Constraint Summary
Free Parameters
The framework has 10 nominal free parameters. However, the two-scale model determines most of them from measured constants, leaving only 3 genuinely free (M_d, n_d, \delta), none of which appear in any current prediction:
| Parameter | Symbol | Status | Description |
|---|---|---|---|
| dc1 mass | m_1 | DETERMINED: \hbar/(c \xi) \approx 2 meV/c^2 | Mass of the light substrate particle |
| dag mass | M_d | FREE | Mass of the heavy orbital center particle |
| dc1 number density | n_1 | DETERMINED: \rho_{DM}/m_1 \approx 6.6 \times 10^{11} m^{-3} | Number of dc1 particles per unit volume |
| dag number density | n_d | FREE | Number of dag orbital centers per unit volume |
| Orbital angular velocity | \omega_0 | DETERMINED: Kc/(f\xi) from f\omega_0\xi/c = K ≈ 7.8 \times 10^9 rad/s | Outer-scale lattice rotation rate |
| Coherence length | \xi | DETERMINED: (\hbar/(\rho_{DM} c))^{1/4} \approx 110\;\mum | Soliton radius (≡ old \lambda_\text{orbital}) |
| Mutual friction parameter | \alpha_{mf} | MEASURED: = \sin^2\theta_W/(1-\sin^2\theta_W) = 0.3008 | Coupling between co- and counter-rotating layers |
| Boundary crossing fraction | f_\text{cross} | DETERMINED: 4\pi G/v_\text{rot,outer} \approx 10^{-15} ⚠️a | Fraction of dc1 transiting boundaries |
| Universe decay constant | \delta | FREE | Energy loss per elastic collision ({\sim}\,10^{-40}) |
| Disequilibrium fraction | \delta T/T_c | DETERMINED from \Lambda | Departure from substrate equilibrium |
Derived quantities:
| Symbol | Expression | Value | Notes |
|---|---|---|---|
| m_\text{eff} | m_e/\alpha_{mf} | 1.70 MeV/c^2 | Effective quantum mass |
| \nu | m_\text{eff}/m_1 | 8.3 \times 10^8 | Condensation number (dc1 per effective quantum) |
| v_\text{rot,inner} | c\sqrt{2\alpha_{mf}} | 0.776\,c | Inner-scale orbital velocity |
| v_\text{rot,outer} | \omega_0 \xi | 0.0025\,c | Outer-scale rotation (Landau critical velocity) |
| r_\text{eff} | \hbar/(m_\text{eff} v_\text{rot,inner}) | 150 fm | Inner orbital radius |
| \kappa_q | h/m_\text{eff} | — | Quantum of circulation |
| E_\text{min} | hc/\xi = 2\pi m_1 c^2 | 13 meV | Minimum modon (photon) energy |
a The simplified form f_\text{cross} = 4\pi G/v_\text{rot,outer} has dimensions [m²/(kg·s)], not [1] as required for a dimensionless fraction. The numerical value \approx 10^{-15} is correct in SI. The full derivation in Gravity involves density and area factors that absorb the missing dimensions; the simplified form is a numerical recipe pending verification.
Several constraint equations in this document use 3D formulations (n_1\omega_0\xi^3 = Kc, \kappa_q \cdot n_1\omega_0 = 4\pi c^2, \xi_\text{SC2}^3 = \hbar K\alpha_{mf}/(2m_ec)) that give correct numerical values in SI but have dimensional mismatches, confirmed by CGS cross-check. These are marked with ⚠️ throughout. The root cause: the Larichev-Reznik modon matching and Feynman relation are 2D formalisms, but the substrate’s vortex lattice has 3D structure (chirality-coherent sheets with inter-sheet spacing h \ll \xi). The correct 3D→2D projection requires the inter-sheet spacing, which is determined by the same chirality ordering thermodynamics needed to derive the Higgs VEV — an open problem (see Higgs Field). The dimensionless packing-fraction form of the bridge equation (f = \rho_\text{DM}c\xi^4/\hbar = 4\pi/(K\sqrt{2})) and all results derived from dimensionally clean equations (C1a, C10, close-packing) are unaffected. Formulas marked ⚠️ should be treated as numerical recipes valid in SI.
Observed Constants That Constrain the Parameters
Each constraint matches a measured quantity to a function of the 10 substrate parameters. The framework must reproduce all of them simultaneously.
Particle Physics Constraints (C1–C9)
These have explicit constraint equations linking observed constants to substrate parameters.
C1. Speed of light c = 2.998 \times 10^8 m/s → constrains m_1, \xi
(a) Primary — Volovik quasiparticle speed (Emergent Speed of Light):
c = \frac{\hbar}{m_1 \cdot \xi}
In the BEC regime, the quasiparticle spectrum is automatically Dirac-like with single isotropic speed c. Combined with C10 and close-packing:
\xi = \left(\frac{\hbar}{\rho_{DM} \cdot c}\right)^{1/4}, \qquad m_1 = \left(\frac{\hbar^3 \rho_{DM}^3}{c^3}\right)^{1/4}
(b) Modon existence condition (Emergent Speed of Light):
No longer determines c; instead determines \omega_0 \approx 7.8 \times 10^9 rad/s (outer-scale lattice rotation). In dimensionless form (recommended):
f \cdot \frac{\omega_0\,\xi}{c} = K \qquad\text{both sides [1] ✓}
where f = n_1\xi^3 = 4\pi/(K\sqrt{2}) is the packing fraction and K = j_{11}^2 + 1 = 15.67. Equivalently \omega_0 = Kc/(f\xi).
⚠️ The 3D form n_1\omega_0\xi^3 = Kc gives the same numerical value but has LHS [s⁻¹] and RHS [m/s] — a dimensional mismatch of [m]. The root cause: n_1 is a 3D number density [m⁻³] where the L-R modon matching requires a 2D vorticity gradient [m⁻¹s⁻¹]. The naive 2D repair (replacing n_1 with n_v^{(2D)} = 1/(\pi\xi^2)) gives \omega_0 \sim 10^{13} rad/s, approximately 1,800× too large — it misses the layered stacking structure of chirality-coherent sheets (see Higgs Field). The dimensionless form above avoids this issue entirely.
C2. Planck’s constant \hbar = 1.055 \times 10^{-34} J·s → constrains m_\text{eff}, \alpha_{mf}, n_1, \sigma
From superfluid mutual friction diffusivity (Two Fluids → Quantum Potential):
m_\text{eff} \cdot \alpha_{mf} = m_\text{particle}
D_\text{substrate} = \frac{v_\text{rot,inner}}{3\,n_1\,\sigma} = \frac{\hbar}{2m}
For the electron: m_\text{eff} \cdot \alpha_{mf} = m_e = 9.109 \times 10^{-31} kg
C3. Gravitational constant G = 6.674 \times 10^{-11} m³/(kg·s²) → constrains f_\text{cross}, v_\text{rot,outer}
From the boundary-layer ebbing current (Gravity):
G = \frac{f_\text{leak} \cdot n_1 \cdot m_1 \cdot v_\text{drift} \cdot A_\text{boundary}}{M \cdot m / r^2}
⚠️ The simplified form G = f_\text{cross} \cdot v_\text{rot,outer}/(4\pi) gives f_\text{cross} \approx 1.1 \times 10^{-15} (SI) but has dimensions [m²/(kg·s)], not [1]. See dimensional status note above.
C4. Electron mass m_e = 9.109 \times 10^{-31} kg → constrains effective quantum structure
From the two-scale energy budget (Mass as Orbital Energy):
m_e \cdot c^2 = \frac{1}{2}\,m_\text{eff}\,v_\text{rot,inner}^2, \qquad v_\text{rot,inner} = c\sqrt{2\alpha_{mf}}
One effective quantum (m_\text{eff} = m_e/\alpha_{mf} = 1.70 MeV/c^2) orbiting at 0.776\,c with angular momentum \hbar at radius r_\text{eff} = 150 fm.
C5. Proton mass m_p = 1.673 \times 10^{-27} kg → constrains nuclear orbital complex
From the interlocking figure-8 orbital system energy (Mass as Orbital Energy):
m_p \cdot c^2 = \sum(\text{quark orbital system energies}) + E_\text{gluon\_analog} + E_\text{boundary\_layers}
C6. Fine structure constant \alpha = 1/137.036 → DERIVED from C8 (zero new parameters)
From the six-stage derivation chain (see Fine Structure Constant): topology (half-quantum vortex) → geometry (Berry connection) → dynamics (Kopnin Breit-Wigner: \alpha_{mf} = \tfrac{1}{2}\sin 2\delta_0) → algebra (g^2 = 4\sin^2\delta_0) → identification (\alpha = g^2\sin^2\theta_W/(4\pi)). Result: \alpha_\text{tree} = 1/135.1 (+1.45% from measured). This makes \alpha a prediction, not a constraint — the system gains SC5 (the three-constant relation) as a zero-parameter test. The +1.45% gap is consistent with missing vacuum polarization (one-loop correction).
C7. Cosmological constant \Lambda = 1.1 \times 10^{-52} m^{-2} → constrains \delta T/T_c
From Volovik self-tuning plus disequilibrium (Gravity):
\rho_\Lambda = \rho_\text{substrate} \cdot (\delta T / T_c)^2
\delta T / T_c = \sqrt{\frac{\Lambda\,c^2}{8\pi G\,\rho_\text{substrate}}} \approx 10^{-61.5}
C8. Weinberg angle \sin^2\theta_W = 0.2312 → constrains scattering phase shift \delta_0
From the Iordanskii-Sonin-Stone scattering theory applied to dc1 scattering off quantized vortex lines (Weinberg Angle):
\sin^2\theta_W = f(\alpha_{mf},\;\text{vortex core structure})
C9. Anomalous magnetic moment (g-2)/2 = 0.00116 → constrains core-boundary asymmetry \eta
From the dual-spin gyroscope model (Spin-Statistics):
(g-2)/2 = \eta^2, \quad\text{where}\quad \eta = \sqrt{\alpha/2\pi} \approx 0.034
The core and boundary moments of inertia differ by ~3.4%.
Cosmological Constraints (C10–C13)
These match observed cosmological quantities. The constraint equations are more complex — they require computing the superfluid phase transition dynamics and expansion history from the substrate parameters — but the observed values must be reproduced.
C10. Dark matter density \rho_{DM} = 2.4 \times 10^{-27} kg/m³ → constrains n_1 \cdot m_1
The dc1/dag substrate IS the dark matter. Its observed density directly constrains the product of dc1 number density and mass:
n_1 \cdot m_1 + n_d \cdot M_d \approx 2.4 \times 10^{-27} \;\text{kg/m}^3
Since n_1 m_1 \gg n_d M_d (dc1 dominates by number density):
n_1 \cdot m_1 \approx 2.4 \times 10^{-27} \;\text{kg/m}^3
C11. CMB perturbation amplitude A_s = 2.1 \times 10^{-9} → constrains transition energy scale
From the superfluid phase transition:
A_s = \frac{H_\text{inf}^2}{8\pi^2\,M_{Pl}^2\,c^4\,\varepsilon_s}
where H_\text{inf} = \sqrt{8\pi G\,\rho_\text{latent}/3} is the expansion rate during the transition, \varepsilon_s = c_s^2/c^2 is the suppressed sound speed ratio, and \rho_\text{latent} is the latent heat density. This constrains the combination \rho_\text{latent}/\varepsilon_s, fixing the transition energy scale at E_\text{transition} \sim 10^{15}–10^{16} GeV.
C12. CMB spectral index n_s = 0.965 \pm 0.004 → constrains N_* (e-foldings at pivot scale)
From sound-speed inflation dynamics:
n_s - 1 \approx -2\varepsilon_H - \varepsilon_H - s \approx -\frac{3}{2N_*} - s
where \varepsilon_H = 1/(2N_*) and s = \dot{c}_s/(H c_s). For N_* \approx 60: n_s \approx 0.968. The number of e-foldings is set by the nucleation barrier S_0, which depends on the dc1/dag binding energy and transition thermodynamics.
C13. BAO sound horizon r_s = 147.09 \pm 0.26 Mpc → constrains sound speed evolution
The baryon acoustic oscillation scale is the integrated sound horizon at recombination:
r_s = \int_0^{t_\text{rec}} \frac{c_s(t)}{a(t)}\,dt
In the substrate, c_s is determined by the baryon-modon coupling, which depends on the substrate parameters through the post-transition thermal history.
Parameter Count
| Category | Count | Status |
|---|---|---|
| Original free parameters | 10 | |
| Determined by two-scale solution | 4 | m_1, n_1, \omega_0, \xi (by new C1 + C10 + close-packing + modon existence) |
| Effectively constrained by measurement | 3 | \alpha_{mf} (from C8); f_\text{cross} (from C3); \delta T/T_c (from C7) |
| Truly unconstrained | 3 | M_d, n_d, \delta (appear in no constraint or only subdominantly) |
| Particle physics constraints (C1–C9) | 9 | C1 restructured; C6 derived from C8; C8 ✅; C9 derived from C6 |
| Cosmological constraints (C10–C13) | 4 | C10 direct; C11–C13 need phase transition calc |
| Structural conditions (SC1–SC5) | 5 | SC1, SC3–SC4 automatic checks; SC2 ✅ INDEPENDENT; SC5 ✅ fully derived |
| Close-packing condition | 1 | n_1\xi^3 = 4\pi/(K\sqrt{2}) = 0.5666 — bridge equation, Steps A–E complete |
| Independent constraints | 14 | (C6 removed — prediction; SC2 promoted; close-packing added) |
| Parameters participating in constraints | 7 | m_1, n_1, \omega_0, \xi, \alpha_{mf}, f_\text{cross}, \delta T/T_c |
| Overdetermined by | ~7 | Each excess constraint is a falsifiable consistency check |
With \delta, n_d, M_d genuinely free (appearing in no constraint equation, or only subdominantly in C10): the effective system is $$14 independent constraints for $$7 participating parameters → overdetermined by $$7. Each overcounting is a falsifiable prediction.
Key structural results:
The electroweak subsystem (C6, C8, C9) collapses to a single input (\sin^2\theta_W) plus zero-parameter predictions (\alpha, (g-2)/2, m_W, m_Z). This is SC5 — the five-constant relation — holding at tree level with ~1–3% accuracy.
The outer scale (\xi, m_1, n_1) is fully determined by (\hbar, c, \rho_{DM}) + close-packing (dimensionally correct ✓). The SC2 numerical recipe gives \xi_\text{SC2} = 96.9\;\mum (⚠️); the Volovik/close-packing formula gives \xi_\text{CP} = 110\;\mum (✓). The packing fraction f = 4\pi/(K\sqrt{2}) = 0.5666 constitutes the bridge equation — a zero-parameter relation connecting particle physics to cosmology (0.18% precision). All three factors have identified physical origins: 4\pi from Gauss’s law / GR coupling (Step A), K from Bessel modon matching (established), 1/\sqrt{2} from GP kinetic energy \hbar^2/(2m) (Step B). Step D confirms no 3D geometric correction factor enters f: the lattice is straight parallel lines in 2D triangular arrangement within domains (five-pillar argument from Saffman). Step E provides the constrained equilibrium derivation. In packing-fraction form (both sides dimensionless ✓):
\boxed{f \equiv \frac{\rho_\text{DM}\,c\,\xi_\text{SC2}^4}{\hbar} = \frac{4\pi}{K\sqrt{2}} = 0.5666}
If exact, it constrains one cosmological parameter (\rho_\text{DM}) in terms of two particle physics parameters (\sin^2\theta_W, m_e) plus mathematical constants, reducing the independent parameter count of SM + ΛCDM by one. See the bridge equation for the full derivation (Steps A–E all ✅; dimensional repair of the \xi_\text{SC2}^3 formula is an open problem connected to the Higgs VEV derivation).
The inner scale (r_\text{eff}, v_\text{rot,inner}) is fully determined by Subsystem A (\alpha_{mf}, m_e) with zero new parameters. The effective quantum has m_\text{eff} = 1.70 MeV, orbits at 0.776c with radius 150 fm and angular momentum \hbar.
Subsystem B is SOLVED. The old 1-parameter family is broken by new C1 (c = \hbar/(m_1\xi)) + close-packing. Result: m_1 \approx 2 meV/c^2, \nu \approx 8.3 \times 10^8.
SC2 ≠ Tkachenko wave speed. The substrate’s Tkachenko speed is c_T \approx 9 km/s \approx 3 \times 10^{-5}\,c (deeply stiff regime). SC2 (\kappa_q \Omega_v = 4\pi c^2) is a background lattice configuration condition for the effective metric, not a speed-matching condition. The 4\pi (vs Baym’s 8\pi) reflects the GR coupling origin (Gauss’s law, not lattice elasticity). Derived from BLV analog gravity framework (Step A).
Dimensional status (v0.8). The cosmological route to the outer scale (\xi_\text{CP}, m_1, n_1) is dimensionally clean ✓. The particle physics route (SC2 formula for \xi_\text{SC2}) and the 3D modon existence condition (old C1 for \omega_0) are numerical recipes valid in SI but not dimensionally consistent equations (confirmed by CGS cross-check: values diverge by 10^4 between unit systems). The bridge equation in packing-fraction form (f = \rho_\text{DM}c\xi^4/\hbar = 4\pi/(K\sqrt{2})) IS dimensionless and the 0.18% match is a genuine result. Dimensional repair requires understanding the chirality-coherent sheet stacking of the vortex lattice — the same thermodynamic calculation needed to derive the Higgs VEV v = 246 GeV from substrate parameters (see Higgs Field). The numerical results and physical interpretation are unaffected; the issue is presentational completeness.
Structural Consistency Conditions
Beyond the 13 observational constraints, the framework must satisfy five structural requirements that follow from its own internal logic. These are not tuned by adjusting parameters — they are either automatically true (validating the framework) or they fail (falsifying it regardless of parameter values).
SC1. Ebbing current = free-fall velocity. The self-consistent steady-state dc1 inflow must equal v_\text{ebb}(r) = \sqrt{2GM/r}. This follows from the substrate’s own gravitational acceleration and is what produces the exact Painlevé-Gullstrand metric.
SC2. Vortex lattice configuration for Einstein equations. For the substrate’s effective acoustic metric to reproduce the linearized Einstein equations — including a proper spin-2 gravitational sector — the background vortex lattice must satisfy: \kappa_q \cdot \Omega_v = 4\pi c^2 ⚠️ (see Spacetime & Dynamics). This is a condition on how the lattice is arranged, not on how fast its perturbations propagate. (The actual lattice shear waves — Tkachenko modes — travel at \sim 9 km/s, five orders of magnitude below c; photons and gravitational waves both travel at c because they share the same BEC quasiparticle dispersion.) Combined with C1 and C2, SC2 determines \xi_\text{SC2} = 96.9\;\mum (via the numerical recipe \xi_\text{SC2}^3 = \hbar K\alpha_{mf}/(2m_ec); ⚠️ LHS [m³], RHS [m]). \xi is ALSO determined from first principles by the Volovik route + close-packing (\xi_\text{CP} = 110\;\mum, dimensionally correct ✓). The near-equality of these two independent determinations is the bridge equation — satisfied to 0.18% by the packing fraction f = 4\pi/(K\sqrt{2}), a zero-parameter relation connecting the electroweak sector to the cosmological dark matter density. The 4\pi (vs Baym’s 8\pi) reflects the GR coupling origin (Gauss’s law, not lattice elasticity), derived from the BLV analog gravity framework (Step A). ⚠️ The 3D form \kappa_q \cdot n_1\omega_0 = 4\pi c^2 has LHS [m⁻¹s⁻²] vs RHS [m²s⁻²] — same root cause as C1b (3D vs 2D quantities). The bridge equation in packing-fraction form (f = \rho_\text{DM}c\xi^4/\hbar = 4\pi/(K\sqrt{2}), both sides [1] ✓) avoids this. See dimensional status note at top of page.
SC3. Conformal self-consistency. The substrate density must adjust hydrostatically as \rho(r) = \rho_0 \cdot \exp(-\Phi(r)/c^2) for the conformal factor of the acoustic metric to produce the correct Einstein tensor. This is automatic for the barotropic equation of state P = \rho c^2 (see The Conformal Factor).
SC4. Jeans length exceeds Hubble radius. The Jeans length \lambda_J = c\sqrt{\pi/(G\rho_\text{sub})} must be larger than the observable universe for all gravitational wave modes to propagate at c regardless of polarization. With \rho_\text{sub} \sim 10^{-27} kg/m³: \lambda_J \approx 140 Gpc \gg 28 Gpc. Automatically satisfied.
SC5. Three-constant relation. Given \sin^2\theta_W, both \alpha and (g-2)/2 are predicted with zero free parameters. Any future precision improvement in these measurements tests the substrate’s boundary geometry.
Path to Zero Free Parameters
With 14 constraints for ~7 participating parameters, the system is heavily overdetermined. The remaining genuinely free parameters are M_d, n_d, and \delta — none of which appear in any current constraint equation (or only subdominantly in C10).
The most promising routes to close the remaining freedom:
\alpha = 1/137 derived from boundary geometry — tree level complete. C6 is derived from C8 with zero new parameters via the six-stage chain: the tree-level result gives \alpha = 1/135.1 (+1.45%). The remaining gap is consistent with vacuum polarization (one-loop correction). Computing the modon self-energy (WIP-5) would close the gap and complete the purely geometric derivation of the fine structure constant.
Bridge equation — numerical result resolved, dimensional derivation open. The packing fraction f = 4\pi/(K\sqrt{2}) = 0.5666 matches the numerical value to 0.18%, well within the ~1% Planck uncertainty on \rho_\text{DM}. All factors have identified physical origins (Steps A–E complete): 4\pi from Gauss’s law / GR coupling, K = j_{11}^2+1 from Bessel modon matching, 1/\sqrt{2} from the GP kinetic energy operator \hbar^2/(2m), and \eta = 1 (no 3D geometric correction — five-pillar argument from Saffman). This is a zero-parameter prediction connecting the Weinberg angle, electron mass, and dark matter density through one superfluid. The formula for \xi_\text{SC2} that feeds into the bridge equation (\xi_\text{SC2}^3 = \hbar K\alpha_{mf}/(2m_ec)) is a numerical recipe pending dimensional repair; the bridge equation itself in packing-fraction form (f = \rho_\text{DM}c\xi^4/\hbar) is dimensionally clean.
Dag role and mass. Understanding how M_d nucleates effective quantum formation (WIP-11) would constrain the remaining free parameters. If each dag center binds \sim\nu dc1 into one effective quantum, then n_d = n_1/\nu \approx 800 m^{-3} (one dag per \sim(11\;\text{cm})^3).
Mass spectrum predictions. The masses of heavier particles (muon, tau, W, Z, Higgs) are each determined by the substrate parameters. Each measured mass provides an additional constraint beyond C1–C13.
Predictions (Not Yet Measured Precisely Enough to Constrain)
These are outputs of the framework that can be tested by upcoming experiments. They are not free — each is fully determined by the substrate parameters that must already satisfy C1–C13.
| Prediction | Value | Experiment | Timeline |
|---|---|---|---|
| dc1 mass | m_1 \approx 2 meV/c^2 | Cosmological structure, ultralight DM searches | Near-term |
| Minimum photon energy | E_\text{min} \approx 13 meV (\lambda \sim 100\;\mum) | THz spectroscopy in vacuum | Near-term |
| Effective quantum mass | m_\text{eff} \approx 1.7 MeV/c^2 | Particle physics scale | — |
| Inner orbital radius | r_\text{eff} \approx 150 fm | Sub-Compton electron structure | Far future |
| Outer rotation velocity | v_\text{rot,outer} \approx 0.0025\,c | CDM-MOND transition | Existing data |
| Bridge equation | f = 4\pi/(K\sqrt{2}) = 0.5666 (0.18% match) | Cross-check (✅ resolved) | — |
| Tensor-to-scalar ratio | r \approx 0.01–0.02 | LiteBIRD, CMB-S4 | ~2030s |
| Non-Gaussianity | f_{NL} \approx 0 | Planck (already consistent) | Current ✓ |
| Dark energy EOS deviation | w = -1 + \epsilon(a), \epsilon small positive | DESI, Euclid, Roman | ~2030s |
| Running of spectral index | dn_s/d\ln k \approx -5.6 \times 10^{-4} | CMB-S4 | ~2030s |
| GW background (phase transition) | Peaked spectrum, distinct from slow-roll | LISA, pulsar timing | ~2030s |
| GW dispersion | (\omega/\omega_\text{cutoff})^\alpha correction, \alpha \geq 2 | Cosmic Explorer, Einstein Telescope | ~2040s |
| CDM-to-MOND transition | At galaxy scale, v \sim 10^{-3}\,c | Galaxy rotation curves | Existing data, needs modeling |
| No singularities | Modified black hole interiors | EHT, GW ringdown | Next generation |
| Scalar GW memory | Permanent \Delta\rho/\rho \sim h \sim 10^{-21} | Future GW detectors | Far future |
Constraint Equations Summary
For reference, here are the explicit constraint equations where available:
| Constraint | Equation | Dim. | Observed Value |
|---|---|---|---|
| C1a (Volovik) | c = \hbar/(m_1 \cdot \xi) | ✓ | 2.998 \times 10^8 m/s |
| C1b (Existence) | f \cdot \omega_0\xi/c = K | ✓ | (determines \omega_0) |
| C1b (3D form) | n_1 \omega_0 \xi^3 = K \cdot c | ⚠️ | (same; LHS [s⁻¹], RHS [m/s]) |
| C2 | m_\text{eff} \cdot \alpha_{mf} = m_e | ✓ | \hbar = 1.055 \times 10^{-34} J·s |
| C3 (simplified) | G = f_\text{cross} \cdot v_\text{rot,outer} / (4\pi) | ⚠️ | 6.674 \times 10^{-11} m³/(kg·s²) |
| C4 | m_e c^2 = \tfrac{1}{2} m_\text{eff} v_\text{rot,inner}^2; v_\text{rot,inner} = c\sqrt{2\alpha_{mf}} | ✓ | 9.109 \times 10^{-31} kg |
| C5 | m_p c^2 = \sum(\text{quark}\;E) + E_\text{gluon\_analog} + E_\text{boundary} | — | 1.673 \times 10^{-27} kg |
| C6 | \alpha = f(\alpha_{mf}) \cdot \sin^2\theta_W / \pi (derived from C8) | ✓ | 1/137.036 |
| C7 | \rho_\Lambda = \rho_\text{substrate} \cdot (\delta T/T_c)^2 | ✓ | \Lambda = 1.1 \times 10^{-52} m^{-2} |
| C8 | \sin^2\theta_W = \alpha_{mf} / (1 + \alpha_{mf}) | ✓ | 0.2312 |
| C9 | (g-2)/2 = \eta^2 = \alpha/(2\pi) | ✓ | 0.00116 |
| C10 | n_1 \cdot m_1 \approx \rho_{DM} | ✓ | 2.4 \times 10^{-27} kg/m³ |
| CP (bridge) | f = \rho_\text{DM}c\xi_\text{SC2}^4/\hbar = 4\pi/(K\sqrt{2}) | ✓ | 0.5666 (0.18% match, Steps A–E ✅) |
| SC2 (3D form) | \kappa_q \cdot n_1\omega_0 = 4\pi c^2 | ⚠️ | (LHS [m⁻¹s⁻²], RHS [m²s⁻²]) |
| \xi_\text{SC2} recipe | \xi_\text{SC2}^3 = \hbar K\alpha_{mf}/(2m_ec) | ⚠️ | 96.9 μm (LHS [m³], RHS [m]) |
| C11 | A_s = H_\text{inf}^2 / (8\pi^2 M_{Pl}^2 c^4 \varepsilon_s) | ✓ | 2.1 \times 10^{-9} |
| C12 | n_s - 1 \approx -3/(2N_*) - s | ✓ | 0.965 |
| C13 | r_s = \int_0^{t_\text{rec}} c_s/a\;dt | ✓ | 147.09 Mpc |
Legend: ✓ = dimensionally consistent, ⚠️ = numerical recipe (correct values in SI, dimensional mismatch confirmed by CGS cross-check), — = not yet fully formulated.