Selection Mechanics Framework (SMF): A Structural Perspective on Variability and Its Transformation in Large Language Model Outputs
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2026
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| _version_ | 1866901856513425408 |
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| author | Kaneda, Mutsumi |
| author_facet | Kaneda, Mutsumi |
| contents | <p>Large language models (LLMs) generate diverse outputs for a given input, typically interpreted as stochastic variability arising from probabilistic sampling. While existing approaches characterize this variability through entropy, dispersion, or uncertainty, they do not explicitly formulate how variability is structurally organized within bounded regions of admissible outputs, nor how this structure transforms under changes in constraints.</p> <p>We introduce the Selection Mechanics Framework (SMF), a complementary interpretive framework that models generation as selection from a structured region</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">A(s;c)={a∣P(a∣s)≥τ(c)},A(s; c) = \{a \mid P(a|s) \ge \tau(c)\},</span><span class="katex-html"><span class="base"><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mrel">=</span></span><span class="base"><span class="mopen">{</span><span class="mord mathnormal">a</span><span class="mrel">∣</span></span><span class="base"><span class="mord mathnormal">P</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord">∣</span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mrel">≥</span></span><span class="base"><span class="mord mathnormal">τ</span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)}</span><span class="mpunct">,</span></span></span></span></span></p> <p>and decomposes outputs as</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">a=b(s;c)+r(a,s;c),a = b(s; c) + r(a, s; c),</span><span class="katex-html"><span class="base"><span class="mord mathnormal">a</span><span class="mrel">=</span></span><span class="base"><span class="mord mathnormal">b</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mbin">+</span></span><span class="base"><span class="mord mathnormal">r</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml">b(s;c)b(s; c)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">b</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span> is a constraint-dependent bias and <span class="katex"><span class="katex-mathml">r(a,s;c)r(a, s; c)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">r</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span> represents structured residual variability.</p> <p>From this formulation, we derive three falsifiable predictions:<br>(i) the persistence of stable structural patterns across repeated trials,<br>(ii) non-proportional structural transformations under constraint changes, and<br>(iii) anisotropic residual variability aligned with bias-dependent directions.</p> <p>SMF provides a structural perspective on how probabilistic mechanisms give rise to organized variability.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19534994 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Selection Mechanics Framework (SMF): A Structural Perspective on Variability and Its Transformation in Large Language Model Outputs Kaneda, Mutsumi Large Language Models, Variability, Sampling, Structure, AI Theory <p>Large language models (LLMs) generate diverse outputs for a given input, typically interpreted as stochastic variability arising from probabilistic sampling. While existing approaches characterize this variability through entropy, dispersion, or uncertainty, they do not explicitly formulate how variability is structurally organized within bounded regions of admissible outputs, nor how this structure transforms under changes in constraints.</p> <p>We introduce the Selection Mechanics Framework (SMF), a complementary interpretive framework that models generation as selection from a structured region</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">A(s;c)={a∣P(a∣s)≥τ(c)},A(s; c) = \{a \mid P(a|s) \ge \tau(c)\},</span><span class="katex-html"><span class="base"><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mrel">=</span></span><span class="base"><span class="mopen">{</span><span class="mord mathnormal">a</span><span class="mrel">∣</span></span><span class="base"><span class="mord mathnormal">P</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord">∣</span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mrel">≥</span></span><span class="base"><span class="mord mathnormal">τ</span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)}</span><span class="mpunct">,</span></span></span></span></span></p> <p>and decomposes outputs as</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml">a=b(s;c)+r(a,s;c),a = b(s; c) + r(a, s; c),</span><span class="katex-html"><span class="base"><span class="mord mathnormal">a</span><span class="mrel">=</span></span><span class="base"><span class="mord mathnormal">b</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mbin">+</span></span><span class="base"><span class="mord mathnormal">r</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml">b(s;c)b(s; c)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">b</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span> is a constraint-dependent bias and <span class="katex"><span class="katex-mathml">r(a,s;c)r(a, s; c)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">r</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">s</span><span class="mpunct">;</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span> represents structured residual variability.</p> <p>From this formulation, we derive three falsifiable predictions:<br>(i) the persistence of stable structural patterns across repeated trials,<br>(ii) non-proportional structural transformations under constraint changes, and<br>(iii) anisotropic residual variability aligned with bias-dependent directions.</p> <p>SMF provides a structural perspective on how probabilistic mechanisms give rise to organized variability.</p> |
| title | Selection Mechanics Framework (SMF): A Structural Perspective on Variability and Its Transformation in Large Language Model Outputs |
| topic | Large Language Models, Variability, Sampling, Structure, AI Theory |
| url | https://doi.org/10.5281/zenodo.19534994 |