Derivation: e as Attractor of Independent BC_open

Step E4.2 — from BC structure to CKM θ₁₃

ArXe Research · July 2026 · Diego Luis Tentor
Derivation document — part of the route toward E4

1. The result to derive

θ₁₃ CKM = arcsin(3×5 / e×11²×13) = 0.201°. The factor e appears in the denominator. The question is why e — not φ, not π, not γ — and what BC process generates it.

2. Reference: how φ emerges from BC

The corpus establishes (Grammar §2.6) that φ emerges from the BC recursion of T⁻¹:

T⁻¹: BC_open = 1
Additive recursion: a_{n+1} = a_n + a_{n-1}
Attractor: a_{n+1}/a_n → φ = 1.6180...

The key to φ: states are dependent — each remembers the two previous ones. There is memory. The ratio converges because BC_open accumulates history.

3. The process that generates e

The fundamental process converging to e is:

lim_{n→∞} (1 + 1/n)^n = e = 2.71828...

Its BC reading is direct:

n states in T⁻¹, each with BC_open = 1
Each state contributes with weight 1/n to the total process
The n states are independent — no coupling, no memory
The product of n independent contributions of size 1/n converges to e

The structural difference from φ:

	φ	e
Dependence between states	YES — complete memory	NO — independent
Type of accumulation	Additive	Multiplicative
Recursion	a_{n+1} = an + a{n-1}	product of (1+1/n)
BC structure	One BC_open with history	n BC_open without coupling

φ is the attractor of recursion with memory. e is the attractor of accumulation without memory.

4. BC structure of the e-process

The e-process requires n independent BC_open without coupling. In the ArXe hierarchy:

T⁻¹ has BC_open = 1: one open degree of freedom.
For e, n copies of T⁻¹ without interaction are needed — the tensor product (T⁻¹)^⊗n.
In the limit n→∞, the attractor of the normalized product is e.

Formal definition:

e-process of order n: let {BCopen^(k)}{k=1}^n be a set of n open boundary conditions where each has weight 1/n and no coupling to the others. The attractor of the product of their contributions as n→∞ is e.

This is the BC version of the Poisson process: n independent events, each with probability 1/n of occurring, in the large-n limit.

5. Why CKM θ₁₃ carries e

5.1 Physical context

θ₁₃ is the mixing angle between the first and third quark generation — the most suppressed of the three CKM angles (0.201°, nearly twenty times smaller than θ₁₂). This extreme suppression reflects that the 1↔3 coupling is very weak.

5.2 The BC reading

The 1↔3 jump in the quark sector operates in a confined medium — T⁻³ (QCD, BC_closed=2) prevents long-range coherence. The 1↔3 coupling does not occur through a direct route but through multiple virtual routes (quantum propagators).

In a confined medium, those virtual routes are independent of each other — confinement prevents them from interfering coherently. Each route is a small-amplitude BC_open. The set of routes is exactly an e-process: n independent BC_open accumulating without memory.

The attractor is e. That is why e appears in the denominator of the θ₁₃ expression: the independent routes suppress the mixing amplitude by a factor e.

5.3 Why in the denominator

In the expression arcsin(15 / e×11²×13):

The numerator (15 = 3×5) encodes the jump structure in pure arities.
The denominator carries e as a suppression factor.
e in the denominator = amplitude suppressed by the independent-routes process.
The greater the independence (more routes, more confinement), the greater the suppression, the smaller the angle.

This is coherent with θ₁₃ being the smallest angle: it is where confinement acts most strongly on route coherence.

6. Comparison with PMNS θ₁₃

θ₁₃ PMNS = arcsin(φ×2×3×13 / 7×11²) = 8.57° — carries φ, not e.

The 1↔3 jump in leptons operates in the continuum (BC_open in T⁻¹). Routes are coherent — the continuous medium allows constructive interference. Accumulation is with memory. The attractor is φ.

Same jump (1↔3), same structure (both carry 13), different medium (discrete vs continuous), different constant (e vs φ), different angle (0.201° vs 8.57°).

7. Residual gap (closed — see update below)

The derivation is structurally complete but has one gap:

What is derived: e emerges from n independent BC_open without coupling. The e-process is the multiplicative attractor of memory-free contributions.

What is missing: formally demonstrating that BC_closed of T⁻³ implies independence of virtual routes. The connection confinement → decoherence → e-process is physically intuitive but not yet formalized in purely BC terms.

This gap is closed in arxe_formal_gap_closure_en.md §2–3 — see the update note in §8 below.

8. Epistemic state

Element	State
e = attractor of n independent BC_open	Derived — process (1+1/n)^n
Distinction φ (dependent) vs e (independent)	Established — structurally clear
Confinement T⁻³ → route independence	Intuitive — not formalized in BC
e in denominator = suppression by independence	Coherent — not demonstrated
θ₁₃ CKM carries e, θ₁₃ PMNS carries φ	Verified pattern — consistent BC reading

Evidential strength: E4 partial — the e-process structure is derived; the connection to T⁻³ confinement is the residual gap.

Update (arxe_formal_gap_closure_en.md, July 2026): the residual gap noted above — the connection between BC_closed(T⁻³) and route independence — is now closed. BC_closed=2 plus BC_open=1 forces serialization of virtual routes (§2 of the gap-closure document), and uniform route length for the d=2 jump gives uniform weights 1/n, i.e. the e-process (§3.2). Status upgraded to E4 complete. See §7 and §9 there for the full chain and the final closure of the amplitude-cost sub-claim.

9. Prediction

If a confined BSM fermionic sector with a 1↔3 jump exists, its 1↔3 mixing angle should carry e in the denominator — regardless of the angle’s magnitude.

If that sector is free (not confined), it should carry φ.

ArXe Research — July 2026
Diego Luis Tentor

“φ remembers. e forgets. The difference is confinement.”

Arxe Theory

arxe_derivation_e_BC_en (1)