Why the Third Axis Is Freedom

# Why the Third Axis Is Freedom
Source: [https://arxiv.org/html/2608.05423](https://arxiv.org/html/2608.05423)
Michael Timothy Bennett Machine Intelligence and Normative Theory Lab, The Australian National University Canberra, ACT 2601, Australia m@michaeltimothybennett\.com

\(1 August 2026\)

###### Abstract

In generative training, a model produces an output and is penalised for its difference from an example\. With one output per comparison, a model that produces one common answer can outperform a model retaining a broader repertoire\. Explorative Modeling \(XM\) producesKKoutputs per comparison and updates on the closest, claiming exploration as a “third pretraining axis” associated with generative expressivity\. Here I show the third axis is actually freedom, meaning the weakness of the constraint implied by a model’s behaviour\. Previous work showed freedom is a property of function rather than form\. Parameters, architecture, minimum\-description\-length \(MDL\), and data can vary while the behavioural constraint remains unchanged\. It was formally proved that weakest models are likeliest to generalise, and freedom selection beat MDL by 110\-500% in induction experiments\. I prove average XM loss depends on the chance a candidate misses an acceptable region, with exploration raising miss probability to powerKK\. ForK\>1K\>1, match probability rises with freedom\. I then demonstrate empirically that XM optimises for freedom\. In a Forward XM experiment, largerKKincreased or saturated measured freedom, and increased freedom at every tested value under context\-dependent targets\. I trained XM candidate pools and compared validation selection with a freedom selector that read unlabelled parent contexts\. Freedom won in 29 of 30 cases\. Generative expressivity is a mode\-count proxy for freedom, that discards the extension structure that gives freedom its generalisation significance\. XM is a means, freedom an end, and selecting for freedom improved XM under distribution shift\.

## 1Introduction

Explorative Modeling drawsKKcandidate generations at each training step, compares them with one datum, and updates only through the closest candidate\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\. It studies this update across image, video, and language models, reports gains that rise with model size, data, and compute, and calls the resulting capacity*generative expressivity*, defined as the greatest mode count retained by a loss minimiser over a class of data distributions\.

The update belongs to an older minimum\-over\-candidates family\. Multiple Choice Learning trained several deterministic structured predictors under an oracle loss\(Guzmán\-Riveraet al\.,[2012](https://arxiv.org/html/2608.05423#bib.bib1)\)\. Stochastic Multiple Choice Learning later routed each example’s gradient through the lowest\-loss member of a deep ensemble\(Leeet al\.,[2016](https://arxiv.org/html/2608.05423#bib.bib3)\)\. Fan, Su, and Guibas introduced Min\-of\-N conditional generation in a December 2016 preprint, drawing several Gaussian latent perturbations for each input and training against the generated point cloud closest to the target\(Fanet al\.,[2017](https://arxiv.org/html/2608.05423#bib.bib2)\)\. Up to notation and output metric, its outer objective has the same conditional stochastic minimum\-over\-candidates form as hard Forward XM\. IMLE followed with a shared global pool of model samples, matching each datum to its nearest generated sample and showing that the resulting estimator coincides with maximum likelihood under stated conditions\(Li and Malik,[2018](https://arxiv.org/html/2608.05423#bib.bib4)\)\. The XM paper classifies IMLE as a shared\-pool instance of end\-to\-end Forward XM\. XM’s contribution lies in the broader Forward and Reverse framework, integration across modern generative architectures, the generative\-expressivity account, and the study ofKKas a scaling variable\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\.

However I will show generative\-expressivity is inferior to a pre\-existing measure of freedom of function\. This notion was first formulated to address the subjective nature of the performance of the AIXI general reinforcement learning agentHutteret al\.\([2024](https://arxiv.org/html/2608.05423#bib.bib438)\)\. Stack Theory measures freedom of function to avoid AIXI’s dependence on form\-dependent measures of complexityBennett \([2023](https://arxiv.org/html/2608.05423#bib.bib214)\)\. A learner has observed a training set or “child task”\. It must choose model parameters, which define a “policy”, and the learner hopes that its chosen policy generalises, to solve a broader parent task\. Many policies remain correct on that child, but only a subset will generalise well\. Which policy is most likely to remain correct when the task expands to the parent task? The answer is the weakest correct policy, meaning the correct policy whose extension contains the most compatible completions\(Bennett,[2025](https://arxiv.org/html/2608.05423#bib.bib224)\)\. Weakness of constraint can be understood more intuitively as freedom\. The process of learning the weakest correct policy is dubbed “w\-maxing” or “free\-maxing” by previous work, and is usually framed as a measure of function in comparison to measures of form, such as flatness or complexityBennett \([2026](https://arxiv.org/html/2608.05423#bib.bib235)\); Hochreiter and Schmidhuber \([1997](https://arxiv.org/html/2608.05423#bib.bib152)\); Solomonoff \([1978](https://arxiv.org/html/2608.05423#bib.bib657)\); Hutteret al\.\([2024](https://arxiv.org/html/2608.05423#bib.bib438)\); Rissanen \([1978](https://arxiv.org/html/2608.05423#bib.bib601)\)\. This result predates Explorative Modeling, separating generalisation from simplicity and from parameter form\(Bennett,[2024b](https://arxiv.org/html/2608.05423#bib.bib218),[2026](https://arxiv.org/html/2608.05423#bib.bib235)\)\.

In addition to being pre\-dated by freedom, I will show generative expressivity several relative weaknesses:

1. 1\.It cannot rank two models trained under the same objective\. Freedom can, and this is necessary for anything we might call an axis\.
2. 2\.It takes a supremum over minimisers, so it does not describe the model training selected\.
3. 3\.It is an objective\-level mode\-count proxy for freedom\. It coincides with a monotone transformation of local freedom only when the embodied language treats output modes as independent symmetric permissions\.
4. 4\.It has no theorem connecting it to optimal generalisation\.
5. 5\.It can remain fixed while freedom varies substantially\.

However the two lines can be complementary\. Over the course of this paper I’m going to show that XM is an excellent means of “free\-maxing”, effectively providing some empirical evidence at scale for a fairly obscure theory that nevertheless has significant implications across not just computer science but philosophy and neurobiologyBennett \([2024a](https://arxiv.org/html/2608.05423#bib.bib219),[2025](https://arxiv.org/html/2608.05423#bib.bib224)\); Soléet al\.\([2026](https://arxiv.org/html/2608.05423#bib.bib655)\)\. That theory in turn shows how to further improve upon XM, and I demonstrate such an improvement empirically\. Exploration records how many candidates training inspects\. The weakness of constraints implied by a model’s function \(freedom\) is how much compatible completion volume the trained policy retainsBennett \([2025](https://arxiv.org/html/2608.05423#bib.bib224)\)\. The first is a budgetary means while the second is an end, a property of function in an embodied language\. A policy can permit many outputs, so that its extension factor is large, while placing almost all sampling mass on one permitted output\. Another policy can permit fewer outputs and spread mass evenly\. Finite best\-of\-KKtraining can prefer either\. The relation therefore passes through probability mass\.

I derive that relation in full\. The central identity is

RK​\(π\)=∫0∞𝔼c,Y​\[\(1−Q~π,c,Y​\(At​\(Y\)\)\)K\]​dt,R\_\{K\}\(\\pi\)=\\int\_\{0\}^\{\\infty\}\\mathbb\{E\}\_\{c,Y\}\\left\[\\left\(1\-\\widetilde\{Q\}\_\{\\pi,c,Y\}\(A\_\{t\}\(Y\)\)\\right\)^\{K\}\\right\]\\,\\mathrm\{d\}t,whereAt​\(Y\)A\_\{t\}\(Y\)is the set of generations within lossttof targetYY, andQ~\\widetilde\{Q\}is the training candidate law\. Exploration applies the transform1−\(1−q\)K1\-\(1\-q\)^\{K\}to acceptable candidate mass\. When the training candidate law equals the deployed law, that mass lies on outputs permitted by the deployed policy\. The count of those permissions determines freedom in the finite language constructed below\.

I establish the following results\.

1. 1\.I derive the exact best\-of\-KKcoverage identity for arbitrary measurable output spaces, nonnegative losses, and target\-dependent training candidate laws\.
2. 2\.I show that the hard objective converges to the essential loss infimum of the training candidate law\. Under deployment consistency and a distance loss, this is distance from deployed support\.
3. 3\.I construct a finite generative Stack Theory language in which a stochastic generator induces a policy whose weakness of implied constraints \(freedom\) has a closed form\.
4. 4\.I show that best\-of\-KKsuccess equals expected distinct output coverage under uniform targets\.
5. 5\.I prove that balanced sampled coverage rises strictly with local freedom for everyK\>1K\>1\. For a fixed target\-support size, local freedom increases the marginal return to another candidate\.
6. 6\.I solve the finite nonuniform problem\. FiniteKKfavours common targets\. The optimum converges to uniform mass over full target support asKKgrows\.
7. 7\.I define independent nonempty unseen demands and prove that future compatibility is exactly proportional to unseen freedom\.
8. 8\.I isolate deterministicKK\-head models as a special case\. Only that special case has a support ceiling ofKK\.
9. 9\.I test whether sample\-based Forward XM converts largerKKinto greater freedom across 144 neural runs\. It does\.
10. 10\.I test whether a prospectively calibrated freedom selector improves balanced\-parent performance when both selectors read the same XM\-trained candidate pool\. It does\.

The result is a formal division of labour\. Parameters and data may enlarge the permitted region a model can embody, while deployment\-consistent exploration controls how strongly training rewards sampled access to that region\. The induced policy freedom is a function\-level quantity in the chosen embodied language\. This yields a testable explanation for the reported scaling interaction without treatingKK, mode count, and freedom as synonyms\.

## 2Weakness, now rebranded as freedom

I use the Stack Theory definitions from my PhD thesis and associated peer\-reviewed papers\(Bennett,[2025](https://arxiv.org/html/2608.05423#bib.bib224)\)\. Because people hated the name weakness, I’ve been changing it to freedom, and changing w\-maxing to “free\-maxing”\.

###### Definition 1\(Environment and embodied language\)\.

An environment is a nonempty setΦ\\Phiof mutually exclusive states\. A program is any setp⊆Φp\\subseteq\\Phi, and𝒫:=2Φ\\mathcal\{P\}:=2^\{\\Phi\}is the set of all programs\. A vocabulary is a finite set𝔳⊆𝒫\\mathfrak\{v\}\\subseteq\\mathcal\{P\}\. It induces the embodied language

L𝔳:=\{l⊆𝔳\|⋂p∈lp≠∅\}\.L\_\{\\mathfrak\{v\}\}:=\\left\\\{l\\subseteq\\mathfrak\{v\}\\ \\middle\|\\ \\bigcap\_\{p\\in l\}p\\neq\\varnothing\\right\\\}\.The members ofL𝔳L\_\{\\mathfrak\{v\}\}are statements\. By convention, the intersection of the empty family isΦ\\Phi, so∅∈L𝔳\\varnothing\\in L\_\{\\mathfrak\{v\}\}\.

Interpretation\.A vocabulary is the finite set of distinctions a body can implement at the chosen layer\. A statement is a bundle of distinctions that can hold together\. The satisfiability condition removes combinations the body cannot instantiate\.

###### Definition 2\(Extension and freedom\)\.

Forl∈L𝔳l\\in L\_\{\\mathfrak\{v\}\}, define

Ext⁡\(l\):=\{y∈L𝔳∣l⊆y\}\.\\operatorname\{Ext\}\(l\):=\\\{y\\in L\_\{\\mathfrak\{v\}\}\\mid l\\subseteq y\\\}\.The freedom ofllis

w​\(l\):=\|Ext⁡\(l\)\|\.w\(l\):=\|\\operatorname\{Ext\}\(l\)\|\.

Interpretation\.A completion adds commitments while retaining every commitment inll\. Freedom counts how many such refinements remain possible\. It counts embodied completions rather than parameter settings or a version space\.

###### Definition 3\(Task and correct policy\)\.

ForX⊆L𝔳X\\subseteq L\_\{\\mathfrak\{v\}\}, writeExt⁡\(X\):=⋃x∈XExt⁡\(x\)\\operatorname\{Ext\}\(X\):=\\bigcup\_\{x\\in X\}\\operatorname\{Ext\}\(x\)\. A task is a pairα=⟨Iα,Oα⟩\\alpha=\\langle I\_\{\\alpha\},O\_\{\\alpha\}\\ranglewhereIα⊆L𝔳I\_\{\\alpha\}\\subseteq L\_\{\\mathfrak\{v\}\}andOα⊆Ext⁡\(Iα\)O\_\{\\alpha\}\\subseteq\\operatorname\{Ext\}\(I\_\{\\alpha\}\)\. A policy is any statementπ∈L𝔳\\pi\\in L\_\{\\mathfrak\{v\}\}\. It is correct forα\\alphaexactly when

Ext⁡\(Iα\)∩Ext⁡\(π\)=Oα\.\\operatorname\{Ext\}\(I\_\{\\alpha\}\)\\cap\\operatorname\{Ext\}\(\\pi\)=O\_\{\\alpha\}\.WriteΠα\\Pi\_\{\\alpha\}for the set of correct policies\.

Interpretation\.Correctness fixes the policy set to the observed task\. Freedom compares the completion volume retained by different policies after that requirement has been met\. A weakest correct policy is one that maximises freedom within the bounds of correctness\. It maintains the integrity of a system without imposing unnecessary constraints\.

For continuous languages, Stack Theory replaces counting by a measureμ\\muand defineswμ​\(π\):=μ​\(Ext⁡\(π\)\)w\_\{\\mu\}\(\\pi\):=\\mu\(\\operatorname\{Ext\}\(\\pi\)\)\(Bennett,[2026](https://arxiv.org/html/2608.05423#bib.bib235),[2025](https://arxiv.org/html/2608.05423#bib.bib224)\)\. The finite theory below uses counting measure\. The general best\-of\-KKidentity in Section[3](https://arxiv.org/html/2608.05423#S3)does not require finiteness\.

### 2\.1A generative language

The map from a finite support profile to freedom is perfected111Well, it becomes “exact” but I am loathe to use that word after the LLMs ruined it for me\.after the language states which outputs a generator permits\.

###### Definition 4\(Permission profiles\)\.

LetCCbe a finite nonempty set of contexts and letYYbe a finite output set with\|Y\|≥2\|Y\|\\geq 2\. A permission profile is a map

F:C⟶2Y∖\{∅\}\.F:C\\longrightarrow 2^\{Y\}\\setminus\\\{\\varnothing\\\}\.The valueF​\(c\)F\(c\)is the nonempty set of outputs permitted at contextcc\.

Interpretation\.A context can be a prompt, a corrupted image, or a masked sequence\. A profile records every output the deployed generator can emit with positive probability at each context\.

###### Definition 5\(Generative permission language\)\.

LetΦ\\Phibe the set of all permission profiles\. For every\(c,y\)∈C×Y\(c,y\)\\in C\\times Y, define the exclusion program

ec,y:=\{G∈Φ∣y∉G​\(c\)\}\.e\_\{c,y\}:=\\\{G\\in\\Phi\\mid y\\notin G\(c\)\\\}\.Let

𝔳:=\{ec,y∣c∈C,y∈Y\}\.\\mathfrak\{v\}:=\\\{e\_\{c,y\}\\mid c\\in C,\\ y\\in Y\\\}\.For a profileFF, define its policy

πF:=\{ec,y∈𝔳∣y∉F​\(c\)\}\.\\pi\_\{F\}:=\\\{e\_\{c,y\}\\in\\mathfrak\{v\}\\mid y\\notin F\(c\)\\\}\.

Interpretation\.The policy states every exclusion the generator enforces\. A generator that permits more outputs asserts fewer exclusions\. It therefore embodies a weaker constraint\.

###### Proposition 1\(Closed\-form freedom\)\.

The mapF↦πFF\\mapsto\\pi\_\{F\}is a bijection from permission profiles toL𝔳L\_\{\\mathfrak\{v\}\}\. Ifac:=\|F​\(c\)\|a\_\{c\}:=\|F\(c\)\|, then

w​\(πF\)=∏c∈C\(2ac−1\)\.w\(\\pi\_\{F\}\)=\\prod\_\{c\\in C\}\\left\(2^\{a\_\{c\}\}\-1\\right\)\.

###### Proof\.

A statementl⊆𝔳l\\subseteq\\mathfrak\{v\}excludes the set

Xc​\(l\):=\{y∈Y∣ec,y∈l\}X\_\{c\}\(l\):=\\\{y\\in Y\\mid e\_\{c,y\}\\in l\\\}at each context\. The programs inllhave nonempty intersection exactly whenY∖Xc​\(l\)Y\\setminus X\_\{c\}\(l\)is nonempty for everycc\. Hence statements correspond bijectively to profiles throughF​\(c\)=Y∖Xc​\(l\)F\(c\)=Y\\setminus X\_\{c\}\(l\)\.

A statement extendsπF\\pi\_\{F\}exactly when it adds exclusions\. Its corresponding profileGGtherefore obeys

∅≠G​\(c\)⊆F​\(c\)\\varnothing\\neq G\(c\)\\subseteq F\(c\)for everycc\. There are2ac−12^\{a\_\{c\}\}\-1nonempty subsets ofF​\(c\)F\(c\), independently across contexts\. Go forth and multiply to get the result\. ∎

Interpretation\.Freedom counts every way the generator can commit further without contradicting its present permissions\. A context with one permitted output has only one possible refinement left unrefined\. A context with three permitted outputs has seven\. The product records joint refinement across contexts\. “Completions” as I so inarticulately put it in earlier work\.

A stochastic generator induces conditional lawsQcQ\_\{c\}onYY\. I identify its permission profile with

FQ​\(c\):=supp⁡Qc=\{y∈Y∣Qc​\(y\)\>0\}\.F\_\{Q\}\(c\):=\\operatorname\{supp\}Q\_\{c\}=\\\{y\\in Y\\mid Q\_\{c\}\(y\)\>0\\\}\.Its Stack Theory policy isπFQ\\pi\_\{F\_\{Q\}\}\. Probability mass determines how often exploration reaches each permitted output\. Support determines the policy statement\. In Proposition[1](https://arxiv.org/html/2608.05423#Thmproposition1)I convert the support cardinalities into an extension count\.

Table 1:Three quantities that answer different questions\.

## 3Best\-of\-KKexploration

I now model hard best\-of\-KKtraining in a form that includes discrete and continuous outputs\. I distinguish the law sampled during training from the law used at deployment\. This distinction is immaterial for end\-to\-end XMs, but essential for target\-dependent coupling searches\.

###### Definition 6\(Stochastic generative setting\)\.

Let\(𝒞,𝒢\)\(\\mathcal\{C\},\\mathcal\{G\}\)be a measurable context space and\(𝒴,𝒜\)\(\\mathcal\{Y\},\\mathcal\{A\}\)a measurable output space\. Draw a contextccfrom a probability lawν\\nu\. LetPPbe a Markov kernel from𝒞\\mathcal\{C\}to𝒴\\mathcal\{Y\}, and draw the targetYYfromPcP\_\{c\}\. LetQ~π\\widetilde\{Q\}\_\{\\pi\}be a Markov kernel from𝒞×𝒴\\mathcal\{C\}\\times\\mathcal\{Y\}to𝒴\\mathcal\{Y\}\. Conditional on\(c,Y=y\)\(c,Y=y\), draw an infinite sequence of training candidatesY^1,Y^2,…\\widehat\{Y\}\_\{1\},\\widehat\{Y\}\_\{2\},\\ldotsindependently fromQ~π,c,y\\widetilde\{Q\}\_\{\\pi,c,y\}\. LetQπQ\_\{\\pi\}be the model’s deployed Markov kernel from𝒞\\mathcal\{C\}to𝒴\\mathcal\{Y\}\. I call the exploration*deployment\-consistent*when

Q~π,c,y=Qπ,c\\widetilde\{Q\}\_\{\\pi,c,y\}=Q\_\{\\pi,c\}forν​\(d​c\)​Pc​\(d​y\)\\nu\(\\,\\mathrm\{d\}c\)P\_\{c\}\(\\,\\mathrm\{d\}y\)\-almost every\(c,y\)\(c,y\)\. Let

J:𝒴×𝒴⟶\[0,∞\)J:\\mathcal\{Y\}\\times\\mathcal\{Y\}\\longrightarrow\[0,\\infty\)be a measurable loss\. The hard best\-of\-KKrisk is

RK​\(π\):=𝔼​\[min1≤i≤K⁡J​\(Y^i,Y\)\]\.R\_\{K\}\(\\pi\):=\\mathbb\{E\}\\left\[\\min\_\{1\\leq i\\leq K\}J\(\\widehat\{Y\}\_\{i\},Y\)\\right\]\.Fort≥0t\\geq 0, define the acceptable neighbourhood

At​\(y\):=\{y^∈𝒴∣J​\(y^,y\)≤t\}A\_\{t\}\(y\):=\\\{\\widehat\{y\}\\in\\mathcal\{Y\}\\mid J\(\\widehat\{y\},y\)\\leq t\\\}and its generator mass

q~π​\(c,y,t\):=Q~π,c,y​\(At​\(y\)\)\.\\widetilde\{q\}\_\{\\pi\}\(c,y,t\):=\\widetilde\{Q\}\_\{\\pi,c,y\}\(A\_\{t\}\(y\)\)\.

Interpretation\.The loss neighbourhood contains every generation that is good enough at tolerancett\. The quantityq~π​\(c,y,t\)\\widetilde\{q\}\_\{\\pi\}\(c,y,t\)is the chance that one training candidate lands there\. Exploration repeats that attemptKKtimes\. Under deployment consistency, the training candidates are draws from the law used to generate at inference\. Otherwise the identity below still holds, but it describes a training\-time coupling search rather than sampled access to the deployed policy\.

###### Theorem 1\(Coverage identity\)\.

Assume𝔼​\[J​\(Y^1,Y\)\]<∞\\mathbb\{E\}\[J\(\\widehat\{Y\}\_\{1\},Y\)\]<\\infty\. Then

RK​\(π\)=∫0∞𝔼c,Y​\[\(1−q~π​\(c,Y,t\)\)K\]​dt\.\\boxed\{R\_\{K\}\(\\pi\)=\\int\_\{0\}^\{\\infty\}\\mathbb\{E\}\_\{c,Y\}\\left\[\\left\(1\-\\widetilde\{q\}\_\{\\pi\}\(c,Y,t\)\\right\)^\{K\}\\right\]\\,\\mathrm\{d\}t\}\.

###### Proof\.

Fixcc,yy, andtt\. The event

min1≤i≤K⁡J​\(Y^i,y\)\>t\\min\_\{1\\leq i\\leq K\}J\(\\widehat\{Y\}\_\{i\},y\)\>toccurs exactly when every candidate lies outsideAt​\(y\)A\_\{t\}\(y\)\. Independence evaluates that conditional probability as

\(1−Q~π,c,y​\(At​\(y\)\)\)K\.\\left\(1\-\\widetilde\{Q\}\_\{\\pi,c,y\}\(A\_\{t\}\(y\)\)\\right\)^\{K\}\.For any nonnegative integrable random variableZZ,

𝔼​\[Z\]=∫0∞ℙ​\(Z\>t\)​dt\.\\mathbb\{E\}\[Z\]=\\int\_\{0\}^\{\\infty\}\\mathbb\{P\}\(Z\>t\)\\,\\mathrm\{d\}t\.Apply this identity toZ=mini⁡J​\(Y^i,Y\)Z=\\min\_\{i\}J\(\\widehat\{Y\}\_\{i\},Y\)and integrate overccandYY\. ∎

Interpretation\.Candidate diversity changes the joint miss probability even when every marginal law is fixed\. The i\.i\.d\. model isolates repeated sampling\. A coupled design can lower the joint miss probability without changing its one\-candidate marginals\.

Interpretation\.Exploration applies the miss function\(1−q\)K\(1\-q\)^\{K\}to every acceptable region\. Equivalently, the chance of at least one hit is

gK​\(q\):=1−\(1−q\)K\.g\_\{K\}\(q\):=1\-\(1\-q\)^\{K\}\.For smallqq,

gK​\(q\)=K​q\+O​\(q2\)\.g\_\{K\}\(q\)=Kq\+O\(q^\{2\}\)\.At a fixed candidate law, exploration amplifies small positive mass\. A compatible region with zero mass remains inaccessible at every finiteKK\. Exploration is leverage on rare permissions rather than a source of new ones\.

###### Corollary 1\(Marginal value of another candidate\)\.

Under the assumptions of Theorem[1](https://arxiv.org/html/2608.05423#Thmtheorem1),

RK​\(π\)−RK\+1​\(π\)=∫0∞𝔼c,Y​\[q~π​\(c,Y,t\)​\(1−q~π​\(c,Y,t\)\)K\]​dt≥0\.R\_\{K\}\(\\pi\)\-R\_\{K\+1\}\(\\pi\)=\\int\_\{0\}^\{\\infty\}\\mathbb\{E\}\_\{c,Y\}\\left\[\\widetilde\{q\}\_\{\\pi\}\(c,Y,t\)\\left\(1\-\\widetilde\{q\}\_\{\\pi\}\(c,Y,t\)\\right\)^\{K\}\\right\]\\,\\mathrm\{d\}t\\geq 0\.

###### Proof\.

Subtract the identity forK\+1K\+1from the identity forKKand factor the integrand\. ∎

Interpretation\.Another candidate helps where acceptable mass is positive yet often missed\. For a fixedKK, the factorq​\(1−q\)Kq\(1\-q\)^\{K\}is largest atq=1/\(K\+1\)q=1/\(K\+1\)\. Regions that are almost certain need little extra exploration\. Regions with zero mass cannot be reached\.

###### Theorem 2\(Essential\-infimum limit\)\.

Define

d~π​\(c,y\):=ess​infy^∼Q~π,c,y⁡J​\(y^,y\)\.\\widetilde\{d\}\_\{\\pi\}\(c,y\):=\\operatorname\*\{ess\\,inf\}\_\{\\widehat\{y\}\\sim\\widetilde\{Q\}\_\{\\pi,c,y\}\}J\(\\widehat\{y\},y\)\.Assumed~π\\widetilde\{d\}\_\{\\pi\}is measurable\. If𝔼​\[J​\(Y^1,Y\)\]<∞\\mathbb\{E\}\[J\(\\widehat\{Y\}\_\{1\},Y\)\]<\\infty, then

min1≤i≤K⁡J​\(Y^i,Y\)⟶d~π​\(c,Y\)\\min\_\{1\\leq i\\leq K\}J\(\\widehat\{Y\}\_\{i\},Y\)\\longrightarrow\\widetilde\{d\}\_\{\\pi\}\(c,Y\)almost surely, and

RK​\(π\)⟶𝔼c,Y​\[d~π​\(c,Y\)\]\\boxed\{R\_\{K\}\(\\pi\)\\longrightarrow\\mathbb\{E\}\_\{c,Y\}\[\\widetilde\{d\}\_\{\\pi\}\(c,Y\)\]\}asK→∞K\\to\\infty\.

###### Proof\.

For fixedccandyy, each sampled loss is at leastd~π​\(c,y\)\\widetilde\{d\}\_\{\\pi\}\(c,y\)outside aQ~π,c,y\\widetilde\{Q\}\_\{\\pi,c,y\}\-null set\. A countable union over the sampled sequence is still null, so the sequence of minima is almost surely nonincreasing and bounded below byd~π​\(c,y\)\\widetilde\{d\}\_\{\\pi\}\(c,y\)\. For everyn≥1n\\geq 1, the set

\{y^∣J​\(y^,y\)<d~π​\(c,y\)\+1n\}\\left\\\{\\widehat\{y\}\\mid J\(\\widehat\{y\},y\)<\\widetilde\{d\}\_\{\\pi\}\(c,y\)\+\\frac\{1\}\{n\}\\right\\\}has positiveQ~π,c,y\\widetilde\{Q\}\_\{\\pi,c,y\}mass by the definition of essential infimum\. The probability that the infinite i\.i\.d\. sequence never visits this set is zero\. Intersecting these probability\-one events overn∈ℕn\\in\\mathbb\{N\}shows that the decreasing sequence of minima converges tod~π​\(c,y\)\\widetilde\{d\}\_\{\\pi\}\(c,y\)almost surely\. Integrating the conditional probability\-one statement over\(c,Y\)\(c,Y\)extends this to joint almost\-sure convergence\. Every minimum is bounded above byJ​\(Y^1,Y\)J\(\\widehat\{Y\}\_\{1\},Y\), so dominated convergence yields the risk limit\. ∎

Interpretation\.Infinite hard exploration retains only the best loss attainable under the training candidate law\. If𝒴\\mathcal\{Y\}is a Polish space222Not invaded\.,Qπ,cQ\_\{\\pi,c\}is a Borel probability law, exploration is deployment\-consistent, andJ​\(y^,y\)J\(\\widehat\{y\},y\)is continuous iny^\\widehat\{y\}, thend~π​\(c,y\)=infy^∈supp⁡Qπ,cJ​\(y^,y\)\\widetilde\{d\}\_\{\\pi\}\(c,y\)=\\inf\_\{\\widehat\{y\}\\in\\operatorname\{supp\}Q\_\{\\pi,c\}\}J\(\\widehat\{y\},y\)\. IfJ​\(y^,y\)=d​\(y^,y\)J\(\\widehat\{y\},y\)=d\(\\widehat\{y\},y\)for the metricdd, this is the distance fromyyto deployed support\. Under those conditions the theorem recovers the support\-coverage limit identified by Gladstone, Ji, and Du\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\. Yay\.

In Sections[4](https://arxiv.org/html/2608.05423#S4)–[6](https://arxiv.org/html/2608.05423#S6), I assume deployment consistency\. I therefore writeQπ,cQ\_\{\\pi,c\}for both the training and deployed law\.

## 4Finite embodied outputs

The general identity is “embodied” and interpreted as a count of completions \(degrees of freedom\), when the embodied vocabulary distinguishes finitely many outputs\.

###### Definition 7\(Identification loss\)\.

LetV​\(c\):=\{y∈Y∣Pc​\(Y=y\)\>0\}V\(c\):=\\\{y\\in Y\\mid P\_\{c\}\(Y=y\)\>0\\\}be the target support at contextcc\. I write

pc​\(y\):=Pc​\(Y=y\),qc​\(y\):=Qπ,c​\(Y^=y\)\.p\_\{c\}\(y\):=P\_\{c\}\(Y=y\),\\qquad q\_\{c\}\(y\):=Q\_\{\\pi,c\}\(\\widehat\{Y\}=y\)\.The identification loss is

J​\(y^,y\):=𝟏​\[y^≠y\]\.J\(\\widehat\{y\},y\):=\\mathbf\{1\}\[\\widehat\{y\}\\neq y\]\.

###### Definition 8\(Soundness, completeness, and correctness\)\.

At contextcc, a generator is sound whensupp⁡Qπ,c⊆V​\(c\)\\operatorname\{supp\}Q\_\{\\pi,c\}\\subseteq V\(c\), complete whenV​\(c\)⊆supp⁡Qπ,cV\(c\)\\subseteq\\operatorname\{supp\}Q\_\{\\pi,c\}, and correct whensupp⁡Qπ,c=V​\(c\)\\operatorname\{supp\}Q\_\{\\pi,c\}=V\(c\)\.

Interpretation\.Soundness limits what a generator may produce, since every output it can emit must be one that actually occurs atcc\. On the other hand completeness limits what it may omit, since every output that occurs must stay within reach\. A generator that always returns the same valid not\-hotdog is sound and incomplete, because everything it produces is correct and almost everything correct is missing\. A generator that reaches every valid not\-hotdog and sometimes emits noise is complete and unsound\. Correctness holds when what a generator can produce is exactly what occurs\. Soundness on its own is easy, and the collapsed generator has it\. What separates that generator from a correct one is how much of the valid set stays available, which is what freedom counts\.

###### Theorem 3\(Exact finite risk\)\.

For identification loss,

RK​\(π\)=𝔼c∼ν​\[∑y∈V​\(c\)pc​\(y\)​\(1−qc​\(y\)\)K\]\.\\boxed\{R\_\{K\}\(\\pi\)=\\mathbb\{E\}\_\{c\\sim\\nu\}\\left\[\\sum\_\{y\\in V\(c\)\}p\_\{c\}\(y\)\\left\(1\-q\_\{c\}\(y\)\\right\)^\{K\}\\right\]\}\.The hit probability is

HK​\(π\):=1−RK​\(π\)=𝔼c∼ν​\[∑y∈V​\(c\)pc​\(y\)​\(1−\(1−qc​\(y\)\)K\)\]\.H\_\{K\}\(\\pi\):=1\-R\_\{K\}\(\\pi\)=\\mathbb\{E\}\_\{c\\sim\\nu\}\\left\[\\sum\_\{y\\in V\(c\)\}p\_\{c\}\(y\)\\left\(1\-\\left\(1\-q\_\{c\}\(y\)\\right\)^\{K\}\\right\)\\right\]\.

###### Proof\.

A targetyyis missed exactly when allKKcandidates differ fromyy\. One candidate differs fromyywith probability1−qc​\(y\)1\-q\_\{c\}\(y\), and the candidates are independent, so allKKdiffer with probability\(1−qc​\(y\)\)K\(1\-q\_\{c\}\(y\)\)^\{K\}\. Weighting that bypc​\(y\)p\_\{c\}\(y\)and taking the expectation overc∼νc\\sim\\nuproduces the displayed risk\. The hit probability is its complement\. ∎

Interpretation\.Three quantities determine the risk\. First the target frequencypc​\(y\)p\_\{c\}\(y\)is how often each valid output is asked for, second the generator massqc​\(y\)q\_\{c\}\(y\)is how often the model produces it, and thirdKKis how many attempts each comparison allows\. AsKKgrows,\(1−qc​\(y\)\)K\(1\-q\_\{c\}\(y\)\)^\{K\}falls to zero for every output with positive mass, so in the limit only the support remains\. At finiteKK, the size of that mass also affects the risk\. A model can permit an output and still produce it so rarely that\(1−qc​\(y\)\)K\(1\-q\_\{c\}\(y\)\)^\{K\}stays near one for a long time\. A count over the support therefore cannot rank finite\-KKperformance on its own\.

###### Corollary 2\(Expected distinct coverage\)\.

Fix one context withm:=\|V​\(c\)\|m:=\|V\(c\)\|and a uniform target distribution onV​\(c\)V\(c\)\. LetNKN\_\{K\}be the number of distinct valid output cells observed among theKKcandidate samples\. Then

HK​\(π∣c\)=𝔼​\[NK\]m\.\\boxed\{H\_\{K\}\(\\pi\\mid c\)=\\frac\{\\mathbb\{E\}\[N\_\{K\}\]\}\{m\}\}\.

###### Proof\.

Each valid cellyyappears among the candidates with probability1−\(1−qc​\(y\)\)K1\-\(1\-q\_\{c\}\(y\)\)^\{K\}\. Therefore

𝔼​\[NK\]=∑y∈V​\(c\)\(1−\(1−qc​\(y\)\)K\)\.\\mathbb\{E\}\[N\_\{K\}\]=\\sum\_\{y\\in V\(c\)\}\\left\(1\-\(1\-q\_\{c\}\(y\)\)^\{K\}\\right\)\.Divide bymmand apply Theorem[3](https://arxiv.org/html/2608.05423#Thmtheorem3)\. ∎

Interpretation\.Best\-of\-KKsuccess equals normalised distinct\-output coverage\. The hit probability is the expected fraction of valid output cells reached by the samples\. The permission language then maps the deployed support count to the local freedom factor2a−12^\{a\}\-1\. Latent draws count only after the decoder converts them into output distinctions\.

###### Proposition 2\(Output counting and recoding\)\.

LetZZbe a latent space andg:Z→Yg:Z\\to Ya decoder\. For latent candidatesz1,…,zKz\_\{1\},\\ldots,z\_\{K\},

\|\{g​\(z1\),…,g​\(zK\)\}\|≤\|\{z1,…,zK\}\|≤K\.\|\\\{g\(z\_\{1\}\),\\ldots,g\(z\_\{K\}\)\\\}\|\\leq\|\\\{z\_\{1\},\\ldots,z\_\{K\}\\\}\|\\leq K\.Every latent or parameter recoding that leaves each conditional output lawQcQ\_\{c\}unchanged also leavesFQF\_\{Q\},πFQ\\pi\_\{F\_\{Q\}\}, andw​\(πFQ\)w\(\\pi\_\{F\_\{Q\}\}\)unchanged\.

###### Proof\.

The image of a finite set under a function cannot contain more elements than the set itself\. Regarding the second claim,FQ​\(c\)F\_\{Q\}\(c\)is the support ofQcQ\_\{c\}, so any recoding that leaves everyQcQ\_\{c\}unchanged leaves everyFQ​\(c\)F\_\{Q\}\(c\)unchanged\. The permission\-profile bijection then fixesπFQ\\pi\_\{F\_\{Q\}\}, and Proposition[1](https://arxiv.org/html/2608.05423#Thmproposition1)computesw​\(πFQ\)w\(\\pi\_\{F\_\{Q\}\}\)from the support cardinalities by themselves\. None of the three quantities refer to a latent or a parameter, so none of them can move when the output law is held fixed\. ∎

Interpretation\.Several latent values can decode to the same output\. Counting latents can therefore overstate the distinctions the model can express at the chosen output layer\. Freedom follows function rather than parameter form\. Hence the third axis claim\.

## 5When exploration ranks freedom

I first hold target frequencies exchangeable\. This isolates the relation between sampled coverage and the support of the policy\.

###### Theorem 4\(Uniform targets\)\.

Fix one context withmmvalid outputs\. Let the target distribution be uniform onVVand letqqrange over all probability distributions onYY\.

1. 1\.ForK=1K=1, every distribution supported onVVminimisesR1R\_\{1\}\.
2. 2\.For everyK≥2K\\geq 2, the unique minimiser is the uniform distribution onVV\.

Consequently, forK≥2K\\geq 2, the optimum has supportVVand maximises the local freedom factor2\|supp⁡q\|−12^\{\|\\operatorname\{supp\}q\|\}\-1among sound policies\.

###### Proof\.

Any mass outsideVVcan be moved to a valid cell and weakly reduce every term in the risk\. Hence an optimum is supported onVV\. ForK=1K=1,

R1​\(q\)=1−1m​∑y∈Vq​\(y\)=1−1mR\_\{1\}\(q\)=1\-\\frac\{1\}\{m\}\\sum\_\{y\\in V\}q\(y\)=1\-\\frac\{1\}\{m\}for every soundqq\.

ForK≥2K\\geq 2, the mapx↦\(1−x\)Kx\\mapsto\(1\-x\)^\{K\}is strictly convex on\[0,1\]\[0,1\]\. Jensen’s inequality then produces the chain

1m​∑y∈V\(1−q​\(y\)\)K≥\(1−1m​∑y∈Vq​\(y\)\)K=\(1−1m\)K\.\\frac\{1\}\{m\}\\sum\_\{y\\in V\}\(1\-q\(y\)\)^\{K\}\\geq\\left\(1\-\\frac\{1\}\{m\}\\sum\_\{y\\in V\}q\(y\)\\right\)^\{K\}=\\left\(1\-\\frac\{1\}\{m\}\\right\)^\{K\}\.Equality holds exactly whenq​\(y\)=1/mq\(y\)=1/mfor everyy∈Vy\\in V\. ∎

Interpretation\.One sampled output cannot distinguish a collapsed generator from a covering one, so atK=1K=1freedom is not visible to training\. However with second or later sample it is, and from that point on the unique population optimum is the weakest sound permission profile\. Exploration is the enabler of that visibility, andK=2K=2is where we can start to see its effect333Thereby further validating years of my life spent in relative obscurity and poverty\.\.

The next theorem holds support size fixed and isolates the effect of balanced mass\.

###### Theorem 5\(Balanced support\)\.

Fixmmuniformly likely valid outputs\. Let a sound generator place uniform mass on a permitted subset of sizeaa, where1≤a≤m1\\leq a\\leq m\. Its hit probability is

HK∗​\(a\)=am​\[1−\(1−1a\)K\]\.\\boxed\{H\_\{K\}^\{\*\}\(a\)=\\frac\{a\}\{m\}\\left\[1\-\\left\(1\-\\frac\{1\}\{a\}\\right\)^\{K\}\\right\]\}\.ForK=1K=1,H1∗​\(a\)=1/mH\_\{1\}^\{\*\}\(a\)=1/mfor everyaa\. For everyK≥2K\\geq 2,HK∗​\(a\)H\_\{K\}^\{\*\}\(a\)is strictly increasing inaaand therefore strictly increasing in the local freedom factor2a−12^\{a\}\-1\.

###### Proof\.

A target outside the permitted subset is always missed\. A target inside it is hit with probability1−\(1−1/a\)K1\-\(1\-1/a\)^\{K\}\. Averaging over themmuniform targets resulting in the formula\.

ForK≥2K\\geq 2, first comparea=1a=1anda=2a=2\. We haveHK∗​\(1\)=1/mH\_\{K\}^\{\*\}\(1\)=1/mand

HK∗​\(2\)=2m​\(1−2−K\)\>1m\.H\_\{K\}^\{\*\}\(2\)=\\frac\{2\}\{m\}\\left\(1\-2^\{\-K\}\\right\)\>\\frac\{1\}\{m\}\.Fora\>1a\>1, extendaato a continuous variable and define

hK​\(a\):=a​\[1−\(1−1a\)K\]\.h\_\{K\}\(a\):=a\\left\[1\-\\left\(1\-\\frac\{1\}\{a\}\\right\)^\{K\}\\right\]\.Its derivative is

hK′​\(a\)=1−\(1−1a\)K−1​\(1\+K−1a\)\.h\_\{K\}^\{\\prime\}\(a\)=1\-\\left\(1\-\\frac\{1\}\{a\}\\right\)^\{K\-1\}\\left\(1\+\\frac\{K\-1\}\{a\}\\right\)\.Letx=1/a∈\(0,1\)x=1/a\\in\(0,1\)andn=K−1n=K\-1\. The logarithm of the product subtracted above is

f​\(x\):=n​log⁡\(1−x\)\+log⁡\(1\+n​x\)\.f\(x\):=n\\log\(1\-x\)\+\\log\(1\+nx\)\.Forx∈\(0,1\)x\\in\(0,1\),

f′​\(x\)=−n​\(n\+1\)​x\(1−x\)​\(1\+n​x\)<0,f^\{\\prime\}\(x\)=\-\\frac\{n\(n\+1\)x\}\{\(1\-x\)\(1\+nx\)\}<0,whilef​\(0\)=0f\(0\)=0\. Hence the product is strictly below one andhK′​\(a\)\>0h\_\{K\}^\{\\prime\}\(a\)\>0fora\>1a\>1\. Together with the directa=1a=1comparison, this proves strict increase on the integers\. ∎

Interpretation\.Under the symmetry assumptions, the objective’s preference order over balanced sound policies is exactly the freedom order\. At everyKKabove one, training pays weaker policies strictly more, and Figure[1](https://arxiv.org/html/2608.05423#S5.F1)is the payment schedule\.

###### Theorem 6\(Complementarity\)\.

Under the assumptions of Theorem[5](https://arxiv.org/html/2608.05423#Thmtheorem5), the gain from one additional candidate is

HK\+1∗​\(a\)−HK∗​\(a\)=1m​\(1−1a\)K\.\\boxed\{H\_\{K\+1\}^\{\*\}\(a\)\-H\_\{K\}^\{\*\}\(a\)=\\frac\{1\}\{m\}\\left\(1\-\\frac\{1\}\{a\}\\right\)^\{K\}\}\.For every fixedK≥1K\\geq 1, this gain is strictly increasing inaa\.

###### Proof\.

Subtract the two expressions from Theorem[5](https://arxiv.org/html/2608.05423#Thmtheorem5), and the remaining term is\(1/m\)​\(1−1/a\)K\(1/m\)\(1\-1/a\)^\{K\}, which rises strictly withaa\. ∎

Interpretation\.Freedom and exploration are complements under the theorem’s symmetry assumptions\. A policy that permits one output gains nothing from repeated sampling\. A policy with many balanced permitted outputs continues to reveal new distinctions\. For fixed target\-support size, local freedom increases the marginal value of exploration\. Exploration then increases the degree to which training distinguishes local freedom\.

###### Proposition 3\(Entropy balances access\)\.

Fix a sound supportA⊆VA\\subseteq Vof sizeaaunder uniform targets\. For everyK≥2K\\geq 2, the uniform law onAAuniquely maximises the best\-of\-KKhit probability among laws supported onAA\. It also uniquely maximises Shannon entropy onAA\. If the support may vary among sound laws, the uniform law on all ofVVuniquely maximises both hit probability and entropy, and its permission profile has maximal local freedom among sound profiles\.

###### Proof\.

On a fixed supportAA, the hit probability is

1m​∑y∈A\[1−\(1−q​\(y\)\)K\]\.\\frac\{1\}\{m\}\\sum\_\{y\\in A\}\\left\[1\-\(1\-q\(y\)\)^\{K\}\\right\]\.The bracketed function is strictly concave forK≥2K\\geq 2, so Jensen’s inequality gives a unique maximum atq​\(y\)=1/aq\(y\)=1/a\. Shannon entropy has the same unique maximiser on a fixed finite support\. Theorem[4](https://arxiv.org/html/2608.05423#Thmtheorem4)establishes the full\-support claim, and Proposition[1](https://arxiv.org/html/2608.05423#Thmproposition1)maximal local freedom\. ∎

Interpretation\.Entropy spreads mass across a permitted output set, while freedom is the count of those completions or refinements of the corresponding policy statement still left to pursue\. Degrees of freedom remaining, so to speak\. Freedom is how loose your straight jacket feels\. Entropy and freedom select the same law in this particular symmetric finite setting, but they remain distinctly different quantities\. In a structured embodied language, supports with equal entropy or equal cardinality are very likely to have different extension structure\.

![Refer to caption](https://arxiv.org/html/2608.05423v1/x1.png)Figure 1:Best\-of\-KKhit probability for a uniform target overm=20m=20valid output cells when the generator is uniform onaapermitted cells\. ForK=1K=1the curve is flat\. For everyK\>1K\>1sampled coverage rises with the local freedom factor2a−12^\{a\}\-1\.###### Example 1\(Freedom alone does not rank finite\-KKaccess\)\.

Letm=3m=3andK=2K=2\. A generator with

q\(1\)=\(1/2,1/2,0\)q^\{\(1\)\}=\(1/2,1/2,0\)has support size two and hit probability

H2​\(q\(1\)\)=12\.H\_\{2\}\(q^\{\(1\)\}\)=\\frac\{1\}\{2\}\.A generator with

q\(2\)=\(0\.998,0\.001,0\.001\)q^\{\(2\)\}=\(0\.998,0\.001,0\.001\)has support size three and hit probability

H2​\(q\(2\)\)≈0\.3347\.H\_\{2\}\(q^\{\(2\)\}\)\\approx 0\.3347\.The second policy has greater freedom in the permission language, yet lower finite\-KKaccess\.

Interpretation\.Probability mass is the intermediary between permission and access\. A generator may hold a commodious permitted set and yet visit scarcely any of it, should almost the whole of its mass repose upon a single output\. The second generator above is the weaker of the two in the permission language, its extension factor standing at seven against three\. It is the poorer performer atK=2K=2notwithstanding\. Finite exploration adjudicates what a generator reaches rather than what it is at liberty to produce\. Freedom resumes its authority only asKKgrows without bound, when even the most meagre allocation is visited in the end\.

## 6Nonuniform targets

The next result shows how finite exploration balances target frequency against support coverage\.

###### Theorem 7\(Optimal mass under nonuniform targets\)\.

Fix one context withm≥2m\\geq 2valid outputs and strictly positive target probabilitiesp1,…,pmp\_\{1\},\\ldots,p\_\{m\}\. For identification loss, let

RK​\(q\)=∑j=1mpj​\(1−qj\)KR\_\{K\}\(q\)=\\sum\_\{j=1\}^\{m\}p\_\{j\}\(1\-q\_\{j\}\)^\{K\}over the probability simplex\.

1. 1\.ForK=1K=1, every optimum is supported on the outputs with maximal target probability\.
2. 2\.ForK\>1K\>1, the optimum is unique and has coordinates qj∗=\[1−\(λK​pj\)1/\(K−1\)\]\+\\boxed\{q\_\{j\}^\{\*\}=\\left\[1\-\\left\(\\frac\{\\lambda\}\{Kp\_\{j\}\}\\right\)^\{1/\(K\-1\)\}\\right\]\_\{\+\}\}where\[x\]\+:=max⁡\(x,0\)\[x\]\_\{\+\}:=\\max\(x,0\)andλ\\lambdais chosen so that∑jqj∗=1\\sum\_\{j\}q\_\{j\}^\{\*\}=1\.

###### Proof\.

ForK=1K=1,

R1​\(q\)=1−∑j=1mpj​qj,R\_\{1\}\(q\)=1\-\\sum\_\{j=1\}^\{m\}p\_\{j\}q\_\{j\},so the optimum concentrates on the largestpjp\_\{j\}values\.

ForK\>1K\>1,RKR\_\{K\}is strictly convex\. Introduce a multiplierλ\\lambdafor∑jqj=1\\sum\_\{j\}q\_\{j\}=1and multipliersμj≥0\\mu\_\{j\}\\geq 0forqj≥0q\_\{j\}\\geq 0\. The stationarity444Is this a word? Well, it is now\.condition is

−K​pj​\(1−qj\)K−1\+λ−μj=0\.\-Kp\_\{j\}\(1\-q\_\{j\}\)^\{K\-1\}\+\\lambda\-\\mu\_\{j\}=0\.For active coordinatesqj\>0q\_\{j\}\>0, complementary slackness results inμj=0\\mu\_\{j\}=0and the displayed formula\. For inactive coordinates,qj=0q\_\{j\}=0whenλ≥K​pj\\lambda\\geq Kp\_\{j\}\. Strict convexity rules out any second minimiser\. ∎

Interpretation\.At finiteKKthe optimum is parsimonious\. It bestows mass only upon targets whose frequency commends them, sinceqj∗q\_\{j\}^\{\*\}is positive exactly whenK​pjKp\_\{j\}exceedsλ\\lambda, and among those so admitted the more frequent are the more generously endowed\. Exploration erodes this parsimony\. Further mass upon a target the candidates reach many times over purchases little, whereas the first mass conferred upon a neglected target purchases a great deal, and the disparity widens as the candidates multiply\. The admitting thresholdλ/K\\lambda/Kfalls in consequence, and at length lies beneath every positive target frequency\. The optima then approach uniform mass across the whole of the target support, and an optimum once governed by frequency comes to be governed by coverage\.

###### Corollary 3\(Full\-support limit\)\.

Under the assumptions of Theorem[7](https://arxiv.org/html/2608.05423#Thmtheorem7), every output is active for all sufficiently largeKK, and

qj∗⟶1mq\_\{j\}^\{\*\}\\longrightarrow\\frac\{1\}\{m\}for everyjjasK→∞K\\to\\infty\.

###### Proof\.

LetAK:=\{j:qj∗\>0\}A\_\{K\}:=\\\{j:q\_\{j\}^\{\*\}\>0\\\}and writerK:=\|AK\|r\_\{K\}:=\|A\_\{K\}\|\. The caserK=1r\_\{K\}=1cannot satisfy the Karush–Kuhn–Tucker conditions forK\>1K\>1, because the active coordinate would force it to be the case thatλ=0\\lambda=0, while every inactive coordinate requiresλ≥K​pj\>0\\lambda\\geq Kp\_\{j\}\>0\. HencerK≥2r\_\{K\}\\geq 2\.

For any fixed active setAAof sizer≥2r\\geq 2, the active\-coordinate formula rearranges to

\(λK\)1/\(K−1\)=r−1∑i∈Api−1/\(K−1\)\.\\left\(\\frac\{\\lambda\}\{K\}\\right\)^\{1/\(K\-1\)\}=\\frac\{r\-1\}\{\\sum\_\{i\\in A\}p\_\{i\}^\{\-1/\(K\-1\)\}\}\.If somej∉Aj\\notin Astayed inactive along an unbounded sequence ofKK, then the right\-hand side would tend to\(r−1\)/r<1\(r\-1\)/r<1\. Thereforeλ/K\\lambda/Kwould tend to zero exponentially, while inactivity requiresλ/K≥pj\>0\\lambda/K\\geq p\_\{j\}\>0\. This is impossible\. Since there are finitely many supports, every coordinate is active for all sufficiently largeKK\.

With full support, write

aK:=\(λK\)1/\(K−1\)=m−1∑i=1mpi−1/\(K−1\)\.a\_\{K\}:=\\left\(\\frac\{\\lambda\}\{K\}\\right\)^\{1/\(K\-1\)\}=\\frac\{m\-1\}\{\\sum\_\{i=1\}^\{m\}p\_\{i\}^\{\-1/\(K\-1\)\}\}\.Then

qj∗=1−aK​pj−1/\(K−1\)\.q\_\{j\}^\{\*\}=1\-a\_\{K\}p\_\{j\}^\{\-1/\(K\-1\)\}\.Every factorpj−1/\(K−1\)p\_\{j\}^\{\-1/\(K\-1\)\}tends to one, soaK→\(m−1\)/ma\_\{K\}\\to\(m\-1\)/mandqj∗→1/mq\_\{j\}^\{\*\}\\to 1/m\. ∎

Interpretation\.Finite exploration trades frequency for support breadth, which determines the local freedom factor in the declared permission language\. AsKKgrows, the unique finite\-KKpopulation optima approach uniform sampling across the positive target support\. The limiting hard objective itself retains only support coverage and does not select a unique density\.

![Refer to caption](https://arxiv.org/html/2608.05423v1/x2.png)Figure 2:The exact optimum from Theorem[7](https://arxiv.org/html/2608.05423#Thmtheorem7)for target probabilities\(0\.50,0\.25,0\.15,0\.07,0\.03\)\(0\.50,0\.25,0\.15,0\.07,0\.03\)\. AtK=1K=1the optimum collapses onto the most frequent target\. AsKKgrows, all target cells receive mass and the optimum approaches the uniform support limit\.
## 7Future compatibility and generalisation

Exploration concerns sampled access to permitted outputs on scored contexts\. Freedom concerns refinements of the whole policy, including unseen contexts\. I now connect the two to future requirements\.

###### Definition 9\(Observed and unseen contexts\)\.

LetCα⊊CC\_\{\\alpha\}\\subsetneq Cbe the observed contexts and letCu:=C∖CαC\_\{u\}:=C\\setminus C\_\{\\alpha\}\. LetVα:Cα→2Y∖\{∅\}V\_\{\\alpha\}:C\_\{\\alpha\}\\to 2^\{Y\}\\setminus\\\{\\varnothing\\\}record the observed valid outputs\. A profileFFis child\-correct when

F​\(c\)=Vα​\(c\)F\(c\)=V\_\{\\alpha\}\(c\)for everyc∈Cαc\\in C\_\{\\alpha\}\.

Interpretation\.Child correctness fixes the permission profile where targets have been observed\. Generalisation depends on what the policy leaves open elsewhere\. Freedom is your remaining freedom to move into whatever requirements the unforeseen stipulates\.

###### Proposition 4\(Canonical Stack correctness\)\.

For each observed contextc∈Cαc\\in C\_\{\\alpha\}, let

𝔳c:=\{ec,y∣y∈Y\}\\mathfrak\{v\}\_\{c\}:=\\\{e\_\{c,y\}\\mid y\\in Y\\\}be the local vocabulary and letπAc:=\{ec,y∣y∉A\}\\pi\_\{A\}^\{c\}:=\\\{e\_\{c,y\}\\mid y\\notin A\\\}for every nonemptyA⊆YA\\subseteq Y\. WriteExtc⁡\(l\):=\{l′∈L𝔳c∣l⊆l′\}\\operatorname\{Ext\}\_\{c\}\(l\):=\\\{l^\{\\prime\}\\in L\_\{\\mathfrak\{v\}\_\{c\}\}\\mid l\\subseteq l^\{\\prime\}\\\}for local statement extension andExtc⁡\(X\):=⋃l∈XExtc⁡\(l\)\\operatorname\{Ext\}\_\{c\}\(X\):=\\bigcup\_\{l\\in X\}\\operatorname\{Ext\}\_\{c\}\(l\)for sets of local statements\. Define the local Stack task

αc:=⟨\{∅\},Extc⁡\(πVα​\(c\)c\)⟩\.\\alpha\_\{c\}:=\\left\\langle\\\{\\varnothing\\\},\\operatorname\{Ext\}\_\{c\}\(\\pi\_\{V\_\{\\alpha\}\(c\)\}^\{c\}\)\\right\\rangle\.ThenπAc\\pi\_\{A\}^\{c\}is correct forαc\\alpha\_\{c\}exactly whenA=Vα​\(c\)A=V\_\{\\alpha\}\(c\)\. Consequently, a profileFFis child\-correct exactly whenπF​\(c\)c∈Παc\\pi\_\{F\(c\)\}^\{c\}\\in\\Pi\_\{\\alpha\_\{c\}\}for every observed context\.

###### Proof\.

The empty statement has every local statement as a completion, so

Extc⁡\(\{∅\}\)=L𝔳c\.\\operatorname\{Ext\}\_\{c\}\(\\\{\\varnothing\\\}\)=L\_\{\\mathfrak\{v\}\_\{c\}\}\.ThereforeπAc\\pi\_\{A\}^\{c\}is correct exactly when

Extc⁡\(πAc\)=Extc⁡\(πVα​\(c\)c\)\.\\operatorname\{Ext\}\_\{c\}\(\\pi\_\{A\}^\{c\}\)=\\operatorname\{Ext\}\_\{c\}\(\\pi\_\{V\_\{\\alpha\}\(c\)\}^\{c\}\)\.Equality of extensions implies equality of the statements\. Each statement contains itself as a completion, so equality implies bothπAc⊆πVα​\(c\)c\\pi\_\{A\}^\{c\}\\subseteq\\pi\_\{V\_\{\\alpha\}\(c\)\}^\{c\}and the reverse inclusion\. The permission\-profile bijection then translates this intoA=Vα​\(c\)A=V\_\{\\alpha\}\(c\)\. ∎

Interpretation\.The profile equality above is the canonical Stack Theory correctness predicate applied context by context\. No new notion of correctness is needed\.

###### Definition 10\(Independent nonempty demands\)\.

For every unseen contextc∈Cuc\\in C\_\{u\}, draw a future demand

Sc∼Unif⁡\(2Y∖\{∅\}\)S\_\{c\}\\sim\\operatorname\{Unif\}\\left\(2^\{Y\}\\setminus\\\{\\varnothing\\\}\\right\)independently across contexts\. A profileFFis compatible with the demands when

Sc⊆F​\(c\)S\_\{c\}\\subseteq F\(c\)for everyc∈Cuc\\in C\_\{u\}\.

Interpretation\.A future task adds one or more positive requirements at each new context\. The profile remains compatible when it has not excluded any required output\. This demand law tests compatibility with future requirements\. It does not declare every permitted output correct, and it does not penalise extra unseen outputs\.

###### Theorem 8\(Future compatibility is proportional to freedom\)\.

Letac:=\|F​\(c\)\|a\_\{c\}:=\|F\(c\)\|\. Under independent nonempty demands,

ℙ​\(F​is compatible\)=∏c∈Cu2ac−12\|Y\|−1\.\\boxed\{\\mathbb\{P\}\(F\\text\{ is compatible\}\)=\\prod\_\{c\\in C\_\{u\}\}\\frac\{2^\{a\_\{c\}\}\-1\}\{2^\{\|Y\|\}\-1\}\}\.Define the unseen freedom factor

wu​\(F\):=∏c∈Cu\(2ac−1\)\.w\_\{u\}\(F\):=\\prod\_\{c\\in C\_\{u\}\}\(2^\{a\_\{c\}\}\-1\)\.Then

ℙ​\(F​is compatible\)=wu​\(F\)\(2\|Y\|−1\)\|Cu\|\.\\mathbb\{P\}\(F\\text\{ is compatible\}\)=\\frac\{w\_\{u\}\(F\)\}\{\(2^\{\|Y\|\}\-1\)^\{\|C\_\{u\}\|\}\}\.Among child\-correct profiles,

ℙ​\(F​is compatible\)=w​\(πF\)\[∏c∈Cα\(2\|Vα​\(c\)\|−1\)\]​\(2\|Y\|−1\)\|Cu\|\.\\boxed\{\\mathbb\{P\}\(F\\text\{ is compatible\}\)=\\frac\{w\(\\pi\_\{F\}\)\}\{\\left\[\\prod\_\{c\\in C\_\{\\alpha\}\}\(2^\{\|V\_\{\\alpha\}\(c\)\|\}\-1\)\\right\]\(2^\{\|Y\|\}\-1\)^\{\|C\_\{u\}\|\}\}\}\.Hence future compatibility probability and total freedom induce exactly the same ordering on child\-correct profiles\.

###### Proof\.

At contextcc, there are2\|Y\|−12^\{\|Y\|\}\-1possible nonempty demands\. Exactly2ac−12^\{a\_\{c\}\}\-1of them are nonempty subsets ofF​\(c\)F\(c\)\. Independence across contexts multiplies these per\-context ratios into the first formula\.

For a child\-correct profile, Proposition[1](https://arxiv.org/html/2608.05423#Thmproposition1)expands the total freedom as

w​\(πF\)=\[∏c∈Cα\(2\|Vα​\(c\)\|−1\)\]​\[∏c∈Cu\(2ac−1\)\]\.w\(\\pi\_\{F\}\)=\\left\[\\prod\_\{c\\in C\_\{\\alpha\}\}\(2^\{\|V\_\{\\alpha\}\(c\)\|\}\-1\)\\right\]\\left\[\\prod\_\{c\\in C\_\{u\}\}\(2^\{a\_\{c\}\}\-1\)\\right\]\.Substitute the unseen product from the first formula\. ∎

Interpretation\.A pooled demand law over all context\-output pairs prices commitments additively and can reverse the total freedom ordering\. Independent nonempty context demands recover the product structure of the embodied language, and under this positive\-demand law freedom determines future compatibility exactly, up to a constant no policy can change\. The theorem concerns future compatibility\. Exact unseen\-output correctness would require a law that also states which outputs are forbidden\.

The measurable Stack Theory result is broader\. LetUUbe the unseen region of a task and let

Bπ:=Ext⁡\(π\)∩UB\_\{\\pi\}:=\\operatorname\{Ext\}\(\\pi\)\\cap Ube the unseen buffer of a correct policy\. Under aμ\\mu\-exchangeable demand law, generalisation probability is a nondecreasing function ofμ​\(Bπ\)\\mu\(B\_\{\\pi\}\), and strict under nondegeneracy\(Bennett,[2026](https://arxiv.org/html/2608.05423#bib.bib235),[2025](https://arxiv.org/html/2608.05423#bib.bib224)\)\. Correctness fixes the observed part of the extension, so ranking by buffer measure agrees with ranking byμ\\mu\-freedom\.

In the finite permission model, under deployment consistency and the independent\-demand assumptions, the full relation is

K⟶1−\(1−q\)K⟶Qπ,c⟶FQ⟶πFQ⟶Ext⁡\(πFQ\)⟶ℙ​\(FQ​remains compatible\)\.K\\longrightarrow 1\-\(1\-q\)^\{K\}\\longrightarrow Q\_\{\\pi,c\}\\longrightarrow F\_\{Q\}\\longrightarrow\\pi\_\{F\_\{Q\}\}\\longrightarrow\\operatorname\{Ext\}\(\\pi\_\{F\_\{Q\}\}\)\\longrightarrow\\mathbb\{P\}\(F\_\{Q\}\\text\{ remains compatible\}\)\.
Interpretation\.The first arrow is the exact sampling transform under deployment consistency\. Training and optimisation connect that transform to the learned lawQπ,cQ\_\{\\pi,c\}\. The permission\-language construction converts the support profile of that law intoExt⁡\(πFQ\)\\operatorname\{Ext\}\(\\pi\_\{F\_\{Q\}\}\)\. The final arrow is Theorem[8](https://arxiv.org/html/2608.05423#Thmtheorem8)\. Under the broader Stack Theory extension model, the corresponding final quantity is generalisation probability\. The training step is empirical\. IncreasingKKneed not enlarge the learned extension in every architecture or dataset\. The reported XM experiments are consistent with that possibility but do not measure freedom directly\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\.

###### Corollary 4\(Exploration accesses freedom\)\.

Assume a uniform target law at each scored context and a sound generator with balanced mass on its permitted outputs\. Assume independent nonempty demands on unseen contexts\. Then for everyK\>1K\>1

1. 1\.local best\-of\-KKhit probability is strictly increasing in the permitted\-output countaca\_\{c\}, and therefore in the local freedom factor2ac−12^\{a\_\{c\}\}\-1,
2. 2\.for fixedm=\|V​\(c\)\|m=\|V\(c\)\|, the gain from increasingKKis strictly larger whenaca\_\{c\}is larger, and
3. 3\.unseen future compatibility probability is exactly proportional to total freedom among child\-correct profiles\.

###### Proof\.

Items 1 and 2 follow from Theorems[5](https://arxiv.org/html/2608.05423#Thmtheorem5)and[6](https://arxiv.org/html/2608.05423#Thmtheorem6)\. Item 3 follows from Theorem[8](https://arxiv.org/html/2608.05423#Thmtheorem8)\. ∎

Interpretation\.At a scored context, exploration ranks the local extension factor through repeated samples under the symmetry assumptions\. Across unseen contexts, future compatibility multiplies those same factors\. Probability allocation mediates the first relation\. Freedom fixes the second under the stated demand law\. Exploration creates the pressure\. Freedom is the quantity it acts on\.

## 8Deterministic heads

The claim thatKKcaps support holds for an enumerated set ofKKdeterministic heads\. It does not hold for stochastic Forward XM\.

###### Proposition 5\(Deterministic special case\)\.

Letf1,…,fK:C→Yf\_\{1\},\\ldots,f\_\{K\}:C\\to Ybe deterministic candidate functions evaluated together\. Their committed set at contextccis

Ff​\(c\):=\{f1​\(c\),…,fK​\(c\)\}\.F\_\{f\}\(c\):=\\\{f\_\{1\}\(c\),\\ldots,f\_\{K\}\(c\)\\\}\.Then

\|Ff​\(c\)\|≤K\.\|F\_\{f\}\(c\)\|\\leq K\.Consequently, the local freedom factor in the permission language obeys

2\|Ff​\(c\)\|−1≤2K−1\.2^\{\|F\_\{f\}\(c\)\|\}\-1\\leq 2^\{K\}\-1\.

###### Proof\.

A set formed fromKKvalues contains at mostKKelements\. Proposition[1](https://arxiv.org/html/2608.05423#Thmproposition1)converts that bound into the freedom factor\. ∎

Interpretation\.A fixed bank ofKKheads has aKK\-output ceiling\. Forward XM drawsKKsamples afresh from one conditional law\. That law can have support larger thanKK\.

###### Example 2\(Two modes and two samples\)\.

Let one context have two equiprobable valid outputs under identification loss\. Two deterministic heads can place one output on each head and attain zero loss\. A stochastic generator with probability1/21/2on each output andK=2K=2misses the target with probability1/41/4\. Its risk is therefore1/41/4under unit identification loss\. With squared distance between distinct one\-hot outputs equal to two, the corresponding expected loss is1/21/2\.

Interpretation\.Deterministic heads and stochastic draws solve different population problems\. The first enumerates candidates\. The second samples from a distribution\. The general XM relation is the coverage identity, not a support ceiling ofKK\.

## 9Generative expressivity

Gladstone, Ji, and Du define generative expressivity as

E:=supp∗,csupθ∗∈arg⁡minθ⁡ℒ​\(θ\)M\(Pθ∗\(⋅∣c\)\),E:=\\sup\_\{p^\{\*\},c\}\\sup\_\{\\theta^\{\*\}\\in\\arg\\min\_\{\\theta\}\\mathcal\{L\}\(\\theta\)\}M\(P\_\{\\theta^\{\*\}\}\(\\cdot\\mid c\)\),whereMMcounts modes\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\. This is an objective\-level capacity\. Under their separated\-mode assumptions, their smooth Forward XM result establishesE≥KE\\geq K\. It does not state that a stochastic generator has at mostKKmodes\.

Freedom answers a different question\. It is evaluated on a particular trained policy in a particular embodied language\. Two models can share the same objective, the sameKK, and the same objective\-levelEEwhile inducing different support profiles and different freedom\. Conversely, two distributions can have the same mode count while their extension structure differs\.

The hard infinite\-KKlimit brings the distinction into focus\. Theorem[2](https://arxiv.org/html/2608.05423#Thmtheorem2)and the XM appendix both show that every support\-covering generator becomes optimal in that limit\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\. The objective then fixes coverage without fixing density\. Freedom can still differ when the embodied language tracks further compatible refinements beyond raw support count\.

Interpretation\.Generative expressivity describes what an objective can retain\. Exploration describes how many draws training examines\. Freedom describes how much functional freedom one trained policy retains\. These quantities can correlate\. They remain distinct\.

## 10Experiments

The theory separates two empirical questions\. The first asks whether gradient training converts a larger exploration budget into a weaker deployed policy\. The second asks whether freedom improves selection among models trained by XM\. Experiment 1 tests the first relation\. Experiment 2 tests the second\. Code, frozen specifications, and result tables accompany this manuscript\.

### 10\.1Experiment 1\. Exploration raises freedom

I used a latent\-conditioned continuous generator with twelve one\-hot output prototypes\. The full profile fixed input dimension eight, latent dimension twelve, network width 128, 16,384 training cases, 4,096 audit contexts, batch size 384, 2,500 updates, twelve paired repetitions, and

K∈\{1,2,4,8,16,32\}\.K\\in\\\{1,2,4,8,16,32\\\}\.At each update, the code drewKKGaussian latent candidates for every target, generatedKKcontinuous output vectors, computed mean squared reconstruction loss, retained the lowest\-loss candidate, and recomputed only that candidate with gradients\. This follows the public Forward XM winner\-only update\. The analytic categorical loss used in the earlier experiment is absent\.

Within each repetition, every value ofKKused the same training set, target law, initial parameters, minibatch stream, audit contexts, and initial candidate random\-number state\. The two target laws were uniform and context\-dependent\. Every target cell was admissible\. A generated vector outside every declared cell was counted as invalid rather than as another output\.

At deployment, I drew 240 fresh latent samples at each audit context\. A generation occupied output celljjwhen its nearest one\-hot prototype lay within radius0\.640\.64\. The active cells defined the deployed permission profile

Fθ​\(x\):=\{j:cell​j​is occupied by at least one deployment draw\}\.F\_\{\\theta\}\(x\):=\\\{j:\\text\{cell \}j\\text\{ is occupied by at least one deployment draw\}\\\}\.Writingaθ​\(x\):=\|Fθ​\(x\)\|a\_\{\\theta\}\(x\):=\|F\_\{\\theta\}\(x\)\|, the measured local freedom factor was2aθ​\(x\)−12^\{a\_\{\\theta\}\(x\)\}\-1\. I report the audit mean oflog⁡\(2aθ​\(x\)−1\)\\log\(2^\{a\_\{\\theta\}\(x\)\}\-1\)\. An empty operational profile lies outside the nonempty permission language, so it received the minimum score zero and its frequency was recorded separately\.

The full study completed 144 of 144 jobs\. Figure[3](https://arxiv.org/html/2608.05423#S10.F3)shows the primary freedom result\. Relative toK=1K=1, the measured permission profile broadened fromK=4K=4onward under uniform targets and fromK=2K=2onward under the context\-dependent law\. The uniform profile reached all twelve cells byK=16K=16and then saturated\.

![Refer to caption](https://arxiv.org/html/2608.05423v1/x3.png)Figure 3:Experiment 1\. Mean log freedom after sample\-based Forward XM training\. Points show means across twelve paired repetitions\. Error bars are 95 percent bootstrap intervals\.Table 2:Experiment 1 cohort means\. The table showsK=1K=1,K=8K=8, andK=32K=32\. Figure[3](https://arxiv.org/html/2608.05423#S10.F3)includes every tested value\.Under uniform targets,K=8K=8increased mean log freedom overK=1K=1by8\.3166478\.316647, with paired 95 percent interval\[8\.316281,8\.316958\]\[8\.316281,8\.316958\]\. AtK=32K=32the increase was8\.3175228\.317522\. Under context\-dependent targets, the corresponding increases were5\.0293435\.029343, with interval\[4\.953404,5\.112625\]\[4\.953404,5\.112625\], and8\.2089228\.208922, with interval\[8\.198358,8\.220584\]\[8\.198358,8\.220584\]\. Every one of the twelve paired differences was positive in these four comparisons\. The uniformK=2K=2condition tiedK=1K=1in log freedom\. FromK=4K=4onward every uniform paired difference was positive\. Under the context\-dependent law every comparison fromK=2K=2onward was positive\.

Best\-of\-eight hit and freedom separated under the context\-dependent law\. Hit rose from0\.3197960\.319796atK=1K=1to0\.7111960\.711196atK=8K=8, then fell to0\.6488700\.648870atK=32K=32, while active support and freedom continued to rise\. This is the probability\-mass mediation predicted by the theory\. A broader permission profile does not guarantee greater finite\-KKaccess when sampling mass becomes less favourable\.

The future\-demand check drew independent nonempty output demands at each audit context\. Observed compatibility differed from the extension\-count prediction by2\.06×10−42\.06\\times 10^\{\-4\}on average and2\.19×10−32\.19\\times 10^\{\-3\}at most\.

Interpretation\.Experiment 1 closes the training arrow in this finite operational language\. The public XM update changed the deployed permission profile\. Freedom measured that change, while the context\-dependent condition also showed why freedom and finite\-KKaccess remain distinct\.

### 10\.2Experiment 2\. Freedom\-selected XM beats child\-validation XM

Experiment 2 tested whether freedom improves selection among models trained by the same sample\-based Forward XM procedure\. Each world used twelve decoded output cells\. I inserted one child example of every output, then sampled the remaining child labels from a randomly permuted Zipf law with exponent1\.61\.6\. The parent extension retained the observed child slice and added new contexts drawn from the same context law, with those twelve outputs weighted uniformly on the added slice\. The design therefore changed demanded output frequency rather than introducing outputs absent from training\.

Each candidate was trained by winner\-only Forward XM\. Every update drewKKGaussian latent candidates, generated one continuous output from each latent, measured reconstruction loss against a one\-hot target, selected the minimum\-loss candidate, and recomputed only that candidate with gradients\. The pool used

K∈\{4,8,16,32\},K\\in\\\{4,8,16,32\\\},four paired families, and checkpoints after 1,250, 1,875, and 2,500 updates\. This yielded 48 candidates per world\. Every continuous generation was decoded to its nearest one\-hot cell, so the twelve Voronoi cells formed a total finite output partition\.

Both selectors read the same XM\-trained candidate pool\. Child\-validation XM selected the candidate with greatest long\-tail child best\-of\-eight hit\. Freedom selection was transductive\. It evaluated each candidate on 256 unlabelled parent contexts, retained candidates whose child hit lay within a prospectively selected fraction of the best child hit, and ranked them using a fitted freedom instrument\. The primary freedom coefficient was fixed at one\. Nonnegative bounded residual terms refined the operational estimate\. Two independent calibration streams fitted and tested in opposite directions across 24 calibration worlds\. Formula\-bank size was selected from 48, 120, and 240\. The child\-fit gate was selected from0\.970\.97,0\.980\.98, and0\.990\.99\. The final instrument was frozen before the thirty final worlds were evaluated\.

The frozen instrument used the 240\-formula bank and a0\.970\.97child\-fit gate\. Its cross\-stream mean calibration gain was0\.0681390\.068139\. Its structural rate was0\.91670\.9167, exceeding the predeclared0\.800\.80requirement that calibration worlds contain at least two eligible candidates separated by one unit of measured log freedom\. The balanced\-parent hit statistic was excluded from final selection\. It was intentionally allowed to read output\-profile features computed on unlabelled parent contexts, including the measured freedom score that defined the intervention\.

The primary endpoint was best\-of\-eight hit on the added, balanced parent slice\. Child hit was reported separately and was not averaged into the parent endpoint\. Across thirty final worlds whose parent\-hit statistic was excluded from selection, freedom selection raised mean parent hit from0\.3169420\.316942to0\.3891810\.389181\. The paired increase was

with paired bootstrap 95 percent interval\[0\.066296,0\.076127\]\[0\.066296,0\.076127\]\. Freedom selection improved 29 of 30 worlds and lost one\. The exact one\-sided paired sign\-flip value was1\.86×10−91\.86\\times 10^\{\-9\}\.

![Refer to caption](https://arxiv.org/html/2608.05423v1/x4.png)Figure 4:Experiment 2\. Paired balanced\-parent improvement from prospectively calibrated freedom selection over child\-validation XM\. Freedom improved 29 of 30 final worlds whose parent\-hit statistic was excluded from selection\.Table 3:Experiment 2 final cohort\. Freedom selection accepted a small reduction in long\-tail child hit and selected XM models with broader valid repertoires and higher balanced\-parent hit\.Freedom selection accepted a0\.0216980\.021698reduction in long\-tail child hit while increasing active parent support by2\.6855472\.685547cells and parent log freedom by1\.8773071\.877307\. Both selectors had access to the same precomputed XM candidate pool, but their selected runs were not individually compute\-matched\. Child validation selectedK=8K=8in all thirty worlds\. Freedom selectedK=16K=16in 29 worlds andK=8K=8in one\. It selected earlier checkpoints on average,1,7081\{,\}708rather than2,3332\{,\}333updates, while the mean quantityK×updatesK\\times\\text\{updates\}was26,66726\{,\}667rather than18,66718\{,\}667\. The primary result is therefore a common\-pool model\-selection comparison, rather than an equal\-training\-compute comparison between two individual runs\.

As a post hoc matched\-setting check, I restricted both selectors to the six combinations withK∈\{16,32\}K\\in\\\{16,32\\\}and one fixed checkpoint in\{1250,1875,2500\}\\\{1250,1875,2500\\\}\. Within each combination, both selectors compared the same four family replicates at identicalKKand update count\. Averaging the six paired gains within each world gave\+0\.006455\+0\.006455, with paired bootstrap 95 percent interval\[0\.005798,0\.007079\]\[0\.005798,0\.007079\]\. The average was positive in all thirty worlds\. This analysis was not predeclared and is reported only as a robustness check\.

Child\-validation XM preferred candidates specialised to frequent child outputs\. Freedom selection chose candidates within three percent of the best child score that retained broader valid repertoires\. The balanced parent rewarded those retained possibilities\.

Interpretation\.Experiment 2 shows that transductive freedom\-enhanced selection can outperform child\-validation selection over the same XM candidate pool under a declared frequency shift\. Experiment 3 tests freedom\-guided selection inside the XM update itself\.555Experiment 3 is in progress using the authors’ public XJumpy ImageNet implementation\. No result from that experiment is included here\.

### 10\.3What the experiments establish

Experiment 1 identifies a function\-level quantity changed byKK\. Experiment 2 uses that quantity to improve selection among XM models\. Together they separate mechanism, measurement, and intervention\.

K⟶sampled access⟶deployed permission profile⟶w​\(π\)⟶balanced\-parent model selection\.K\\longrightarrow\\text\{sampled access\}\\longrightarrow\\text\{deployed permission profile\}\\longrightarrow w\(\\pi\)\\longrightarrow\\text\{balanced\-parent model selection\}\.The first experiment measures the middle relation under sample\-based gradient training\. The second shows that, among models trained by the same XM procedure, prospectively calibrated freedom selection can outperform child\-validation selection when future output frequencies differ\. The primary comparison is transductive and common\-pool rather than individually compute\-matched\. Exploration is the mechanism\. Freedom is the axis it exposes\.

## 11Further predictions

The theory and experiments suggest several further tests\.

1. 1\.MeasureKK, output support, probability allocation, and freedom separately\. A mode count alone cannot test the relation in Example[1](https://arxiv.org/html/2608.05423#Thmexample1)\.
2. 2\.At fixed correctness, fixed valid\-output count, uniform targets, and matched probability balance, models with greater measured local freedom should gain more from increasingKK\. This follows from Theorem[6](https://arxiv.org/html/2608.05423#Thmtheorem6)\.
3. 3\.At fixedKK, mass allocation should approach theKK\-dependent optimum in Theorem[7](https://arxiv.org/html/2608.05423#Thmtheorem7)\. Entropy can balance access across a fixed permitted set, but entropy and freedom agree only under the symmetry conditions of Proposition[3](https://arxiv.org/html/2608.05423#Thmproposition3)\.
4. 4\.AsKKgrows under nonuniform targets, the nonparametric population optimum should broaden toward full target support\. Figure[2](https://arxiv.org/html/2608.05423#S6.F2)plots the exact finite identification prediction\. Optimisation error can prevent a trained model from reaching that optimum\.
5. 5\.The interaction with model and data scale should depend on whether those axes enlarge the correct permitted region the model can embody\. Exploration has little to distinguish when that region is small\. It gains value when the model can retain many correct outputs\.
6. 6\.A mediation test can distinguish the explanation\. If exploration improves generalisation through freedom, the direct association betweenKKand held\-out performance should fall after measured freedom is included\.

Experiment 1 and Experiment 2 test these relations in a finite output partition where support and completion counts are tractable\. Dense softmax, image, video, and language models require a declared operational output vocabulary, a predeclared admissibility rule, or a measured completion proxy\. The language and admissibility rule must be fixed before comparisons are drawn\.

## 12Related work

Minimum\-over\-candidates training has several distinct antecedents\. Lloyd quantisation gives the classical centroid geometry behind deterministic squared\-loss heads\(Lloyd,[1982](https://arxiv.org/html/2608.05423#bib.bib5)\)\. Multiple Choice Learning trained several deterministic structured predictors under an oracle loss\(Guzmán\-Riveraet al\.,[2012](https://arxiv.org/html/2608.05423#bib.bib1)\)\. Stochastic Multiple Choice Learning adapted this winner\-takes\-gradient rule to deep ensembles\(Leeet al\.,[2016](https://arxiv.org/html/2608.05423#bib.bib3)\)\.

Fan, Su, and Guibas introduced Min\-of\-N for conditional point\-cloud generation in a December 2016 preprint, later published at CVPR 2017\(Fanet al\.,[2017](https://arxiv.org/html/2608.05423#bib.bib2)\)\. For each input image, they drewnnGaussian latent perturbations, generatednncandidate point clouds, and minimised the distance from the target to the closest candidate\. This has the same conditional stochastic minimum\-over\-candidates form as hard Forward XM, though it was developed for one conditional generation problem rather than as a general scaling framework\.

IMLE later drew a shared global pool of model samples, matched every datum to its nearest sample, and adjusted the generator through those pairs\(Li and Malik,[2018](https://arxiv.org/html/2608.05423#bib.bib4)\)\. It shares Forward XM’s data\-to\-nearest\-generated\-sample direction while differing in sample organisation\. IMLE also established a maximum\-likelihood interpretation under stated conditions\. The XM paper retrospectively classifies IMLE as a shared\-global\-pool instance of end\-to\-end Forward XM\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\.

The XM paper explicitly states that it does not claim the minimum\-over\-candidates update itself\. Its contribution is the broader Explorative Modeling framework\. It integrates the update across modern generative architectures, distinguishes Forward and Reverse XM, defines generative expressivity, and studies exploration as a scaling variable across modalities, model sizes, data scales, compute budgets, and generalisation\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\. The present paper identifies conditions under which sampled output coverage orders Stack Theory freedom and separates extension size from probability allocation\.

Freedom was introduced as a generalisation criterion in 2023\(Bennett,[2023](https://arxiv.org/html/2608.05423#bib.bib214)\)\. Later work separated it from simplicity\(Bennett,[2024b](https://arxiv.org/html/2608.05423#bib.bib218)\), repaired and restated the finite optimality result and consolidated the framework in the associated thesis\(Bennett,[2025](https://arxiv.org/html/2608.05423#bib.bib224)\), and extended freedom to measured continuous languages and neural functions\(Bennett,[2026](https://arxiv.org/html/2608.05423#bib.bib235)\)\. The chronology is relevant\. The least\-commitment explanation of generalisation came first\. Explorative Modeling is a new framework and scaling study built around an older stochastic minimum\-over\-candidates update\. Its deployment\-consistent form can reward sampled coverage of policy permissions\.

## 13Limitations

The power\-form coverage identity concerns candidates that are conditionally i\.i\.d\. given the context and target\. Remark 1 sets out the corresponding joint miss probability for coupled candidates\. The identity permits a target\-dependent training kernel\. The bridge from that kernel to deployed freedom additionally requires deployment consistency\. End\-to\-end XMs meet that condition by construction\. The noise\-searching continuous hybrids in the XM paper condition each candidate on a different corruption of the datum, so their training kernel need not equal the deployed law\. Gladstone, Ji, and Du also leave that coupling\-search case for later analysis\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\.

The smooth kernel objective studied in the XM appendix has a mixture\-likelihood interpretation and retains density information\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\. The support limit does not transfer unchanged to that objective\.

The closed\-form freedom result uses a finite exclusion vocabulary\. It treats output cells symmetrically\. Richer embodied languages can distinguish two supports of equal cardinality through their refinement structure\. In those languages, mode count and freedom separate further\.

Raw support can saturate\. A finite softmax assigns positive probability to every token\. Many continuous generators assign positive density throughout their ambient output space\. In either case the raw permission profile is constant and cannot rank models\. Support freedom is also discontinuous when an output moves between zero and arbitrarily small positive mass\. Empirical work therefore needs a declared finite vocabulary with a predeclared operational admissibility rule, a task\-relevant measurable language, or a measured extension proxy\. Changing that language or admissibility rule changes the numerical freedom value, so comparison requires one fixed embodied language and rule\.

Experiments 1 and 2 measure freedom in fixed finite output partitions\. They establish the training and model\-selection effects in those settings\. Architecture, coupling, optimisation, data, and the chosen embodied language can alter the effect elsewhere\. Dense image, video, and language generators still need a fixed operational vocabulary or a measured completion proxy before the same claim can be tested\.

Experiment 2 was designed to expose a long\-tail\-to\-uniform frequency shift\. Its freedom selector read unlabelled parent contexts, while the child\-validation selector did not\. The same 240 deployment draws were used to form the selector’s parent\-context features and the reported parent\-hit estimate, so selection and endpoint noise were not independent\. The primary selectors could also choose different values ofKKand different checkpoints\. The result therefore establishes a transductive common\-pool selection advantage in this synthetic shift\. It does not establish an equal\-compute improvement to the XM training update\. The post hoc matched\-setting check reduces the compute concern, but an independent replication should split selector\-probe draws from audit draws and predeclare fixed\-KK, fixed\-checkpoint comparisons\.

Finally, best\-of\-KKcan alter optimisation even when support coverage is no longer limiting\. Gladstone, Ji, and Du report gains under weak multimodality and hypothesise reduced gradient conflict\(Gladstoneet al\.,[2026](https://arxiv.org/html/2608.05423#bib.bib6)\)\. The present theory does not exclude that route\. It isolates the completion\-volume route and states when it is exact\.

## 14Conclusion

Exploration and freedom occupy different levels of explanation\. Exploration is repeated candidate sampling\. Freedom is compatible completion volume\. Probability mass connects them only after a deployed policy and embodied language have been fixed\.

Best\-of\-KKtraining transforms the mass of every acceptable region by

q⟼1−\(1−q\)K\.q\\longmapsto 1\-\(1\-q\)^\{K\}\.Under identification loss and uniform finite targets, this transform equals expected distinct valid\-output coverage\. Under uniform targets and balanced mass, coverage rises strictly with the local freedom factor\. For fixed target\-support size, weaker local policies gain more from another sample\. Under nonuniform targets, the sequence of finite\-KKpopulation optima trades frequency against support and approaches uniform mass over full target support\. Under independent uniform nonempty demands, future compatibility probability is exactly proportional to unseen freedom\.

The experiments complete the argument within their declared finite languages\. Across 144 faithful sample\-based Forward XM runs, largerKKincreased or saturated the measured deployed permission profile under uniform targets and increased it under the context\-dependent law\. In thirty final long\-tail\-to\-balanced worlds, prospectively calibrated freedom selection raised balanced\-parent hit from0\.3169420\.316942to0\.3891810\.389181over child\-validation XM\. It won in 29 worlds, selected 2\.686 more active parent outputs, and increased mean parent log freedom by 1\.877 while accepting a 0\.0217 reduction in long\-tail child hit\. This was a transductive common\-pool selector result\. The individual selected runs were not compute\-matched\.

The third pretraining axis therefore scales sampled access to functional freedom when training and deployment use the same candidate law\. Parameters and data may enlarge the permitted region a model can embody\. Exploration rewards sampled coverage of that region\. The fixed embodied language converts the resulting permission profile into freedom\. Parameters decide representation\. Data decide learning\. Freedom decides future compatibility, and exploration is how training comes to care about it\.

Explorative Modeling operationalises an older invariant\. The third axis has always been freedom\.

## Appendix AVerification

The accompanying scriptfig\.pyverifies the finite formulas by enumeration or numerical optimisation and regenerates Figures[1](https://arxiv.org/html/2608.05423#S5.F1)and[2](https://arxiv.org/html/2608.05423#S6.F2)\. It checks balanced\-support monotonicity, marginal complementarity, entropy balancing, the Karush–Kuhn–Tucker solution against random simplex points, the expected\-distinct identity, compatibility proportionality, and Example[1](https://arxiv.org/html/2608.05423#Thmexample1)\. The scriptcheck\.pyindependently recomputes every numerical claim added in the experiment section, including the exact sign\-flip value and the matched\-setting robustness check\. The folderseaw1andeaw2666Codenames for experiments 1 and 2, now meaningless letter salad\.contain the Colab notebooks, code, and result tables for Experiments 1 and 2\.

## Acknowledgement

A large language model assisted with literature search, simulated reviews, LaTeX prep and project organisation\.

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