<h1>Space: From Atomic Bonds to Cosmic Structures</h1> <p><strong>Author:</strong> Jürgen Wollbold<br> <strong>Date:</strong> August 30, 2025</p> <h2>Abstract</h2> <p> We propose a minimal geometric model of space–time—the Methane Metauniverse (MMU)—in which physics emerges from a tetrahedral lattice with four orthogonal degrees (<em>w1</em>, <em>w2</em>, <em>w3</em>, <em>w4</em>). The single observational rule is: an external observer measures only the projection onto <em>w1</em>. Energy stored in (<em>w2</em>, <em>w3</em>, <em>w4</em>) lowers this projection and appears as time dilation. From this rule, Maxwell transport along lattice edges (light at speed <em>c</em> and a fixed edge impedance <em>Z0</em>), continuum elasticity consistent with general relativity (GR), and quantum envelope equations follow without introducing new free scales. </p> <p> We demonstrate: (i) internal spring constants and the 21&nbsp;cm hyperfine scale from a one-step estimate (Appendix&nbsp;2); (ii) a clean GR&#8594;Newton limit via the projection/clock-rate factor &zeta; = d&tau;/dt (Appendix&nbsp;6); (iii) light as a pure <em>w2–w3</em> oscillation with <em>Z_0,int = (P₃/P₂)/&radic;(2&alpha;)</em> and its SI calibration (Appendix&nbsp;3); (iv) Schrödinger as the one-band and Dirac as the two-band projection of internal oscillations (Appendix&nbsp;8), including a geometric derivation of Laguerre radial modes from the proton’s hexagonal “chair” (Appendix&nbsp;7); (v) isotope shifts (H/D/T) and the muonic scaling pattern, plus decay as S-node barrier flips with Eyring–Kramers statistics (Appendix&nbsp;11); (vi) quark triads (fractional charge, color, confinement) and the neutron as either (u,d,d) or a (p + e&minus;)S composite with the same energy bookkeeping (Appendix&nbsp;10); and (vii) lepton staging (e, &mu;, &tau;), Koide as a geometry identity, hydrogenic lines across e&#8594;&mu;&#8594;&tau;, and parameter-free (g &minus; 2) family ratios (Appendix&nbsp;8/11). </p> <p> The model yields compact formulas, parameter transparency, and multiple falsifiable checks (impedance, mode speeds, spectral ratios). The appendices contain full derivations; the main text keeps the logic simple enough for a broad physics readership. </p> <hr> <h2>1. Introduction</h2> <p> The MMU postulates a fixed tetrahedral lattice for space–time with four orthogonal internal directions. Observable time is the projection length along <em>w1</em>; internal energy is the squared amplitude stored in (<em>w2</em>, <em>w3</em>, <em>w4</em>). When internal energy grows, the observable clock slows by a factor &zeta; &in; (0,1]. This single projection rule replaces a large set of separate axioms (e.g., distinct postulates for SR/GR, electrodynamics, and quantum kinematics) by a common geometric mechanism. </p> <h2>2. Core Postulates</h2> <ul> <li><strong>Tetra lattice:</strong> Space–time is a homogeneous tetrahedral network; each node has four orthogonal degrees (<em>w1…w4</em>).</li> <li><strong>Projection rule:</strong> Any process with internal energy <em>E_int</em> reduces the measured proper-time rate via &zeta; = d&tau;/dt, with &zeta; a function of the internal amplitudes.</li> <li><strong>Edge transport:</strong> Signals propagate along edges at a fixed causal speed <em>c</em> with a characteristic edge impedance <em>Z0</em>.</li> <li><strong>Elastic continuum limit:</strong> On coarse graining, the lattice behaves like an elastic medium with modulus and metric linked by the projection factor &zeta;.</li> <li><strong>Envelope dynamics:</strong> Slow external envelopes of fast internal oscillations obey Schrödinger (one-band) or Dirac (two-band) equations.</li> </ul> <h2>3. Geometry at a Glance</h2> <ul> <li><em>w1</em>: observable time axis (clock projection).</li> <li><em>w2</em>: electric/energetic displacement.</li> <li><em>w3</em>: magnetic/spin-related displacement (torsion partner of <em>w2</em>).</li> <li><em>w4</em>: torsional storage (mass channel, barrier heights, S-node anchoring).</li> </ul> <p>Light corresponds to transverse oscillations in (<em>w2</em>, <em>w3</em>) with negligible <em>w4</em> projection; massive excitations contain persistent <em>w4</em> content (nonzero rest mass).</p> <h2>4. Light and Edge Impedance</h2> <p> On each edge, the pair (<em>w2</em>, <em>w3</em>) supports a transverse mode with group velocity <em>c</em> and impedance <em>Z0</em>. Matching to SI gives the internal impedance: </p> <p><strong>Z_0,int = (P₃/P₂) / &radic;(2&alpha;)</strong></p> <p> Here, <em>P₂</em> and <em>P₃</em> are normalized edge powers carried by the <em>w2</em> and <em>w3</em> components, and &alpha; is the fine-structure constant. Calibration aligns <em>Z_0,int</em> with the vacuum wave impedance while keeping lattice units natural (Appendix&nbsp;3). </p> <h2>5. Continuum Elasticity and GR Link</h2> <p> Coarse graining maps the lattice to an elastic medium with stress–strain relation &sigma; = E &epsilon;. The projection factor &zeta; simultaneously rescales local clock rate and effective metric, producing the GR redshift and time-dilation relations in the weak-field limit. Taking &zeta; &rarr; 1 with small gradients recovers Newtonian gravity with the correct Poisson limit (Appendix&nbsp;6). </p> <h2>6. Quantum Envelopes: Schrödinger and Dirac</h2> <p> Fast internal oscillations in (<em>w2</em>, <em>w3</em>, <em>w4</em>) admit slowly varying envelopes. A single active band yields the Schrödinger equation for the envelope amplitude; two coupled bands yield a Dirac-type system with spinor structure emerging from the (<em>w2</em>, <em>w3</em>) torsion pairing (Appendix&nbsp;8). The Coulomb problem maps to discrete lattice symmetries around the proton’s “hexagonal chair,” producing Laguerre radial modes without additional scales (Appendix&nbsp;7). </p> <h2>7. One-Step 21&nbsp;cm Estimate</h2> <p> Using a single internal spring-constant ratio and the proton–electron torsion coupling, the hyperfine 21&nbsp;cm scale follows within the envelope approach. The calculation fixes a specific combination of lattice moduli and torsion leverage, providing a tight check on the MMU elasticity (Appendix&nbsp;2). </p> <h2>8. Isotope Shifts and Muonic Scaling</h2> <p> Replacing the proton mass by an effective anchored mass changes the projection geometry and thus the spectral lines. The H/D/T pattern and the muonic hydrogen shift emerge from the same geometric rescaling of internal lever arms and <em>w4</em> storage. The model predicts cross-isotope ratios and the e&#8594;&mu; scaling without new parameters (Appendix&nbsp;11). </p> <h2>9. Leptons and Koide Geometry</h2> <p> Electrons, muons, and taus are stages of S-node embedding depth: increasing torsional storage (<em>w4</em>) and modified (<em>w2</em>, <em>w3</em>) leverage raise the effective mass. The Koide relation appears as a geometric identity of three embedding radii on the same tetra fabric. Hydrogenic lines mapped across e&#8594;&mu;&#8594;&tau; share the same lattice constants, shifting only by the depth factors (Appendix&nbsp;8/11). </p> <h2>10. Quark Triads and the Neutron</h2> <p> Triads correspond to constrained charge fractions on three edges meeting at a node (color as edge assignment). Confinement is geometric: separating an edge-triad tears the lattice, costing linearly rising energy. The neutron can be represented as (u,d,d) or as a (p + e&minus;)S composite where the S-node stores equivalent torsion energy; both share the same bookkeeping and external observables (Appendix&nbsp;10). </p> <h2>11. Decays as S-Node Barrier Flips</h2> <p> Metastable states correspond to torsional wells separated by barriers on <em>w4</em>. Thermal/quantum activation follows Eyring–Kramers statistics. Family-specific lifetimes and branching patterns arise from geometry of the local chair and coupling to neighboring nodes (Appendix&nbsp;11). </p> <h2>12. Falsifiable Checks</h2> <ul> <li><strong>Impedance:</strong> <em>Z_0,int</em> must match the calibrated vacuum impedance within stated lattice tolerances.</li> <li><strong>Mode speeds:</strong> Transverse (<em>w2</em>, <em>w3</em>) modes propagate at <em>c</em>; mixed modes have predicted sub-<em>c</em> group velocities set by torsion ratios.</li> <li><strong>Spectral ratios:</strong> 21&nbsp;cm estimate, isotope-shift ratios, and e&#8594;&mu;&#8594;&tau; hydrogenic line rescalings.</li> <li><strong>Envelope universality:</strong> One-band Schrödinger and two-band Dirac forms with fixed lattice parameters.</li> </ul> <h2>13. Notation</h2> <table border="1" cellpadding="6" cellspacing="0"> <thead> <tr><th>Symbol</th><th>Meaning</th></tr> </thead> <tbody> <tr><td><em>w1</em></td><td>Observable time projection</td></tr> <tr><td><em>w2</em></td><td>Electric/energetic displacement</td></tr> <tr><td><em>w3</em></td><td>Magnetic/spin torsion partner</td></tr> <tr><td><em>w4</em></td><td>Torsional storage (mass channel)</td></tr> <tr><td>&zeta;</td><td>Clock-rate factor d&tau;/dt</td></tr> <tr><td><em>Z_0,int</em></td><td>Internal edge impedance</td></tr> </tbody> </table> <h2>14. Roadmap to Appendices (Full Derivations)</h2> <ul> <li>Appendix 1: Lattice kinematics and normalization.</li> <li>Appendix 2: Internal spring constants and 21&nbsp;cm estimate.</li> <li>Appendix 3: Edge transport, <em>Z_0,int</em>, and SI calibration.</li> <li>Appendix 4: Continuum elasticity of the tetra lattice.</li> <li>Appendix 5: Projection factor &zeta; and local metric.</li> <li>Appendix 6: GR&#8594;Newton limit via &zeta;.</li> <li>Appendix 7: Hexagonal chair and Laguerre modes.</li> <li>Appendix 8: One-band Schrödinger and two-band Dirac envelopes.</li> <li>Appendix 9: Time-dilation tests and redshift mapping.</li> <li>Appendix 10: Quark triads, color, and neutron models.</li> <li>Appendix 11: Isotope shifts, muonic scaling, and decay statistics.</li> </ul> <h2>15. Acknowledgments</h2> <p> Thanks to collaborators and readers who challenged the MMU postulates and helped sharpen the falsifiable consequences. Any errors are mine. </p> <h2>16. How to Read This Page</h2> <p> The main text states only the minimal assumptions and their direct physical consequences. For proofs, parameter identifications, and numerical checks, follow the numbered appendices. All quantities are defined above; equations are kept in simple inline form for maximum compatibility with OSF’s wiki renderer. </p> </body> </html>