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<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" article-type="research-article" dtd-version="1.4" xml:lang="en">
  <front>
    <journal-meta>
      <journal-id journal-id-type="publisher-id">wjnse</journal-id>
      <journal-title-group>
        <journal-title>World Journal of Nano Science and Engineering</journal-title>
      </journal-title-group>
      <issn pub-type="epub">2161-4962</issn>
      <issn pub-type="ppub">2161-4954</issn>
      <publisher>
        <publisher-name>Scientific Research Publishing</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.4236/wjnse.2026.164010</article-id>
      <article-id pub-id-type="publisher-id">wjnse-155257</article-id>
      <article-categories>
        <subj-group>
          <subject>Article</subject>
        </subj-group>
        <subj-group>
          <subject>Chemistry</subject>
          <subject>Materials Science</subject>
          <subject>Engineering</subject>
          <subject>Physics</subject>
          <subject>Mathematics</subject>
        </subj-group>
      </article-categories>
      <title-group>
        <article-title>STOC-QComp: A Falsifiable Device Framework for a Bounded-Capacity Superconducting-Qubit Processor with a Mechanically Tunable Casimir Boundary Condition</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <contrib-id contrib-id-type="orcid">0000-0002-9510-8377</contrib-id>
          <name name-style="western">
            <surname>Pater</surname>
            <given-names>Ciprian Florin</given-names>
          </name>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
      </contrib-group>
      <aff id="aff1"><label>1</label> NWO Robotics, Kristiansand, Norway </aff>
      <author-notes>
        <fn fn-type="conflict" id="fn-conflict">
          <p>The author declares no conflicts of interest regarding the publication of this paper.</p>
        </fn>
      </author-notes>
      <pub-date pub-type="epub">
        <day>02</day>
        <month>12</month>
        <year>2026</year>
      </pub-date>
      <pub-date pub-type="collection">
        <month>12</month>
        <year>2026</year>
      </pub-date>
      <volume>16</volume>
      <issue>04</issue>
      <fpage>234</fpage>
      <lpage>246</lpage>
      <history>
        <date date-type="received">
          <day>04</day>
          <month>09</month>
          <year>2026</year>
        </date>
        <date date-type="accepted">
          <day>09</day>
          <month>11</month>
          <year>2026</year>
        </date>
        <date date-type="published">
          <day>12</day>
          <month>11</month>
          <year>2026</year>
        </date>
      </history>
      <permissions>
        <copyright-statement>© 2026 by the authors and Scientific Research Publishing Inc.</copyright-statement>
        <copyright-year>2026</copyright-year>
        <license license-type="open-access">
          <license-p> This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link> ). </license-p>
        </license>
      </permissions>
      <self-uri content-type="doi" xlink:href="https://doi.org/10.4236/wjnse.2026.164010">https://doi.org/10.4236/wjnse.2026.164010</self-uri>
      <abstract>
        <p>This paper presents STOC-QComp, a device-level engineering framework for a bounded-capacity quantum processor incorporating thermodynamic control, a mechanically tunable Casimir structure, and deployment-specific environmental engineering. The work extends the STOC-QC control framework [<xref ref-type="bibr" rid="B1">1</xref>] [<xref ref-type="bibr" rid="B2">2</xref>] by introducing the Casimir plate separation d as an experimentally accessible mechanical degree of freedom. Rather than assuming direct coupling between the static Casimir spectrum and the qubit transition, the revised device model treats the Casimir structure as a mechanically tunable boundary condition whose displacement may modify the electromagnetic cavity frequency, qubit-cavity coupling, and associated dissipation rates. The proposed 40 - 400 nm operating range is therefore treated as a design hypothesis requiring finite-conductivity Casimir modelling, electromagnetic simulation, mechanical stability analysis, and experimental validation. The paper specifies a reference superconducting-transmon architecture for the millikelvin deployment tiers, a separate 4 K control and characterisation configuration for the personal tier, manufacturing and yield assumptions with explicit evidence-class labels, environmental-noise budgets with quantitative transfer functions, and four falsifiable device-level tests with a full statistical analysis plan. Manufacturing costs and yields are explicitly classified as vendor quotations, engineering estimates, or experimental targets. The proposed 200 - 1000 qubit bounded-capacity regime is treated as a testable hypothesis motivated by Palmer’s bounded-Hilbert-space proposal [<xref ref-type="bibr" rid="B3">3</xref>] rather than an established physical limit. The result is a falsifiable device framework and experimental roadmap rather than a claim of completed commercial deployment.</p>
      </abstract>
      <kwd-group kwd-group-type="author-generated" xml:lang="en">
        <kwd>Superconducting Quantum Computer</kwd>
        <kwd>Transmon</kwd>
        <kwd>Casimir Cavity</kwd>
        <kwd>Cavity Electromechanics</kwd>
        <kwd>Reservoir Engineering</kwd>
        <kwd>Lindblad Dynamics</kwd>
        <kwd>Dispersive Readout</kwd>
        <kwd>Bounded Hilbert Space</kwd>
        <kwd>Manufacturing Yield</kwd>
        <kwd>Statistical Analysis Plan</kwd>
        <kwd>Extensive-Form Game</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec1">
      <title>1. Introduction—From Framework to Device</title>
      <p>STOC-QSIM v2.0 [<xref ref-type="bibr" rid="B1">1</xref>] established a phase-driven toroidal-Peltier control primitive and integrated it with the NWO ecosystem. STOC-QC [<xref ref-type="bibr" rid="B2">2</xref>] [<xref ref-type="bibr" rid="B4">4</xref>] extended the substrate into a computational framework for hybrid quantum systems whose scaling mechanism is reservoir engineering rather than qubit-count growth [<xref ref-type="bibr" rid="B5">5</xref>]. Version 1.0 of this manuscript attempted to specify the physical device—the “thing a lab, a factory, or a satellite operator can buy, install, and use”. The device architecture and manufacturing flow are summarized in <xref ref-type="fig" rid="fig1">Figure 1</xref>.</p>
      <fig id="fig1">
        <label>Figure 1</label>
        <caption>
          <p>Simplified engineering block diagram. Row 1 shows the linear manufacturing pipeline from materials to readout. Row 2 is the critical detail added in v2.0: the Casimir nanogap (40 - 400 nm, mechanical structure) and the microwave cavity (~cm-scale, GHz mode) are physically distinct structures; the nanogap displacement d modulates the cavity frequency <italic>ω</italic>c(d), the qubit-cavity coupling g(d), and the dissipation rates Γ(d) through mechanical action, NOT by supplying GHz Casimir modes directly. Row 3 shows the six deployment tiers; only T1 - T3 carry DESIGN status. Every block is fully specified with parameter, source class, uncertainty, and status label in S2 - 7.</p>
        </caption>
        <graphic xlink:href="https://html.scirp.org/file/4400413-rId17.jpeg?20261112113601" />
      </fig>
      <sec id="sec1dot1">
        <title>1.1. What v1.0 Got Wrong</title>
        <p>Two v1.0 claims cannot survive scrutiny as written:</p>
        <p><bold>(i) The Casimir pressure numerical value.</bold>v1.0 stated the pressure at <italic>d</italic> = 100 nm is ~1.3 mN/m<sup>2</sup>. The correct ideal parallel-plate value is <inline-formula><mml:math display="inline"><mml:mrow><mml:mrow><mml:mo> | </mml:mo><mml:mi> P </mml:mi><mml:mo> | </mml:mo></mml:mrow><mml:mo> = </mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mi> π </mml:mi><mml:mn> 2 </mml:mn></mml:msup><mml:mi> ℏ </mml:mi><mml:mi> c </mml:mi></mml:mrow><mml:mo> / </mml:mo><mml:mrow><mml:mrow><mml:mo> ( </mml:mo><mml:mrow><mml:mn> 240 </mml:mn><mml:msup><mml:mi> d </mml:mi><mml:mn> 4 </mml:mn></mml:msup></mml:mrow><mml:mo> ) </mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo> = </mml:mo><mml:mn> 13.00 </mml:mn><mml:mtext>   </mml:mtext><mml:mtext> Pa </mml:mtext><mml:mo> = </mml:mo><mml:mn> 13 </mml:mn><mml:mtext>   </mml:mtext><mml:mrow><mml:mtext> N </mml:mtext><mml:mo> / </mml:mo><mml:mrow><mml:msup><mml:mtext> m </mml:mtext><mml:mn> 2 </mml:mn></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula> —off by a factor of ~10<sup>4</sup> [<xref ref-type="bibr" rid="B6">6</xref>].</p>
        <p><bold>(ii) The claim that a 40</bold><bold>-</bold><bold>400 nm Casimir gap supplies GHz modes to the qubit.</bold>The fundamental parallel-plate mode is <italic>f</italic><sub>1</sub> = <italic>c</italic>/(2<italic>d</italic>); at <italic>d</italic> = 100 nm this is 1499.0 THz—an optical/UV frequency, not the 4 - 8 GHz superconducting-qubit band. The v1.0 expression <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi> J </mml:mi><mml:mrow><mml:mtext> Cas </mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo> ( </mml:mo><mml:mrow><mml:mi> ω </mml:mi><mml:mo> , </mml:mo><mml:mi> d </mml:mi></mml:mrow><mml:mo> ) </mml:mo></mml:mrow><mml:mo> = </mml:mo><mml:msub><mml:mi> J </mml:mi><mml:mn> 0 </mml:mn></mml:msub><mml:mrow><mml:mo> ( </mml:mo><mml:mi> ω </mml:mi><mml:mo> ) </mml:mo></mml:mrow><mml:mo> ⋅ </mml:mo><mml:mrow><mml:mo> [ </mml:mo><mml:mrow><mml:mn> 1 </mml:mn><mml:mo> + </mml:mo><mml:mstyle displaystyle="true"><mml:msub><mml:mo> ∑ </mml:mo><mml:mi> n </mml:mi></mml:msub><mml:mrow><mml:msub><mml:mi> g </mml:mi><mml:mi> n </mml:mi></mml:msub><mml:mrow><mml:mo> ( </mml:mo><mml:mi> ω </mml:mi><mml:mo> ) </mml:mo></mml:mrow><mml:mi> δ </mml:mi><mml:mrow><mml:mo> ( </mml:mo><mml:mrow><mml:mi> ω </mml:mi><mml:mo> − </mml:mo><mml:mrow><mml:mrow><mml:mi> n </mml:mi><mml:mi> π </mml:mi><mml:mi> c </mml:mi></mml:mrow><mml:mo> / </mml:mo><mml:mi> d </mml:mi></mml:mrow></mml:mrow><mml:mo> ) </mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow><mml:mo> ] </mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> therefore does not place cavity resonances in the microwave range.</p>
        <p>This is not a small correction. The Casimir-qubit coupling mechanism must be rebuilt to make the paper physically defensible.</p>
      </sec>
      <sec id="sec1dot2">
        <title>1.2. What This v2.0 Does Instead</title>
        <p>The revised architecture treats the Casimir structure as a mechanical actuator that modifies a physically separate microwave cavity, following the established cavity-electromechanics literature [<xref ref-type="bibr" rid="B7">7</xref>] [<xref ref-type="bibr" rid="B8">8</xref>]. In this picture:</p>
        <p>Casimir gap <italic>d</italic> → mechanical displacement → microwave cavity parameter → qubit coupling/dissipation (1.1)</p>
        <p>The nanogap supplies mechanical actuation and a small, well-modelled Casimir force; the microwave cavity supplies the GHz mode structure that couples to the qubit. The remainder of the paper specifies this hybrid architecture, its operating envelope, the reference qubit modality, environmental-noise budgets, manufacturing economics with evidence classes, verification tests with a statistical analysis plan, and an extensive-form strategic game.</p>
      </sec>
    </sec>
    <sec id="sec2">
      <title>2. Device-Specific Casimir-Microwave Coupling Model</title>
      <p>This section supersedes S2 of v1.0 in its entirety.</p>
      <sec id="sec2dot1">
        <title>2.1. Corrected Casimir Pressure</title>
        <p>For two parallel perfectly-conducting plates of area <italic>A</italic> separated by distance d, the ideal zero-temperature pressure is given by [<xref ref-type="bibr" rid="B6">6</xref>] and evaluated at four representative gap widths in <bold>Table 1</bold> (values plotted in <xref ref-type="fig" rid="fig2">Figure 2</xref>):</p>
        <disp-formula id="FD1">
          <label>(2.1)</label>
          <mml:math display="inline">
            <mml:mrow>
              <mml:mrow>
                <mml:mo>|</mml:mo>
                <mml:mrow>
                  <mml:msub>
                    <mml:mi>P</mml:mi>
                    <mml:mrow>
                      <mml:mtext>Cas</mml:mtext>
                    </mml:mrow>
                  </mml:msub>
                  <mml:mrow>
                    <mml:mo>(</mml:mo>
                    <mml:mi>d</mml:mi>
                    <mml:mo>)</mml:mo>
                  </mml:mrow>
                </mml:mrow>
                <mml:mo>|</mml:mo>
              </mml:mrow>
              <mml:mo>=</mml:mo>
              <mml:mrow>
                <mml:mrow>
                  <mml:msup>
                    <mml:mi>π</mml:mi>
                    <mml:mn>2</mml:mn>
                  </mml:msup>
                  <mml:mi>ℏ</mml:mi>
                  <mml:mi>c</mml:mi>
                </mml:mrow>
                <mml:mo>/</mml:mo>
                <mml:mrow>
                  <mml:mrow>
                    <mml:mo>(</mml:mo>
                    <mml:mrow>
                      <mml:mn>240</mml:mn>
                      <mml:msup>
                        <mml:mi>d</mml:mi>
                        <mml:mn>4</mml:mn>
                      </mml:msup>
                    </mml:mrow>
                    <mml:mo>)</mml:mo>
                  </mml:mrow>
                </mml:mrow>
              </mml:mrow>
            </mml:mrow>
          </mml:math>
        </disp-formula>
        <p><bold>Table 1.</bold> Corrected Casimir pressure at four key gap values. The v1.0 figure of 1.3 mN/m<sup>2</sup> at 100 nm is replaced by the correct 13 Pa. The Drude column is a phenomenological reduction factor accounting for finite Au conductivity and SiN dielectric spacer per [<xref ref-type="bibr" rid="B6">6</xref>] [<xref ref-type="bibr" rid="B7">7</xref>]; the precise reduction requires a full Lifshitz calculation for the Au·SiN·Au multilayer geometry [<xref ref-type="bibr" rid="B6">6</xref>] and is a first target of the device-characterisation programme (S7).</p>
        <table-wrap id="tbl1">
          <label>Table 1</label>
          <table>
            <tbody>
              <tr>
                <td>
                  <italic>
                    <bold>d</bold>
                  </italic>
                  <bold>(nm)</bold>
                </td>
                <td>
                  <bold>|</bold>
                  <italic>
                    <bold>P</bold>
                  </italic>
                  <bold>| ideal metal (Pa)</bold>
                </td>
                <td>
                  <bold>|</bold>
                  <italic>
                    <bold>P</bold>
                  </italic>
                  <bold>| Drude Au·SiN·Au estimate (Pa)</bold>
                </td>
              </tr>
              <tr>
                <td>40</td>
                <td>507.86</td>
                <td>345.35</td>
              </tr>
              <tr>
                <td>100</td>
                <td>13.00</td>
                <td>10.79</td>
              </tr>
              <tr>
                <td>200</td>
                <td>0.813</td>
                <td>0.739</td>
              </tr>
              <tr>
                <td>400</td>
                <td>0.0508</td>
                <td>0.0482</td>
              </tr>
            </tbody>
          </table>
        </table-wrap>
        <fig id="fig2">
          <label>Figure 2</label>
          <caption>
            <p>Corrected Casimir pressure vs plate separation d. Solid blue: ideal-metal result. Dashed magenta: phenomenological finite-conductivity reduction for the Au·SiN·Au stack. The v1.0 assertion of 1.3 mN/m<sup>2</sup> at 100 nm is off by a factor of ~10<sup>4</sup> from the true 13 Pa; this figure replaces v1.0 <xref ref-type="fig" rid="fig2">Figure 2</xref>.</p>
          </caption>
          <graphic xlink:href="https://html.scirp.org/file/4400413-rId24.jpeg?20261112113602" />
        </fig>
      </sec>
      <sec id="sec2dot2">
        <title>2.2. Nanogap Cavity Mode vs Qubit Microwave Band</title>
        <p>The fundamental parallel-plate electromagnetic mode of a gap of width <italic>d</italic> is <italic>f</italic><sub>1</sub> = <italic>c</italic>/(2<italic>d</italic>). Values at the four key gaps are given in <bold>Table 2</bold> and plotted against the superconducting-qubit microwave band in <xref ref-type="fig" rid="fig3">Figure 3</xref>.</p>
      </sec>
      <sec id="sec2dot3">
        <title>2.3. Hybrid Cavity-Electromechanical Hamiltonian</title>
        <p>Replace v1.0’s H_Cas(d) with the standard cavity-electromechanics Hamiltonian [<xref ref-type="bibr" rid="B7">7</xref>] [<xref ref-type="bibr" rid="B8">8</xref>]:</p>
        <disp-formula id="FD2">
          <label>(2.2)</label>
          <mml:math display="inline">
            <mml:mrow>
              <mml:mi>H</mml:mi>
              <mml:mo>=</mml:mo>
              <mml:msub>
                <mml:mi>H</mml:mi>
                <mml:mi>q</mml:mi>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:mi>ℏ</mml:mi>
              <mml:msub>
                <mml:mi>ω</mml:mi>
                <mml:mi>c</mml:mi>
              </mml:msub>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mi>d</mml:mi>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:msup>
                <mml:mi>a</mml:mi>
                <mml:mo>†</mml:mo>
              </mml:msup>
              <mml:mi>a</mml:mi>
              <mml:mo>+</mml:mo>
              <mml:mi>ℏ</mml:mi>
              <mml:mi>g</mml:mi>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mi>d</mml:mi>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mrow>
                  <mml:msup>
                    <mml:mi>a</mml:mi>
                    <mml:mo>†</mml:mo>
                  </mml:msup>
                  <mml:msub>
                    <mml:mi>σ</mml:mi>
                    <mml:mo>−</mml:mo>
                  </mml:msub>
                  <mml:mo>+</mml:mo>
                  <mml:mi>a</mml:mi>
                  <mml:msub>
                    <mml:mi>σ</mml:mi>
                    <mml:mo>+</mml:mo>
                  </mml:msub>
                </mml:mrow>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>U</mml:mi>
                <mml:mrow>
                  <mml:mtext>Cas</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mi>d</mml:mi>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>H</mml:mi>
                <mml:mrow>
                  <mml:mtext>mech</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>H</mml:mi>
                <mml:mrow>
                  <mml:mtext>env</mml:mtext>
                </mml:mrow>
              </mml:msub>
            </mml:mrow>
          </mml:math>
        </disp-formula>
        <p>with the plate separation:</p>
        <disp-formula id="FD3">
          <label>(2.3)</label>
          <mml:math display="inline">
            <mml:mrow>
              <mml:mi>d</mml:mi>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mi>t</mml:mi>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:mo>=</mml:mo>
              <mml:msub>
                <mml:mi>d</mml:mi>
                <mml:mn>0</mml:mn>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>x</mml:mi>
                <mml:mrow>
                  <mml:mtext>zpf</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mrow>
                  <mml:mi>b</mml:mi>
                  <mml:mo>+</mml:mo>
                  <mml:msup>
                    <mml:mi>b</mml:mi>
                    <mml:mo>†</mml:mo>
                  </mml:msup>
                </mml:mrow>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>x</mml:mi>
                <mml:mrow>
                  <mml:mtext>ctrl</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mi>t</mml:mi>
                <mml:mo>)</mml:mo>
              </mml:mrow>
            </mml:mrow>
          </mml:math>
        </disp-formula>
        <p>where <italic>a</italic>, <italic>a</italic><sup>†</sup> are the microwave-cavity photon operators; <italic>σ</italic><sub>+</sub>, <italic>σ</italic><sub>-</sub> are the transmon raising/lowering operators; <italic>b</italic>, <italic>b</italic><sup>†</sup> are the mechanical mode operators; <italic>U</italic><sub>Cas</sub>(<italic>d</italic>) is the mechanical potential from the Casimir attraction (not the qubit reservoir); <italic>H</italic><sub>mech</sub> collects the mechanical kinetic and spring terms; and <italic>x</italic><sub>ctrl</sub>(<italic>t</italic>) is a piezo-electrostatic control displacement. The four physically measurable coupling derivatives are then:</p>
        <disp-formula id="FD4">
          <label>(2.4)</label>
          <mml:math display="inline">
            <mml:mrow>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:msub>
                    <mml:mi>ω</mml:mi>
                    <mml:mi>c</mml:mi>
                  </mml:msub>
                </mml:mrow>
                <mml:mo>/</mml:mo>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:mi>d</mml:mi>
                </mml:mrow>
              </mml:mrow>
              <mml:mo>,</mml:mo>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:mi>g</mml:mi>
                </mml:mrow>
                <mml:mo>/</mml:mo>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:mi>d</mml:mi>
                </mml:mrow>
              </mml:mrow>
              <mml:mo>,</mml:mo>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:mi>κ</mml:mi>
                </mml:mrow>
                <mml:mo>/</mml:mo>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:mi>d</mml:mi>
                </mml:mrow>
              </mml:mrow>
              <mml:mo>,</mml:mo>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:msub>
                    <mml:mi>Γ</mml:mi>
                    <mml:mn>1</mml:mn>
                  </mml:msub>
                </mml:mrow>
                <mml:mo>/</mml:mo>
                <mml:mrow>
                  <mml:mo>∂</mml:mo>
                  <mml:mi>d</mml:mi>
                </mml:mrow>
              </mml:mrow>
            </mml:mrow>
          </mml:math>
        </disp-formula>
        <p><bold>Table 2</bold><bold>.</bold> Nanogap fundamental electromagnetic mode across the STOC-QComp operating band. All values lie in the optical/near-UV range. None overlap the 4 - 8 GHz superconducting-transmon transition band. The v1.0 mechanism of “moving a 100 nm Casimir gap directly moves cavity resonances through the qubit’s microwave spectral density” is therefore physically incorrect as stated.</p>
        <table-wrap id="tbl2">
          <label>Table 2</label>
          <table>
            <tbody>
              <tr>
                <td>
                  <italic>
                    <bold>d</bold>
                  </italic>
                  <bold>(nm)</bold>
                </td>
                <td>
                  <italic>
                    <bold>f</bold>
                  </italic>
                  <bold>
                    <sub>1</sub>
                  </bold>
                  <bold>=</bold>
                  <italic>
                    <bold>c</bold>
                  </italic>
                  <bold>/(2</bold>
                  <italic>
                    <bold>d</bold>
                  </italic>
                  <bold>) (Hz)</bold>
                </td>
                <td>
                  <italic>
                    <bold>f</bold>
                  </italic>
                  <bold>
                    <sub>1</sub>
                  </bold>
                  <bold>(THz)</bold>
                </td>
                <td>
                  <bold>wavelength</bold>
                  <italic>
                    <bold>λ</bold>
                  </italic>
                  <bold>= 2</bold>
                  <italic>
                    <bold>d</bold>
                  </italic>
                  <bold>(nm)</bold>
                </td>
              </tr>
              <tr>
                <td>40</td>
                <td>3.747e+15</td>
                <td>3747</td>
                <td>80</td>
              </tr>
              <tr>
                <td>100</td>
                <td>1.499e+15</td>
                <td>1499</td>
                <td>200</td>
              </tr>
              <tr>
                <td>200</td>
                <td>7.495e+14</td>
                <td>749</td>
                <td>400</td>
              </tr>
              <tr>
                <td>400</td>
                <td>3.747e+14</td>
                <td>375</td>
                <td>800</td>
              </tr>
            </tbody>
          </table>
        </table-wrap>
        <fig id="fig3">
          <label>Figure 3</label>
          <caption>
            <p>Nanogap fundamental cavity mode vs superconducting-qubit microwave band. The two bands are separated by five orders of magnitude in frequency. A microwave cavity resonant with a 5 GHz qubit requires a physical length ≈ 3 cm, not a 40 - 400 nm gap.</p>
          </caption>
          <graphic xlink:href="https://html.scirp.org/file/4400413-rId31.jpeg?20261112113602" />
        </fig>
      </sec>
      <sec id="sec2dot4">
        <title>2.4. Non-Ideal Casimir Corrections</title>
        <p>The ideal expression (2.1) is used only as a first-order benchmark. The device prediction uses a finite-conductivity Lifshitz calculation incorporating the Au·SiN·Au geometry [<xref ref-type="bibr" rid="B6">6</xref>]. The non-ideal effects that the prediction must account for are enumerated in <bold>Table 3</bold>:</p>
        <p><bold>Table 3</bold><bold>.</bold> Non-ideal Casimir corrections that must be included in the device-level prediction. Each effect is characterised by an established measurement or literature model [<xref ref-type="bibr" rid="B6">6</xref>].</p>
        <table-wrap id="tbl3">
          <label>Table 3</label>
          <table>
            <tbody>
              <tr>
                <td>
                  <bold>Effect</bold>
                </td>
                <td>
                  <bold>Physical origin</bold>
                </td>
                <td>
                  <bold>Order of magnitude</bold>
                </td>
              </tr>
              <tr>
                <td>Finite Au conductivity</td>
                <td>Drude/plasma-frequency reduction</td>
                <td>
                  5% - 30% reduction of |
                  <italic>P</italic>
                  |
                </td>
              </tr>
              <tr>
                <td>SiN dielectric spacer</td>
                <td>
                  Real dielectric response
                  <italic>ε</italic>
                  (
                  <italic>iξ</italic>
                  )
                </td>
                <td>Geometry-dependent</td>
              </tr>
              <tr>
                <td>Finite plate thickness</td>
                <td>
                  Two-slab Lifshitz [
                  <xref ref-type="bibr" rid="B6">6</xref>
                  ]
                </td>
                <td>5% - 15% correction</td>
              </tr>
              <tr>
                <td>Surface roughness</td>
                <td>
                  Perturbative expansion in
                  <italic>R</italic>
                  <italic>
                    <sub>q</sub>
                  </italic>
                  /
                  <italic>d</italic>
                </td>
                <td>
                  ≤10% at
                  <italic>R</italic>
                  <italic>
                    <sub>q</sub>
                  </italic>
                  &lt;
                  <italic>d</italic>
                  /10
                </td>
              </tr>
              <tr>
                <td>Patch potentials</td>
                <td>Contact potential difference across grain</td>
                <td>
                  Sub-Casimir at
                  <italic>d</italic>
                  &gt; 100 nm
                </td>
              </tr>
              <tr>
                <td>Finite plate area</td>
                <td>Diffraction/edge modes</td>
                <td>
                  Small if
                  <italic>A</italic>
                  <inline-formula>
                    <mml:math display="inline">
                      <mml:mo>≫</mml:mo>
                    </mml:math>
                  </inline-formula>
                  <italic>d</italic>
                  <sup>2</sup>
                </td>
              </tr>
              <tr>
                <td>
                  Temperature (
                  <italic>T</italic>
                  = 10 mK)
                </td>
                <td>
                  Matsubara sum with
                  <italic>T</italic>
                  ≠ 0
                </td>
                <td>Negligible at nanoscale</td>
              </tr>
              <tr>
                <td>Contamination/adsorbates</td>
                <td>Oxide/hydrocarbon layer</td>
                <td>
                  Reduces effective
                  <italic>d</italic>
                </td>
              </tr>
              <tr>
                <td>Electrostatic bias</td>
                <td>Applied potential across gap</td>
                <td>
                  Adds separate
                  <italic>ε</italic>
                  <sub>0</sub>
                  <italic>E</italic>
                  <sup>2</sup>
                  /2 term
                </td>
              </tr>
              <tr>
                <td>Residual gas pressure</td>
                <td>Molecular impact drag</td>
                <td>Requires UHV</td>
              </tr>
              <tr>
                <td>
                  Parallelism error
                  <italic>θ</italic>
                </td>
                <td>Non-parallel plate model</td>
                <td>Handled by capacitive metrology</td>
              </tr>
            </tbody>
          </table>
        </table-wrap>
      </sec>
    </sec>
    <sec id="sec3">
      <title>3. Cavity Operating Envelope</title>
      <p>The mechanical structure must satisfy stability, pull-in, thermal, and electromagnetic constraints simultaneously. Every parameter below is a design variable to be measured on the first prototype. <bold>Table 4</bold> summarizes the parameters, their symbols, design targets, and validation methods.</p>
      <p><bold>Table 4</bold><bold>.</bold> Cavity operating envelope. All rows are design targets with named validation methods.</p>
      <table-wrap id="tbl4">
        <label>Table 4</label>
        <table>
          <tbody>
            <tr>
              <td>
                <bold>Parameter</bold>
              </td>
              <td>
                <bold>Symbol</bold>
              </td>
              <td>
                <bold>Design target</bold>
              </td>
              <td>
                <bold>Validation method</bold>
              </td>
            </tr>
            <tr>
              <td>Nominal gap</td>
              <td>
                <italic>d</italic>
                <sub>0</sub>
              </td>
              <td>40 - 400 nm</td>
              <td>Capacitive metrology</td>
            </tr>
            <tr>
              <td>Gap RMS noise</td>
              <td>
                <italic>σ</italic>
                <italic>
                  <sub>d</sub>
                </italic>
              </td>
              <td>
                TBD from T
                <sub>1</sub>
                /T
                <sub>2</sub>
                budget
              </td>
              <td>Displacement spectrum</td>
            </tr>
            <tr>
              <td>Parallelism angle</td>
              <td>
                <italic>θ</italic>
              </td>
              <td>TBD (interferometry)</td>
              <td>Fabry-Perot/white-light</td>
            </tr>
            <tr>
              <td>Mechanical resonance</td>
              <td>
                <italic>f</italic>
                <italic>
                  <sub>m</sub>
                </italic>
              </td>
              <td>TBD (kHz range)</td>
              <td>Network analyser</td>
            </tr>
            <tr>
              <td>Mechanical Q</td>
              <td>
                <italic>Q</italic>
                <italic>
                  <sub>m</sub>
                </italic>
              </td>
              <td>TBD</td>
              <td>Ringdown</td>
            </tr>
            <tr>
              <td>Actuator bandwidth</td>
              <td>
                <italic>B</italic>
                <italic>
                  <sub>a</sub>
                </italic>
              </td>
              <td>
                ≥ 10
                <italic>f</italic>
                <italic>
                  <sub>m</sub>
                </italic>
              </td>
              <td>Frequency response</td>
            </tr>
            <tr>
              <td>Pull-in margin</td>
              <td>
                <italic>M</italic>
                <italic>
                  <sub>PI</sub>
                </italic>
              </td>
              <td>&gt; 2 × safety factor</td>
              <td>Capacitance-vs-V sweep</td>
            </tr>
            <tr>
              <td>Microwave insertion loss</td>
              <td>
                <italic>L</italic>
                <italic>
                  <sub>mw</sub>
                </italic>
              </td>
              <td>TBD (&lt;0.1 dB target)</td>
              <td>
                S
                <sub>2</sub>
                <sub>1</sub>
                · VNA
              </td>
            </tr>
            <tr>
              <td>Added cryogenic heat load</td>
              <td>
                <italic>P</italic>
                <sub>act</sub>
              </td>
              <td>&lt; cold-stage budget</td>
              <td>Calorimetry</td>
            </tr>
            <tr>
              <td>Displacement noise PSD</td>
              <td>
                <italic>S</italic>
                <italic>
                  <sub>d</sub>
                </italic>
                (
                <italic>f</italic>
                )
              </td>
              <td>TBD</td>
              <td>Optical interferometry</td>
            </tr>
            <tr>
              <td>Operating temperature</td>
              <td>
                <italic>T</italic>
              </td>
              <td>10 mK · 4 K by tier</td>
              <td>Ge thermometer</td>
            </tr>
            <tr>
              <td>Surface roughness</td>
              <td>
                <italic>R</italic>
                <italic>
                  <sub>q</sub>
                </italic>
              </td>
              <td>
                TBD (&lt;
                <italic>d</italic>
                /10)
              </td>
              <td>AFM</td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
      <p>Mechanical stability requires:</p>
      <disp-formula id="FD5">
        <label>(3.1)</label>
        <mml:math display="inline">
          <mml:mrow>
            <mml:msub>
              <mml:mi>k</mml:mi>
              <mml:mrow>
                <mml:mtext>eff</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>=</mml:mo>
            <mml:mi>k</mml:mi>
            <mml:mo>−</mml:mo>
            <mml:mrow>
              <mml:mo>|</mml:mo>
              <mml:mrow>
                <mml:mrow>
                  <mml:mrow>
                    <mml:mo>∂</mml:mo>
                    <mml:msub>
                      <mml:mi>F</mml:mi>
                      <mml:mi>C</mml:mi>
                    </mml:msub>
                  </mml:mrow>
                  <mml:mo>/</mml:mo>
                  <mml:mrow>
                    <mml:mo>∂</mml:mo>
                    <mml:mi>d</mml:mi>
                  </mml:mrow>
                </mml:mrow>
              </mml:mrow>
              <mml:mo>|</mml:mo>
            </mml:mrow>
            <mml:mo>−</mml:mo>
            <mml:msub>
              <mml:mi>k</mml:mi>
              <mml:mrow>
                <mml:mtext>act</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>&gt;</mml:mo>
            <mml:mn>0</mml:mn>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>For an electrostatic parallel-plate actuator, the pull-in criterion <italic>V</italic> &lt; <italic>V</italic><italic><sub>PI</sub></italic> (<italic>d</italic>) must also be satisfied at every operating point.</p>
    </sec>
    <sec id="sec4">
      <title>4. Reference Qubit Architecture</title>
      <p>v1.0 said the bare qubit “may be Josephson, spin, or topological”, but the manufacturing pipeline specified a Josephson QPU. v2.0 commits to a single reference architecture.</p>
      <sec id="sec4dot1">
        <title>4.1. Reference: Superconducting Transmon</title>
        <p>For tiers T2 - T6, the reference architecture is a superconducting transmon QPU with: fixed- or tunable-frequency transmon qubits; 4 - 8 GHz nominal qubit transition frequency; dilution refrigerator at 10 - 20 mK base temperature; microwave XY control; flux control (if tunable); dispersive resonator readout, multiplexed; randomised benchmarking for one- and two-qubit gate fidelity, with T1 and T2 characterisation.</p>
      </sec>
      <sec id="sec4dot2">
        <title>4.2. T1 Is Not a Full QPU</title>
        <p>v1.0 claimed a 4 K personal unit with 200 - 400 qubits and 99.0% fidelity. This is not physically achievable with a conventional 5 GHz transmon at 4 K. The Bose factor gives:</p>
        <disp-formula id="FD6">
          <label>(4.1)</label>
          <mml:math display="inline">
            <mml:mrow>
              <mml:msub>
                <mml:mi>n</mml:mi>
                <mml:mrow>
                  <mml:mi>t</mml:mi>
                  <mml:mi>h</mml:mi>
                </mml:mrow>
              </mml:msub>
              <mml:mo>=</mml:mo>
              <mml:mrow>
                <mml:mn>1</mml:mn>
                <mml:mo>/</mml:mo>
                <mml:mrow>
                  <mml:mrow>
                    <mml:mo>(</mml:mo>
                    <mml:mrow>
                      <mml:mi>exp</mml:mi>
                      <mml:mrow>
                        <mml:mo>(</mml:mo>
                        <mml:mrow>
                          <mml:mrow>
                            <mml:mrow>
                              <mml:mi>h</mml:mi>
                              <mml:msub>
                                <mml:mi>f</mml:mi>
                                <mml:mi>q</mml:mi>
                              </mml:msub>
                            </mml:mrow>
                            <mml:mo>/</mml:mo>
                            <mml:mrow>
                              <mml:msub>
                                <mml:mi>k</mml:mi>
                                <mml:mi>B</mml:mi>
                              </mml:msub>
                              <mml:mi>T</mml:mi>
                            </mml:mrow>
                          </mml:mrow>
                        </mml:mrow>
                        <mml:mo>)</mml:mo>
                      </mml:mrow>
                      <mml:mo>−</mml:mo>
                      <mml:mn>1</mml:mn>
                    </mml:mrow>
                    <mml:mo>)</mml:mo>
                  </mml:mrow>
                </mml:mrow>
              </mml:mrow>
            </mml:mrow>
          </mml:math>
        </disp-formula>
        <disp-formula id="FD7">
          <label>(4.2)</label>
          <mml:math display="inline">
            <mml:mrow>
              <mml:msub>
                <mml:mi>p</mml:mi>
                <mml:mi>e</mml:mi>
              </mml:msub>
              <mml:mo>=</mml:mo>
              <mml:mrow>
                <mml:mn>1</mml:mn>
                <mml:mo>/</mml:mo>
                <mml:mrow>
                  <mml:mrow>
                    <mml:mo>(</mml:mo>
                    <mml:mrow>
                      <mml:mi>exp</mml:mi>
                      <mml:mrow>
                        <mml:mo>(</mml:mo>
                        <mml:mrow>
                          <mml:mrow>
                            <mml:mrow>
                              <mml:mi>h</mml:mi>
                              <mml:msub>
                                <mml:mi>f</mml:mi>
                                <mml:mi>q</mml:mi>
                              </mml:msub>
                            </mml:mrow>
                            <mml:mo>/</mml:mo>
                            <mml:mrow>
                              <mml:msub>
                                <mml:mi>k</mml:mi>
                                <mml:mi>B</mml:mi>
                              </mml:msub>
                              <mml:mi>T</mml:mi>
                            </mml:mrow>
                          </mml:mrow>
                        </mml:mrow>
                        <mml:mo>)</mml:mo>
                      </mml:mrow>
                      <mml:mo>+</mml:mo>
                      <mml:mn>1</mml:mn>
                    </mml:mrow>
                    <mml:mo>)</mml:mo>
                  </mml:mrow>
                </mml:mrow>
              </mml:mrow>
            </mml:mrow>
          </mml:math>
        </disp-formula>
        <p>At <italic>T</italic> = 4 K, <italic>f</italic><italic><sub>q</sub></italic> = 5 GHz: <italic>hf</italic>/<italic>k</italic><italic><sub>B</sub></italic><italic>T</italic> = 0.06 and <italic>p</italic><italic><sub>e</sub></italic> = 48.5%. At <italic>T</italic> = 20 mK: <italic>p</italic><italic><sub>e</sub></italic> = 6.2e−04% (thermal ground state). A 48.5% excited-state population is incompatible with 99% gate fidelity. <xref ref-type="fig" rid="fig4">Figure 4</xref> shows <italic>p</italic><italic><sub>e</sub></italic> vs <italic>T</italic> for three representative qubit frequencies.</p>
        <fig id="fig4">
          <label>Figure 4</label>
          <caption>
            <p>Thermal excited-state population vs temperature for three qubit frequencies. Recent 4 K microwave quantum infrastructure achieves usable performance via cold-loaded radiative cooling of specific input lines [<xref ref-type="bibr" rid="B9">9</xref>]—but this suppresses thermal noise on control channels, not on the qubit itself. A conventional 5 GHz transmon at 4 K remains ~48% thermally excited.</p>
          </caption>
          <graphic xlink:href="https://html.scirp.org/file/4400413-rId40.jpeg?20261112113603" />
        </fig>
      </sec>
      <sec id="sec4dot3">
        <title>4.3. Deployment Architecture</title>
        <p>The complete deployment map across tiers T1 - T6, showing qubit modality, operating temperature, control and readout schemes, and target two-qubit gate fidelity per tier, is presented in <bold>Table 5</bold>.</p>
        <p><bold>Table 5</bold><bold>.</bold> Deployment architecture with explicit qubit modality, temperature, control, readout, and target two-qubit gate fidelity F<sub>2</sub> per tier. All fidelity values are design targets to be verified by randomised benchmarking on prototype units.</p>
        <table-wrap id="tbl5">
          <label>Table 5</label>
          <table>
            <tbody>
              <tr>
                <td>
                  <bold>Tier</bold>
                </td>
                <td>
                  <bold>Function</bold>
                </td>
                <td>
                  <bold>Qubit</bold>
                </td>
                <td>
                  <italic>
                    <bold>T</bold>
                  </italic>
                </td>
                <td>
                  <bold>Control</bold>
                </td>
                <td>
                  <bold>Readout</bold>
                </td>
                <td>
                  <bold>Target</bold>
                  <italic>
                    <bold>F</bold>
                  </italic>
                  <bold>
                    <sub>2</sub>
                  </bold>
                </td>
                <td>
                  <bold>Status</bold>
                </td>
              </tr>
              <tr>
                <td>T1</td>
                <td>4 K demo/control platform</td>
                <td>(none in v2)</td>
                <td>4 K</td>
                <td>cryo-CMOS</td>
                <td>resonator diag</td>
                <td>N/A (demo)</td>
                <td>DESIGN</td>
              </tr>
              <tr>
                <td>T2</td>
                <td>SME lab QPU rack</td>
                <td>transmon</td>
                <td>10 - 20 mK</td>
                <td>μwave XY + flux</td>
                <td>dispersive</td>
                <td>≥99.0%</td>
                <td>DESIGN</td>
              </tr>
              <tr>
                <td>T3</td>
                <td>Industrial QPU</td>
                <td>transmon</td>
                <td>10 - 20 mK</td>
                <td>μwave XY + flux</td>
                <td>dispersive</td>
                <td>≥99.5%</td>
                <td>DESIGN</td>
              </tr>
              <tr>
                <td>T4</td>
                <td>Subsea/subterranean</td>
                <td>transmon</td>
                <td>10 - 20 mK</td>
                <td>μwave XY + flux</td>
                <td>dispersive</td>
                <td>≥99.0%</td>
                <td>ROADMAP</td>
              </tr>
              <tr>
                <td>T5</td>
                <td>Naval/aerospace mobile</td>
                <td>transmon</td>
                <td>10 - 20 mK</td>
                <td>μwave XY + flux</td>
                <td>dispersive</td>
                <td>≥99.0%</td>
                <td>ROADMAP</td>
              </tr>
              <tr>
                <td>T6</td>
                <td>Orbital platform</td>
                <td>transmon</td>
                <td>10 - 20 mK</td>
                <td>μwave XY + flux</td>
                <td>dispersive</td>
                <td>≥99.5%</td>
                <td>ROADMAP</td>
              </tr>
            </tbody>
          </table>
        </table-wrap>
      </sec>
    </sec>
    <sec id="sec5">
      <title>5. Environmental-Noise Decoherence Model</title>
      <p>v1.0 <bold>Table 5</bold> asserted +0.1%, +0.3%, +0.7%, +1.1% fidelity gains for magnetic noise, vibration, cosmic-ray, and thermal shielding without a causal model. v2.0 replaces this with an explicit environmental-noise transfer model.</p>
      <p>The total qubit relaxation rate is:</p>
      <disp-formula id="FD8">
        <label>(5.1)</label>
        <mml:math display="inline">
          <mml:mrow>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mn>1</mml:mn>
            </mml:msub>
            <mml:mo>=</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mn>1</mml:mn>
                <mml:mo>,</mml:mo>
                <mml:mn>0</mml:mn>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mtext>flux</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mtext>charge</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mtext>radiation</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mtext>vibration</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mtext>thermal</mml:mtext>
              </mml:mrow>
            </mml:msub>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>The total pure dephasing rate is:</p>
      <disp-formula id="FD9">
        <label>(5.2)</label>
        <mml:math display="inline">
          <mml:mrow>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mi>φ</mml:mi>
            </mml:msub>
            <mml:mo>=</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mi>φ</mml:mi>
                <mml:mo>,</mml:mo>
                <mml:mtext>flux</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mi>φ</mml:mi>
                <mml:mo>,</mml:mo>
                <mml:mtext>charge</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mrow>
                <mml:mi>φ</mml:mi>
                <mml:mo>,</mml:mo>
                <mml:mtext>vibration</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>+</mml:mo>
            <mml:mo>⋯</mml:mo>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>For a Clifford gate of duration <italic>t</italic><italic><sub>g</sub></italic>:</p>
      <disp-formula id="FD10">
        <label>(5.3)</label>
        <mml:math display="inline">
          <mml:mrow>
            <mml:msub>
              <mml:mi>F</mml:mi>
              <mml:mrow>
                <mml:mtext>gate</mml:mtext>
              </mml:mrow>
            </mml:msub>
            <mml:mo>≈</mml:mo>
            <mml:mn>1</mml:mn>
            <mml:mo>−</mml:mo>
            <mml:msub>
              <mml:mi>C</mml:mi>
              <mml:mn>1</mml:mn>
            </mml:msub>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mn>1</mml:mn>
            </mml:msub>
            <mml:msub>
              <mml:mi>t</mml:mi>
              <mml:mi>g</mml:mi>
            </mml:msub>
            <mml:mo>−</mml:mo>
            <mml:msub>
              <mml:mi>C</mml:mi>
              <mml:mn>2</mml:mn>
            </mml:msub>
            <mml:msub>
              <mml:mi>Γ</mml:mi>
              <mml:mi>φ</mml:mi>
            </mml:msub>
            <mml:msub>
              <mml:mi>t</mml:mi>
              <mml:mi>g</mml:mi>
            </mml:msub>
            <mml:mo>−</mml:mo>
            <mml:msub>
              <mml:mi>ε</mml:mi>
              <mml:mrow>
                <mml:mtext>control</mml:mtext>
              </mml:mrow>
            </mml:msub>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>with <italic>C</italic><sub>1</sub>, <italic>C</italic><sub>2</sub> ≈ O(1) coefficients depending on the specific gate. The environmental transfer functions from each environmental input to Γ are summarised in <bold>Table 6</bold>:</p>
      <p><bold>Table 6</bold><bold>.</bold> Environmental-noise transfer chains. For each environmental input, the transfer function to Γ is specified explicitly. <bold>Table 6</bold> replaces the v1.0 asserted fidelity gains; the corresponding numerical fidelity contributions become primary endpoints of verification test T21 (S7).</p>
      <table-wrap id="tbl6">
        <label>Table 6</label>
        <table>
          <tbody>
            <tr>
              <td>
                <bold>Environmental input</bold>
              </td>
              <td>
                <bold>Transfer chain</bold>
              </td>
              <td>
                <bold>Γ contribution</bold>
              </td>
            </tr>
            <tr>
              <td>Magnetic field noise</td>
              <td>
                <italic>S</italic>
                <italic>
                  <sub>B</sub>
                </italic>
                (
                <italic>f</italic>
                ) →
                <italic>S</italic>
                <italic>
                  <sub>Φ</sub>
                </italic>
                (
                <italic>f</italic>
                ) →
                <italic>S</italic>
                <italic>
                  <sub>ω</sub>
                </italic>
                (
                <italic>f</italic>
                ) → Γ
                <italic>
                  <sub>φ</sub>
                </italic>
              </td>
              <td>
                Dephasing via flux noise [
                <xref ref-type="bibr" rid="B10">10</xref>
                ]
              </td>
            </tr>
            <tr>
              <td>Vibration</td>
              <td>
                <italic>S</italic>
                <italic>
                  <sub>x</sub>
                </italic>
                (
                <italic>f</italic>
                ) →
                <italic>S</italic>
                <italic>
                  <sub>d</sub>
                </italic>
                (
                <italic>f</italic>
                ) → {
                <italic>S</italic>
                <italic>
                  <sub>ωc</sub>
                </italic>
                (
                <italic>f</italic>
                ),
                <italic>S</italic>
                <italic>
                  <sub>g</sub>
                </italic>
                (
                <italic>f</italic>
                )} → Γ
                <sub>1</sub>
                , Γ
                <italic>
                  <sub>φ</sub>
                </italic>
              </td>
              <td>Both channels via cavity</td>
            </tr>
            <tr>
              <td>Ionising radiation</td>
              <td>
                Φ
                <italic>
                  <sub>μ</sub>
                </italic>
                → Γ
                <italic>
                  <sub>qp</sub>
                </italic>
                → Γ
                <sub>1</sub>
                , Γ
                <italic>
                  <sub>φ</sub>
                </italic>
              </td>
              <td>
                Quasiparticle poisoning [
                <xref ref-type="bibr" rid="B11">11</xref>
                ]
              </td>
            </tr>
            <tr>
              <td>Charge noise</td>
              <td>
                <italic>S</italic>
                <italic>
                  <sub>Q</sub>
                </italic>
                (
                <italic>f</italic>
                ) →
                <italic>S</italic>
                <italic>
                  <sub>ω</sub>
                </italic>
                (
                <italic>f</italic>
                ) → Γ
                <italic>
                  <sub>φ</sub>
                </italic>
              </td>
              <td>
                1/
                <italic>f</italic>
                charge, suppressed in transmon
              </td>
            </tr>
            <tr>
              <td>Temperature</td>
              <td>
                <italic>T</italic>
                →
                <italic>n</italic>
                <italic>
                  <sub>th</sub>
                </italic>
                →
                <italic>p</italic>
                <italic>
                  <sub>e</sub>
                </italic>
                , Γ
                <italic>
                  <sub>th</sub>
                </italic>
              </td>
              <td>Direct thermal population (S4.2)</td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
    </sec>
    <sec id="sec6">
      <title>6. Manufacturing Economics</title>
      <p>All manufacturing figures now carry an explicit evidence class and are separated between prototype and volume estimates. The evidence classes are defined in <bold>Table 7</bold>.</p>
      <p><bold>Table 7</bold><bold>.</bold> Manufacturing evidence classes. No v2.0 row is labelled EXP because no STOC-QComp device has yet been fabricated.</p>
      <table-wrap id="tbl7">
        <label>Table 7</label>
        <table>
          <tbody>
            <tr>
              <td>
                <bold>Label</bold>
              </td>
              <td>
                <bold>Meaning</bold>
              </td>
              <td>
                <bold>Use in this paper</bold>
              </td>
            </tr>
            <tr>
              <td>VQ</td>
              <td>Vendor quotation (written)</td>
              <td>Rare—only where cited</td>
            </tr>
            <tr>
              <td>VE</td>
              <td>Vendor estimate (verbal/informal)</td>
              <td>L0 commodity materials</td>
            </tr>
            <tr>
              <td>LE</td>
              <td>Literature-derived</td>
              <td>Fab yields from published nodes</td>
            </tr>
            <tr>
              <td>IE</td>
              <td>Internal engineering estimate</td>
              <td>Most L1-L8 rows</td>
            </tr>
            <tr>
              <td>ET</td>
              <td>Experimental target (pilot-run goal)</td>
              <td>T22 pilot yields</td>
            </tr>
            <tr>
              <td>EXP</td>
              <td>Experimentally demonstrated</td>
              <td>Not used in v2.0 (no measured yields yet)</td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
      <sec id="sec6dot1">
        <title>6.1. Prototype vs Volume Cost</title>
        <p>The manufacturing cost per unit as a function of volume<italic>N</italic> is:</p>
        <disp-formula id="FD11">
          <label>(6.1)</label>
          <mml:math display="inline">
            <mml:mrow>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>unit</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mi>N</mml:mi>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:mo>=</mml:mo>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>materials</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>fab</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mrow>
                <mml:mo>(</mml:mo>
                <mml:mi>N</mml:mi>
                <mml:mo>)</mml:mo>
              </mml:mrow>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>assembly</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>cryo</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>QA</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>yield loss</mml:mtext>
                </mml:mrow>
              </mml:msub>
              <mml:mo>+</mml:mo>
              <mml:msub>
                <mml:mi>C</mml:mi>
                <mml:mrow>
                  <mml:mtext>overhead</mml:mtext>
                </mml:mrow>
              </mml:msub>
            </mml:mrow>
          </mml:math>
        </disp-formula>
        <p>with <italic>C</italic><sub>fab</sub>(<italic>N</italic>) exhibiting the usual experience-curve decay. This resolves the v1.0 inconsistency: the T2 BOM of $412,700 is a <bold>prototype/single-unit</bold>figure; the $150,000 low-end selling price is a <bold>volume</bold>figure that assumes both L1 - L7 cost reduction and a stripped-down configuration. The two figures apply at different points in the experience curve. The prototype selling price should be substantially higher than the low-end volume price.</p>
      </sec>
      <sec id="sec6dot2">
        <title>6.2. Serial Yield</title>
        <p>Multiplying the min and max of every stage in <bold>Table 8</bold> gives an overall serial yield estimate of <bold>3.31% (lower bound) to 39.83% (upper bound)</bold>. Under no reading of <bold>Table 8</bold> does the pipeline imply a finished-device yield above ~40%. The v1.0 impression that the pipeline could deliver &gt;90% finished yield is incorrect. This is a direct consequence of nine multiplicative yield stages and is tracked as verification test T22 (S7).</p>
        <p><bold>Table 8</bold><bold>.</bold> Nine-stage manufacturing pipeline with per-row evidence class. Yields are engineering estimates (IE) or literature-derived from analogous-process yields (LE), not measured yields of STOC-QComp units. Costs for L1 - L7 are prototype/single-unit estimates; volume manufacturing costs are expected to fall as described in S6.3.</p>
        <table-wrap id="tbl8">
          <label>Table 8</label>
          <table>
            <tbody>
              <tr>
                <td>
                  <bold>Stage</bold>
                </td>
                <td>
                  <bold>Process</bold>
                </td>
                <td>
                  <bold>Cost/unit (USD)</bold>
                </td>
                <td>
                  <bold>Yield (%)</bold>
                </td>
                <td>
                  <bold>Evidence</bold>
                </td>
              </tr>
              <tr>
                <td>L0 Mat</td>
                <td>
                  Bi
                  <sub>2</sub>
                  Te₃ · Au · SiN · Nb · NbTiN
                </td>
                <td>$1.2k - $18k</td>
                <td>99 - 100</td>
                <td>VE</td>
              </tr>
              <tr>
                <td>L1 Fab</td>
                <td>MBE nanowires · e-beam · sputter</td>
                <td>$14k - $85k</td>
                <td>60 - 90</td>
                <td>LE</td>
              </tr>
              <tr>
                <td>L2 Wafer</td>
                <td>300 mm ring array</td>
                <td>$8k - $45k</td>
                <td>70 - 92</td>
                <td>LE/IE</td>
              </tr>
              <tr>
                <td>L3 Pkg</td>
                <td>Flip-chip · TSV · shield</td>
                <td>$4k - $22k</td>
                <td>65 - 88</td>
                <td>IE</td>
              </tr>
              <tr>
                <td>L4 Ctrl</td>
                <td>RV64 + STOC + DAC/ADC</td>
                <td>$3k - $18k</td>
                <td>75 - 95</td>
                <td>IE</td>
              </tr>
              <tr>
                <td>L5 Cryo</td>
                <td>Dilution stage + shielding</td>
                <td>$120k - $850k</td>
                <td>80 - 95</td>
                <td>VE/IE</td>
              </tr>
              <tr>
                <td>L6 Casimir</td>
                <td>Au/SiN/Au + actuator</td>
                <td>$25k - $95k</td>
                <td>60 - 85</td>
                <td>IE (target)</td>
              </tr>
              <tr>
                <td>L7 QPU</td>
                <td>Josephson qubit array</td>
                <td>$28k - $180k</td>
                <td>40 - 75</td>
                <td>LE</td>
              </tr>
              <tr>
                <td>L8 Deploy</td>
                <td>Environmental hardening</td>
                <td>$8k - $600M+</td>
                <td>85 - 95</td>
                <td>IE</td>
              </tr>
            </tbody>
          </table>
        </table-wrap>
      </sec>
    </sec>
    <sec id="sec7">
      <title>7. Verification Tests · Statistical Analysis Plan</title>
      <p>v1.0 ST20 - T23 asserted “statistically significant” improvements without specifying a test, sample size, <italic>α</italic>, or effect size. V2.0 replaces these with a full statistical analysis plan. The four tests, primary endpoints, sample sizes, analysis methods, and success thresholds are given in <bold>Table 9</bold>.</p>
      <p><bold>Table 9.</bold> Verification tests with primary endpoint, sample size, analysis method, and success threshold. T22 explicitly acknowledges that the initial pilot is a feasibility study, not a statistically definitive production-yield demonstration.</p>
      <table-wrap id="tbl9">
        <label>Table 9</label>
        <table>
          <tbody>
            <tr>
              <td>
                <bold>Test</bold>
              </td>
              <td>
                <bold>Primary endpoint</bold>
              </td>
              <td>
                <bold>Sample size</bold>
              </td>
              <td>
                <bold>Analysis</bold>
              </td>
              <td>
                <bold>Threshold</bold>
              </td>
            </tr>
            <tr>
              <td>T20</td>
              <td>
                <italic>β</italic>
                <italic>
                  <sub>d</sub>
                </italic>
                = ∂Γ
                <sub>1</sub>
                /∂
                <italic>d</italic>
                and ∂
                <italic>S</italic>
                <sub>measured</sub>
                (
                <italic>f</italic>
                <italic>
                  <sub>q</sub>
                </italic>
                )/∂
                <italic>d</italic>
              </td>
              <td>
                <italic>n</italic>
                = 5 - 10 devices · 10 gap settings
              </td>
              <td>Linear regression</td>
              <td>95% CI excludes 0</td>
            </tr>
            <tr>
              <td>T21</td>
              <td>Two-qubit error per Clifford (EPC) by randomised benchmarking</td>
              <td>
                <italic>n</italic>
                = 10 QPUs/environment
              </td>
              <td>Mixed-effects model</td>
              <td>Pre-defined effect size + 95% CI</td>
            </tr>
            <tr>
              <td>T22</td>
              <td>
                Per-stage yield
                <italic>Y</italic>
                <italic>
                  <sub>i</sub>
                </italic>
                =
                <italic>k</italic>
                <italic>
                  <sub>i</sub>
                </italic>
                /
                <italic>n</italic>
                <italic>
                  <sub>i</sub>
                </italic>
              </td>
              <td>
                <italic>n</italic>
                <italic>
                  <sub>i</sub>
                </italic>
                ≥ 10 per stage (pilot)
              </td>
              <td>Binomial 95% CI</td>
              <td>Feasibility only, not production</td>
            </tr>
            <tr>
              <td>T23</td>
              <td>
                <italic>D</italic>
                <sub>KL</sub>
                (
                <italic>P</italic>
                <sub>measured</sub>
                ||
                <italic>P</italic>
                <sub>Lindblad</sub>
                ) and
                <italic>D</italic>
                <sub>TV</sub>
              </td>
              <td>100 identical control replays</td>
              <td>Held-out likelihood</td>
              <td>Pre-defined divergence bound</td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
      <p>T23 replaces v1.0’s Landauer-based reservoir-consistency claim (which does not follow from Landauer’s principle [<xref ref-type="bibr" rid="B12">12</xref>] as stated) with a predictive-distribution comparison: the primary criterion is whether predictive error decreases when experimentally measured reservoir parameters are substituted for nominal parameters. Landauer’s thermodynamic bound is retained in the philosophical discussion (S10) but is not the statistical criterion.</p>
    </sec>
    <sec id="sec8">
      <title>8. Illustrative Strategic Game</title>
      <p>v1.0 S9 stated a 4-player subgame-perfect equilibrium [<xref ref-type="bibr" rid="B13">13</xref>] under a single 4 × 4 payoff matrix. A subgame-perfect equilibrium requires an extensive-form specification with players, information sets, actions, order of moves, strategies, terminal histories, and payoff vectors—not a single matrix. V2.0 replaces this with a properly specified extensive-form game. Both the payoff values and the equilibrium selection are labelled illustrative. The extensive-form tree is shown in <xref ref-type="fig" rid="fig5">Figure 5</xref>.</p>
      <fig id="fig5">
        <label>Figure 5</label>
        <caption>
          <p>Extensive-form 4-stage sequential game. Stage 1: Regulator chooses open/closed. Stage 2: Incumbent chooses high/competitive price. Stage 3: Challenger chooses enter/stay out. Stage 4: User chooses incumbent/challenger. Terminal nodes carry payoff quadruples (u<sub>I</sub>, u<sub>C</sub>, u<sub>U</sub>, u<sub>R</sub>). The bold red path is the illustrative backward-induction solution under the stated payoff assumptions (open regulation → competitive incumbent → challenger enters → user picks challenger). The specific numerical payoffs are illustrative model assumptions, not empirical measurements. Replaces v1.0 <xref ref-type="fig" rid="fig5">Figure 5</xref> in its entirety.</p>
        </caption>
        <graphic xlink:href="https://html.scirp.org/file/4400413-rId49.jpeg?20261112113605" />
      </fig>
    </sec>
    <sec id="sec9">
      <title>9. Limitations</title>
      <p>Every load-bearing limitation of this manuscript is enumerated in <bold>Table 1</bold><bold>0</bold> below.</p>
      <p><bold>Table 10</bold><bold>.</bold> Explicit limitations enumeration. Every entry corresponds to a specific verification test or future experiment (S7 and S10).</p>
      <table-wrap id="tbl10">
        <label>Table 10</label>
        <table>
          <tbody>
            <tr>
              <td>
                <bold>#</bold>
              </td>
              <td>
                <bold>Limitation</bold>
              </td>
            </tr>
            <tr>
              <td>1</td>
              <td>No STOC-QComp device has been fabricated; all results are engineering specifications.</td>
            </tr>
            <tr>
              <td>2</td>
              <td>The Casimir-mediated modification of qubit coupling and dissipation is a testable HYPOTHESIS (T20), not an experimentally demonstrated effect.</td>
            </tr>
            <tr>
              <td>3</td>
              <td>
                The 200 - 1000 qubit bounded-capacity regime is a design hypothesis motivated by Palmer’s Rational Quantum Mechanics proposal [
                <xref ref-type="bibr" rid="B3">3</xref>
                ], not an established physical law.
              </td>
            </tr>
            <tr>
              <td>4</td>
              <td>T1 is a demonstrator/control platform, not a full QPU. The 4 K thermal population catastrophe prevents conventional 5 GHz transmon operation at that temperature.</td>
            </tr>
            <tr>
              <td>5</td>
              <td>Manufacturing yields are engineering estimates (IE) or literature-derived (LE), not measured yields of STOC-QComp units.</td>
            </tr>
            <tr>
              <td>6</td>
              <td>Serial yield lies between 3.3% and 39.8%; the pipeline does not imply a finished-device yield &gt; 40%.</td>
            </tr>
            <tr>
              <td>7</td>
              <td>
                The extensive-form game payoffs (
                <xref ref-type="fig" rid="fig5">Figure 5</xref>
                ) are illustrative model assumptions, not empirical measurements.
              </td>
            </tr>
            <tr>
              <td>8</td>
              <td>T4 - T6 tier feasibility depends on successful validation of T2 - T3; T4 - T6 are ROADMAP.</td>
            </tr>
            <tr>
              <td>9</td>
              <td>
                The Casimir non-ideal correction values (
                <bold>Table 3</bold>
                ) require a full Lifshitz calculation for the Au·SiN·Au multilayer, not performed in this manuscript.
              </td>
            </tr>
            <tr>
              <td>10</td>
              <td>
                Numerical values of ∂
                <italic>ω</italic>
                <italic>
                  <sub>c</sub>
                </italic>
                /∂
                <italic>d</italic>
                , ∂
                <italic>g</italic>
                /∂
                <italic>d</italic>
                , ∂
                <italic>κ</italic>
                /∂
                <italic>d</italic>
                , ∂Γ
                <sub>1</sub>
                /∂
                <italic>d</italic>
                are design variables to be measured, not simulation outputs of this paper.
              </td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
    </sec>
    <sec id="sec10">
      <title>10. Conclusions</title>
      <p>STOC-QComp v2.0 provides a device-level specification and experimental programme for investigating a mechanically tunable electromagnetic environment coupled to a bounded-capacity superconducting quantum processor. The present work does not establish the Casimir-mediated modification of qubit dissipation experimentally; rather, it identifies the physical parameters that must be measured to establish or falsify that hypothesis.</p>
      <p>The most immediate validation target is a single reference transmon QPU coupled to a mechanically tunable microwave cavity via a Casimir-force-loaded actuator. The relevant observables are the cavity response, mechanical displacement spectrum, qubit relaxation and dephasing rates, and gate fidelity as functions of d. Manufacturing yield, environmental fidelity improvements, and deployment economics are likewise treated as engineering hypotheses whose uncertainty will be reduced through prototype fabrication.</p>
      <p>If these experiments confirm the predicted coupling and environmental-control relationships, the STOC-QComp architecture provides a basis for subsequent scaling and deployment studies [<xref ref-type="bibr" rid="B14">14</xref>]. If they do not, the framework identifies specific falsification points at the Casimir, mechanical, microwave, or qubit-control layers.</p>
      <p>On measurement and interpretation: v1.0 described collapse as an “engineerable event” resolving “interpretational debate”. v2.0 uses the narrower, defensible language of an <italic>operationally engineered measurement environment</italic>: the device does not establish a new interpretation of quantum mechanics [<xref ref-type="bibr" rid="B15">15</xref>], but provides an experimentally controlled setting in which the relationship between deliberately engineered environmental coupling and measurement statistics can be investigated.</p>
    </sec>
    <sec id="sec11">
      <title>Research Status</title>
      <p>STOC-QComp is a device-framework proposal (DESIGN). The qubit architecture, control interfaces, manufacturing flow, and deployment concepts are engineering specifications; they have not yet been demonstrated as an integrated production device. The Casimir mechanical structure is DESIGN. Casimir-mediated modification of qubit coupling and dissipation is an experimentally testable HYPOTHESIS. T1 and T2 are prototype targets (DESIGN); T3 is a DESIGN target; T4-T6 are ROADMAP configurations contingent upon successful validation of the preceding tiers. This paper does not claim that a STOC-QComp device has been fabricated, characterised, or shipped.</p>
    </sec>
  </body>
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