group as a first\-person persona followed by the question and lettered options, then requests one option letter; Tables[4](https://arxiv.org/html/2609.16454#A8.T4),[5](https://arxiv.org/html/2609.16454#A8.T5), and[6](https://arxiv.org/html/2609.16454#A8.T6)show the templates\. At the generation position, we take the first token identifier of each letter’s tokenizer encoding, assert that the candidate identifiers are distinct, and apply a softmax only across their logits\. The resultingqopt​\(Y∣X\)q\_\{\\mathrm\{opt\}\}\(Y\\mid X\)is exact conditional on the candidate set and has no sampling noise, but it is not the full next\-token distribution\. We fine\-tune four models of at most44B parameters—gemma\-2\-2b\-it,Qwen3\.5\-2B,Qwen3\.5\-4B, andgemma\-3\-4b\-it—with shared hyperparameters and model\-specific LoRA target scopes\. A respondent\-level split prevents leakage between the fine\-tuning and validation splits, while the empirical target intentionally uses both splits\. The pipeline saves an early\-stopped adapter and the final adapter; callback state is not persisted across requeues, so the former is not guaranteed to be the global minimum\-validation\-loss checkpoint\. The sample\-size grid halves each survey’s fine\-tuning poolNNdown to the smallestn≥8n\\geq 8\(GSS:2020points,4,706,9404\{,\}706\{,\}940to88; WVS:1818,1,462,6451\{,\}462\{,\}645to1111; ANES:1818,1,662,7881\{,\}662\{,\}788to1212\)\. For multimodalgemma\-3\-4b\-it, LoRA and measurement use only the text decoder\.

Table 4:GSS prompt format\.The persona template \(top\) is instantiated from a group’s variables; the model then receives the question with its lettered \(graded\) options and must reply with one letter\. Five curated items are shown with their graded options and an illustrative answer; the full measurable set of376376items is evaluated\.It is the \{decade\}\. You are a \{age\} year\-old \{race\} \{sex\} living in the \{region\} region of the United States\. Your highest education credential is: \{degree\}\. \{question\} Options: A\) \{option A\} B\) \{option B\} … Respond with only the single letter of the option that best matches your view\.

Table 5:WVS prompt format\.Same template as the GSS up to the persona variables \(country / year / urban\-rural / age / sex / education\)\. Five curated items are shown with their graded options and an illustrative answer; a random third of the measurable items \(5353of159159, seed4242\) is evaluated\.It is \{year\}\. You live in \{urban/rural area\} of \{country\}\. You are a \{age\} year\-old \{sex\}\. Your highest level of education is \{education\}\. \{question\} Options: A\) \{option A\} B\) \{option B\} … Respond with only the single letter of the option that best matches your view\.

Table 6:ANES prompt format\.Same template up to the persona variables \(decade / age / sex / race / region / education\)\. Five illustrative items are shown with their graded options and an answer; the full measurable set of4444items is evaluated\. Endpoint\-anchored issue scales keep numbered interior points with labelled poles\.It is the \{decade\}s\. You are a \{age\} year\-old \{sex\} living in \{region\} of the United States\. You identify as \{race\} and your highest level of education is \{education\}\. \{question\} Options: A\) \{option A\} B\) \{option B\} … Respond with only the single letter of the option that best matches your view\.

Fine\-tuning hyperparameters\.All four models use LoRA adapters with shared hyperparameters and model\-specific target scope\. Fine\-tuning and validation use the same shifted mean cross\-entropy over the encoded answer\-letter token\(s\) followed by EOS; prompt and padding positions are masked\. Runs usebf16, gradient checkpointing, and baseuse\_cachedisabled\. Tables[7](https://arxiv.org/html/2609.16454#A8.T7)and[8](https://arxiv.org/html/2609.16454#A8.T8)give shared settings and per\-model scope\. The text models apply LoRA to seven attention and multilayer perceptron \(MLP\) projections in every decoder layer;gemma\-3\-4b\-itapplies the same projections only within itslanguage\_model, leaving the vision tower fixed\. The adapters modify0\.47%0\.47\\%–0\.79%0\.79\\%of parameters \(10\.910\.9–29\.829\.8million\)\.

Table 7:Experiment 2 shared fine\-tuning hyperparameters\.Table 8:Experiment 2 per\-model LoRA configuration\.The shared hyperparameters of Table[7](https://arxiv.org/html/2609.16454#A8.T7)apply to all; the LoRA target scope differs\. Target projections areq\_proj,k\_proj,v\_proj,o\_proj,gate\_proj,up\_proj,down\_proj; forgemma\-3\-4b\-itthese are matched only within thelanguage\_modeldecoder \(vision tower excluded\)\. Trainable counts are fromget\_peft\_model\.
### H\.3CodeNet

Dataset and estimand\.Project CodeNet contains programming\-contest problems, submitted programs, and metadata including language, user, timestamp, and judge status\([Puri et al\., 2021](https://arxiv.org/html/2609.16454#bib.bib30)\)\. We restrict the target to parseable, accepted Python submissions and compare implementation structure rather than lexical form or execution output\. We measure expected similarity under the normalized Zhang–Shasha kernel in Equation \([10](https://arxiv.org/html/2609.16454#S4.E10)\), conditional on both programs parsing successfully\. For each problem and distribution, we average the kernel over a deterministic seed\-4242sample of up to3232distinct unordered pairs, using every pair when fewer are available\.

The metadata contain13,916,86813\{,\}916\{,\}868submissions, including3,286,3143\{,\}286\{,\}314labeled Python and1,796,5631\{,\}796\{,\}563both Python and accepted, across3,1133\{,\}113problems\. We impose no restriction on users, source length, or revisions per user; multiple accepted revisions remain separate observations\. A submission enters the usable target only if its description and source are available and its Python AST parses successfully\.

We split solutions deterministically9090–1010at the submission level\. A problem may appear in both partitions, but a submission identifier cannot\. The90%90\\%partition supplies SFT data and the10%10\\%partition validation loss; the structural target draws up to3232parseable accepted solutions from both\. Problems with one usable solution remain available for SFT, while target collision requires at least two\.Fine\-tuning configuration\.We usegoogle/gemma\-2\-2b\-it,Qwen/Qwen3\.5\-2B,Qwen/Qwen3\.5\-4B, andgoogle/gemma\-3\-4b\-it\. For fine\-tuning\-pool sizeNN, the grid successively halves nested prefixes of a deterministic submission order down ton≥8n\\geq 8\. Each model runs for three epochs with the survey experiment’s LoRA rank, scaling, dropout, learning rate, cosine schedule,4,0004\{,\}000\-row validation subset,1212scheduled validation evaluations, and effective batch size1616\. Fine\-tuning and validation use shifted mean cross\-entropy over code plus the tokenizer’s assistant terminator, with the prompt masked\. We report the final adapter after three epochs; validation loss monitors fine\-tuning but does not select the reported checkpoint, and loss magnitudes are not compared across tokenizers\.

Decoding and parse conditioning\.For each problem and checkpoint, we draw3232samples with temperature1\.01\.0, top\-p=1\.0p=1\.0, and at most1,0241\{,\}024new tokens\. The generation call does not set top\-kkorenable\_thinking, so loaded model and chat\-template defaults supply them; model revisions and the effective generation configuration were not logged\. The loaded Qwen templates are asymmetric:Qwen3\.5\-2Bdefaults to non\-thinking output, whereasQwen3\.5\-4Bdefaults to thinking output\. Decoding is therefore not matched across models\. Seeds derive deterministically from seed4242, problem identifier, fine\-tuning size, and checkpoint\. Outputs that fail AST parsing are recorded but excluded from collision; a ratio is finite only when at least two outputs parse\. Thus cross\-model diversity and parseability comparisons are confounded by decoding defaults, and all Zhang–Shasha diversity results are conditional on parsing\.

Comparison on the same problems\.For each model, we retain problems with at least two parseable outputs both before fine\-tuning and after fine\-tuning on the full dataset\. In Figure[4](https://arxiv.org/html/2609.16454#S4.F4)panel order, these sets contain2,6462\{,\}646,2,5692\{,\}569,986986, and2,6462\{,\}646problems\. Their median collision ratios change from1\.0351\.035to0\.9710\.971,0\.7400\.740to0\.9150\.915,0\.8440\.844to1\.0031\.003, and1\.6371\.637to0\.9740\.974, respectively\. Every model’s median therefore moves closer to11when evaluated on the same problems at both stages\. Both estimates remain conditional on parsing\.

Code\-generation prompt\.The problem description is inserted after the following instruction in one user turn; each model’s chat template supplies its own control tokens\.

> You are an expert competitive programmer\. Return only a complete Python 3 program that solves the problem\. Do not use Markdown fences or explanations\. \{problem description\}

Two accepted\-solution examples\.The following pairs illustrate the primary metric using accepted Python submissions\. Similarities are recomputed with exactly the normalized Zhang–Shasha kernel in Equation \([10](https://arxiv.org/html/2609.16454#S4.E10)\)\. Each table has four cells in the requested order: source code A, canonical AST A, source code B, and canonical AST B\. The AST cells are complete automatic exports from the metric representation\. They contain every retained node and every ordered, field\-typed child edge; nothing is manually pruned\. Source locations andLoad/Storecontext nodes are absent because the metric itself excludes them\.

High structural similarity: divisor rule \(p03125\)\.The problem gives positive integersAAandBB\. IfAAdividesBB, the program printsA\+BA\+B; otherwise, it printsB−AB\-A\. Both accepted submissions use the same conditional implementation\. Their normalized Zhang–Shasha similarity is0\.9705880\.970588\(edit distance22across two3434\-node trees\)\.

Table 9:High\-similarity accepted pair for problemp03125\.The four cells contain the complete source and complete automatically exported canonical AST for submissionss613176273ands053364634\. Node classes in the AST are highlighted in purple; Python keywords, comments, and strings use distinct syntax colors\.Code A:s613176273[⬇](data:text/plain;base64,QSxCPW1hcChpbnQsaW5wdXQoKS5zcGxpdCgpKQppZiBCJUE9PTA6CiAgcHJpbnQoQStCKQplbHNlOgogIHByaW50KEItQSkK)A,B=map\(int,input\(\)\.split\(\)\)ifB%A==0:print\(A\+B\)else:print\(B\-A\)Complete canonical AST A[⬇](data:text/plain;base64,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)root:Modulebody\[\]:Assigntargets\[\]:Tupleelts\[\]:Name:VAR\_0\_0elts\[\]:Name:VAR\_0\_1value:Callfunc:Name:mapargs\[\]:Name:intargs\[\]:Callfunc:Attribute:splitvalue:Callfunc:Name:inputbody\[\]:Iftest:Compareleft:BinOpleft:Name:VAR\_0\_1op:Modright:Name:VAR\_0\_0ops\[\]:Eqcomparators\[\]:Constant:INTbody\[\]:Exprvalue:Callfunc:Name:printargs\[\]:BinOpleft:Name:VAR\_0\_0op:Addright:Name:VAR\_0\_1orelse\[\]:Exprvalue:Callfunc:Name:printargs\[\]:BinOpleft:Name:VAR\_0\_1op:Subright:Name:VAR\_0\_0Code B:s053364634[⬇](data:text/plain;base64,YSxiID0gbWFwKGludCwgaW5wdXQoKS5zcGxpdCgpKQoKaWYgYiAlIGEgPT0gMDoKICBwcmludChiICthKQplbHNlOgogIHByaW50KGIgLSBhKQo=)a,b=map\(int,input\(\)\.split\(\)\)ifb%a==0:print\(b\+a\)else:print\(b\-a\)Complete canonical AST B[⬇](data:text/plain;base64,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)root:Modulebody\[\]:Assigntargets\[\]:Tupleelts\[\]:Name:VAR\_0\_0elts\[\]:Name:VAR\_0\_1value:Callfunc:Name:mapargs\[\]:Name:intargs\[\]:Callfunc:Attribute:splitvalue:Callfunc:Name:inputbody\[\]:Iftest:Compareleft:BinOpleft:Name:VAR\_0\_1op:Modright:Name:VAR\_0\_0ops\[\]:Eqcomparators\[\]:Constant:INTbody\[\]:Exprvalue:Callfunc:Name:printargs\[\]:BinOpleft:Name:VAR\_0\_1op:Addright:Name:VAR\_0\_0orelse\[\]:Exprvalue:Callfunc:Name:printargs\[\]:BinOpleft:Name:VAR\_0\_1op:Subright:Name:VAR\_0\_0Table 10:Low\-similarity accepted pair for problemp03135\.The problem asks how much time passes in World A when a student studies forTThours in World B, where time passesXXtimes as fast\. Both accepted submissions computeT/XT/X\. Solution A converts a split input list in a loop and formats the result; solution B uses nested generator loops, helper functions, and type annotations\. Their normalized Zhang–Shasha similarity is0\.4074070\.407407\(edit distance6464across trees with4444and6464nodes\)\. The four cells contain the source \(comments removed\) and the complete automatically exported canonical AST for submissionss227737564ands089174134; the AST difference is not a manually prepared summary\.Code A:s227737564[⬇](data:text/plain;base64,TCA9IGlucHV0KCkKTCA9IEwuc3BsaXQoKQpmb3IgaSwgdmFsIGluIGVudW1lcmF0ZShMKToKICAgIExbaV0gPSBpbnQodmFsKQp0ID0gTFswXS9MWzFdCnQgPSBmb3JtYXQodCwgJzAuNGYnKQpwcmludCh0KQo=)L=input\(\)L=L\.split\(\)fori,valinenumerate\(L\):L\[i\]=int\(val\)t=L\[0\]/L\[1\]t=format\(t,’0\.4f’\)print\(t\)Complete canonical AST A[⬇](data:text/plain;base64,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)root:Modulebody\[\]:Assigntargets\[\]:Name:VAR\_0\_0value:Callfunc:Name:inputbody\[\]:Assigntargets\[\]:Name:VAR\_0\_0value:Callfunc:Attribute:splitvalue:Name:VAR\_0\_0body\[\]:Fortarget:Tupleelts\[\]:Name:VAR\_0\_1elts\[\]:Name:VAR\_0\_2iter:Callfunc:Name:enumerateargs\[\]:Name:VAR\_0\_0body\[\]:Assigntargets\[\]:Subscriptvalue:Name:VAR\_0\_0slice:Name:VAR\_0\_1value:Callfunc:Name:intargs\[\]:Name:VAR\_0\_2body\[\]:Assigntargets\[\]:Name:VAR\_0\_3value:BinOpleft:Subscriptvalue:Name:VAR\_0\_0slice:Constant:INTop:Divright:Subscriptvalue:Name:VAR\_0\_0slice:Constant:INTbody\[\]:Assigntargets\[\]:Name:VAR\_0\_3value:Callfunc:Name:formatargs\[\]:Name:VAR\_0\_3args\[\]:Constant:STRINGbody\[\]:Exprvalue:Callfunc:Name:printargs\[\]:Name:VAR\_0\_3Code B:s089174134[⬇](data:text/plain;base64,aW1wb3J0IHN5cwoKCmRlZiBzb2x2ZShUOiBpbnQsIFg6IGludCk6CiAgICBwcmludChUL1gpCiAgICByZXR1cm4KCgpkZWYgbWFpbigpOgogICAgZGVmIGl0ZXJhdGVfdG9rZW5zKCk6CiAgICAgICAgZm9yIGxpbmUgaW4gc3lzLnN0ZGluOgogICAgICAgICAgICBmb3Igd29yZCBpbiBsaW5lLnNwbGl0KCk6CiAgICAgICAgICAgICAgICB5aWVsZCB3b3JkCiAgICB0b2tlbnMgPSBpdGVyYXRlX3Rva2VucygpCiAgICBUID0gaW50KG5leHQodG9rZW5zKSkKICAgIFggPSBpbnQobmV4dCh0b2tlbnMpKQogICAgc29sdmUoVCwgWCkKCmlmIF9fbmFtZV9fID09ICdfX21haW5fXyc6CiAgICBtYWluKCkK)importsysdefsolve\(T:int,X:int\):print\(T/X\)returndefmain\(\):defiterate\_tokens\(\):forlineinsys\.stdin:forwordinline\.split\(\):yieldwordtokens=iterate\_tokens\(\)T=int\(next\(tokens\)\)X=int\(next\(tokens\)\)solve\(T,X\)if\_\_name\_\_==’\_\_main\_\_’:main\(\)Complete canonical AST 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## Appendix ITabular intuition for finite\-sample collision

The classical per\-context histogram provides a deliberately restricted comparison to the neural decomposition in Section[3\.1](https://arxiv.org/html/2609.16454#S3.SS1)\. Fix a contexthhwithnhn\_\{h\}independent and identically distributed \(i\.i\.d\.\) samples fromphp\_\{h\}and empirical maximum\-likelihood estimate \(MLE\)p^h​\(a\)=nh,a/nh\\hat\{p\}\_\{h\}\(a\)=n\_\{h,a\}/n\_\{h\}\. Then𝔼⁡\[p^h​\(a\)\]=ph​\(a\)\\mathbb\{E\}\[\\hat\{p\}\_\{h\}\(a\)\]=p\_\{h\}\(a\): finite samples favor no token in expectation, although any realized dataset overweights some tokens\.

Sincenh,a∼Binomial⁡\(nh,ph​\(a\)\)n\_\{h,a\}\\sim\\mathrm\{Binomial\}\(n\_\{h\},p\_\{h\}\(a\)\),𝔼⁡\[p^h​\(a\)\]=𝔼⁡\[nh,a\]/nh=ph​\(a\)\\mathbb\{E\}\[\\hat\{p\}\_\{h\}\(a\)\]=\\mathbb\{E\}\[n\_\{h,a\}\]/n\_\{h\}=p\_\{h\}\(a\)\. ∎

Conditioning on one realized histogram makes its sampling fluctuations shared across all subsequent generations\.

Simple example\.Forp=\(1/3,1/3,1/3\)p=\(1/3,1/3,1/3\)andn=6n=6, realized counts might be\(4,1,1\)\(4,1,1\)or\(1,4,1\)\(1,4,1\)\. The favored token changes across datasets, but both histograms have higher collision thanpp\.

### I\.1Expected collision inflation

For arbitraryphp\_\{h\},𝔼⁡\[p^h​\(a\)2\]=ph​\(a\)2\+ph​\(a\)​\(1−ph​\(a\)\)nh\\mathbb\{E\}\[\\hat\{p\}\_\{h\}\(a\)^\{2\}\]=p\_\{h\}\(a\)^\{2\}\+\\frac\{p\_\{h\}\(a\)\(1\-p\_\{h\}\(a\)\)\}\{n\_\{h\}\}\. Summing overaa,

𝔼⁡\[C⁡\(p^h\)\]=C⁡\(ph\)\+1−C⁡\(ph\)nh,\\mathbb\{E\}\[C\(\\hat\{p\}\_\{h\}\)\]=C\(p\_\{h\}\)\+\\frac\{1\-C\(p\_\{h\}\)\}\{n\_\{h\}\},\(26\)so the local expected collision ratio is

𝔼⁡\[C⁡\(p^h\)\]C⁡\(ph\)=1\+1−C⁡\(ph\)nh​C​\(ph\)=1\+N2​\(ph\)−1nh\.\\frac\{\\mathbb\{E\}\[C\(\\hat\{p\}\_\{h\}\)\]\}\{C\(p\_\{h\}\)\}=1\+\\frac\{1\-C\(p\_\{h\}\)\}\{n\_\{h\}\\,C\(p\_\{h\}\)\}=1\+\\frac\{N\_\{2\}\(p\_\{h\}\)\-1\}\{n\_\{h\}\}\.\(27\)This exact identity is the local scaling law for the tabular MLE:

local expected collision ratio=1\+target effective diversity​N2​\(ph\)−1sample count​nh\.\\text\{local expected collision ratio\}=1\+\\frac\{\\text\{target effective diversity \}N\_\{2\}\(p\_\{h\}\)\-1\}\{\\text\{sample count \}n\_\{h\}\}\.\(28\)The expected collision ratio is also exactly𝔼⁡\[N2​\(ph\)/N2​\(p^h\)\]\\mathbb\{E\}\[N\_\{2\}\(p\_\{h\}\)/N\_\{2\}\(\\hat\{p\}\_\{h\}\)\], becauseN2​\(ph\)/N2​\(p^h\)=C⁡\(p^h\)/C⁡\(ph\)N\_\{2\}\(p\_\{h\}\)/N\_\{2\}\(\\hat\{p\}\_\{h\}\)=C\(\\hat\{p\}\_\{h\}\)/C\(p\_\{h\}\)for every dataset\. It generally differs fromN2​\(ph\)/𝔼⁡\[N2​\(p^h\)\]N\_\{2\}\(p\_\{h\}\)/\\mathbb\{E\}\[N\_\{2\}\(\\hat\{p\}\_\{h\}\)\]\.

Fix the contexthhand abbreviatep​\(a\)=ph​\(a\)p\(a\)=p\_\{h\}\(a\),n=nhn=n\_\{h\}\. The observed counts\(nh,a\)a\(n\_\{h,a\}\)\_\{a\}areMultinomial⁡\(n,ph\)\\mathrm\{Multinomial\}\(n,p\_\{h\}\); in particular eachnh,a∼Binomial⁡\(n,p⁡\(a\)\)n\_\{h,a\}\\sim\\mathrm\{Binomial\}\(n,p\(a\)\), so the empirical MLEp^h​\(a\)=nh,a/n\\hat\{p\}\_\{h\}\(a\)=n\_\{h,a\}/nhas

𝔼⁡\[p^h​\(a\)\]=p⁡\(a\),Var⁡\(p^h​\(a\)\)=p​\(a\)​\(1−p​\(a\)\)n\.\\mathbb\{E\}\[\\hat\{p\}\_\{h\}\(a\)\]=p\(a\),\\qquad\\Var\(\\hat\{p\}\_\{h\}\(a\)\)=\\frac\{p\(a\)\\big\(1\-p\(a\)\\big\)\}\{n\}\.\(29\)By the variance–mean\-square identity𝔼⁡\[X2\]=Var⁡\(X\)\+\(𝔼​X\)2\\mathbb\{E\}\[X^\{2\}\]=\\Var\(X\)\+\(\\mathbb\{E\}X\)^\{2\},

𝔼⁡\[p^h​\(a\)2\]=p​\(a\)​\(1−p​\(a\)\)n\+p​\(a\)2\.\\mathbb\{E\}\[\\hat\{p\}\_\{h\}\(a\)^\{2\}\]=\\frac\{p\(a\)\\big\(1\-p\(a\)\\big\)\}\{n\}\+p\(a\)^\{2\}\.\(30\)Summing overaaand using linearity of expectation \(which needs no independence across tokens, so the within\-multinomial correlations are irrelevant\),

𝔼⁡\[C⁡\(p^h\)\]=∑a𝔼⁡\[p^h​\(a\)2\]=∑ap​\(a\)2\+1n​∑ap⁡\(a\)​\(1−p⁡\(a\)\)=C⁡\(ph\)\+1−C⁡\(ph\)n,\\mathbb\{E\}\[C\(\\hat\{p\}\_\{h\}\)\]=\\sum\_\{a\}\\mathbb\{E\}\[\\hat\{p\}\_\{h\}\(a\)^\{2\}\]=\\sum\_\{a\}p\(a\)^\{2\}\+\\frac\{1\}\{n\}\\sum\_\{a\}p\(a\)\\big\(1\-p\(a\)\\big\)=C\(p\_\{h\}\)\+\\frac\{1\-C\(p\_\{h\}\)\}\{n\},\(31\)where the last equality uses∑ap⁡\(a\)=1\\sum\_\{a\}p\(a\)=1, hence∑ap⁡\(a\)​\(1−p⁡\(a\)\)=1−∑ap​\(a\)2=1−C⁡\(ph\)\\sum\_\{a\}p\(a\)\(1\-p\(a\)\)=1\-\\sum\_\{a\}p\(a\)^\{2\}=1\-C\(p\_\{h\}\)\. Dividing byC⁡\(ph\)C\(p\_\{h\}\)and substitutingN2​\(ph\)=1/C⁡\(ph\)N\_\{2\}\(p\_\{h\}\)=1/C\(p\_\{h\}\),

𝔼⁡\[C⁡\(p^h\)\]C⁡\(ph\)=1\+1−C⁡\(ph\)n​C​\(ph\)=1\+N2​\(ph\)−1n,\\frac\{\\mathbb\{E\}\[C\(\\hat\{p\}\_\{h\}\)\]\}\{C\(p\_\{h\}\)\}=1\+\\frac\{1\-C\(p\_\{h\}\)\}\{n\\,C\(p\_\{h\}\)\}=1\+\\frac\{N\_\{2\}\(p\_\{h\}\)\-1\}\{n\},\(32\)because\(1−C⁡\(ph\)\)/C⁡\(ph\)=N2​\(ph\)−1\(1\-C\(p\_\{h\}\)\)/C\(p\_\{h\}\)=N\_\{2\}\(p\_\{h\}\)\-1\. Restoringn=nhn=n\_\{h\}gives the stated law\. ∎

The critical factor is not the overall size of the SFT dataset, but rather the local effective sample size,nhn\_\{h\}, representing the number of fine\-tuning examples that contribute meaningfully to contexthh\.

###### Example 1\(Why any finite sample inflates collision \(Jensen\)\)\.

CollisionC⁡\(r\)=∑ar​\(a\)2C\(r\)=\\sum\_\{a\}r\(a\)^\{2\}is convex\. Since𝔼⁡\[p^h\]=ph\\mathbb\{E\}\[\\hat\{p\}\_\{h\}\]=p\_\{h\}, Jensen’s inequality gives𝔼⁡\[C⁡\(p^h\)\]≥C⁡\(ph\)\\mathbb\{E\}\[C\(\\hat\{p\}\_\{h\}\)\]\\geq C\(p\_\{h\}\)\. The excess\(1−C⁡\(ph\)\)/nh\(1\-C\(p\_\{h\}\)\)/n\_\{h\}is the summed multinomial variance and vanishes asnh→∞n\_\{h\}\\to\\infty\.

Homogeneous\-tree illustration\.If every context has a uniformK=100K=100continuation distribution, each context hasnh=1000n\_\{h\}=1000independent observations, and continuation collision is identical across sibling branches, the local factor is1\+99/1000=1\.0991\+99/1000=1\.099\. Under these restrictive assumptions,5050decisions yield the product1\.09950≈1121\.099^\{50\}\\approx 112\. This is an illustration, not a neural or generic sequence law\.

2002004004006006008008001,0001\{,\}0001,2001\{,\}2001,4001\{,\}4001,6001\{,\}6001,8001\{,\}8002,0002\{,\}00010010^\{0\}100\.510^\{0\.5\}10110^\{1\}local effective sample sizenhn\_\{h\}expected collision ratio𝔼⁡\[R\]\\mathbb\{E\}\[R\]N2=10N\_\{2\}=10N2=50N\_\{2\}=50N2=100N\_\{2\}=100Figure 6:Tabular collision inflation decays with local coverage\.The exact expected collision ratio1\+\(N2−1\)/nh1\+\(N\_\{2\}\-1\)/n\_\{h\}forN2=10,50,100N\_\{2\}=10,50,100\.0010102020303040405050606010010^\{0\}10110^\{1\}10210^\{2\}10310^\{3\}10410^\{4\}10510^\{5\}sequence lengthLL\(number of nontrivial decisions\)expected sequence collision rationh=2000n\_\{h\}=2000nh=1000n\_\{h\}=1000nh=500n\_\{h\}=500Figure 7:Homogeneous\-tree illustration\.With independent tabular estimates,N2=100N\_\{2\}=100, and identical continuation collision at every branch, the per\-step factor compounds multiplicatively\. The generic recursion need not factor this way\.
### I\.2Amplification over sequences

For variable\-length autoregressive models, sequence collision is not a simple product of probabilities over fixed positions, due to the dependence of contexts on sampled prefixes\. Analysis of these models therefore requires a branching recursive framework\.

Cp​\(h\)=∑ap​\(a∣h\)2​Cp​\(h​a\),C\_\{p\}\(h\)=\\sum\_\{a\}p\(a\\mid h\)^\{2\}\\,C\_\{p\}\(ha\),\(33\)Each summand is the probability that two draws both choose branchaaand then collide fromh​aha\. Thus the collision\-biased next\-token weight is proportional top​\(a∣h\)2​Cp​\(h​a\)p\(a\\mid h\)^\{2\}C\_\{p\}\(ha\), not top​\(a∣h\)2p\(a\\mid h\)^\{2\}alone; the recursion is generally a sum of branch products rather than a product of aggregate one\-step and continuation collisions\.

LetDDdenote the random tabular dataset and letRp​\(y∣x\)=p​\(y∣x\)2/Cp​\(x\)R\_\{p\}\(y\\mid x\)=p\(y\\mid x\)^\{2\}/C\_\{p\}\(x\)be the exact\-match collision\-biased path distribution\. For small local terms and approximately independent per\-context estimates, a first\-order expansion motivates the homogeneous\-tree heuristic

log𝔼D​\[Cp^​\(x\)\]Cp​\(x\)≈𝔼Y∼Rp\(⋅∣x\)\[∑t=1T⁡\(Y\)\+1N2​\(pht\)−1nht\],\\log\\frac\{\\mathbb\{E\}\_\{D\}\[C\_\{\\hat\{p\}\}\(x\)\]\}\{C\_\{p\}\(x\)\}\\approx\\mathbb\{E\}\_\{Y\\sim R\_\{p\}\(\\cdot\\mid x\)\}\\\!\\left\[\\sum\_\{t=1\}^\{T\(Y\)\+1\}\\frac\{N\_\{2\}\(p\_\{h\_\{t\}\}\)\-1\}\{n\_\{h\_\{t\}\}\}\\right\],\(34\)This approximation additionally treatsRpR\_\{p\}as fixed and requiresCp​\(h​a\)C\_\{p\}\(ha\)to be approximately constant across sibling branches\. Without those conditions, theRpR\_\{p\}\-average of local factors is not the expected sequence collision ratio\.

### I\.3Entropy analogue

For a fixed finite support of sizeSS, the classical plug\-in bias is𝔼⁡\[H⁡\(p^h\)\]=H⁡\(ph\)−\(S−1\)/\(2​nh\)\+o⁡\(nh−1\)\\mathbb\{E\}\[H\(\\hat\{p\}\_\{h\}\)\]=H\(p\_\{h\}\)\-\(S\-1\)/\(2n\_\{h\}\)\+o\(n\_\{h\}^\{\-1\}\)under its regularity conditions\([Paninski, 2003](https://arxiv.org/html/2609.16454#bib.bib28);[Jiao et al\., 2015](https://arxiv.org/html/2609.16454#bib.bib21)\)\. ReplacingSSbyexp⁡H⁡\(ph\)\\exp H\(p\_\{h\}\)or summing this local approximation along generated paths is heuristic; neither substitution is used in the empirical claims\.

## Appendix JLimits of temperature scaling

Temperature can change a model’s collision value without generally calibrating its distribution\. To make the global sequence transformation well\-defined for everyT\>0T\>0, fix a distributionq\(⋅∣x\)q\(\\cdot\\mid x\)with finite support and writeβ=1/T\\beta=1/T,qT​\(y∣x\)=q​\(y∣x\)β/Zβ​\(x\)q\_\{T\}\(y\\mid x\)=q\(y\\mid x\)^\{\\beta\}/Z\_\{\\beta\}\(x\), andZβ​\(x\)=∑yq​\(y∣x\)βZ\_\{\\beta\}\(x\)=\\sum\_\{y\}q\(y\\mid x\)^\{\\beta\}\.

###### Proposition 2\(Global temperature monotonically increases order\-2 diversity\)\.

CT​\(x\)=∑yqT​\(y∣x\)2=Z2​β​\(x\)Zβ​\(x\)2=Z2/T​\(x\)Z1/T​\(x\)2\.C\_\{T\}\(x\)=\\sum\_\{y\}q\_\{T\}\(y\\mid x\)^\{2\}=\\frac\{Z\_\{2\\beta\}\(x\)\}\{Z\_\{\\beta\}\(x\)^\{2\}\}=\\frac\{Z\_\{2/T\}\(x\)\}\{Z\_\{1/T\}\(x\)^\{2\}\}\.\(35\)Moreover,CT​\(x\)C\_\{T\}\(x\)is nonincreasing andN2T​\(x\)=1/CT​\(x\)N\_\{2\}^\{T\}\(x\)=1/C\_\{T\}\(x\)is nondecreasing inTT\.

WithqT​\(y\)=q​\(y\)β/Zβq\_\{T\}\(y\)=q\(y\)^\{\\beta\}/Z\_\{\\beta\}andZβ=∑yq​\(y\)βZ\_\{\\beta\}=\\sum\_\{y\}q\(y\)^\{\\beta\},CT=∑yqT​\(y\)2=∑yq​\(y\)2​β/Zβ2=Z2​β/Zβ2C\_\{T\}=\\sum\_\{y\}q\_\{T\}\(y\)^\{2\}=\\sum\_\{y\}q\(y\)^\{2\\beta\}/Z\_\{\\beta\}^\{2\}=Z\_\{2\\beta\}/Z\_\{\\beta\}^\{2\}\. For monotonicity inTT, letψ⁡\(β\)=log⁡Zβ\\psi\(\\beta\)=\\log Z\_\{\\beta\}\. Thenψ′​\(β\)=𝔼qβ​\[log⁡q\]\\psi^\{\\prime\}\(\\beta\)=\\mathbb\{E\}\_\{q\_\{\\beta\}\}\[\\log q\]whereqβ∝qβq\_\{\\beta\}\\propto q^\{\\beta\}, andψ′′​\(β\)=Varqβ⁡\(log⁡q\)≥0\\psi^\{\\prime\\prime\}\(\\beta\)=\\Var\_\{q\_\{\\beta\}\}\(\\log q\)\\geq 0, soψ′\\psi^\{\\prime\}is nondecreasing\. Sincelog⁡CT=ψ⁡\(2​β\)−2​ψ​\(β\)\\log C\_\{T\}=\\psi\(2\\beta\)\-2\\psi\(\\beta\),dd​β​log⁡CT=2​\[ψ′​\(2​β\)−ψ′​\(β\)\]≥0\\frac\{d\}\{d\\beta\}\\log C\_\{T\}=2\\big\[\\psi^\{\\prime\}\(2\\beta\)\-\\psi^\{\\prime\}\(\\beta\)\\big\]\\geq 0forβ\>0\\beta\>0\. ThusCTC\_\{T\}is nondecreasing inβ=1/T\\beta=1/T, i\.e\. nonincreasing inTT, with equality only in the degenerate cases wherelog⁡q\\log qisqβq\_\{\\beta\}\-a\.s\. constant \(uniform or point mass\); equivalently the order\-2 diversityN2T​\(x\)=1/CT​\(x\)N\_\{2\}^\{T\}\(x\)=1/C\_\{T\}\(x\)is nondecreasing inTT\. ∎The reachable values depend only on the fixed probability profile ofqq\.

### J\.1The temperature\-reachable diversity interval

Raising global sequence temperature increases diversity, but only within a fixed interval determined by the model distribution\. For a given promptxx, letS⁡\(x\)S\(x\)denote the number of sequences with non\-zero probability underqqgivenxx\(S\(x\):=\|suppq\(⋅∣x\)\|S\(x\):=\|\\mathrm\{supp\}\\,q\(\\cdot\\mid x\)\|\), and letg⋆​\(x\)g\_\{\\star\}\(x\)denote the number of modes ofqqgivenxx\(g⋆​\(x\):=\|arg⁡maxy⁡q⁡\(y∣x\)\|g\_\{\\star\}\(x\):=\|\\arg\\max\_\{y\}q\(y\\mid x\)\|\)\.

###### Lemma 1\(Temperature\-reachable diversity interval\)\.

For a fixed model distributionq\(⋅∣x\)q\(\\cdot\\mid x\)with finite support,N2T​\(x\)=Zβ​\(x\)2/Z2​β​\(x\)N\_\{2\}^\{T\}\(x\)=Z\_\{\\beta\}\(x\)^\{2\}/Z\_\{2\\beta\}\(x\)is continuous and nondecreasing inTT, with limits

limT→0\+N2T​\(x\)=g⋆​\(x\),limT→∞N2T​\(x\)=S⁡\(x\)\.\\lim\_\{T\\to 0^\{\+\}\}N\_\{2\}^\{T\}\(x\)=g\_\{\\star\}\(x\),\\qquad\\lim\_\{T\\to\\infty\}N\_\{2\}^\{T\}\(x\)=S\(x\)\.\(36\)Hence asTTranges over\(0,∞\)\(0,\\infty\),N2T​\(x\)N\_\{2\}^\{T\}\(x\)takes every value in the open interval\(g⋆​\(x\),S​\(x\)\)\(\\,g\_\{\\star\}\(x\),\\,S\(x\)\\,\); the endpointsg⋆​\(x\)g\_\{\\star\}\(x\)andS⁡\(x\)S\(x\)are the limits asT→0\+T\\to 0^\{\+\}andT→∞T\\to\\inftyand are not attained at any finiteTTexcept in degenerate cases \(qqalready a point mass, resp\. already uniform on its support\)\. Its closure is\[g⋆​\(x\),S​\(x\)\]\[\\,g\_\{\\star\}\(x\),\\,S\(x\)\\,\]\(generically\[1,S⁡\(x\)\]\[1,S\(x\)\]\)\. One degenerate case also affects the lower endpoint: ifq\(⋅∣x\)q\(\\cdot\\mid x\)is uniform on its entire support, then every token is a maximizer, sog⋆​\(x\)=S​\(x\)g\_\{\\star\}\(x\)=S\(x\)andN2T​\(x\)=S⁡\(x\)N\_\{2\}^\{T\}\(x\)=S\(x\)at every finiteTT; the reachable interval then collapses to the single point\{S⁡\(x\)\}\\\{S\(x\)\\\}\. The “point mass / uniform limit” description above is thus the generic non\-degenerate picture, but is incomplete for this collapsed \(uniform\-qq\) case\.

Monotonicity is the previous result \(CTC\_\{T\}nonincreasing inTT, soN2T=1/CTN\_\{2\}^\{T\}=1/C\_\{T\}nondecreasing\)\. ForT→0\+T\\to 0^\{\+\}\(β→∞\\beta\\to\\infty\): writingqmax=maxy⁡q⁡\(y\)q\_\{\\max\}=\\max\_\{y\}q\(y\)andg⋆g\_\{\\star\}for the number of maximizers,Zβ=g⋆​qmaxβ​\(1\+o⁡\(1\)\)Z\_\{\\beta\}=g\_\{\\star\}q\_\{\\max\}^\{\\beta\}\(1\+o\(1\)\)andZ2​β=g⋆​qmax2​β​\(1\+o⁡\(1\)\)Z\_\{2\\beta\}=g\_\{\\star\}q\_\{\\max\}^\{2\\beta\}\(1\+o\(1\)\), soN2T=Zβ2/Z2​β→g⋆2​qmax2​β/\(g⋆​qmax2​β\)=g⋆N\_\{2\}^\{T\}=Z\_\{\\beta\}^\{2\}/Z\_\{2\\beta\}\\to g\_\{\\star\}^\{2\}q\_\{\\max\}^\{2\\beta\}/\(g\_\{\\star\}q\_\{\\max\}^\{2\\beta\}\)=g\_\{\\star\}\. ForT→∞T\\to\\infty\(β→0\\beta\\to 0\):q\(y\)β→𝟏\{q\(y\)\>0\}q\(y\)^\{\\beta\}\\to\\mathbf\{1\}\\\{q\(y\)\>0\\\}, soZβ→SZ\_\{\\beta\}\\to SandZ2​β→SZ\_\{2\\beta\}\\to S, givingN2T→S2/S=SN\_\{2\}^\{T\}\\to S^\{2\}/S=S\. Continuity inTTand the intermediate value theorem give every value in the open interval; the endpoints are limits, attained only in the stated degenerate cases\. ∎

WhenS​\(x\)\>g⋆​\(x\)S\(x\)\>g\_\{\\star\}\(x\), the fraction of this interval reached atTTis

ρT​\(x\):=N2T​\(x\)−g⋆​\(x\)S​\(x\)−g⋆​\(x\)∈\[0,1\],\\rho\_\{T\}\(x\):=\\frac\{N\_\{2\}^\{T\}\(x\)\-g\_\{\\star\}\(x\)\}\{S\(x\)\-g\_\{\\star\}\(x\)\}\\in\[0,1\],\(37\)which is a normalized diversity location, not a temperature parameter\. It is undefined when the interval collapses\.

###### Theorem 2\(Temperature only redistributes existing support\)\.

Assume a finite candidate space \(e\.g\. a finite vocabulary together with a maximum sequence length\), so thatS\(x\)=\|suppq\(⋅∣x\)\|<∞S\(x\)=\|\\mathrm\{supp\}\\,q\(\\cdot\\mid x\)\|<\\infty\. For every temperatureT\>0T\>0the temperature\-scaled modelqTq\_\{T\}has the same support asqq, and its diversity is bounded by that support size:

N2T​\(x\)≤S⁡\(x\),supT\>0N2T​\(x\)=S⁡\(x\),N\_\{2\}^\{T\}\(x\)\\leq S\(x\),\\qquad\\sup\_\{T\>0\}N\_\{2\}^\{T\}\(x\)=S\(x\),\(38\)the supremum corresponding to the uniform distribution onsuppq\(⋅∣x\)\\mathrm\{supp\}\\,q\(\\cdot\\mid x\)and approached only asT→∞T\\to\\infty\(not attained at any finiteTTunlessqqis already uniform on its support\)\. Consequently, relative to the true conditionalp\(⋅∣x\)p\(\\cdot\\mid x\):

1. \(i\)ifN2\(p\(⋅∣x\)\)\>S\(x\)N\_\{2\}\\big\(p\(\\cdot\\mid x\)\\big\)\>S\(x\), no temperature reaches the true diversity—the model’s support is simply too small;
2. \(ii\)ifN2\(p\(⋅∣x\)\)N\_\{2\}\\big\(p\(\\cdot\\mid x\)\\big\)lies in the open interval\(g⋆​\(x\),S​\(x\)\)\(g\_\{\\star\}\(x\),S\(x\)\), some finite temperature attains the true diversity value by the intermediate value theorem; the boundary valuesg⋆​\(x\)g\_\{\\star\}\(x\)andS⁡\(x\)S\(x\)are only approached asT→0\+T\\to 0^\{\+\}andT→∞T\\to\\inftyand are not attained at any finiteTTexcept in the degenerate cases of the interval Lemma; but
3. \(iii\)attaining that value does not reproducepp:qT=pq\_\{T\}=pholds only ifp\(⋅∣x\)∝q\(⋅∣x\)βp\(\\cdot\\mid x\)\\propto q\(\\cdot\\mid x\)^\{\\beta\}for someβ\>0\\beta\>0\.

Temperature scaling preserves support:qT​\(y\)\>0⇔q⁡\(y\)\>0q\_\{T\}\(y\)\>0\\iff q\(y\)\>0for every finiteT\>0T\>0, soqTq\_\{T\}is supported on a set of sizeS⁡\(x\)S\(x\)\. For any distributionrroverSSoutcomes, Cauchy–Schwarz gives1=\(∑ar⁡\(a\)\)2≤S​∑ar​\(a\)21=\\big\(\\sum\_\{a\}r\(a\)\\big\)^\{2\}\\leq S\\sum\_\{a\}r\(a\)^\{2\}, i\.e\.N2​\(r\)=1/∑ar​\(a\)2≤SN\_\{2\}\(r\)=1/\\sum\_\{a\}r\(a\)^\{2\}\\leq S, with equality iffrris uniform on its support\. Applying this tor=qTr=q\_\{T\}yieldsN2T​\(x\)≤S⁡\(x\)N\_\{2\}^\{T\}\(x\)\\leq S\(x\); the bound is approached asqT→Unif⁡\(supp​q\)q\_\{T\}\\to\\mathrm\{Unif\}\(\\mathrm\{supp\}\\,q\)whenT→∞T\\to\\inftyand is not attained at finiteTTunlessqqis already uniform on its support\. This gives the displayed bounds and case \(i\): ifN2​\(p\)\>S​\(x\)N\_\{2\}\(p\)\>S\(x\), noqTq\_\{T\}can reach it\. For case \(ii\),N2T​\(x\)N\_\{2\}^\{T\}\(x\)is continuous inTTand takes every value in the open interval\(g⋆​\(x\),S​\(x\)\)\(g\_\{\\star\}\(x\),S\(x\)\)\(Lemma\), so wheneverN2​\(p\)N\_\{2\}\(p\)lies in this open interval the intermediate value theorem supplies a finiteTTwithN2T​\(x\)=N2​\(p\)N\_\{2\}^\{T\}\(x\)=N\_\{2\}\(p\); the endpoints are limits, not attained at finiteTToutside the degenerate cases\. For case \(iii\),\{qT:T\>0\}\\\{q\_\{T\}:T\>0\\\}is the one\-parameter familylog⁡qT​\(y\)=β​log⁡q⁡\(y\)−log⁡Zβ\\log q\_\{T\}\(y\)=\\beta\\log q\(y\)\-\\log Z\_\{\\beta\}; matchingppexactly requireslog⁡p⁡\(y\)=β​log⁡q⁡\(y\)−log⁡Zβ\\log p\(y\)=\\beta\\log q\(y\)\-\\log Z\_\{\\beta\}for allyy, i\.e\.p∝qβp\\propto q^\{\\beta\}\. ∎

Interpretation\.Temperature preserves support and moves the global sequence distribution from the uniform distribution over its maximizers asT→0\+T\\to 0^\{\+\}toward the uniform distribution over its support asT→∞T\\to\\infty\. It can therefore match a target collision value while missing the target distribution, and the high\-temperature limit recovers the target only when the target itself is uniform on the model’s support\.

###### Lemma 2\(Temperature scaling is an order\-preserving one\-parameter family\)\.

Fix a promptxxand writeqT\(⋅∣x\)∝q\(⋅∣x\)1/Tq\_\{T\}\(\\cdot\\mid x\)\\propto q\(\\cdot\\mid x\)^\{1/T\}\.

1. \(i\)Log\-odds identity\. For everyT\>0T\>0and any outcomesa,ba,bwithq⁡\(b∣x\)\>0q\(b\\mid x\)\>0, log⁡qT​\(a∣x\)qT​\(b∣x\)=1T​log⁡q⁡\(a∣x\)q⁡\(b∣x\)\.\\log\\frac\{q\_\{T\}\(a\\mid x\)\}\{q\_\{T\}\(b\\mid x\)\}=\\frac\{1\}\{T\}\\,\\log\\frac\{q\(a\\mid x\)\}\{q\(b\\mid x\)\}\.\(39\)
2. \(ii\)Order preservation\. Hence temperature scaling is an order\-preserving reweighting: the rank ordering of\{qT​\(y∣x\)\}y\\\{q\_\{T\}\(y\\mid x\)\\\}\_\{y\}is the same for allT\>0T\>0\. The reachable set is a one\-parameter curve from the uniform distribution overarg​maxy⁡q​\(y∣x\)\\argmax\_\{y\}q\(y\\mid x\)asT→0\+T\\to 0^\{\+\}to the uniform distribution oversuppq\(⋅∣x\)\\operatorname\{supp\}q\(\\cdot\\mid x\)asT→∞T\\to\\infty\.
3. \(iii\)Unreachability\. ConsequentlyqT=p\(⋅∣x\)q\_\{T\}=p\(\\cdot\\mid x\)for someT\>0T\>0only ifp\(⋅∣x\)∝q\(⋅∣x\)βp\(\\cdot\\mid x\)\\propto q\(\\cdot\\mid x\)^\{\\beta\}, i\.e\.log⁡p\\log pis an affine function oflog⁡q\\log q\. In particular, if the rank ordering ofp\(⋅∣x\)p\(\\cdot\\mid x\)differs from that ofq\(⋅∣x\)q\(\\cdot\\mid x\), no temperature matchespp\.

For \(i\),qT​\(y\)=q​\(y\)1/T/Z1/Tq\_\{T\}\(y\)=q\(y\)^\{1/T\}/Z\_\{1/T\}, so the normalizer cancels in the ratio andqT​\(a\)/qT​\(b\)=\(q⁡\(a\)/q⁡\(b\)\)1/Tq\_\{T\}\(a\)/q\_\{T\}\(b\)=\(q\(a\)/q\(b\)\)^\{1/T\}; take logarithms\. For \(ii\),u↦u1/Tu\\mapsto u^\{1/T\}is strictly increasing on\(0,∞\)\(0,\\infty\), soqT​\(a\)≥qT​\(b\)q\_\{T\}\(a\)\\geq q\_\{T\}\(b\)exactly whenq⁡\(a\)≥q⁡\(b\)q\(a\)\\geq q\(b\); this common ordering holds for allTT, with the two endpoints given by the interval Lemma above\. For \(iii\), a fixed ordering rules out anyqTq\_\{T\}with a different one, whileqT=pq\_\{T\}=pforcesp∝q1/Tp\\propto q^\{1/T\}\. ∎

Temperature rescales every log\-odds ratio by the same factor\. It can flatten or sharpen relative\-probability errors, but it cannot reorder outcomes or independently correct them\. Figure[8](https://arxiv.org/html/2609.16454#A10.F8)gives a six\-outcome example\.

### J\.2Effective support

The finite\-support Theorem[2](https://arxiv.org/html/2609.16454#Thmtheorem2)requires a finite candidate space to ensureS⁡\(x\)<∞S\(x\)<\\infty\. While standard softmax assigns strictly positive probability to all tokens and, subject to a length constraint, to most sequences, the Theorem’s condition of finite support may be violated without a maximum length, rendering the claim about temperature’s inability to recover missing support inapplicable\. A more broadly applicable statement is probabilistic: rare, valid continuations are not entirely absent fromqTq\_\{T\}, but are effectively absent when sampling from a feasible number of samples\.

###### Proposition 3\(Effective absence of valid missing mass under sampling\)\.

Fix a promptxxand consider a sequence of fitted modelsqnq\_\{n\}indexed by dataset sizenn, with temperature\-scaled tail massεn,T​\(x\)=qn,T​\(U⁡\(x\)∣x\)\\varepsilon\_\{n,T\}\(x\)=q\_\{n,T\}\(U\(x\)\\mid x\)on a setU⁡\(x\)U\(x\)of valid but unobserved or severely underweighted continuations\. DrawingMnM\_\{n\}independent generations fromqn,T\(⋅∣x\)q\_\{n,T\}\(\\cdot\\mid x\),

Pr⁡\(at least one of the​Mn​samples lands in​U​\(x\)\)=1−\(1−εn,T​\(x\)\)Mn≤Mn​εn,T​\(x\)\.\\Pr\\big\(\\text\{at least one of the \}M\_\{n\}\\text\{ samples lands in \}U\(x\)\\big\)=1\-\\big\(1\-\\varepsilon\_\{n,T\}\(x\)\\big\)^\{M\_\{n\}\}\\leq M\_\{n\}\\,\\varepsilon\_\{n,T\}\(x\)\.\(40\)Hence ifεn,T​\(x\)→0\\varepsilon\_\{n,T\}\(x\)\\to 0andMn​εn,T​\(x\)→0M\_\{n\}\\varepsilon\_\{n,T\}\(x\)\\to 0along a sequence of fitted models, these continuations are effectively absent from the generated samples\. This is an assumed asymptotic regime over models, not a consequence of changingTTat fixedqnq\_\{n\}\.

TheMnM\_\{n\}generations are i\.i\.d\. draws fromqn,T\(⋅∣x\)q\_\{n,T\}\(\\cdot\\mid x\), so the probability that none lands inU⁡\(x\)U\(x\)is\(1−εn,T​\(x\)\)Mn\\big\(1\-\\varepsilon\_\{n,T\}\(x\)\\big\)^\{M\_\{n\}\}, giving the stated equality\. Bernoulli’s inequality\(1−ε\)Mn≥1−Mn​ε\(1\-\\varepsilon\)^\{M\_\{n\}\}\\geq 1\-M\_\{n\}\\varepsilonforε∈\[0,1\]\\varepsilon\\in\[0,1\]yields1−\(1−εn,T​\(x\)\)Mn≤Mn​εn,T​\(x\)1\-\(1\-\\varepsilon\_\{n,T\}\(x\)\)^\{M\_\{n\}\}\\leq M\_\{n\}\\,\\varepsilon\_\{n,T\}\(x\)\. IfMn​εn,T​\(x\)→0M\_\{n\}\\,\\varepsilon\_\{n,T\}\(x\)\\to 0this probability vanishes, so with high probability no sample realizes any continuation inU⁡\(x\)U\(x\)\. ∎

No validity\-aware guarantee\.Split an underweighted tail into valid and invalid sets,Uvalid​\(x\)U\_\{\\mathrm\{valid\}\}\(x\)andUinvalid​\(x\)U\_\{\\mathrm\{invalid\}\}\(x\)\. Temperature applies the same mapu↦u1/Tu\\mapsto u^\{1/T\}to both:

qT​\(Uvalid​\(x\)∣x\)qT​\(Uinvalid​\(x\)∣x\)=∑y∈Uvalid​\(x\)q​\(y∣x\)1/T∑y∈Uinvalid​\(x\)q​\(y∣x\)1/T\.\\frac\{q\_\{T\}\(U\_\{\\mathrm\{valid\}\}\(x\)\\mid x\)\}\{q\_\{T\}\(U\_\{\\mathrm\{invalid\}\}\(x\)\\mid x\)\}=\\frac\{\\sum\_\{y\\in U\_\{\\mathrm\{valid\}\}\(x\)\}q\(y\\mid x\)^\{1/T\}\}\{\\sum\_\{y\\in U\_\{\\mathrm\{invalid\}\}\(x\)\}q\(y\\mid x\)^\{1/T\}\}\.\(41\)The ratio can vary withTT, but the transformation contains no validity signal\. Temperature can incidentally increase valid mass; it cannot guarantee selective recovery\. Proposition[3](https://arxiv.org/html/2609.16454#Thmproposition3)implies effective absence only when its stated conditionMn​εn,T​\(x\)→0M\_\{n\}\\varepsilon\_\{n,T\}\(x\)\\to 0holds\.

###### Example 2\(A two\-peaked target temperature cannot reach\)\.

Take a fixed six\-outcome context in which the modelq=\(0\.50,0\.25,0\.12,0\.07,0\.04,0\.02\)q=\(0\.50,0\.25,0\.12,0\.07,0\.04,0\.02\)is monotone decreasing while the targetp=\(0\.30,0\.05,0\.05,0\.05,0\.25,0\.30\)p=\(0\.30,0\.05,0\.05,0\.05,0\.25,0\.30\)is two\-peaked: hereN2​\(p\)=4≤S=6N\_\{2\}\(p\)=4\\leq S=6, so the diversity value is attainable \(Theorem, case \(ii\)\), yet the shape ofppis not, because its ordering differs from that ofqq\(case \(iii\)\)\.

112233445566000\.20\.20\.40\.40\.60\.60\.80\.8outcomeprobabilityT=0\.5T=0\.5qTq\_\{T\}pp112233445566outcomeT=1T=1112233445566outcomeT=2T=2112233445566outcomeT=∞T=\\inftyFigure 8:Temperature preserves outcome order\.Forq=\(0\.50,0\.25,0\.12,0\.07,0\.04,0\.02\)q=\(0\.50,0\.25,0\.12,0\.07,0\.04,0\.02\), increasingTTflattens the distribution but cannot reproduce the off\-order targetp=\(0\.30,0\.05,0\.05,0\.05,0\.25,0\.30\)p=\(0\.30,0\.05,0\.05,0\.05,0\.25,0\.30\), even thoughN2​\(p\)=4<S=6N\_\{2\}\(p\)=4<S=6\.Two\-sided miscalibration\.Temperature can leave diversity below the target or flatten beyond it\. Neither direction alone establishes output validity: low\-probability continuations can be valid or invalid\. Even when a temperature matchesN2​\(p\)N\_\{2\}\(p\), it matchespponly in the power\-family case of Theorem[2](https://arxiv.org/html/2609.16454#Thmtheorem2)\.

The rate of temperature\-driven diversity change is determined by the spread of the model’s log\-probabilities\. Definingψ⁡\(β\)=log⁡Zβ\\psi\(\\beta\)=\\log Z\_\{\\beta\}, we haveψ′​\(β\)=𝔼qβ​\[log⁡q\]\\psi^\{\\prime\}\(\\beta\)=\\mathbb\{E\}\_\{q\_\{\\beta\}\}\[\\log q\]andψ′′​\(β\)=Varqβ⁡\(log⁡q\)≥0\\psi^\{\\prime\\prime\}\(\\beta\)=\\Var\_\{q\_\{\\beta\}\}\(\\log q\)\\geq 0, whereqβq\_\{\\beta\}is proportional toqβq^\{\\beta\}\. Given thatlog⁡CT=ψ⁡\(2​β\)−2​ψ​\(β\)\\log C\_\{T\}=\\psi\(2\\beta\)\-2\\psi\(\\beta\), the derivative with respect tolog⁡T\\log Tis−β\-\\betatimes the derivative with respect toβ\\beta\.

d​log⁡N2T​\(x\)d​log⁡T=−d​log⁡CTd​log⁡T=2​β​∫β2​βVarqs⁡\(log⁡q\)​𝑑s≥0\.\\frac\{d\\,\\log N\_\{2\}^\{T\}\(x\)\}\{d\\,\\log T\}=\-\\frac\{d\\,\\log C\_\{T\}\}\{d\\,\\log T\}=2\\beta\\\!\\int\_\{\\beta\}^\{2\\beta\}\\Var\_\{q\_\{s\}\}\(\\log q\)\\,ds\\;\\geq\\;0\.\(42\)The derivative is governed by the log\-probability variance along the power family\. For nonuniform finite\-supportqq, finite temperatures trace the open interval\(g⋆​\(x\),S​\(x\)\)\(g\_\{\\star\}\(x\),S\(x\)\)and approach its endpoints only in the limits\.

10−110^\{\-1\}10010^\{0\}10110^\{1\}11223344T=1:N2≈1\.87T=1:\\ N\_\{2\}\\approx 1\.87S=4S=4g⋆=1g\_\{\\star\}=1decoding temperatureTT\(log scale\)effective number of choicesN2TN\_\{2\}^\{T\}N2TN\_\{2\}^\{T\}forq=\(0\.7,0\.2,0\.07,0\.03\)q=\(0\.7,0\.2,0\.07,0\.03\)Figure 9:Global temperature traces a fixed diversity interval\.Forq=\(0\.7,0\.2,0\.07,0\.03\)q=\(0\.7,0\.2,0\.07,0\.03\),N2TN\_\{2\}^\{T\}increases from the mode count11toward the support size44; finite temperatures attain neither endpoint\.LLM decoding employs temperature locally, applyingqT​\(a∣h\)=q​\(a∣h\)1/T/∑bq​\(b∣h\)1/Tq\_\{T\}\(a\\mid h\)=q\(a\\mid h\)^\{1/T\}/\\sum\_\{b\}q\(b\\mid h\)^\{1/T\}, which influences the future contexts reached, as well as the probabilities of EOS, token length, format, and refusal, and the mixture of rare\-valid and rare\-invalid tokens\. The monotonicity result for global sequence temperature therefore need not hold for token\-by\-token decoding\.

Simple example\.Suppose the root probabilities are0\.40\.4for EOS and0\.60\.6for CONTINUE, followed by100100uniform endings after CONTINUE\. Sequence collision is0\.42\+0\.62​\(0\.01\)=0\.16360\.4^\{2\}\+0\.6^\{2\}\(0\.01\)=0\.1636\. Raising local temperature moves the root toward\(0\.5,0\.5\)\(0\.5,0\.5\)while leaving the uniform branch unchanged, so collision approaches0\.52\+0\.52​\(0\.01\)=0\.25250\.5^\{2\}\+0\.5^\{2\}\(0\.01\)=0\.2525\. Thus local temperature can reduce sequence diversity even though global temperature of a fixed sequence distribution cannot\.

0\.50\.5111\.51\.5222\.52\.533000\.10\.10\.20\.20\.30\.3T=1T=1decoding temperatureTTsequence collisionCT=1/N2TC\_\{T\}=1/N\_\{2\}^\{T\}\(lower = more diverse\)CTC\_\{T\}for the EOS/continue exampleFigure 10:Local temperature can increase sequence collision\.In the two\-stage EOS/continue example, raisingTTmoves mass toward the short deterministic branch and reduces sequence diversity\.