Framework Parameters

The Standard Model of particle physics has roughly 25 free parameters (including neutrino masses and gravity) that must be measured experimentally. This page tracks how the observer-centrism framework addresses each one — whether it derives, constrains, explains the mechanism of, or leaves free each parameter.

Honest accounting. “Derived” means the framework produces the value from axioms (plus structural postulates) with no empirical input beyond unit conventions. “Constrained” means boundary conditions or ranges are derived but the exact value requires further computation (typically RG running or bootstrap dynamics). “Mechanism only” means the physical mechanism is explained but the numerical value remains empirical. Parameters listed as “free” are genuine degrees of freedom. “Non-viable” means the framework currently cannot explain the observed value.

Derived

Constrained

Mechanism only

Free

Non-viable

25 of 28 parameters are determined in principle (derived, constrained, or mechanism identified). Only 3 remain genuinely free or non-viable.

Comparison with Other Frameworks

How does this compare to the parameter-reduction claims of other major theoretical frameworks? The table below compares the number of free parameters each framework leaves unexplained out of the Standard Model's ~25.

Framework	Free parameters	Determined in principle	Key mechanism
Standard Model	~25	0	All measured experimentally
MSSM (minimal SUSY)	~124	0 (adds ~100 new parameters)	Soft SUSY-breaking terms
SU(5) GUT	~24	~1 (predicts coupling unification)	Gauge unification at high scale
SO(10) GUT	~18–22	~3–7 (constrains Yukawa textures)	Single spinor representation for fermions
String/M-theory	0 in principle; ~10⁵⁰⁰ landscape vacua	All in principle (vacuum selection unsolved)	Geometry of extra dimensions
Observer-Centrism	3	25	Division algebras + coherence geometry + bootstrap hierarchy

Caveats. “Determined in principle” for this framework means the derivation chain from axioms to the parameter exists and is identified, even where the quantitative computation (e.g. full RG flow, bootstrap fixed-point) is not yet completed. String theory similarly claims all parameters are determined once the vacuum is selected, but the vacuum selection problem remains open. GUTs reduce parameters through group-theoretic constraints but do not explain Yukawa couplings. No framework has yet computed all SM parameters from first principles.

Fundamental Constants

$c$ Speed of light Derived

SM value $2.998 \times 10^8$ m/s (defines metre)

Framework Structurally determined as universal phase propagation speed through coherence geometry. $L = cT$ from loop closure.

Numerical value reflects human unit conventions. Framework determines $c$ uniquely from axioms + S1 (pseudo-Riemannian structure).

Speed of Light →

$\hbar$ Reduced Planck constant Derived

SM value $1.055 \times 10^{-34}$ J·s

Framework Minimum coherence cost of one observer cycle on $S^1$ . Identified with arc length of minimal closed geodesic via Lyusternik–Fet theorem.

Structure (minimum action, quantization, uncertainty principle) all rigorously derived. Numerical value is a unit convention.

Remaining gaps: Absolute numerical value not computed from first principles

Action and Planck's Constant →

$G$ Newton's gravitational constant Constrained

SM value $6.674 \times 10^{-11}\;\text{m}^3\text{kg}^{-1}\text{s}^{-2}$

Framework Constrained: $G = \ell_{\min}^2 c^3/\hbar$ where $\ell_{\min}$ is the minimum resolvable scale. Three convergent arguments (Jacobson thermodynamic, minimal observer self-consistency, curvature-spacetime bridge) establish this relationship. Full derivation requires independent determination of $\ell_{\min}$ .

Three independent arguments converge on $G = \ell_{\min}^2 c^3/\hbar$ . The coupling strength is structurally constrained by the coherence Lagrangian, not fully free. Draft status: the curvature-spacetime bridge conjecture (Conjecture 5.3) would complete the derivation.

Remaining gaps: Independent derivation of $\ell_{\min}$ from axioms (curvature-spacetime bridge conjecture)

Gravitational Coupling from Coherence Geometry →

Gauge Couplings

$\alpha_{\text{em}}$ Fine structure constant Derived 0.1% (2-loop)

SM value $\alpha^{-1}(M_Z) = 127.95 \pm 0.02$

Framework Determined by the same constraint as $\sin^2\!\theta_W$ : once $\Lambda_{\text{EW}}$ is fitted, $\alpha_{\text{em}}^{-1} = \alpha_Y^{-1} + \alpha_2^{-1}$ is fixed. Two-loop: $\alpha_{\text{em}}^{-1}(M_Z) = 128.1$ .

Not an independent prediction from $\sin^2\!\theta_W$ — both follow from the same 1-parameter fit ( $\Lambda_{\text{EW}}$ ). The 0.1% accuracy reflects two-loop RG precision, not a separate test.

Remaining gaps: Independent prediction of $\Lambda_{\text{EW}}$ from bootstrap dynamics

Coupling Constants →

$\alpha_s$ Strong coupling constant Mechanism Only ~6% (suggestive, not a precision prediction)

SM value $\alpha_s(M_Z) = 0.1179 \pm 0.0009$

Framework Suggestive: $1/\dim(\mathbb{O}) = 1/8 = 0.125 \approx \alpha_s(M_Z)$ ( $\sim 6\%$ agreement). Requires $\text{SU}(3)$ crystallization near $M_Z$ , which is not independently derived. The ratio $\alpha_3/\alpha_2 = 2$ at a common EW scale fails (Landau pole).

The near-coincidence $1/8 \approx 0.118$ is intriguing but cannot be promoted to a prediction without an independent $\Lambda_3$ . No gauge unification predicted — falsifiable.

Remaining gaps: Independent derivation of $\text{SU}(3)$ crystallization scale $\Lambda_3$ Connecting $\alpha_3/\alpha_2 = 2$ ratio to measurable $\alpha_s$

Coupling Constants →

$\theta_W$ Weinberg angle Constrained Fitted (1 parameter determines $\Lambda_{\text{EW}}$ )

SM value $\sin^2\!\theta_W(M_Z) = 0.23122 \pm 0.00004$

Framework $\sin^2\!\theta_W = 1/3$ at algebraic scale (from $\mathbb{C} \subset \mathbb{H}$ embedding). Fitted to experiment by choosing $\Lambda_{\text{EW}}$ . Two-loop: $\Lambda_{\text{EW}} = 1.26 \times 10^{10}$ GeV (3.3% shift from 1-loop).

Algebraic boundary condition $\sin^2\!\theta_W = 1/3$ is a genuine structural prediction. The low-energy value is fitted (1 parameter). Two-loop analysis confirms 1-loop is reliable (3.3% shift in $\Lambda_{\text{EW}}$ ).

Remaining gaps: Independent derivation of $\Lambda_{\text{EW}}$ from bootstrap dynamics

Weinberg Angle →

Electroweak Sector

$v$ Higgs VEV (electroweak scale) Mechanism Only

SM value $v = 246.22 \pm 0.04$ GeV

Framework Mechanism derived (Coleman-Weinberg coherence crystallization, top-quark loop dominance). Numerical value $v \approx 246$ GeV requires full RG flow of coherence Lagrangian.

The instability condition ( $6y_t^2 > 9g^2/4 + 3\lambda$ ) is rigorous. Hierarchy protection via bootstrap dimensional transmutation — no SUSY needed.

Remaining gaps: Computing $v$ from coherence Lagrangian RG flow

Electroweak Breaking →

$m_h$ Higgs boson mass Mechanism Only

SM value $m_h = 125.10 \pm 0.14$ GeV

Framework $m_h = \sqrt{2\lambda}\,v$ . Both $\lambda$ and $v$ follow from the coherence potential shape near crystallization, but neither is computed from first principles.

The formula is exact; the inputs ( $\lambda$ , $v$ ) are not independently predicted.

Remaining gaps: Higgs self-coupling from coherence potential

Electroweak Breaking →

$\lambda$ Higgs self-coupling Free

SM value $\lambda \approx 0.13$ (from $m_h = 125$ GeV)

Framework Depends on shape of coherence potential near crystallization. Not independently predicted.

Requires non-perturbative treatment of the coherence potential at the crystallization transition.

Remaining gaps: Detailed effective potential computation

Electroweak Breaking →

$\hat{n}_{\text{EW}}$ Electroweak crystallization direction Free

SM value (not a SM parameter — implicit in Yukawa couplings)

Framework A point on $S^2$ specifying the Higgs crystallization direction relative to the three generation winding axes. Two free real parameters satisfying $\cos^2\!\alpha_1 + \cos^2\!\alpha_2 + \cos^2\!\alpha_3 = 1$ .

The crystallization direction determines the intra-type mass hierarchy: the three masses within each fermion type (e.g., $u, c, t$ ) follow from $y_k \propto \exp(-\alpha_k/g_{\text{eff}}^2)$ with the same three angles $\alpha_k$ but a type-specific effective coupling $g_{\text{eff}}$ . The inter-type mass splittings (why $m_t \neq m_b \neq m_\tau$ within each generation) require the CKM and PMNS rotations, which relate the distinct winding-axis triples for up-type, down-type, and lepton sectors (Proposition 5.2 of Three Generations). Numerical testing confirms that the three fermion types have incompatible geometric ratios $R = \ln(m_3/m_2)/\ln(m_2/m_1)$ : $R_{\text{up}} \approx 0.91$ , $R_{\text{down}} \approx 1.31$ , $R_{\text{lepton}} \approx 0.53$ . A shared crystallization direction with type-specific couplings fits each mass to within a factor of ~1.4 (RMS log error 0.33). The original claim of '2 parameters replacing 9 Yukawa couplings' is overstated; the actual parameter content is $\hat{n}_{\text{EW}}$ (2 DOF) + CKM/PMNS rotations (7–9 DOF) + effective coupling ratios.

Remaining gaps: Deriving $\hat{n}_{\text{EW}}$ from bootstrap dynamics or crystallization energetics

Three Fermion Generations →

$m_W$ W boson mass Derived Exact given $v$ and $\theta_W$

SM value $m_W = 80.36 \pm 0.01$ GeV

Framework $m_W = gv/2$ — follows exactly from electroweak breaking once $v$ and $g$ are known.

Not an independent parameter — determined by $v$ and gauge couplings.

Electroweak Breaking →

$m_Z$ Z boson mass Derived Exact given $m_W$ and $\theta_W$

SM value $m_Z = 91.188 \pm 0.002$ GeV

Framework $m_Z = m_W/\cos\theta_W$ — follows exactly from electroweak breaking.

Not an independent parameter — determined by $v$ and gauge couplings.

Electroweak Breaking →

Quark Masses

$m_t$ Top quark mass Mechanism Only

SM value $m_t = 172.69 \pm 0.30$ GeV

Framework $m_t = y_t v/\sqrt{2}$ where $y_t \propto \exp(-\alpha_3/g_{\text{EW}}^2)$ . Third generation most aligned with EW axis, hence heaviest.

All quark masses share the same mechanism: $y_k \propto \exp(-\alpha_k/g^2)$ , with $\alpha_1 > \alpha_2 > \alpha_3$ . The exponential hierarchy is rigorous; the specific angles are not.

Remaining gaps: Winding geometry computation of $\alpha_3$

Three Fermion Generations →

$m_c$ Charm quark mass Mechanism Only

SM value $m_c = 1.27 \pm 0.02$ GeV

Framework Second generation: intermediate alignment with EW axis. Ratio $m_t/m_c \sim \exp((\alpha_2 - \alpha_3)/g^2) \sim 130$ .

See top quark entry for shared mechanism.

Three Fermion Generations →

$m_u$ Up quark mass Mechanism Only

SM value $m_u = 2.16 \pm 0.07$ MeV

Framework First generation: least aligned with EW axis, hence lightest. Ratio $m_c/m_u \sim 600$ consistent with mechanism.

See top quark entry for shared mechanism.

Three Fermion Generations →

$m_b$ Bottom quark mass Mechanism Only

SM value $m_b = 4.18 \pm 0.03$ GeV

Framework Third generation down-type. Yukawa from winding geometry overlap with EW axis in down-type sector.

See top quark entry for shared mechanism.

Three Fermion Generations →

$m_s$ Strange quark mass Mechanism Only

SM value $m_s = 93 \pm 5$ MeV

Framework Second generation down-type.

See top quark entry for shared mechanism.

Three Fermion Generations →

$m_d$ Down quark mass Mechanism Only

SM value $m_d = 4.67 \pm 0.07$ MeV

Framework First generation down-type.

See top quark entry for shared mechanism.

Three Fermion Generations →

Lepton Masses

$m_\tau$ Tau mass Mechanism Only

SM value $m_\tau = 1776.86 \pm 0.12$ MeV

Framework Third generation charged lepton. Same winding geometry mechanism as quarks.

See top quark entry for shared mechanism.

Three Fermion Generations →

$m_\mu$ Muon mass Mechanism Only

SM value $m_\mu = 105.66$ MeV

Framework Second generation charged lepton.

See top quark entry for shared mechanism.

Three Fermion Generations →

$m_e$ Electron mass Mechanism Only

SM value $m_e = 0.511$ MeV

Framework First generation charged lepton. Lightest because most misaligned from EW axis.

See top quark entry for shared mechanism.

Three Fermion Generations →

$m_\nu$ Neutrino masses (3) Constrained Ordering derived; absolute scale open

SM value $\Delta m^2_{21} = 7.5 \times 10^{-5}$ eV², $|\Delta m^2_{32}| = 2.5 \times 10^{-3}$ eV²

Framework Type-I seesaw: $m_\nu = m_D^2/M_R$ with $M_R \sim y_R v$ at EW scale (not GUT). Normal ordering $m_1 < m_2 < m_3$ rigorously derived. Majorana nature from pseudo-real $\text{SU}(2)$ . Absolute scale requires $\varepsilon_\nu$ overlap coefficient.

Falsifiable: predicts Majorana nature (testable via $0\nu\beta\beta$ ), normal ordering (JUNO/DUNE), and $M_R \sim 100$ – $1000$ GeV (colliders).

Remaining gaps: Absolute mass scale from winding geometryMajorana phases from $A_5$ breaking

Neutrino Masses →

Mixing Parameters

$\theta_{\text{CKM}}$ CKM mixing angles (3) Mechanism Only Hierarchy (small angles) derived; values not computed

SM value $\theta_{12} \approx 13°$ , $\theta_{23} \approx 2.4°$ , $\theta_{13} \approx 0.2°$

Framework Small angles explained by steep quark mass hierarchy: $\theta_{ij} \sim V_{ij}/(m_i - m_j)$ . $A_5$ symmetry breaking determines structure. Quantitative values require winding geometry.

Near-diagonal CKM structure is a rigorous consequence of the steep mass hierarchy.

Remaining gaps: $A_5$ channel selection for quark sectorRG running from symmetry scale

Mixing Angles →

$\delta_{\text{CKM}}$ CKM CP-violating phase Constrained Existence derived; value from discrete set

SM value $\delta \approx 69°$ ( $1.20 \pm 0.08$ rad)

Framework Nonzero phase geometrically unavoidable with 3 generations (Kobayashi-Maskawa). Discrete value from $A_5$ breaking channel. Existence rigorous; specific value requires channel selection.

CP violation requires $\geq 3$ generations — this is a theorem, not a postulate.

Remaining gaps: $A_5$ channel selection

Mixing Angles →

$\theta_{\text{PMNS}}$ PMNS mixing angles (3) Constrained $\theta_{12}$ : ~2° discrepancy; $\theta_{23}$ : ~4° discrepancy

SM value $\theta_{12} \approx 33.4°$ , $\theta_{23} \approx 49°$ , $\theta_{13} \approx 8.5°$

Framework Large angles from mild neutrino mass hierarchy. Golden ratio prediction: $\tan\theta_{12} = 1/\varphi \to \theta_{12} \approx 31.7°$ . $\theta_{23} \approx \pi/4 \approx 45°$ . $\theta_{13}$ small (controlled by $A_5$ breaking parameter).

Golden ratio prediction ( $\tan\theta_{12} = 1/\varphi$ ) is the most precise semi-quantitative result. Falsifiable by precision PMNS measurements.

Remaining gaps: Higher-order $A_5$ correctionsRG running from symmetry scale to measurement scale

Mixing Angles →

$\delta_{\text{PMNS}}$ PMNS CP-violating phase Constrained Discrete prediction (5 values); channel selection open

SM value $\delta \sim -90°$ (poorly constrained)

Framework One of five discrete values from $A_5$ breaking channels. Current data favor $\delta \sim -\pi/2$ (maximal CP violation), consistent with one channel.

Directly testable by late 2020s. Five discrete values is a strong structural prediction vs. continuous SM parameter.

Remaining gaps: $A_5$ channel selectionDUNE/Hyper-Kamiokande precision measurement

Mixing Angles →

QCD Vacuum

$\theta_{\text{QCD}}$ QCD vacuum angle Derived Exact (predicts zero)

SM value $|\theta| < 10^{-10}$ (from neutron EDM)

Framework $\theta_{\text{QCD}} = 0$ exactly. Non-associativity of octonions constrains QCD vacuum topology; associator obstruction prevents non-trivial instanton sectors.

Predicts no QCD axion — all axion searches (ADMX, CASPEr, IAXO) should return null. Falsifiable.

Remaining gaps: Formal proof that octonionic bundles have trivial instanton moduli space

Strong CP →

Cosmological

$\Lambda$ Cosmological constant Non-Viable Sign and bound correct; value not derived

SM value $\Lambda \approx 10^{-56}\;\text{cm}^{-2}$

Framework Sign: $\Lambda \geq 0$ (rigorous, from coherence conservation). Upper bound: $\Lambda < 3/\ell_P^2$ (rigorous). Observed tiny value: 120 orders below Planck bound — qualitatively explained but not quantitatively derived.

The observed $10^{-122}$ hierarchy is the framework's main quantitative failure, shared with all known approaches.

Remaining gaps: Quantitative $\Lambda$ (requires geometry functor — hard-open)Sharpening to $\Lambda > 0$

Observer-Loop Viability →