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Main Author:	Scretching, Daniel
Format:	Recurso digital
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Published:	Zenodo 2026
Online Access:	https://doi.org/10.5281/zenodo.19931300
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author	Scretching, Daniel
author_facet	Scretching, Daniel
contents	<div> <div class="flex h-svh w-screen flex-col"> <div class="relative z-0 flex min-h-0 w-full flex-1"> <div class="relative flex min-h-0 w-full flex-1"> <div class="@container/main relative flex min-w-0 flex-1 flex-col -translate-y-[calc(env(safe-area-inset-bottom,0px)/2)] pt-[calc(env(safe-area-inset-bottom,0px)/2)]"> <div class="@w-sm/main:[scrollbar-gutter:var(--stage-scroll-gutter)] touch:[scrollbar-width:none] group/scroll-root relative flex min-h-0 min-w-0 flex-1 flex-col [scrollbar-gutter:stable] not-print:overflow-x-clip not-print:overflow-y-auto group-data-stream-active/scroll-root:[overflow-anchor:none] scroll-pt-(--header-height) [--sticky-padding-top:var(--header-height)] [--sticky-padding-bottom:0px] [--scroll-root-safe-area-inset-top:calc(var(--sticky-padding-top)+env(safe-area-inset-top,0px))] [--scroll-root-safe-area-inset-bottom:calc(var(--sticky-padding-bottom)+var(--screen-keyboard-height,0px)+env(safe-area-inset-bottom,0px))] [--scroll-root-safe-area-height:calc(100lvh-var(--scroll-root-safe-area-inset-top)-var(--scroll-root-safe-area-inset-bottom))] has-data-[fixed-header=less-than-xl]:@w-xl/main:scroll-pt-0 has-data-[fixed-header=less-than-xl]:@w-xl/main:[--sticky-padding-top:0px] has-data-[fixed-header=less-than-xxl]:@w-2xl/main:scroll-pt-0 has-data-[fixed-header=less-than-xxl]:@w-2xl/main:[--sticky-padding-top:0px]"> <div class="group/thread flex flex-col min-h-full"> <div class="composer-parent flex flex-1 flex-col focus-visible:outline-0"> <div class="relative basis-auto flex-col -mb-(--composer-overlap-px) pb-(--composer-overlap-px) [--composer-overlap-px:28px] grow flex"> <div class="flex flex-col text-sm"> <div class="text-base my-auto mx-auto pb-10 [--thread-content-margin:var(--thread-content-margin-xs,calc(var(--spacing)4))] @w-sm/main:[--thread-content-margin:var(--thread-content-margin-sm,calc(var(--spacing)6))] @w-lg/main:[--thread-content-margin:var(--thread-content-margin-lg,calc(var(--spacing)*16))] px-(--thread-content-margin)"> <div class="[--thread-content-max-width:40rem] @w-lg/main:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn"> <div class="flex max-w-full flex-col gap-4 grow"> <div class="min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal outline-none keyboard-focused:focus-ring [.text-message+&]:mt-1"> <div class="flex w-full flex-col gap-1 empty:hidden"> <div class="markdown prose dark:prose-invert w-full wrap-break-word light markdown-new-styling"> <p>This paper reformulates the proposed relationship between the <strong>Scretching–Schrödinger Equation (SSE)</strong>, the <strong>Scretching Quantum Chain (SQC)</strong>, and <strong>non-Abelian Yang–Mills theory</strong>. The central correction made in this work is that the SSE/SQC framework is <strong>not presented as a completed proof</strong> of the Clay Mathematics Institute Yang–Mills existence and mass-gap problem. Instead, the framework is developed as a mathematically motivated <strong>spectroscopic constraint architecture</strong> that may be coupled to Yang–Mills fields and studied as a candidate positive functional within gauge-field theory.</p> <p>The corrected analysis preserves the verified SQC invariant,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">Kν=2πe2ε0mec3=7.4216560×10−22 s,K_{\nu} = \frac{2\pi e^2}{\varepsilon_0 m_e c^3} = 7.4216560\times10^{-22}\ \mathrm{s},</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ν</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="mord mathnormal">ε</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span><span class="vlist-s"></span></span><span class="mord mathnormal">m</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span><span class="vlist-s"></span></span><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span>2<span class="mord mathnormal">π</span><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">7.4216560</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−22</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord mathrm">s</span><span class="mpunct">,</span></span></span></span></span> <p>and the SSE closure constant,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">KSSE=3ℏ(4.32×10−9)2me=7.50174281×10−13 m2 s−1.K_{\mathrm{SSE}} = \frac{3\hbar(4.32\times10^{-9})}{2m_e} = 7.50174281\times10^{-13}\ \mathrm{m^2\,s^{-1}}.</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist">2<span class="mord mathnormal">m</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span><span class="vlist-s"></span></span>3ℏ<span class="mopen">(</span>4.32<span class="mbin">×</span>10<span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−9</span></span></span></span><span class="mclose">)</span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">7.50174281</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−13</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span></span></span><span class="mord mathrm">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−<span class="mord mathrm mtight">1</span></span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span> <p>These constants remain central to the SQC–SSE framework because they link spectroscopic transition strength, oscillator-strength closure, and wavefunction-level admissibility. In the proposed Yang–Mills extension, they are not used to replace the Yang–Mills action or the known gauge-field structure. Rather, they are introduced as additional closure constraints that may help define measurable or computable positive scales associated with gauge-field excitations.</p> <p>The earlier claim that a hydrogen-derived expression,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">Δmin⁡=34hcRH,\Delta_{\min} = \frac{3}{4}hcR_H,</span><span class="katex-html"><span class="base"><span class="mord">Δ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">i</span><span class="mtight">n</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist">43</span><span class="vlist-s"></span></span></span></span></span><span class="mord mathnormal">h</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">H</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span></span></span></span></span> <p>produces a Yang–Mills-scale mass gap of approximately <span class="katex copyable-equation"><span class="katex-mathml">1.2 GeV1.2\ \mathrm{GeV}</span><span class="katex-html"><span class="base"><span class="mord">1.2</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span></span></span></span> is corrected. Direct verification shows that this expression instead gives the Lyman-<span class="katex copyable-equation"><span class="katex-mathml">α\alpha</span><span class="katex-html"><span class="base"><span class="mord mathnormal">α</span></span></span></span> energy,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ΔLyα=10.19871545 eV=1.019871545×10−8 GeV.\Delta_{\mathrm{Ly}\alpha} = 10.19871545\ \mathrm{eV} = 1.019871545\times10^{-8}\ \mathrm{GeV}.</span><span class="katex-html"><span class="base"><span class="mord">Δ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">Ly</span><span class="mord mathnormal mtight">α</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">10.19871545</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">eV</span></span><span class="mrel">=</span></span><span class="base"><span class="mord">1.019871545</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−8</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span><span class="mord">.</span></span></span></span></span> <p>This value is an atomic hydrogen transition scale, not a non-Abelian Yang–Mills confinement-scale mass gap. Therefore, it cannot be interpreted as a proof of the required positive Yang–Mills spectral gap.</p> <p>A separate dimensional expression based on the SQC invariant gives the corrected scale,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ΔKν=0.05839157 eV=5.8391571×10−11 GeV.\Delta_{K_{\nu}} = 0.05839157\ \mathrm{eV} = 5.8391571\times10^{-11}\ \mathrm{GeV}.</span><span class="katex-html"><span class="base"><span class="mord">Δ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">K</span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">ν</span></span><span class="vlist-s"></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">0.05839157</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">eV</span></span><span class="mrel">=</span></span><span class="base"><span class="mord">5.8391571</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−11</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span><span class="mord">.</span></span></span></span></span> <p>This is also far below the hadronic scale associated with QCD confinement and cannot be equated with a <span class="katex copyable-equation"><span class="katex-mathml">1.2 GeV1.2\ \mathrm{GeV}</span><span class="katex-html"><span class="base"><span class="mord">1.2</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span></span></span></span> Yang–Mills mass gap. These corrections are important because they distinguish verified spectroscopic scales from unsupported claims about non-Abelian mass generation.</p> <p>The official Clay Mathematics Institute problem remains unsolved. A valid solution must construct a mathematically rigorous quantum Yang–Mills theory on <span class="katex copyable-equation"><span class="katex-mathml">R4\mathbb{R}^4</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span>, satisfy the required axiomatic standards, and prove the existence of a strictly positive spectral gap. The SSE/SQC framework, as corrected here, does not yet meet those requirements.</p> <p>The purpose of the present paper is therefore more precise and defensible: to formulate the SSE/SQC structure as a possible <strong>spectroscopic constraint model</strong> for investigating positive energy scales in gauge-field systems. In this interpretation, the SQC invariant and SSE closure constant may be used to build candidate positive functionals, closure conditions, and numerical diagnostics that can be applied to Yang–Mills-type fields. Such a program may be valuable for studying how spectroscopic constraints interact with gauge curvature, field strength tensors, vacuum structure, and excitation spectra.</p> <p>This corrected framing preserves the useful mathematical content of the SSE/SQC approach while avoiding an overstatement of its current status. The framework may suggest new ways to analyze gauge-field excitations, positive-energy constraints, and spectroscopy-inspired field functionals, but it does not presently constitute a constructive proof of Yang–Mills existence or the Clay mass gap.</p> </div> </div> </div> </div> </div> </div> </div> </div> <div class="sticky bottom-0 z-10 group/thread-bottom-container relative isolate w-full basis-auto has-data-has-thread-error:pt-2 has-data-has-thread-error:[box-shadow:var(--sharp-edge-bottom-shadow)] md:border-transparent md:pt-0 dark:border-white/20 md:dark:border-transparent print:hidden content-fade single-line flex flex-col"> <div class="relative mx-auto h-0"> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> <p> </p> <p> </p> <p> </p> <p></p>
format	Recurso digital
id	zenodo_https___doi_org_10_5281_zenodo_19931300
institution	Zenodo
language
publishDate	2026
publisher	Zenodo
record_format	zenodo
spellingShingle	Scretching–Schrödinger Spectroscopic Constraint Framework for Yang–Mills MassGap Modeling Scretching, Daniel <div> <div class="flex h-svh w-screen flex-col"> <div class="relative z-0 flex min-h-0 w-full flex-1"> <div class="relative flex min-h-0 w-full flex-1"> <div class="@container/main relative flex min-w-0 flex-1 flex-col -translate-y-[calc(env(safe-area-inset-bottom,0px)/2)] pt-[calc(env(safe-area-inset-bottom,0px)/2)]"> <div class="@w-sm/main:[scrollbar-gutter:var(--stage-scroll-gutter)] touch:[scrollbar-width:none] group/scroll-root relative flex min-h-0 min-w-0 flex-1 flex-col [scrollbar-gutter:stable] not-print:overflow-x-clip not-print:overflow-y-auto group-data-stream-active/scroll-root:[overflow-anchor:none] scroll-pt-(--header-height) [--sticky-padding-top:var(--header-height)] [--sticky-padding-bottom:0px] [--scroll-root-safe-area-inset-top:calc(var(--sticky-padding-top)+env(safe-area-inset-top,0px))] [--scroll-root-safe-area-inset-bottom:calc(var(--sticky-padding-bottom)+var(--screen-keyboard-height,0px)+env(safe-area-inset-bottom,0px))] [--scroll-root-safe-area-height:calc(100lvh-var(--scroll-root-safe-area-inset-top)-var(--scroll-root-safe-area-inset-bottom))] has-data-[fixed-header=less-than-xl]:@w-xl/main:scroll-pt-0 has-data-[fixed-header=less-than-xl]:@w-xl/main:[--sticky-padding-top:0px] has-data-[fixed-header=less-than-xxl]:@w-2xl/main:scroll-pt-0 has-data-[fixed-header=less-than-xxl]:@w-2xl/main:[--sticky-padding-top:0px]"> <div class="group/thread flex flex-col min-h-full"> <div class="composer-parent flex flex-1 flex-col focus-visible:outline-0"> <div class="relative basis-auto flex-col -mb-(--composer-overlap-px) pb-(--composer-overlap-px) [--composer-overlap-px:28px] grow flex"> <div class="flex flex-col text-sm"> <div class="text-base my-auto mx-auto pb-10 [--thread-content-margin:var(--thread-content-margin-xs,calc(var(--spacing)4))] @w-sm/main:[--thread-content-margin:var(--thread-content-margin-sm,calc(var(--spacing)6))] @w-lg/main:[--thread-content-margin:var(--thread-content-margin-lg,calc(var(--spacing)*16))] px-(--thread-content-margin)"> <div class="[--thread-content-max-width:40rem] @w-lg/main:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn"> <div class="flex max-w-full flex-col gap-4 grow"> <div class="min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal outline-none keyboard-focused:focus-ring [.text-message+&]:mt-1"> <div class="flex w-full flex-col gap-1 empty:hidden"> <div class="markdown prose dark:prose-invert w-full wrap-break-word light markdown-new-styling"> <p>This paper reformulates the proposed relationship between the <strong>Scretching–Schrödinger Equation (SSE)</strong>, the <strong>Scretching Quantum Chain (SQC)</strong>, and <strong>non-Abelian Yang–Mills theory</strong>. The central correction made in this work is that the SSE/SQC framework is <strong>not presented as a completed proof</strong> of the Clay Mathematics Institute Yang–Mills existence and mass-gap problem. Instead, the framework is developed as a mathematically motivated <strong>spectroscopic constraint architecture</strong> that may be coupled to Yang–Mills fields and studied as a candidate positive functional within gauge-field theory.</p> <p>The corrected analysis preserves the verified SQC invariant,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">Kν=2πe2ε0mec3=7.4216560×10−22 s,K_{\nu} = \frac{2\pi e^2}{\varepsilon_0 m_e c^3} = 7.4216560\times10^{-22}\ \mathrm{s},</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ν</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="mord mathnormal">ε</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span><span class="vlist-s"></span></span><span class="mord mathnormal">m</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span><span class="vlist-s"></span></span><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span>2<span class="mord mathnormal">π</span><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">7.4216560</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−22</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord mathrm">s</span><span class="mpunct">,</span></span></span></span></span> <p>and the SSE closure constant,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">KSSE=3ℏ(4.32×10−9)2me=7.50174281×10−13 m2 s−1.K_{\mathrm{SSE}} = \frac{3\hbar(4.32\times10^{-9})}{2m_e} = 7.50174281\times10^{-13}\ \mathrm{m^2\,s^{-1}}.</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist">2<span class="mord mathnormal">m</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span><span class="vlist-s"></span></span>3ℏ<span class="mopen">(</span>4.32<span class="mbin">×</span>10<span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−9</span></span></span></span><span class="mclose">)</span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">7.50174281</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−13</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span></span></span><span class="mord mathrm">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−<span class="mord mathrm mtight">1</span></span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span> <p>These constants remain central to the SQC–SSE framework because they link spectroscopic transition strength, oscillator-strength closure, and wavefunction-level admissibility. In the proposed Yang–Mills extension, they are not used to replace the Yang–Mills action or the known gauge-field structure. Rather, they are introduced as additional closure constraints that may help define measurable or computable positive scales associated with gauge-field excitations.</p> <p>The earlier claim that a hydrogen-derived expression,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">Δmin⁡=34hcRH,\Delta_{\min} = \frac{3}{4}hcR_H,</span><span class="katex-html"><span class="base"><span class="mord">Δ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">i</span><span class="mtight">n</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist">43</span><span class="vlist-s"></span></span></span></span></span><span class="mord mathnormal">h</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">H</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span></span></span></span></span> <p>produces a Yang–Mills-scale mass gap of approximately <span class="katex copyable-equation"><span class="katex-mathml">1.2 GeV1.2\ \mathrm{GeV}</span><span class="katex-html"><span class="base"><span class="mord">1.2</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span></span></span></span> is corrected. Direct verification shows that this expression instead gives the Lyman-<span class="katex copyable-equation"><span class="katex-mathml">α\alpha</span><span class="katex-html"><span class="base"><span class="mord mathnormal">α</span></span></span></span> energy,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ΔLyα=10.19871545 eV=1.019871545×10−8 GeV.\Delta_{\mathrm{Ly}\alpha} = 10.19871545\ \mathrm{eV} = 1.019871545\times10^{-8}\ \mathrm{GeV}.</span><span class="katex-html"><span class="base"><span class="mord">Δ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">Ly</span><span class="mord mathnormal mtight">α</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">10.19871545</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">eV</span></span><span class="mrel">=</span></span><span class="base"><span class="mord">1.019871545</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−8</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span><span class="mord">.</span></span></span></span></span> <p>This value is an atomic hydrogen transition scale, not a non-Abelian Yang–Mills confinement-scale mass gap. Therefore, it cannot be interpreted as a proof of the required positive Yang–Mills spectral gap.</p> <p>A separate dimensional expression based on the SQC invariant gives the corrected scale,</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ΔKν=0.05839157 eV=5.8391571×10−11 GeV.\Delta_{K_{\nu}} = 0.05839157\ \mathrm{eV} = 5.8391571\times10^{-11}\ \mathrm{GeV}.</span><span class="katex-html"><span class="base"><span class="mord">Δ<span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">K</span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">ν</span></span><span class="vlist-s"></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">0.05839157</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">eV</span></span><span class="mrel">=</span></span><span class="base"><span class="mord">5.8391571</span><span class="mbin">×</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">−11</span></span></span></span></span></span></span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span><span class="mord">.</span></span></span></span></span> <p>This is also far below the hadronic scale associated with QCD confinement and cannot be equated with a <span class="katex copyable-equation"><span class="katex-mathml">1.2 GeV1.2\ \mathrm{GeV}</span><span class="katex-html"><span class="base"><span class="mord">1.2</span><span class="mspace"> </span><span class="mord"><span class="mord mathrm">GeV</span></span></span></span></span> Yang–Mills mass gap. These corrections are important because they distinguish verified spectroscopic scales from unsupported claims about non-Abelian mass generation.</p> <p>The official Clay Mathematics Institute problem remains unsolved. A valid solution must construct a mathematically rigorous quantum Yang–Mills theory on <span class="katex copyable-equation"><span class="katex-mathml">R4\mathbb{R}^4</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span>, satisfy the required axiomatic standards, and prove the existence of a strictly positive spectral gap. The SSE/SQC framework, as corrected here, does not yet meet those requirements.</p> <p>The purpose of the present paper is therefore more precise and defensible: to formulate the SSE/SQC structure as a possible <strong>spectroscopic constraint model</strong> for investigating positive energy scales in gauge-field systems. In this interpretation, the SQC invariant and SSE closure constant may be used to build candidate positive functionals, closure conditions, and numerical diagnostics that can be applied to Yang–Mills-type fields. Such a program may be valuable for studying how spectroscopic constraints interact with gauge curvature, field strength tensors, vacuum structure, and excitation spectra.</p> <p>This corrected framing preserves the useful mathematical content of the SSE/SQC approach while avoiding an overstatement of its current status. The framework may suggest new ways to analyze gauge-field excitations, positive-energy constraints, and spectroscopy-inspired field functionals, but it does not presently constitute a constructive proof of Yang–Mills existence or the Clay mass gap.</p> </div> </div> </div> </div> </div> </div> </div> </div> <div class="sticky bottom-0 z-10 group/thread-bottom-container relative isolate w-full basis-auto has-data-has-thread-error:pt-2 has-data-has-thread-error:[box-shadow:var(--sharp-edge-bottom-shadow)] md:border-transparent md:pt-0 dark:border-white/20 md:dark:border-transparent print:hidden content-fade single-line flex flex-col"> <div class="relative mx-auto h-0"> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> <p> </p> <p> </p> <p> </p> <p></p>
title	Scretching–Schrödinger Spectroscopic Constraint Framework for Yang–Mills MassGap Modeling
url	https://doi.org/10.5281/zenodo.19931300

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