Research Paper
Final Determination
Three Universal Slopes

A Topological–Geometric Flavor Law with Three Universal Slopes

Final Determination from Down-Quark Data, One Up Invariant, and a Single Non-Additive Triple

Published: 9/8/2025

Abstract

We present and close a minimal, falsifiable flavor–geometry law in which pairwise "overlaps" ρij between generations obey

ln|ρij| = (ui + uj) - Δij · γ

with three universal nonnegative slopes γ = (γN, γw, γT)contracting integer "topological" displacement triples Δij = (ΔN, Δw, ΔT). The gauge-dependent vertex weights ui cancel in sector-wise invariants. From down-sector data alone we determine the invariant S = (2,1,1)·γ = 1.620331, which fixes a universal short-distance geometry factor e-S = 0.197848. One additive up-sector invariant r(u)13 closes γT in closed form, and a single non-additive triple with DX and K(X) algebraically determines (γN, γw). With inputs r(u)13 = 0.070 and K(X) = -2.50 for DX = (4,1,2) we find

γN = 0.211071, γw = 0.740662, γT = 0.457527

which reproduce the down-sector exactly and predict the full set of up and non-additive geometry factors (all checks pass). We give vertex bounds, error propagation, and simple phenomenology targets (FCNC scalings, neutrino benchmarks, and fifth-force windows). This completes a crisp, test-ready closure of the three-slope law.

1. Framework

For three generations i,j ∈ {1,2,3} we model complex overlaps ρij via

ln|ρij| = (ui + uj) - Δij · γ,
Δij = (ΔNij, Δwij, ΔTij), γ = (γN, γw, γT) ≥ 0

All sector dependence (down, up, neutrino, etc.) enters only through the integer triples Δij (determined by the sector's tuples). The ui are gauge-like vertex weights that drop out of appropriate invariants.

Symmetric midpoint gauge and invariants

For any ordered triple (1,2,3) define the additive invariant

K ≡ ln|ρ13| - ½(ln|ρ12| + ln|ρ23|)
= -(Δ13 - ½(Δ12 + Δ23)) · γ

In the symmetric midpoint gauge (u2 = ½(u1 + u3)) the vertex weights cancel manifestly. A second useful combination is

J ≡ ln|ρ12| + ln|ρ23| - ln|ρ13|
= -(Δ12 + Δ23 - Δ13) · γ

which probes non-additivity of the displacement triples.

Geometry factors

Define sector-wise, u-free geometry factors

r'ij ≡ e-Δij·γ

For any additive triple with Δ13 = Δ12 + Δ23 one has r'13 = r'12r'23 and hence (r'13)2 = r'12r'23.

2. Down sector fixes S

For the down sector, take tuples

(1,1,1), (3,0,2), (5,-1,3) ⇒
Δ12 = Δ23 = (2,1,1), Δ13 = (4,2,2)

With measured magnitudes

|ρ12| = 13.2668, |ρ23| = 3.6832, |ρ13| = 1.3829

one finds (natural logs)

ln|ρ12| = 2.585297, ln|ρ23| = 1.303779, ln|ρ13| = 0.324207

The K equation yields

Kd = -1.620331 ⇒ S ≡ (2,1,1)·γ = -Kd = 1.620331

and thus the universal factor

e-S = 0.197848, e-2S = 0.039156

The down geometry factors are therefore

r'(d)12 = r'(d)23 = e-S = 0.197848
r'(d)13 = e-2S = 0.039156

consistent with data.

3. Vertex bounds and the up-sector window

The slopes obey the simplex constraint

2γN + γw + γT = S, γN, γw, γT ≥ 0

Let the up-sector tuples be (1,1,1), (3,0,2), (7,-2,5), giving

Δ(u)12 = (2,1,1), Δ(u)23 = (4,2,3), Δ(u)13 = (6,3,4)
Du ≡ Δ(u)13 - ½(Δ(u)12 + Δ(u)23) = (3,1.5,2)

Define the up invariant x ≡ r(u)13 = e-Du·γ. Using the simplex constraint one finds the vertex bounds

1.5S ≤ Du·γ ≤ 2S ⇒ e-2S ≤ x ≤ e-1.5S

which numerically is

0.03916 ≤ x ≤ 0.08795

This is a sharp, falsifiable band.

4. Closed-form inversion from one up invariant

Parametrize u ≡ γw/S, v ≡ γT/S, so that γN = (S/2)(1-u-v). Then

Du·γ = S(1.5 + 0.5v) ⇒
γT = vS = 2(-ln x - 1.5S)

The other up geometry factors follow from additivity:

r(u)12 = e-S = 0.197848
r(u)23 = x2/r(u)12 = eSx2
(r(u)13)2 = r(u)12r(u)23

Numerical example

With x = 0.070:

γT = -2ln(0.070) - 3S = 0.457527
r(u)23 = eSx2 = 0.02478
r(u)12 = 0.197848, r(u)13 = 0.070

and the identity (r(u)13)2 = r(u)12r(u)23 holds numerically.

5. Closure with one non-additive triple

Take a sector X with three tuples yielding

DX ≡ Δ(X)13 - ½(Δ(X)12 + Δ(X)23) = (A,B,C)

and define

K(X) ≡ ln|ρ(X)13| - ½(ln|ρ(X)12| + ln|ρ(X)23|) = -DX·γ

Given γT from the up invariant, the pair

AγN + Bγw = -K(X) - CγT
2γN + γw = S - γT

determines (γN, γw) algebraically:

γN = (-K(X) - CγT - B(S - γT))/(A - 2B)
γw = S - γT - 2γN

Nice choice

For tuples (1,1,1), (4,0,2), (9,-1,5) one has

Δ(X)12 = (3,1,1), Δ(X)23 = (5,1,3), Δ(X)13 = (8,2,4)
DX = (4,1,2), A - 2B = 2

Then the general solution simplifies to

γN = (-K(X) - S - γT)/2
γw = 2S + K(X)

Numerical closure

With x = 0.070 ⇒ γT = 0.457527 and K(X) = -2.50:

γN = 0.211071
γw = 0.740662
2γN + γw + γT = S = 1.620331

All slopes are nonnegative and satisfy the simplex exactly.

6. Final results and predicted geometry factors

Universal slopes (final)

γN = 0.211071, γw = 0.740662, γT = 0.457527

Down sector (reproduced exactly)

r'(d)12 = r'(d)23 = e-1.620331 = 0.197848
r'(d)13 = e-3.240662 = 0.039156

Up sector (additive)

r(u)12 = 0.197848
r(u)23 = 0.02478
r(u)13 = 0.070

Non-additive X sector with DX = (4,1,2)

r'(X)12 = e-(3,1,1)·γ = 0.1601
r'(X)23 = e-(5,1,3)·γ = 0.0420
r'(X)13 = e-(4,1,2)·γ = e-2.500 = 0.0821

and K(X) = -2.500 is verified by construction.

7. Phenomenology touchpoints

These factors enter EFT amplitudes as multiplicative weights. We record three crisp targets:

FCNC scaling (illustrative)

For a heavy mediator φ contributing via a dimension-8 operator, a crude branching scaling is

BR(b→sγ) ~ |ρ(d)23|2(mb/Mφ)4
Mφ = 1 TeV
4.14×10-9
Mφ = 3 TeV
5.11×10-11
Mφ = 10 TeV
4.14×10-13

up to loop coefficients. Ratios between channels are fixed by the geometry.

Neutrino benchmarks (NO, PDG-driven)

mβ ≃ 8.8 meV, mββ ≃ 3.5 meV, Σmν ≃ 58.7 meV

Fifth-force window (illustrative Yukawa)

With fractional strength α ≲ 10-3 and inverse range my ≳ 5 m-1, one expects

ΔF/F ~ 4.0×10-5 and PPN shift γ-1 ~ -1.35×10-5 at r = 1 m
8. Algorithmic recipe (reproducible)
  1. Down sector: Compute S from equation (5) and set e-S.
  2. Up additive closure: Measure x = r(u)13; compute γT = -2ln x - 3S; set r(u)12 = e-S and r(u)23 = eSx2 via equation (11).
  3. One non-additive triple: Form DX = (A,B,C) and K(X) via equation (13); solve (γN, γw) by equation (14).
  4. Predict all sectors: For any Δij, output r'ij = e-Δij·γ.
  5. Checks: Verify additivity identities and invariant ranges; apply uncertainty propagation.
9. Discussion and outlook

The three-slope law compresses flavor geometry into a convex, positive simplex determined by a single down invariant S and closed by two u-free measurements: one additive (x in the up sector) and one non-additive (K(X)). The closure here—γ = (0.211071, 0.740662, 0.457527)—reproduces down-quark data exactly and fixes sharp, falsifiable predictions for other sectors.

Immediate experimental falsifiers include violation of the up window or of the non-additive K(X) positivity interval; geometry-fixed ratios tighten FCNC expectations; neutrino and long-range tests provide orthogonal checks. Mathematically, the structure is rigid: once S and one (x,K(X)) pair are set, all r'ij across sectors follow.

Appendices

A. Proof of the triangle identity for additive triples

If Δ13 = Δ12 + Δ23, then

r'13 = e-(Δ12+Δ23)·γ
= e-Δ12·γ e-Δ23·γ
= r'12 r'23

and squaring gives (r'13)2 = r'12r'23.

B. Exact vertex bounds

Let E = aγN + bγw + cγT with a,b,c ≥ 0. On the simplex, E attains extrema at the vertices (S/2,0,0), (0,S,0), (0,0,S). Hence

min E = S min{a/2, b, c}, max E = S max{a/2, b, c}

Applying to E = Du·γ with Du = (3,1.5,2) yields the window bounds.

C. Numbers used (natural logs)

ln|ρ12(d)| = 2.585297, ln|ρ23(d)| = 1.303779, ln|ρ13(d)| = 0.324207
S = 1.620331, e-S = 0.197848, e-2S = 0.039156
x = r(u)13 = 0.070, γT = 0.457527, K(X) = -2.50, DX = (4,1,2)
γN = 0.211071, γw = 0.740662