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| Format: | Recurso digital |
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Zenodo
2026
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| Online adgang: | https://doi.org/10.5281/zenodo.19712686 |
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Indholdsfortegnelse:
- <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">We derive the propagation of mechanical waves (sound) within the Structured Vacuum Theory with 23 channels (TVS23). Three results follow from the geometry of the V23 network with zero free parameters. First, sound velocity is identically zero in the vacuum: the V23 network contains no topological knots, and mechanical transmission requires knots. Second, sound velocity in any material is bounded above by the torsional wave speed <span class="katex"><span class="katex-mathml">c/23≈13,043c/23 \approx 13{,}043 </span><span class="katex-html"><span class="base"><span class="mord mathnormal">c</span><span class="mord">/23</span><span class="mrel">≈</span></span><span class="base"><span class="mord">13</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">043</span></span></span></span> km/s. Third, in gases the adiabatic index <span class="katex"><span class="katex-mathml">γ=(N+2)/N\gamma = (N+2)/N </span><span class="katex-html"><span class="base"><span class="mord mathnormal">γ</span><span class="mrel">=</span></span><span class="base"><span class="mopen">(</span><span class="mord mathnormal">N</span><span class="mbin">+</span></span><span class="base"><span class="mord">2</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathnormal">N</span></span></span></span> is derived from the number of active V23 modes per molecular knot, reproducing experimental sound speeds for monatomic, diatomic, and polyatomic gases with errors below 0.5%. For elastic solids, a geometric estimate gives <span class="katex"><span class="katex-mathml">vs∼(c/23)4/23≈5,440v_s \sim (c/23)\sqrt{4/23} \approx 5{,}440 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">v</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">s</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">∼</span></span><span class="base"><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mord">/23</span><span class="mclose">)</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="svg-align"><span class="mord">4/23</span></span></span><span class="vlist-s"></span></span></span></span><span class="mrel">≈</span></span><span class="base"><span class="mord">5</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">440</span></span></span></span> m/s consistent with observed metal values; for the hardest solid (diamond), the Pell solution <span class="katex"><span class="katex-mathml">(x,y)=(24,5)(x,y)=(24,5) </span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mord mathnormal">y</span><span class="mclose">)</span><span class="mrel">=</span></span><span class="base"><span class="mopen">(</span><span class="mord">24</span><span class="mpunct">,</span><span class="mord">5</span><span class="mclose">)</span></span></span></span> of <span class="katex"><span class="katex-mathml">x2−23y2=1x^2-23y^2=1 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">x</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">2</span></span></span></span></span></span></span><span class="mbin">−</span></span><span class="base"><span class="mord">23</span><span class="mord"><span class="mord mathnormal">y</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">2</span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord">1</span></span></span></span> gives <span class="katex"><span class="katex-mathml">vs=c/24≈12,491v_s = c/24 \approx 12{,}491 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">v</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">s</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mrel">=</span></span><span class="base"><span class="mord mathnormal">c</span><span class="mord">/24</span><span class="mrel">≈</span></span><span class="base"><span class="mord">12</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">491</span></span></span></span> m/s (observed: 12,000 m/s, error 4.1%), with equilibrium distance <span class="katex"><span class="katex-mathml">req=λ0⋅ρ≈1.40r_{\rm eq} = \lambda_0 \cdot \rho \approx 1.40 </span><span class="katex-html"><span class="base"><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 mtight"><span class="mord mathrm mtight">eq</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">λ</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">0</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mbin">⋅</span></span><span class="base"><span class="mord mathnormal">ρ</span><span class="mrel">≈</span></span><span class="base"><span class="mord">1.40</span></span></span></span> Å (observed: 1.54 Å, error 9%), where <span class="katex"><span class="katex-mathml">ρ=1.3247…\rho = 1.3247\ldots </span><span class="katex-html"><span class="base"><span class="mord mathnormal">ρ</span><span class="mrel">=</span></span><span class="base"><span class="mord">1.3247</span><span class="minner">…</span></span></span></span> is the plastic constant — the real root of <span class="katex"><span class="katex-mathml">x3−x−1=0x^3-x-1=0 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">x</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">3</span></span></span></span></span></span></span><span class="mbin">−</span></span><span class="base"><span class="mord mathnormal">x</span><span class="mbin">−</span></span><span class="base"><span class="mord">1</span><span class="mrel">=</span></span><span class="base"><span class="mord">0</span></span></span></span>.</p>