MARC21: :: Library Catalog

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Dettagli Bibliografici
Autore principale:	Scretching, Daniel
Natura:	Recurso digital
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Pubblicazione:	Zenodo 2026
Accesso online:	https://doi.org/10.5281/zenodo.20102200
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_version_	1866901552534388736
author	Scretching, Daniel Scretching, Daniel
author_facet	Scretching, Daniel Scretching, Daniel
contents	<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=""> <div class="relative w-full overflow-visible"> <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 wrap-break-word w-full light markdown-new-styling"> <p><strong>The <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-Algebraic Maxwell–Scretching–SQC–SSE Formalism: A Unified Operator Algebra for Hydrogen, DNA/RNA Nucleobases, and Quantum Molecular Spectroscopy</strong> develops the Scretching framework into a single operator-algebraic system. Instead of treating the Maxwell–Scretching Chain, the Scretching Quantum Chain, and the Scretching–Schrödinger Equation as separate scalar equation systems, this paper combines them inside a complex <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-algebra where measurable spectroscopic quantities become observables or observable-generated functions.</p> <p>The central idea is that the experimentally measurable quantities</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ε, A, αabs, κ, σ, εr′′, f, ∣μ12∣2, A21, B21, B12\varepsilon,\ A,\ \alpha_{\mathrm{abs}},\ \kappa,\ \sigma,\ \varepsilon_r'',\ f,\ \|\mu_{12}\|^2,\ A_{21},\ B_{21},\ B_{12}</span><span class="katex-html"><span class="base"><span class="mord mathnormal">ε</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">α</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">abs</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">κ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">ε</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">r</span></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′′</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">f</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">μ</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">12</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mord">∣<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">2</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">A</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">21</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">B</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">21</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">B</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">12</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span></span> <p>can be organized as elements of an operator algebra. These quantities describe molar absorptivity, absorbance, Napierian absorption coefficient, extinction coefficient, absorption cross-section, dielectric-loss response, oscillator strength, squared transition-dipole moment, and Einstein transition coefficients. Once placed into an operator-algebraic framework, they form a mathematical bridge from laboratory spectroscopy to quantum dynamics.</p> <p>The main transformation chain is</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ε→f→∣μ12∣2→A21→V^SQC→H^SSE→αtSSE(A)\boxed{ \varepsilon \rightarrow f \rightarrow \|\mu_{12}\|^2 \rightarrow A_{21} \rightarrow \widehat{V}_{\mathrm{SQC}} \rightarrow \widehat{H}_{\mathrm{SSE}} \rightarrow \alpha_t^{\mathrm{SSE}}(A) }</span><span class="katex-html"><span class="base"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="boxpad"><span class="mord mathnormal">ε</span><span class="mrel">→</span><span class="mord mathnormal">f</span><span class="mrel">→</span>∣<span class="mord mathnormal">μ</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">12</span></span><span class="vlist-s"></span></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 class="mrel">→</span><span class="mord mathnormal">A</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">21</span></span><span class="vlist-s"></span></span><span class="mrel">→</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">V</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SQC</span></span></span><span class="vlist-s"></span></span><span class="mrel">→</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">H</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mrel">→</span><span class="mord mathnormal">α</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span> <p>This chain shows how molar absorptivity can be converted into oscillator strength, oscillator strength into transition-dipole magnitude, transition-dipole structure into Einstein coefficients, Einstein coefficients into the SQC constraint operator, and the SQC correction into the Scretching–Schrödinger Hamiltonian. The final object, <span class="katex copyable-equation"><span class="katex-mathml">αtSSE(A)\alpha_t^{\mathrm{SSE}}(A)</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">α</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">t</span></span><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="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>, represents the time evolution of an observable <span class="katex copyable-equation"><span class="katex-mathml">AA</span><span class="katex-html"><span class="base"><span class="mord mathnormal">A</span></span></span></span> under the SSE-modified Hamiltonian.</p> <p>In this formulation, the Maxwell–Scretching Chain supplies the measurable optical and spectroscopic observables; the Scretching Quantum Chain supplies the invariant transition-coefficient bridge; and the Scretching–Schrödinger Equation supplies the Hamiltonian operator and time-evolution law. Together, they define a unified algebraic system:</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">AMS-SQC-SSE=C&lowast;(ε, A, αabs, κ, σ, εr′′, f, ∣μ12∣2, A21, B21, B12,V^SQC,H^SSE)\boxed{ \mathcal{A}_{\mathrm{MS\text{-}SQC\text{-}SSE}} = C^ \left( \varepsilon,\ A,\ \alpha_{\mathrm{abs}},\ \kappa,\ \sigma,\ \varepsilon_r'',\ f,\ \|\mu_{12}\|^2,\ A_{21},\ B_{21},\ B_{12},\widehat{V}_{\mathrm{SQC}},\widehat{H}_{\mathrm{SSE}} \right) }</span><span class="katex-html"><span class="base"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="boxpad"><span class="mord mathcal">A</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">MS</span><span class="mord text mtight">-</span><span class="mord mathrm mtight">SQC</span><span class="mord text mtight">-</span><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mrel">=</span><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span><span class="minner"><span class="mopen delimcenter"><span class="delimsizing size2">(</span></span><span class="mord mathnormal">ε</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">α</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">abs</span></span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">κ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">ε</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">r</span></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′′</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">f</span><span class="mpunct">,</span><span class="mspace"> </span>∣<span class="mord mathnormal">μ</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">12</span></span><span class="vlist-s"></span></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 class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">A</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">21</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">B</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">21</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">B</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">12</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">V</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SQC</span></span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">H</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mclose delimcenter"><span class="delimsizing size2">)</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span> <p>The formalism uses the corrected Scretching–Schrödinger bridge constant</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">KSSE=3ℏ(4.32×10−9)2me=7.502×10−13 m2 s,K_{\mathrm{SSE}} = \frac{3\hbar(4.32\times10^{-9})}{2m_e} = 7.502\times10^{-13}\ \mathrm{m^2\,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 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">7.502</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><span class="mpunct">,</span></span></span></span></span> <p>with the corrected transition-dipole constraint</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">∣μ∣2e2=KSSEε(λ)Δν~effω.\frac{\|\mu\|^2}{e^2} = K_{\mathrm{SSE}} \frac{\varepsilon(\lambda)\Delta\tilde{\nu}_{\mathrm{eff}}}{\omega}.</span><span class="katex-html"><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">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 class="mord mathnormal">μ</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"><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="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="mord mathnormal">ω</span><span class="mord mathnormal">ε</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span>Δ<span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">ν</span><span class="accent-body">~</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">eff</span></span></span><span class="vlist-s"></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mord">.</span></span></span></span></span> <p>It also fixes the hydrogen verification convention</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">g1g2=26=13\frac{g_1}{g_2} = \frac{2}{6} = \frac{1}{3}</span><span class="katex-html"><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">g</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span><span class="vlist-s"></span></span><span class="mord mathnormal">g</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span><span class="vlist-s"></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">62</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">31</span><span class="vlist-s"></span></span></span></span></span></span></span></span></span> <p>for the <span class="katex copyable-equation"><span class="katex-mathml">1s→2p1s\rightarrow2p</span><span class="katex-html"><span class="base"><span class="mord">1</span><span class="mord mathnormal">s</span><span class="mrel">→</span></span><span class="base"><span class="mord">2</span><span class="mord mathnormal">p</span></span></span></span> transition, with <span class="katex copyable-equation"><span class="katex-mathml">f=0.4162f=0.4162</span><span class="katex-html"><span class="base"><span class="mord mathnormal">f</span><span class="mrel">=</span></span><span class="base"><span class="mord">0.4162</span></span></span></span>, and the SQC invariant</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">Kν=2πe2ε0mec3=7.421656×10−22 s.K_\nu = \frac{2\pi e^2}{\varepsilon_0 m_e c^3} = 7.421656\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 mathnormal mtight">ν</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">7.421656</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="mord">.</span></span></span></span></span> <p>These constants become structural constants of the <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-algebraic Scretching formalism.</p> <p>Hydrogen serves as the reference atomic system because its <span class="katex copyable-equation"><span class="katex-mathml">1s→2p1s\rightarrow2p</span><span class="katex-html"><span class="base"><span class="mord">1</span><span class="mord mathnormal">s</span><span class="mrel">→</span></span><span class="base"><span class="mord">2</span><span class="mord mathnormal">p</span></span></span></span> transition provides a clean test of oscillator strength, degeneracy ratio, Einstein coefficients, and SQC closure. DNA and RNA nucleobases serve as molecular-biological systems because their ultraviolet absorption properties can be mapped from molar absorptivity to oscillator strength, transition-dipole magnitude, dielectric response, and SQC/SSE correction terms.</p> <p>Therefore, this paper describes a unified mathematical framework in which atomic spectroscopy, nucleobase spectroscopy, DNA/RNA ultraviolet absorption, and quantum molecular biological observables are treated through one common operator-algebraic language. The resulting theory converts the Scretching framework from a collection of scalar transformations into a <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-algebraic dynamical system capable of modeling measurement, observables, transition coefficients, Hamiltonian corrections, and quantum time evolution.</p> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div>
format	Recurso digital
id	zenodo_https___doi_org_10_5281_zenodo_20102200
institution	Zenodo
language
publishDate	2026
publisher	Zenodo
record_format	zenodo
spellingShingle	The C-Algebraic Maxwell–Scretching–SQC–SSE Formalism: A Unified Operator Algebra for Hydrogen, DNA/RNA Nucleobases, and Quantum Molecular Spectroscopy Scretching, Daniel Scretching, Daniel <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=""> <div class="relative w-full overflow-visible"> <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 wrap-break-word w-full light markdown-new-styling"> <p><strong>The <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-Algebraic Maxwell–Scretching–SQC–SSE Formalism: A Unified Operator Algebra for Hydrogen, DNA/RNA Nucleobases, and Quantum Molecular Spectroscopy</strong> develops the Scretching framework into a single operator-algebraic system. Instead of treating the Maxwell–Scretching Chain, the Scretching Quantum Chain, and the Scretching–Schrödinger Equation as separate scalar equation systems, this paper combines them inside a complex <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-algebra where measurable spectroscopic quantities become observables or observable-generated functions.</p> <p>The central idea is that the experimentally measurable quantities</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ε, A, αabs, κ, σ, εr′′, f, ∣μ12∣2, A21, B21, B12\varepsilon,\ A,\ \alpha_{\mathrm{abs}},\ \kappa,\ \sigma,\ \varepsilon_r'',\ f,\ \|\mu_{12}\|^2,\ A_{21},\ B_{21},\ B_{12}</span><span class="katex-html"><span class="base"><span class="mord mathnormal">ε</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">α</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">abs</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">κ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">ε</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">r</span></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′′</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">f</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">μ</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">12</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mord">∣<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">2</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">A</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">21</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">B</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">21</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal">B</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">12</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span></span> <p>can be organized as elements of an operator algebra. These quantities describe molar absorptivity, absorbance, Napierian absorption coefficient, extinction coefficient, absorption cross-section, dielectric-loss response, oscillator strength, squared transition-dipole moment, and Einstein transition coefficients. Once placed into an operator-algebraic framework, they form a mathematical bridge from laboratory spectroscopy to quantum dynamics.</p> <p>The main transformation chain is</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">ε→f→∣μ12∣2→A21→V^SQC→H^SSE→αtSSE(A)\boxed{ \varepsilon \rightarrow f \rightarrow \|\mu_{12}\|^2 \rightarrow A_{21} \rightarrow \widehat{V}_{\mathrm{SQC}} \rightarrow \widehat{H}_{\mathrm{SSE}} \rightarrow \alpha_t^{\mathrm{SSE}}(A) }</span><span class="katex-html"><span class="base"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="boxpad"><span class="mord mathnormal">ε</span><span class="mrel">→</span><span class="mord mathnormal">f</span><span class="mrel">→</span>∣<span class="mord mathnormal">μ</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">12</span></span><span class="vlist-s"></span></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 class="mrel">→</span><span class="mord mathnormal">A</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">21</span></span><span class="vlist-s"></span></span><span class="mrel">→</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">V</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SQC</span></span></span><span class="vlist-s"></span></span><span class="mrel">→</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">H</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mrel">→</span><span class="mord mathnormal">α</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span> <p>This chain shows how molar absorptivity can be converted into oscillator strength, oscillator strength into transition-dipole magnitude, transition-dipole structure into Einstein coefficients, Einstein coefficients into the SQC constraint operator, and the SQC correction into the Scretching–Schrödinger Hamiltonian. The final object, <span class="katex copyable-equation"><span class="katex-mathml">αtSSE(A)\alpha_t^{\mathrm{SSE}}(A)</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">α</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">t</span></span><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="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>, represents the time evolution of an observable <span class="katex copyable-equation"><span class="katex-mathml">AA</span><span class="katex-html"><span class="base"><span class="mord mathnormal">A</span></span></span></span> under the SSE-modified Hamiltonian.</p> <p>In this formulation, the Maxwell–Scretching Chain supplies the measurable optical and spectroscopic observables; the Scretching Quantum Chain supplies the invariant transition-coefficient bridge; and the Scretching–Schrödinger Equation supplies the Hamiltonian operator and time-evolution law. Together, they define a unified algebraic system:</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">AMS-SQC-SSE=C&lowast;(ε, A, αabs, κ, σ, εr′′, f, ∣μ12∣2, A21, B21, B12,V^SQC,H^SSE)\boxed{ \mathcal{A}_{\mathrm{MS\text{-}SQC\text{-}SSE}} = C^* \left( \varepsilon,\ A,\ \alpha_{\mathrm{abs}},\ \kappa,\ \sigma,\ \varepsilon_r'',\ f,\ \|\mu_{12}\|^2,\ A_{21},\ B_{21},\ B_{12},\widehat{V}_{\mathrm{SQC}},\widehat{H}_{\mathrm{SSE}} \right) }</span><span class="katex-html"><span class="base"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="boxpad"><span class="mord mathcal">A</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">MS</span><span class="mord text mtight">-</span><span class="mord mathrm mtight">SQC</span><span class="mord text mtight">-</span><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mrel">=</span><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span><span class="minner"><span class="mopen delimcenter"><span class="delimsizing size2">(</span></span><span class="mord mathnormal">ε</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">α</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">abs</span></span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">κ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">ε</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">r</span></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′′</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">f</span><span class="mpunct">,</span><span class="mspace"> </span>∣<span class="mord mathnormal">μ</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">12</span></span><span class="vlist-s"></span></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 class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">A</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">21</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">B</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">21</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mord mathnormal">B</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">12</span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">V</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SQC</span></span></span><span class="vlist-s"></span></span><span class="mpunct">,</span><span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">H</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">SSE</span></span></span><span class="vlist-s"></span></span><span class="mclose delimcenter"><span class="delimsizing size2">)</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span> <p>The formalism uses the corrected Scretching–Schrödinger bridge constant</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">KSSE=3ℏ(4.32×10−9)2me=7.502×10−13 m2 s,K_{\mathrm{SSE}} = \frac{3\hbar(4.32\times10^{-9})}{2m_e} = 7.502\times10^{-13}\ \mathrm{m^2\,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 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">7.502</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><span class="mpunct">,</span></span></span></span></span> <p>with the corrected transition-dipole constraint</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">∣μ∣2e2=KSSEε(λ)Δν~effω.\frac{\|\mu\|^2}{e^2} = K_{\mathrm{SSE}} \frac{\varepsilon(\lambda)\Delta\tilde{\nu}_{\mathrm{eff}}}{\omega}.</span><span class="katex-html"><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">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 class="mord mathnormal">μ</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"><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="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="mord mathnormal">ω</span><span class="mord mathnormal">ε</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span>Δ<span class="mord accent"><span class="vlist-t"><span class="mord mathnormal">ν</span><span class="accent-body">~</span></span></span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">eff</span></span></span><span class="vlist-s"></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mord">.</span></span></span></span></span> <p>It also fixes the hydrogen verification convention</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">g1g2=26=13\frac{g_1}{g_2} = \frac{2}{6} = \frac{1}{3}</span><span class="katex-html"><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">g</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span><span class="vlist-s"></span></span><span class="mord mathnormal">g</span><span class="msupsub"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span><span class="vlist-s"></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">62</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">31</span><span class="vlist-s"></span></span></span></span></span></span></span></span></span> <p>for the <span class="katex copyable-equation"><span class="katex-mathml">1s→2p1s\rightarrow2p</span><span class="katex-html"><span class="base"><span class="mord">1</span><span class="mord mathnormal">s</span><span class="mrel">→</span></span><span class="base"><span class="mord">2</span><span class="mord mathnormal">p</span></span></span></span> transition, with <span class="katex copyable-equation"><span class="katex-mathml">f=0.4162f=0.4162</span><span class="katex-html"><span class="base"><span class="mord mathnormal">f</span><span class="mrel">=</span></span><span class="base"><span class="mord">0.4162</span></span></span></span>, and the SQC invariant</p> <span class="katex-display"><span class="katex copyable-equation"><span class="katex-mathml">Kν=2πe2ε0mec3=7.421656×10−22 s.K_\nu = \frac{2\pi e^2}{\varepsilon_0 m_e c^3} = 7.421656\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 mathnormal mtight">ν</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">7.421656</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="mord">.</span></span></span></span></span> <p>These constants become structural constants of the <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-algebraic Scretching formalism.</p> <p>Hydrogen serves as the reference atomic system because its <span class="katex copyable-equation"><span class="katex-mathml">1s→2p1s\rightarrow2p</span><span class="katex-html"><span class="base"><span class="mord">1</span><span class="mord mathnormal">s</span><span class="mrel">→</span></span><span class="base"><span class="mord">2</span><span class="mord mathnormal">p</span></span></span></span> transition provides a clean test of oscillator strength, degeneracy ratio, Einstein coefficients, and SQC closure. DNA and RNA nucleobases serve as molecular-biological systems because their ultraviolet absorption properties can be mapped from molar absorptivity to oscillator strength, transition-dipole magnitude, dielectric response, and SQC/SSE correction terms.</p> <p>Therefore, this paper describes a unified mathematical framework in which atomic spectroscopy, nucleobase spectroscopy, DNA/RNA ultraviolet absorption, and quantum molecular biological observables are treated through one common operator-algebraic language. The resulting theory converts the Scretching framework from a collection of scalar transformations into a <span class="katex copyable-equation"><span class="katex-mathml">C&lowast;C^</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">&lowast;</span></span></span></span></span></span></span></span></span></span>-algebraic dynamical system capable of modeling measurement, observables, transition coefficients, Hamiltonian corrections, and quantum time evolution.</p> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div>
title	The C*-Algebraic Maxwell–Scretching–SQC–SSE Formalism: A Unified Operator Algebra for Hydrogen, DNA/RNA Nucleobases, and Quantum Molecular Spectroscopy
url	https://doi.org/10.5281/zenodo.20102200

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