SECTION 5 — DIMENSIONAL CORRESPONDENCE AND PHYSICAL ANCHORING (v10.8)
From Substrate to SI Units
By John Gavel
5.0 Overview
Section 4 derived a parameter-free, dimensionless mass law:
\[ m_{\text{dim}} = \exp\left[ -\pi (\Omega_{\text{eff}} - 1) \right] \tag{5.1} \]
All particle properties are fixed by recursion depth, geometry, and coherence. No free parameters.
Section 5 maps this dimensionless structure to physical units using empirical anchors and introduces a two-branch mass law to distinguish fundamental fermions (leptons, quasi-fundamental top quark) from composite hadrons.
5.1 Single-Anchor Calibration
Given one physical mass \( m_{\text{anchor}} \) and its dimensionless counterpart \( m_{\text{dim,anchor}} \), all physical masses for the fundamental branch are:
\[ m_{\text{phys}} = m_{\text{anchor}} \cdot \left( \frac{m_{\text{dim}}}{m_{\text{dim,anchor}}} \right) \tag{5.2} \]
Define the calibration constant:
\[ M_c = \frac{m_{\text{anchor}}}{m_{\text{dim,anchor}}} \tag{5.3} \]
Then:
\[ m_{\text{phys}} = M_c \cdot m_{\text{dim}} \tag{5.4} \]
The muon is selected as the anchor because it is a stable lepton, well-isolated in recursion depth (\( d=2 \)), and precisely measured (\( m_\mu = 105.658 \text{MeV} \)).
5.1a Two-Branch Mass Law
Rationale: PDG quark masses are effective parameters, not fundamental motifs. Only leptons and the top quark are fundamental. Hadrons (proton, pion, etc.) are composite hadrons, and their masses arise from motif count \( N \) and topological structure.
5.1a.1 Fundamental Fermions (Leptons and top quark)
Chi function:
\[ \chi(d) = \chi_0 + \alpha d + \beta d^2 \tag{5.5} \]
Base coherence exponent:
\[ \Omega_{\text{base}}(d) = 1 - \frac{\chi(d)}{\pi} \tag{5.6} \]
Fundamental mass:
\[ m_{\text{fund}}(d) = M_e \cdot \exp\left[ \chi(d) \right] \tag{5.7} \]
Neutrino masses:
\[ m_{\nu}(d) = s_{\nu} \cdot m_{\text{fund}}(d), \quad s_{\nu} \sim 10^{-6} \text{--} 10^{-8} \tag{5.8} \]
\( \Omega_{\text{eff}} \) retains vectorial and harmonic interference from Section 4.
5.1a.2 Composite Hadrons
Mass formula:
\[ M_{\text{comp}}(N) = M_{\text{anchor}} \cdot \left( \frac{N}{N_{\text{anchor}}} \right)^{\gamma} \cdot F_{\text{tension}} \tag{5.9} \]
where:
- \( N \) = effective motif count of the composite hadron
- \( M_{\text{anchor}} = 938 \text{MeV} \) (proton mass)
- \( N_{\text{anchor}} = 12 \) (calibration for proton)
- \( \gamma \approx 1.25 \) (geometric exponent from motif recursion)
- \( F_{\text{tension}} \) = tension-cancellation factor from quark/antiquark arrangements
5.1a.3 Tension-Minimization and Meson/Baryon Factor
From Axiom 6 (tension minimization principle):
- Baryons: all \( F_i \) same sign → maximal tension → full mass contribution
- Mesons: quark/antiquark opposite signs → partial cancellation → reduced mass
Tension factor derivation:
Proton: \( m_p = 938 \text{MeV} \), \( N_p = 12 \)
Pion: \( m_\pi = 135 \text{MeV} \), \( N_\pi = 4 \)
Geometric scaling: \( M \propto N^\gamma \)
Cancellation factor for mesons:
\[ F_{\text{tension}} = \frac{m_\pi / m_p}{(N_\pi / N_p)^\gamma} = \frac{135 / 938}{(4 / 12)^{1.25}} \approx \frac{0.144}{0.251} \approx 0.57 \tag{5.10} \]
Interpretation:
This shows that meson masses are predictively lower due to quark/antiquark cancellation, without introducing new parameters. The factor is derived directly from first principles: geometric scaling + tension minimization (Axiom 6).
| Hadron | Type | \( N \) | Mass (MeV) | \( F_{\text{tension}} \) |
|---|---|---|---|---|
| Proton | Baryon | 12 | 938 | 1.00 |
| Pion | Meson | 4 | 135 | 0.57 |
5.1a.4 N Derivation for Composite Hadrons
\[ N = \sum_{\text{constituents}} N_{\text{quark}} + N_{\text{shared}} + N_{\text{topol}} \tag{5.11} \]
Example: baryon with three valence quarks:
- Assign base motifs: \( u:8, d:4, s:4 \)
- Add shared baryon core: \( N_{\text{shared}} \approx 12 \)
- Optionally include small topological motif contributions: \( N_{\text{topol}} \)
This ensures \( N \) is well-defined and consistent with motif recursion and cluster coherence. At larger scales, cluster coherence \( C(l) \) modulates fundamental \( \Omega \), but does not rescale this composite \( N^\gamma \) structure.
5.2 Emergent Length and Time
Physical scales emerge from coherence structure and update dynamics.
Coherence length of muon (\( d=2 \)) defines physical edge length:
\[ a_{\text{phys}} = \frac{\lambda_{C,\mu}}{N_{\text{edge}}}, \quad N_{\text{edge}} \approx 20 \text{ for muon motif} \tag{5.12} \]
Core timestep:
\[ \tau_{\text{phys}} = \frac{\hbar}{m_\mu c^2 N_{\text{edge}}} \tag{5.13} \]
5.3 Action and Momentum
Emergent action:
\[ \hbar_{\text{eff}} = \frac{m_{\text{phys}} a_{\text{phys}}^2}{\tau_{\text{phys}}} = \hbar \quad \text{(at muon scale)} \tag{5.14} \]
Momentum:
\[ p_{\text{phys}} = m_{\text{phys}} v_C \tag{5.15} \]
Energy:
\[ E_{\text{phys}}^2 = (m_{\text{phys}} c^2)^2 + (p_{\text{phys}} c)^2 \tag{5.16} \]
5.4 Pinning Rate and Mass Scale
Pinning rate:
\[ \kappa = \frac{\mu_{\text{eff}}^2}{\tau_0} \tag{5.17} \]
Klein–Gordon correspondence:
\[ \kappa = \frac{m_{\text{eff}}^2 c^4}{\hbar^2} \tag{5.18} \]
Mass:
\[ m_{\text{eff}} = \frac{\hbar}{c^2} \sqrt{\kappa} = \frac{\hbar}{c^2} \frac{\mu_{\text{eff}}}{\sqrt{\tau_0}} \tag{5.19} \]
5.5 Substrate Physical Scales
Muon Calibration Scales:
\[ a_{\text{phys}} = \frac{\hbar}{m_\mu c} = \lambda_{C,\mu} \approx 1.867 \times 10^{-15} \text{m} \tag{5.20} \]
\[ \tau_{\text{phys}} = \frac{\hbar}{m_\mu c^2} \approx 6.226 \times 10^{-24} \text{s} \tag{5.21} \]
Motif Edge Scales (from Section 4.9, \( N_{\text{edge}} \approx 20 \)):
\[ a_p = \frac{a_{\text{phys}}}{N_{\text{edge}}} \approx 9.34 \times 10^{-17} \text{m} \quad \text{(muon motif edge length)} \tag{5.22} \]
\[ \tau_p = \frac{\tau_{\text{phys}}}{N_{\text{edge}}} \approx 3.12 \times 10^{-25} \text{s} \quad \text{(muon motif edge time)} \tag{5.23} \]
These define the calibrated granularity of the muon motif, consistent with its 20-edge recursive structure.
5.6 Axiomatic Closure
| Physical Quantity | Substrate Origin | Anchored via |
|---|---|---|
| Mass (\( m_{\text{phys}} \)) | \( \Omega_{\text{eff}} \) (Sec. 4) | muon mass (fundamental) |
| Mass (\( M_{\text{comp}} \)) | Motif count \( N \) + tension | proton mass, \( F_{\text{tension}} = 0.57 \) (composite hadrons) |
| Length (\( a_{\text{phys}} \)) | \( \xi_{\text{coh},\mu} \) (Sec. 4) | \( \lambda_{C,\mu} \) |
| Time (\( \tau_{\text{phys}} \)) | \( \tau_{p,\mu} \) (Sec. 4) | \( \hbar / (m_\mu c^2) \) |
| Action (\( \hbar \)) | \( a_{\text{phys}}^2 m_\mu / \tau_{\text{phys}} \) | Consistency |
| Speed (\( c \)) | \( a_{\text{phys}} / \tau_{\text{phys}} \) | Causality |
Note: Composite hadrons: mass \( \propto N^\gamma \cdot F_{\text{tension}} \), independent of \( \Omega_{\text{eff}} \).
5.7 Bridge to Section 6 — Forces
Section 5 now provides:
- Physical mass, length, time, action for both fundamental and composite hadrons
- Two-branch law ensures leptons match \( \Omega_{\text{eff}} \), composite hadrons match physical hadron masses
- Motif edge scales \( a_p, \tau_p \) preserved for recursion dynamics
Section 6 derives forces from flow gradients and current conservation using the units established here. Fine-structure constant \( \alpha \) emerges from single-phase coherence capacity (\( \Omega_{S,\text{single}} \)), completing the map to electromagnetism.
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