A τ–ε Phase Framework Across Physical Regimes
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2026
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| author | Thomas Gøtzsche |
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| contents | <p>This paper introduces a regime-dependent framework based on the quantities <span class="katex"><span class="katex-mathml">τ\tau</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span></span></span></span> and <span class="katex"><span class="katex-mathml">ϵ\epsilon</span><span class="katex-html"><span class="base"><span class="mord mathnormal">ϵ</span></span></span></span>, providing a unified structural interpretation of phase evolution across multiple domains of physics. Rather than proposing a new fundamental theory, the framework reorganizes established physical effects in terms of phase accumulation and deviation fields, while remaining consistent with existing theoretical descriptions.</p> <p>A central feature of the formulation is the interpretation of time as a quantity derived from phase accumulation, expressed operationally as <span class="katex"><span class="katex-mathml">dT=dϕ/ω0dT = d\phi / \omega_0</span><span class="katex-html"><span class="base"><span class="mord mathnormal">d</span><span class="mord mathnormal">T</span><span class="mrel">=</span></span><span class="base"><span class="mord mathnormal">d</span><span class="mord mathnormal">ϕ</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">ω</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></span></span>. Within this perspective, differences in observed temporal behavior arise from differences in accumulated phase, governed locally by a co-scaling parameter <span class="katex"><span class="katex-mathml">τ=1+ϵ\tau = 1 + \epsilon</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span><span class="mrel">=</span></span><span class="base"><span class="mord">1</span><span class="mbin">+</span></span><span class="base"><span class="mord mathnormal">ϵ</span></span></span></span>. The deviation field <span class="katex"><span class="katex-mathml">ϵ\epsilon</span><span class="katex-html"><span class="base"><span class="mord mathnormal">ϵ</span></span></span></span> is decomposed into regime-dependent components, including gravitational and kinematic contributions (<span class="katex"><span class="katex-mathml">ϵgrav,ϵkin\epsilon_{\text{grav}}, \epsilon_{\text{kin}}</span><span class="katex-html"><span class="base"><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 mtight"><span class="mord text mtight">grav</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</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 mtight"><span class="mord text mtight">kin</span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>) and path-dependent contributions (<span class="katex"><span class="katex-mathml">ϵpath\epsilon_{\text{path}}</span><span class="katex-html"><span class="base"><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 mtight"><span class="mord text mtight">path</span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>).</p> <p>The framework is applied consistently across several physical regimes:</p> <ul> <li>In relativistic systems, <span class="katex"><span class="katex-mathml">τ\tau</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span></span></span></span> describes local co-scaling of physical processes and provides a phase-based interpretation of time dilation.</li> <li>In propagation-dominated systems such as GNSS, <span class="katex"><span class="katex-mathml">ϵpath\epsilon_{\text{path}}</span><span class="katex-html"><span class="base"><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 mtight"><span class="mord text mtight">path</span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span> accounts for signal delay and distortion, while local <span class="katex"><span class="katex-mathml">τ\tau</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span></span></span></span> governs intrinsic clock behavior.</li> <li>In extended quantum systems, including giant atoms and giant superatoms, path-dependent phase contributions enter dynamically, giving rise to interference, delayed feedback, and effective non-Markovian behavior.</li> <li>In cosmological contexts, <span class="katex"><span class="katex-mathml">τ(t)\tau(t)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> is interpreted as a global evolution parameter, leading to a Friedmann-class description of large-scale phase dynamics.</li> </ul> <p>The framework emphasizes a clear separation between intrinsic phase evolution and propagation-induced phase accumulation, allowing diverse phenomena—relativistic time dilation, signal propagation effects, quantum coherence, and large-scale evolution—to be expressed within a common phase-based structure.</p> <p>A worked GNSS example demonstrates how standard relativistic clock corrections (on the order of 38.5 microseconds per day) arise directly from differences in phase accumulation rates, providing a concrete link between the framework and operational systems.</p> <p>This work is interpretative in scope and does not introduce new dynamical equations or empirical predictions. Instead, it provides a consistent conceptual language that connects multiple physical regimes through phase accumulation and co-scaling, and serves as a foundation for future development of dynamical formulations.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19650610 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A τ–ε Phase Framework Across Physical Regimes Thomas Gøtzsche phase evolution time from phase relativistic time dilation GNSS timing quantum coherence decoherence non-Markovian dynamics Physical cosmology cosmology <p>This paper introduces a regime-dependent framework based on the quantities <span class="katex"><span class="katex-mathml">τ\tau</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span></span></span></span> and <span class="katex"><span class="katex-mathml">ϵ\epsilon</span><span class="katex-html"><span class="base"><span class="mord mathnormal">ϵ</span></span></span></span>, providing a unified structural interpretation of phase evolution across multiple domains of physics. Rather than proposing a new fundamental theory, the framework reorganizes established physical effects in terms of phase accumulation and deviation fields, while remaining consistent with existing theoretical descriptions.</p> <p>A central feature of the formulation is the interpretation of time as a quantity derived from phase accumulation, expressed operationally as <span class="katex"><span class="katex-mathml">dT=dϕ/ω0dT = d\phi / \omega_0</span><span class="katex-html"><span class="base"><span class="mord mathnormal">d</span><span class="mord mathnormal">T</span><span class="mrel">=</span></span><span class="base"><span class="mord mathnormal">d</span><span class="mord mathnormal">ϕ</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">ω</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></span></span>. Within this perspective, differences in observed temporal behavior arise from differences in accumulated phase, governed locally by a co-scaling parameter <span class="katex"><span class="katex-mathml">τ=1+ϵ\tau = 1 + \epsilon</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span><span class="mrel">=</span></span><span class="base"><span class="mord">1</span><span class="mbin">+</span></span><span class="base"><span class="mord mathnormal">ϵ</span></span></span></span>. The deviation field <span class="katex"><span class="katex-mathml">ϵ\epsilon</span><span class="katex-html"><span class="base"><span class="mord mathnormal">ϵ</span></span></span></span> is decomposed into regime-dependent components, including gravitational and kinematic contributions (<span class="katex"><span class="katex-mathml">ϵgrav,ϵkin\epsilon_{\text{grav}}, \epsilon_{\text{kin}}</span><span class="katex-html"><span class="base"><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 mtight"><span class="mord text mtight">grav</span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</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 mtight"><span class="mord text mtight">kin</span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>) and path-dependent contributions (<span class="katex"><span class="katex-mathml">ϵpath\epsilon_{\text{path}}</span><span class="katex-html"><span class="base"><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 mtight"><span class="mord text mtight">path</span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>).</p> <p>The framework is applied consistently across several physical regimes:</p> <ul> <li>In relativistic systems, <span class="katex"><span class="katex-mathml">τ\tau</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span></span></span></span> describes local co-scaling of physical processes and provides a phase-based interpretation of time dilation.</li> <li>In propagation-dominated systems such as GNSS, <span class="katex"><span class="katex-mathml">ϵpath\epsilon_{\text{path}}</span><span class="katex-html"><span class="base"><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 mtight"><span class="mord text mtight">path</span></span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span> accounts for signal delay and distortion, while local <span class="katex"><span class="katex-mathml">τ\tau</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span></span></span></span> governs intrinsic clock behavior.</li> <li>In extended quantum systems, including giant atoms and giant superatoms, path-dependent phase contributions enter dynamically, giving rise to interference, delayed feedback, and effective non-Markovian behavior.</li> <li>In cosmological contexts, <span class="katex"><span class="katex-mathml">τ(t)\tau(t)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">τ</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> is interpreted as a global evolution parameter, leading to a Friedmann-class description of large-scale phase dynamics.</li> </ul> <p>The framework emphasizes a clear separation between intrinsic phase evolution and propagation-induced phase accumulation, allowing diverse phenomena—relativistic time dilation, signal propagation effects, quantum coherence, and large-scale evolution—to be expressed within a common phase-based structure.</p> <p>A worked GNSS example demonstrates how standard relativistic clock corrections (on the order of 38.5 microseconds per day) arise directly from differences in phase accumulation rates, providing a concrete link between the framework and operational systems.</p> <p>This work is interpretative in scope and does not introduce new dynamical equations or empirical predictions. Instead, it provides a consistent conceptual language that connects multiple physical regimes through phase accumulation and co-scaling, and serves as a foundation for future development of dynamical formulations.</p> |
| title | A τ–ε Phase Framework Across Physical Regimes |
| topic | phase evolution time from phase relativistic time dilation GNSS timing quantum coherence decoherence non-Markovian dynamics Physical cosmology cosmology |
| url | https://doi.org/10.5281/zenodo.19650610 |