Constraint Summary
Free Parameters
The framework has 12 free parameters (10 original + 2 from the DESI crust model). The two-scale model determines most of the original set from measured constants, leaving 3 genuinely free (\delta, B, z_s). Of these \delta appears in no current prediction, while B and z_s describe the previous cycle’s remnant boundary (inherently cycle-dependent):
| Parameter | Symbol | Status | Description |
|---|---|---|---|
| dc1 mass | m_1 | DETERMINED: \hbar/(c \xi) \approx 2 meV/c^2 | Mass of the light substrate particle, never observed individually, always vortex lines in a collective state |
| 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 |
| Orbital angular velocity | \omega_0 | DETERMINED (gravity sector): v_\text{rot,outer}/\xi ≈ 7.8 \times 10^9 rad/s; not fixed by C1b/SC2 (see Open Problems WIP-15) | Outer-scale lattice rotation rate |
| Coherence length | \xi | DETERMINED: (\hbar/(\rho_{DM} c))^{1/4} \approx 110\;\mum | Perturbation envelope |
| Mutual friction parameter | \alpha_{mf} | MEASURED: = \sin^2\theta_W/(1-\sin^2\theta_W) = 0.30078 | 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 |
| Crust amplitude | B | FIT to DESI DR2 (P11) | Energy density of previous cycle’s remnant boundary; 1.88 |
| Crust redshift | z_s | FIT to DESI DR2 (P12) | Location of crust encounter (previous cycle geometry); 0.63 |
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 |
| d_\text{GJO} | \xi\sqrt{\ln(\xi/\xi_\text{GP})/(4\pi)} | \approx 16\;\mum | Inter-sheet period (GJO instability wavelength); counter-rotating boundary at d_\text{GJO}/2 \approx 8\;\mum; zero new parameters |
| \varepsilon | d_\text{GJO}/\xi = \sqrt{\ln(\xi/\xi_\text{GP})/(4\pi)} | 0.166 | Anisotropy parameter; \varepsilon \ll 1 confirms strongly layered regime |
| \tau_\text{cr} | \varepsilon^{-1} \cdot (d/\xi) | \approx 2 | Blatter crossover parameter; right at 2D/3D boundary |
| \Lambda | d_\text{GJO}/\varepsilon = \xi | \approx 100\;\mum | Phase screening length (Blatter); \Lambda = \xi maps 2D ↔︎ 3D |
| E_\text{min} | hc/\xi = 2\pi m_1 c^2 | 12.6 meV | Minimum modon (photon) energy |
| C | = 1.0 (predicted) | — | Bulk deficit depth; Volovik: equilibrium DE = 0. C \neq 1 would falsify self-tuning. |
| z_b | = 2.20 (derivable from \tau_\text{relax}) | — | Bulk deficit onset; H(z_b) \cdot \tau_\text{relax} \sim 1 |
| f(z) | = 1 + Bze^{-z/z_s} - Cz^2/(z^2+z_b^2) | — | Normalized DE density profile |
| w(z) | = -1 + (1+z)f'/(3f) | — | Dark energy equation of state |
| \tau_\text{relax} | To be derived (WIP-16) | — | Substrate cosmological relaxation timescale |
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 period d_\text{GJO} \approx 16\;\mum \ll \xi, a counter-rotating boundary every half-period \approx 8\;\mum).
Blatter mapping (WIP-15, dimensional repair closed in form): The Blatter et al. framework for layered superconductors provides the mathematical template for the 3D→2D crossover. The identification \Lambda = d_\text{GJO}/\varepsilon = \xi (where \varepsilon = d_\text{GJO}/\xi \approx 0.17 is the anisotropy parameter) shows that the phase screening length equals the in-plane coherence length — below \Lambda, each sheet’s physics is purely 2D. The Feynman relation and L-R modon matching operate at scale \sim \xi = \Lambda, right at the 2D boundary, while the modon (wavelength \sim \xi \gg d_\text{GJO}) sees the long-wavelength 3D-isotropic regime. The crossover parameter \tau_\text{cr} \approx 2 sits right at the 2D/3D crossover — automatically from the mapping. This resolves why 2D math works for a 3D medium. The dimensional repair itself (replacing n_1\omega_0 with properly structured 2D quantities involving n_v^{(2D)} = 1/\xi^2, \Omega_\text{sheet}, d_\text{GJO}, and \varepsilon_\text{chirality}) is now closed in form: the inter-sheet spacing is the GJO instability wavelength d_\text{GJO} \approx 16\;\mum and the chirality factor is pinned to \varepsilon_\text{chirality} = \sqrt{\pi\ln 2}/K by packing-fraction self-consistency; only the functional shape of the chirality term and the Higgs VEV remain (see Higgs Field, Open Problems WIP-15).
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 12 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 an independent constraint, and it does not determine \omega_0. Written honestly — the Larichev-Reznik dispersion with the modon riding the dc1-circulation gradient \kappa_1 = 2\pi\hbar/m_1 — it collapses to the Volovik speed c = \hbar/(m_1\xi), i.e. C1a itself, carrying no \omega_0 (see Open Problems WIP-15).
⚠️ The old 3D form n_1\omega_0\xi^3 = Kc — and its dimensionless rewrite f\cdot\omega_0\xi/c = K, which is dimensionally balanced — were both artifacts of inserting the outer rotation \omega_0 into a relation that does not contain it. Evaluated at the gravity-sector \omega_0 \approx 7.8\times10^9 rad/s, f\cdot\omega_0\xi/c \approx 1.4\times10^{-3} \ne K; solving instead for \omega_0 returns \approx 8.6\times10^{13} rad/s, \sim 10^4 above the physical value. That \sim 10^4 gap was the long-standing “projection factor”; it dissolves once \omega_0 is removed from modon matching, since \omega_0 is fixed in the gravity sector (C3), not here.
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 Leaking Rotational Kinetic 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 Leaking Rotational Kinetic 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})
\alpha_{mf} is taken from the measured \sin^2\theta_W here, but is now the target of a staged vortex-core BdG computation (Weinberg Angle § A Numerical Program for \alpha_{mf}) that lands the value at the close-packing spacing of the vortex lattice to \sim10\% — not yet a precision derivation, but no longer purely phenomenological.
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 & Galactic Constraints (C10)
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 substrate IS the dark matter. Its observed density directly constrains the product of dc1 number density and mass:
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 fixed geometrically — N_* \approx \ln(c/H_0\xi) \approx 69, eased to ~60 by the subluminal bubble wall — with no free parameter (see Why \sim 60 E-folds).
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 | 4 now determined (P1, P3, P5, P6 by two-scale solution) |
| DESI crust parameters | 2 | P11 (B), P12 (z_s) — fit to data |
| Total free parameters | 12 | |
| Remaining free parameters | 8 | P2, P4, P7, P8, P9, P10, P11, P12 |
| Effectively constrained by measurement | 3 | P7 (\alpha_{mf} from C8); P8 (f_\text{cross} from C3); P10 (\delta T/T_c from C7) |
| Truly unconstrained | 3 | P9 (\delta), P11 (B), P12 (z_s) |
| Particle physics constraints (C1–C9) | 9 | C1 restructured; C6 derived from C8; C8 ✅; C9 derived from C6 |
| Cosmological & galactic constraints (C10) | 6 | C10 direct; C11–C13 need phase transition calc; C14 ✅ (a_0); |
| 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 (WIP-10 ✅) |
| Independent constraints | 16 | (C6 removed — prediction; SC2 promoted; close-packing added; C14 added;) |
| Parameters participating in constraints | 9 | P1, P3, P5, P6 (determined), P7, P8, P10 (constrained), P11, P12 (fit) |
| Overdetermined by | ~7 | 16 constraints for ~9 participating parameters; P2, P4, P9 appear in no constraint |
| Effectively unconstrained | 1 | \delta (no constraint) |
| Cycle-dependent (inherently free) | 2 | B, z_s (previous cycle properties) |
| Zero-parameter predictions confirmed by data | 2 | a_0 = c\sqrt{G\rho_\text{DM}} (C14, ~3%); C = 1 (C15, DESI) |
With \delta free (appearing in no constraint equation, or only subdominantly in C10), and B, z_s inherently cycle-dependent: the effective system is ~16 independent constraints for ~7 participating parameters → overdetermined by ~9. 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 (0.20% 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.
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 (v0.9). 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 0.20% match is a genuine result. New (WIP-15): The Blatter mapping resolves why the 2D math works — the phase screening length \Lambda = \xi places the Feynman relation and L-R matching at the exact 2D/3D crossover (\tau_\text{cr} = 2). The inter-sheet spacing d_\text{GJO}/\xi = \sqrt{\ln(\xi/\xi_\text{GP})/(4\pi)} = 0.166 (d_\text{GJO} \approx 16\;\mum) is fixed in closed form with zero new parameters as the GJO instability wavelength (the earlier Lawrence-Doniach saddle at \approx 7\;\mum is reread as an upper bound only); the \varepsilon \approx 0.17 anisotropy places the system in the strongly layered regime where Lawrence-Doniach applies. Completing the dimensional repair is now closed in form — with d_\text{GJO}, n_v^{(2D)} = 1/\xi^2, and \varepsilon_\text{chirality} = \sqrt{\pi\ln 2}/K all fixed; the functional shape of the chirality term and the Higgs VEV v = 246 GeV remain (see Higgs Field).
Galactic dynamics from boundary parity (C14). The counter-rotating boundary’s parity symmetry forces a quadratic current-phase relation → MOND field equation in the deep low-acceleration regime. The MOND acceleration scale a_0 = c\sqrt{G\rho_\text{DM}} = 1.16 \times 10^{-10} m/s² matches McGaugh et al. (2016) to \sim 3\% (well within systematic uncertainty) with zero free parameters. Flat rotation curves and the baryonic Tully-Fisher relation (M_b \propto v^4) follow as consequences. The value of a_0 \approx cH_0/6 with a 3-5% uncertainty is explained: a_0/(cH_0) = \sqrt{3\Omega_\text{DM}/(8\pi)}. This extends the bridge equation’s reach to five domains: electroweak → QM → GR → cosmology → galactic dynamics. See Galactic Dynamics.
DESI dark energy profile (C15). The dark energy density evolves as f(z) = 1 + Bze^{-z/z_s} - z^2/(z^2+z_b^2) with C = 1 predicted by Volovik self-tuning and confirmed by DESI DR2 fit (1\sigma match in w_0–w_a plane). The crust component (B = 1.88, z_s = 0.63) is the first observational evidence for cyclic cosmology — energy absorbed from the previous cycle’s remnant boundary. The bulk deficit (C = 1, z_b = 2.20) says dark energy was zero in the deep past, exactly as the Gibbs-Duhem argument (G4) requires. If z_b is derived from relaxation dynamics (WIP-16), the dark energy background becomes a zero-parameter prediction, and only the crust (B, z_s) remains free — a 2-parameter model for the entire dark energy evolution. This extends the bridge equation’s reach to six domains: electroweak → QM → GR → cosmology → galactic dynamics → dark energy evolution. See Dark Energy and the Crust.
Structural Consistency Conditions
Beyond the 15 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 run as a length recipe returns \xi_\text{SC2} = 96.9\;\mum (via \xi_\text{SC2}^3 = \hbar K\alpha_{mf}/(2m_ec); ⚠️ LHS [m³], RHS [m]) — a mnemonic for the value, not an independent determination of the length: the electroweak sector cannot build a 100\;\mum length on its own (see WIP-30). The length is fixed from first principles by the Volovik route + close-packing (\xi_\text{CP} = 110\;\mum, dimensionally correct ✓); what the electroweak scaffold adds is one pure number — the condensation number \nu = m_\text{eff}/m_1 \approx 9.6\times10^8. Their agreement, stated dimensionlessly, is the bridge equation — satisfied to 0.20% 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). Written honestly with a genuine rotation rate, SC2 is the effective-quantum Compton clock: \kappa_q\,\Omega_v = 4\pi c^2 \Rightarrow \Omega_v = 2m_\text{eff}c^2/\hbar, an identity satisfied automatically by \kappa_q = 2\pi\hbar/m_\text{eff} (both sides [m²s⁻²] ✓; see Open Problems WIP-15). It contains no \omega_0 — the outer rotation is gravity-fixed (C3). ⚠️ The old 3D form \kappa_q \cdot n_1\omega_0 = 4\pi c^2 has LHS [m⁻¹s⁻²] vs RHS [m²s⁻²] — the spurious [m³] of the 3D vorticity density, same root cause as C1b. The Blatter mapping (\Lambda = \xi, \varepsilon = d_\text{GJO}/\xi \approx 0.17) confirms this is the expected behavior of a strongly layered system: in-plane lattice relations are 2D at scale R < \Lambda, while modons experience 3D isotropy at R \gg \Lambda. 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 16 constraints for ~7 participating parameters, the system is heavily overdetermined. The genuinely free parameters are \delta (appearing in no constraint), plus B and z_s (cycle-dependent, describing the previous cycle’s crust).
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 advancing. The packing fraction f = 4\pi/(K\sqrt{2}) = 0.5666 matches the numerical value to 0.20%, 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. New (WIP-15): The Blatter layered-superconductor mapping shows the formula’s 2D accuracy in a 3D medium is physically correct: the phase screening length \Lambda = \xi, so Feynman/L-R matching operates in the purely 2D regime while modons propagate in the 3D-isotropic regime. The inter-sheet spacing d_\text{GJO}/\xi = 0.166 (\approx 16\;\mum, the GJO instability wavelength; zero parameters) completes the layered picture. The remaining Step F — restructuring the dimensional recipes into properly separated 2D and stacking factors — is now closed in form: the Glaberson–Johnson–Ostermeier instability fixes the inter-sheet spacing d_\text{GJO} and packing-fraction self-consistency pins the chirality factor \varepsilon_\text{chirality} = \sqrt{\pi\ln 2}/K, leaving only the chirality term’s functional shape and the Higgs VEV.
The need for a heavier particle to pin the lattice is no longer part of the framework: dc1’s logarithmic equation of state self-binds the lattice and self-organizes it to close-packing, so no scaffold is needed.
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–C15.
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–C15.
| 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) | Sub-floor speed = c to 7\times10^{-16} ✓ (CHIME/FRB Catalog-2 +\nu^2 refit, 896 bursts, 2026 — floor changes quantization character, not speed); floor crossing adiabatic to \sim10^{-28} (\propto\nu^{-3/2}) ✓ (COBE/FIRAS axis = crossing-epoch map z\approx4–50, 23 orders under \mu,y); lab THz band-edge spectroscopy outstanding | Current ✓ + lab |
| 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 (0.20% 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 | w(z) \neq -1: f(z) = 1 + Bze^{-z/z_s} - z^2/(z^2+z_b^2); C = 1 confirmed | DESI DR2 ✅ (1\sigma); Euclid/Roman for w(z) shape | Current + ~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) | \to c = \hbar/(m_1\xi) (= C1a) | ✓ | identity; does not fix \omega_0 (WIP-15) |
| C1b (3D form) | n_1 \omega_0 \xi^3 = K \cdot c | ⚠️ | artifact; \omega_0 here is spurious |
| 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 (0.20% match, Steps A–E ✅) |
| SC2 (clean form) | \kappa_q\,\Omega_v = 4\pi c^2, \Omega_v = 2m_\text{eff}c^2/\hbar | ✓ | Compton clock; \kappa_q = 2\pi\hbar/m_\text{eff} satisfies it identically |
| SC2 (3D form) | \kappa_q \cdot n_1\omega_0 = 4\pi c^2 | ⚠️ | artifact of n_1\omega_0; 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 |
| C14 | a_0 = c\sqrt{G\rho_\text{DM}} | ✓ | 1.16 \times 10^{-10} m/s² (~3% match, zero parameters) |
Legend: ✓ = dimensionally consistent, ⚠️ = numerical recipe (correct values in SI, dimensional mismatch confirmed by CGS cross-check), — = not yet fully formulated.