GIFT Observable Reference

Version: 3.4 (top-level snapshot; see Supplement S3 for the full 95-observable dataset). The legacy v3.3.24 sector breakdown below is preserved for traceability of the NuFIT 6.0 analysis. Status: Reference documentation Date: May 2026

Executive Summary (v3.4)

Metric	Value
Total observables	95 (33 Type I + 19 Type II + 21 Type III + 22 Type IV)
Lean-certified	55 / 95 (Type I: 33/33, Type III: 14/21, Type IV: 8/22)
With experimental comparison	66 / 95
Mean deviation (Type I, 33 obs)	0.73%
Mean deviation (Type II, 19 obs)	0.17%
Mean deviation (Type III, 21 obs)	3.4%
Exact matches (< 0.01%)	11
Within 1%	53
Free parameters (continuously adjustable)	0
Lean certificate	213 conjuncts, 4 main-chain axioms, 0 sorry, 134 .lean files
Statistical null	uniform P=10⁻³⁴⁶, algebraic null (4.2M random formulas) P=10⁻¹³³, overdetermination 2.13×

Legacy v3.3.24 sector breakdown (preserved for traceability)

Metric	Value (v3.3.24)
Core dimensionless predictions	18
Extended dimensionless	15
Cosmological parameters	11
Structural constants	18
Total cataloged quantities	51
Mean deviation (32 well-measured)	0.24% (0.57% incl. δ_CP; PDG 2024 / NuFIT 6.0)
Exact matches (< 0.1%)	14 (42%)
Multiply determined (>=3 expr.)	92%
Total equivalent expressions	280+
Free parameters	0

1. GIFT Topological Constants

1.1 Primary Constants

Symbol	Value	Definition	mod 7	Factor
b_0	1	Zeroth Betti number	1	-
p_2	2	Duality parameter	2	-
N_gen	3	Number of generations	3	-
Weyl	5	Weyl factor	5	-
dim(K_7)	7	Compact manifold dimension	0	7
rank(E_8)	8	E_8 Cartan rank	1	-
D_bulk	11	Bulk dimension	4	-
alpha_sum	13	Anomaly sum	6	-
dim(G_2)	14	G_2 holonomy dimension	0	2x7
b_2	21	Second Betti number	0	3x7
dim(J_3(O))	27	Exceptional Jordan algebra	6	-
det(g)_den	32	Metric determinant denominator	4	2^5
2b_2	42	Structural constant (= p₂ × b₂)	0	6x7
dim(F_4)	52	F_4 dimension	3	-
fund(E_7)	56	E_7 fundamental representation	0	8x7
kappa_T^-1	61	Inverse torsion capacity	5	prime
det(g)_num	65	Metric determinant numerator	2	5x13
b_3	77	Third Betti number	0	11x7
dim(E_6)	78	E_6 dimension	1	-
H*	99	Total cohomology (b_2+b_3+1)	1	9x11
PSL(2,7)	168	Fano symmetry order	0	24x7
dim(E_8)	248	E_8 dimension	3	-
dim(E_8xE_8)	496	Gauge group dimension	6	-

1.2 Master Algebraic Identities

dim(G_2)       = p_2 x dim(K_7)           = 2 x 7   = 14
b_2            = N_gen x dim(K_7)         = 3 x 7   = 21
b_3 + dim(G_2) = dim(K_7) x alpha_sum     = 7 x 13  = 91
alpha_sum      = rank(E_8) + Weyl         = 8 + 5   = 13
D_bulk         = rank(E_8) + N_gen        = 8 + 3   = 11
2b_2           = p_2 x b_2                = 2 x 21  = 42  (structural constant)
H*             = b_2 + b_3 + 1            = 21+77+1 = 99

PSL(2,7) = 168 = rank(E_8) x b_2          = 8 x 21
               = N_gen x fund(E_7)        = 3 x 56
               = (b_3 + dim(G_2)) + b_3   = 91 + 77

2. Structural Inevitability Classification

Each observable receives a classification based on the number of independent algebraic expressions:

Classification	Criteria	Interpretation
CANONICAL	>=20 expressions	Maximally over-determined; value emerges from algebraic web
ROBUST	10-19 expressions	Highly constrained; multiple independent derivations
SUPPORTED	5-9 expressions	Multiply derived; structural redundancy
DERIVED	2-4 expressions	At least dual derivation
SINGULAR	1 expression	Unique derivation (possible numerical coincidence)

Cross-reference with GIFT_ATLAS.json status labels:

This document	Atlas equivalent	Mapping rationale
CANONICAL	VERIFIED	Lean-proven, maximally over-determined
ROBUST	VERIFIED	Multiple independent derivations confirm
SUPPORTED	TOPOLOGICAL	Direct topological consequence
DERIVED	TOPOLOGICAL	Algebraic consequence of topological invariants
SINGULAR	TOPOLOGICAL	Single derivation, but topologically grounded

3. Core 18 Dimensionless Predictions

3.1 Structural

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
1	N_gen	Atiyah-Singer index	3	3	0.00%	24+	CANONICAL

3.2 Electroweak Sector

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
2	sin^2 theta_W	b_2/(b_3+dim_G_2)	3/13 = 0.2308	0.23122	0.20%	19	ROBUST
3	alpha_s(M_Z)	sqrt(2)/(dim_G2 - p_2)	sqrt(2)/12 = 0.1179	0.1179	0.042%	9	TOPOLOGICAL
4	lambda_H	sqrt(17)/32	0.1288	0.129	0.12%	4	DERIVED
5	alpha^-1(M_Z)	128+9+corr	137.033	137.036	0.002%	3	DERIVED

3.3 Lepton Sector

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
6	Q_Koide	dim_G_2/b_2	2/3	0.666661	0.001%	27	CANONICAL
7	m_tau/m_e	7+10x248+10x99	3477	3477.15	0.004%	3	DERIVED
8	m_mu/m_e	27^phi	207.01	206.768	0.12%	2	DERIVED

3.4 Quark Sector

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
9	m_s/m_d	p_2^2 x Weyl	4 x 5 = 20	20.0	0.00%	14	VERIFIED
10	m_c/m_s	(dim_E8-p_2)/b_2	246/21 = 11.71	11.7	0.12%	5	SUPPORTED
11	m_b/m_t	1/(2b₂)	1/42 = 0.0238	0.024	0.79%	12	ROBUST
12	m_u/m_d	(1+dim_E6)/PSL_27	79/168 = 0.470	0.47	0.05%	4	DERIVED

3.5 Neutrino/PMNS Sector

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
13	delta_CP	dim_K7 x dim_G2 + H*	197 deg	197 deg +/- 24 deg	0.00%	3	DERIVED
14	theta_13^PMNS	pi/b_2	8.57 deg	8.54 deg	0.37%	3	DERIVED
15	theta_23^PMNS	arcsin((b_3-p_2)/H*) = arcsin(25/33)	49.25 deg	49.3 deg	0.10%	2	TOPOLOGICAL
16	theta_12^PMNS	arctan(sqrt(delta/gamma))	33.40 deg	33.41 deg	0.03%	2	DERIVED

3.6 Cosmological Sector

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
17	Omega_DE	ln(2)x(b_2+b_3)/H*	0.6861	0.6847	0.21%	2	DERIVED
18	n_s	zeta(11)/zeta(5)	0.9649	0.9649	0.004%	2	DERIVED

4. Extended Dimensionless Predictions (15)

4.1 PMNS sin^2 Form

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
19	sin^2 theta_12^PMNS	(1+N_gen)/alpha_sum	4/13 = 0.308	0.307	0.23%	21	CANONICAL
20	sin^2 theta_23^PMNS	(D_bulk-Weyl)/D_bulk	6/11 = 0.545	0.546	0.10%	13	ROBUST
21	sin^2 theta_13^PMNS	D_bulk/dim_E8^2	11/496 = 0.022	0.0220	0.81%	5	SUPPORTED

4.2 CKM Matrix

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
22	sin^2 theta_12^CKM	fund_E7/dim_E8	56/248 = 0.2258	0.2250	0.36%	16	ROBUST
23	A_Wolfenstein	(Weyl+dim_E6)/H*	83/99 = 0.838	0.836	0.29%	7	SUPPORTED
24	sin^2 theta_23^CKM	dim_K7/PSL_27	7/168 = 0.042	0.0412	1.13%	4	DERIVED

4.3 Boson Mass Ratios

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
25	m_H/m_t	fund_E7/b_3	56/77 = 0.7273	0.725	0.31%	16	ROBUST
26	m_H/m_W	(N_gen+dim_E6)/dim_F4	81/52 = 1.5577	1.558	0.02%	3	DERIVED
27	m_W/m_Z	(chi-Weyl)/chi	37/42 = 0.8810	0.8815	0.06%	8	SUPPORTED

Note: m_W/m_Z = 37/42 is a v3.3 correction. Previous formula (23/26) had 0.35% deviation; new formula achieves 0.06%.

4.4 Lepton Ratios Extended

#	Observable	GIFT Formula	Value	Exp.	Dev.	# Expr.	Status
28	m_mu/m_tau	(b_2-D_bulk)/PSL_27	10/168 = 0.0595	0.0595	0.04%	9	SUPPORTED

5. Cosmological Parameters (Complete)

5.1 Universe Composition

#	Observable	Planck 2018	GIFT	Value	Dev	# Expr
29	Omega_DM/Omega_b	5.375 +/- 0.1	(b_0+chi)/rank	43/8 = 5.375	0.00%	3
30	Omega_c/Omega_Lambda	0.387 +/- 0.01	det_g_num/PSL_27	65/168 = 0.3869	0.01%	5
31	Omega_Lambda/Omega_m	2.175 +/- 0.05	(dim_G2+H*)/dim_F4	113/52 = 2.173	0.07%	6
32	h (Hubble)	0.674 +/- 0.005	(PSL_27-b_0)/dim_E8	167/248 = 0.6734	0.09%	4
33	Omega_b/Omega_m	0.156 +/- 0.003	Weyl/det_g_den	5/32 = 0.1562	0.16%	7
34	Omega_c/Omega_m	0.841 +/- 0.01	(dim_E8^2-dim_E6)/dim_E8^2	0.8427	0.17%	4
35	sigma_8	0.811 +/- 0.006	(p_2+det_g_den)/chi	34/42 = 0.8095	0.18%	3
36	Omega_m/Omega_Lambda	0.460 +/- 0.01	(b_0+dim_J3O)/kappa_T	28/61 = 0.459	0.18%	5
37	Y_p (primordial He)	0.245 +/- 0.003	(b_0+dim_G2)/kappa_T	15/61 = 0.2459	0.37%	4
38	Omega_Lambda/Omega_b	13.9 +/- 0.3	(dim_E8^2-dim_F4)/det_g_den	13.875	0.14%	3
39	Omega_b/Omega_Lambda	0.072 +/- 0.002	b_0/dim_G2	1/14 = 0.0714	0.75%	2

5.2 The 42 in Cosmology

Notable result:

\[\frac{\Omega_{DM}}{\Omega_b} = \frac{b_0 + 2b_2}{\text{rank}(E_8)} = \frac{1 + 42}{8} = \frac{43}{8} = 5.375\]

The ratio of dark matter to baryonic matter explicitly contains the structural constant 2b₂ = 42.

Note: The Euler characteristic χ(K₇) = 0 for any compact odd-dimensional manifold like K₇. The value 42 = p₂ × b₂ is a distinct structural constant derived from Betti numbers.

6. Structural Constants (18)

6.1 E_8 Structure

#	Constant	Value	Definition	# Expr.	Status
40	dim(E_8)	248	E_8 Lie algebra dimension	5+	SUPPORTED
41	rank(E_8)	8	Cartan subalgebra	3+	DERIVED
42	dim(E_8 x E_8)	496	Product group	2	DERIVED
43	**	W(E_8)	**	696,729,600	Weyl group order	1	SINGULAR

6.2 G_2 & K_7 Topology

#	Constant	Value	Definition	# Expr.	Status
44	dim(G_2)	14	Holonomy group	4+	DERIVED
45	dim(K_7)	7	Compact manifold	5+	SUPPORTED
46	b_2(K_7)	21	Second Betti (gauge moduli)	3+	DERIVED
47	b_3(K_7)	77	Third Betti (matter modes)	3+	DERIVED
48	H*	99	b_2+b_3+1 (total cohomology)	5+	SUPPORTED
49	2b₂	42	Structural constant (p₂ × b₂)	3+	DERIVED

6.3 Exceptional Algebras

#	Constant	Value	Definition	# Expr.	Status
50	dim(J_3(O))	27	Exceptional Jordan	2+	DERIVED
51	dim(F_4)	52	F_4 dimension	3+	DERIVED

7. Over-Determination Analysis

7.1 Top Equivalent Expressions by Fraction

Fraction	Observable	# Expressions
2/3	Q_Koide	27
21/7 = 3	N_gen	24
4/13	sin^2 theta_12^PMNS	21
3/13	sin^2 theta_W	19
8/11 = 56/77	m_H/m_t	16
56/248	sin^2 theta_12^CKM	16
1/42	m_b/m_t	12
6/11	sin^2 theta_23^PMNS	13
37/42	m_W/m_Z	8

Total: 280+ expressions for major observables

7.2 Example: Q_Koide = 2/3 (27 expressions)

#	Expression	Computation
1	p_2 / N_gen	2/3
2	dim_G_2 / b_2	14/21 = 2/3
3	dim_F_4 / dim_E_6	52/78 = 2/3
4	rank_E_8 / (Weyl + dim_K_7)	8/12 = 2/3
5	chi / (b_2 + chi)	42/63 = 2/3
…	…	…

7.3 Statistical Significance

For random numerology with ~20 constants:

Expected expressions per fraction: ~1-2
Observed: ~16 average

Probability of this by chance: p < 10^-12

The structure is real, not coincidental.

8. Statistical Distribution

8.1 By Deviation (33 observables)

Range	Count	%	Examples
Exact (0%)	2	6%	N_gen, Omega_DM/Omega_b
< 0.1%	12	36%	Q_Koide, m_H/m_W, m_W/m_Z, h
0.1-0.5%	12	36%	sin^2 theta_W, m_mu/m_e, m_H/m_t
0.5-1%	5	15%	m_b/m_t, sin^2 theta_13^PMNS
> 1%	2	6%	sin^2 theta_23^CKM

8.2 By Category

Category	Observables	Mean Dev	Best Match
Electroweak	4	0.27%	m_W/m_Z (0.06%)
PMNS	4	0.29%	sin^2 theta_23 (0.10%)
Quark masses	5	0.35%	m_s/m_d (0.00%)
Lepton masses	2	0.04%	m_mu/m_tau (0.04%)
Boson masses	3	0.13%	m_H/m_W (0.02%)
CKM	4	0.59%	A_Wolf (0.29%)
Cosmology	11	0.16%	Omega_DM/Omega_b (0.00%)
Total	32+1	0.24% (excl. δ_CP)	-

8.3 By Structural Classification

Classification	Count	%
CANONICAL	4	12%
ROBUST	8	24%
SUPPORTED	12	36%
DERIVED	8	24%
SINGULAR	1	3%

9. Unique Expressions (Caution)

Observables with only one GIFT expression (possible numerical coincidence):

Observable	Expression	Value	Status
	W(E_8)		696,729,600	-	Definition

10. Uniqueness Analysis

10.1 Gauge Group Uniqueness

E₈×E₈ is optimal among all tested physically motivated gauge groups.

Rank	Gauge Group	Mean Dev	N_gen	Status
1	E₈×E₈	0.24%	3.000	✓ OPTIMAL
2	E₇×E₈	3.06%	2.625	✗
3	E₆×E₈	5.72%	2.250	✗
4	E₇×E₇	6.05%	2.625	✗
5	SO(32)	6.82%	6.000	✗
6	E₆×E₆	14.52%	2.250	✗

Improvement factor: E₈×E₈ is 12.8× better than the next best (E₇×E₈).

Why rank=8 is special:

N_gen = (rank × b₂) / (b₃ - b₂) = (rank × 21) / 56

For N_gen = 3 exactly: rank = 168/21 = 8 ✓
Note: 168 = |PSL(2,7)| = Fano plane symmetry order

Only E₈ (rank 8) gives exactly 3 generations.

10.2 Holonomy Uniqueness

G₂ holonomy achieves significantly better agreement. Calabi-Yau manifolds show poor results.

Rank	Holonomy	dim_K	SUSY	Mean Dev	Status
1	G₂	7	N=1	0.24%	✓
2	SU(4)	8	N=1	0.71%	✗
3	SU(3)	6	N=2	3.12%	✗✗
4	Spin(7)	8	N=0	3.56%	✗✗

Calabi-Yau penalty: SU(3) holonomy fails by factor 13×.

10.3 The PSL(2,7) Connection

N_gen = |PSL(2,7)| / fund(E₇) = 168 / 56 = 3
      = |Fano_symmetry| / E₇_fundamental

The number of generations equals Fano plane symmetry order / E₇ representation dimension.

This is not numerology, it’s the octonionic Fano structure manifesting in particle generations.

10.4 Validation Script

Full analysis available: publications/validation/validation_v33.py

python publications/validation/validation_v33.py

Results: publications/references/observables.csv

11. Falsification Schedule

Prediction	Current	Target	Experiment	Year
delta_CP = 197 deg	+/- 24 deg	+/- 10 deg	DUNE (first results)	2028-2030
delta_CP = 197 deg	+/- 10 deg	+/- 5 deg	DUNE (precision)	2034-2039
sin^2 theta_W = 3/13	+/- 0.00004	+/- 0.00001	FCC-ee	2040s
N_gen = 3	3	4th gen?	LHC/FCC	ongoing
m_s/m_d = 20	+/- 1.0	+/- 0.3	Lattice QCD	2030
Q_Koide	+/- 0.000007	+/- 0.000001	tau factories	2030s

Note: DUNE timeline follows Snowmass 2021 projections. First beam ~2028; +/- 5 deg precision requires extended operation through late 2030s.

12. The Balmer Analogy

Aspect	Balmer (1885)	GIFT
Empirical formula	lambda = B x n^2/(n^2-4)	sin^2 theta_W = 3/13
Fits experiment	Yes	Yes
Unique formula	Yes	Yes (up to equivalence)
Derivation came later	Bohr (1913), QM (1926)	?

13. References

Harvey, R., Lawson, H.B. “Calibrated geometries.” Acta Math. 148 (1982)
Joyce, D.D. Compact Manifolds with Special Holonomy. Oxford (2000)
Koide, Y. “Fermion-boson two-body model.” Lett. Nuovo Cim. 34 (1982)
Particle Data Group (2024), Review of Particle Physics
Planck Collaboration (2020), Cosmological parameters
GIFT Publications: Paper Main Framework, Paper S2 Derivations

GIFT Framework v3.4 - Observable Reference