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Derivation: γ as Attractor of BC_open with Decreasing Memory

Derivation: γ as Attractor of BC_open with Decreasing Memory

Step E4.3 — from BC structure to CKM θ₂₃

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

1. The result to derive

θ₂₃ CKM = arctan(3²×γ / 5³) = 2.38°. The factor γ (Euler-Mascheroni constant) appears in the numerator. The question is why γ — and what BC process generates it.

2. Definition and convergence of γ

H_n = 1 + 1/2 + 1/3 + ... + 1/n   (harmonic series)
γ = lim_{n→∞} (H_n - ln(n)) = 0.5772156649...

γ is the difference between the discrete accumulation of harmonic weights and its continuous limit (the logarithm). It is the residual irregularity that does not cancel when the discrete sum is compared with its integral.

n=10:      H_n - ln(n) = 0.62638
n=100:     H_n - ln(n) = 0.58221
n=1000:    H_n - ln(n) = 0.57772
n=100000:  H_n - ln(n) = 0.57722
γ =                      0.57722

3. The memory spectrum — φ, γ, e

The three constants are attractors of BC_open accumulation processes that differ in one parameter: the memory structure between states.

Constant Accumulation type Memory between states Attractor
φ Additive Complete — a_{n+1} = an + a{n-1} φ = 1.618…
γ Additive Decreasing — weight 1/k for the k-th γ = 0.577…
e Multiplicative None — independent states e = 2.718…

The axis of the spectrum is memory:

  • φ: each state completely remembers the two previous ones. The ratio between successive states converges to φ.
  • γ: each state remembers the previous ones, but with weight 1/k — the past matters but progressively less. The residual irregularity between the discrete sum and its continuous limit is γ.
  • e: states do not remember each other. Each contribution is autonomous. The product of independent contributions converges to e.

4. BC reading of the γ-process

Definition: γ-process

Let {BCopen^(k)}{k=1}^n be a set of n open boundary conditions where the k-th BC_open operates with weight w_k = 1/k.

Accumulation: S_n = Σ_{k=1}^n (1/k) = H_n
Continuous limit: ∫₁^n (1/x) dx = ln(n)
Residual irregularity: H_n - ln(n) → γ

The BC_open of the γ-process are not independent (as in e) but neither do they have complete memory (as in φ). They have decreasing memory: the k-th BC_open “sees” the previous ones but with weight inversely proportional to its position.

Geometric interpretation: γ is what remains when discrete space (sum) attempts to approximate continuous space (integral) through a ladder of harmonic weights. It is not error — it is structure. It is the irreducible difference between the discrete and the continuous under 1/k weights.

5. The γ-process as superposition of scales

Unlike φ (one level, T⁻¹) and e (n independent copies of T⁻¹), the γ-process is not a process at a single level but the superposition of processes at different scales where each scale k contributes with weight 1/k.

In the ArXe hierarchy, this corresponds to a system where levels T⁻¹, T⁻², T⁻³… contribute simultaneously, with weights inversely proportional to their position. There is no dominant scale — the contribution distributes through the hierarchy with harmonic decay.

6. Why CKM θ₂₃ carries γ

6.1 Physical context

θ₂₃ is the mixing angle between the second and third quark generation — the larger of the two suppressed angles (2.38°, adjacent in the Cayley graph but confined).

6.2 The intermediate regime

The 2↔3 jump is adjacent (distance 1 in the Cayley graph of S₃). Unlike the 1↔3 jump (long-range, totally incoherent → e), the adjacent 2↔3 jump maintains partial correlation between mixing routes.

In the confined sector:

  • Routes are not completely independent (there is correlation between adjacent generations).
  • But they do not have complete memory either (confinement degrades coherence).
  • The weight of each contribution decays with “distance” in the space of virtual routes — harmonically.

That regime is exactly the γ-process: correlated contributions with decreasing memory.

6.3 The complete physical spectrum

φ: complete memory   → PMNS (continuous, free leptons)
γ: decreasing memory → CKM θ₂₃ (confined, adjacent d=1)
e: no memory         → CKM θ₁₃ (confined, long-range d=2)
—: no constant       → CKM order-one (closed BC, arithmetic)

The spectrum φ→γ→e is the spectrum of memory degradation under increasing confinement and generational distance:

  • Leptons (continuous): complete memory → φ
  • Adjacent quarks (confined, d=1): decreasing memory → γ
  • Long-range quarks (confined, d=2): no memory → e

6.4 Why γ in the numerator

In the expression arctan(3²×γ / 5³):

  • γ in the numerator = harmonic irregularity slightly amplifies mixing over the pure arity ratio.
  • Contrasts with e in the denominator for θ₁₃ = route independence suppresses mixing.
  • Coherent with θ₂₃ > θ₁₃: the γ-regime produces greater mixing than the e-regime, because there is residual correlation not completely cancelled.

7. Unified spectrum table

Angle Constant Position Medium Memory Effect
θ₁₂ PMNS=33° — implicit Continuous Complete Pure golden ratio
θ₂₃ PMNS=49° — idem Continuous Complete Pure golden ratio
θ₁₃ PMNS=8.6° φ Numerator Continuous Complete φ amplifies
θ₁₂ CKM=13° — — Discrete — No constant
δ CKM=65° — — Discrete — No constant
θ₂₃ CKM=2.4° γ Numerator Confined, d=1 Decreasing Residual irregularity
θ₁₃ CKM=0.2° e Denominator Confined, d=2 None Independent suppression

The pattern is complete and without exceptions.

8. Residual gaps (closed — see update in §9 below)

Gap 1: The connection between BC_closed of T⁻³ and the memory structure of mixing routes is not formalized in purely BC terms. The argument confinement → memory degradation is physically intuitive but requires formal derivation.

Gap 2: Why the decay is specifically harmonic (1/k) and not exponential (e^{-k}) or polynomial (1/k²). The harmonic weight is what produces γ — other weights would produce other constants. Why the ArXe hierarchy generates harmonic weights in this context is not yet derived.

Both gaps are closed in arxe_formal_gap_closure_en.md §2–3 and §9 — see the update note in §9 below.

9. Epistemic state

Element State
γ = attractor of decreasing-memory accumulation Established mathematically
φ→γ→e spectrum as memory degradation Coherent — emergent, not sought
γ in θ₂₃ CKM = intermediate correlation regime Consistent interpretation
Harmonic weight 1/k as natural BC cascade structure Hypothesis — not derived
Connection confinement T⁻³ → decreasing memory Intuitive — formal gap

Evidential strength: E3+ → E4 partial — the φ/γ/e spectrum as memory degradation is the strongest conceptual result; complete BC formalization is the residual gap.

Update (arxe_formal_gap_closure_en.md, July 2026): both residual gaps above are now closed. BC_closed(T⁻³)=2 with BC_open=1 forces serialization of virtual routes (§2); for the adjacent d=1 jump, routes of all lengths k=1,2,3,… contribute with weight 1/k (unit amplitude cost per channel use, §9.2), giving the harmonic series whose residual H_n − ln(n) → γ, verified numerically to 5 significant figures (§9.3). The “why harmonic and not exponential” question (Gap 2 above) is answered: BC_open=1 has unit transmission amplitude by canonical normalization, so a route of length k costs exactly 1/k — no other decay law is consistent with BC_open=1. Status upgraded to E4 complete.

10. Joint result — the three processes

With the three derivations (13, e, γ), the complete map of BC attractors for fermionic mixing is:

ΦPROCESS (complete memory, golden recursion of T⁻¹):
  a_{n+1} = a_n + a_{n-1},  ratio → φ
  → PMNS, all angles — continuous space, free leptons

γ-PROCESS (decreasing memory, harmonic cascade across scales):
  H_n - ln(n) → γ
  → CKM θ₂₃ — confined, adjacent (d=1), partial correlation

e-PROCESS (no memory, independent BC_open):
  lim(1+1/n)^n → e
  → CKM θ₁₃ — confined, long-range (d=2), total decoherence

NO CONSTANT (closed BC, rational ratio):
  → CKM θ₁₂ and δ — discrete, order-one, no irrational attractor

The four categories cover all seven angles without exception.

ArXe Research — July 2026
Diego Luis Tentor

“φ remembers everything. γ remembers less and less. e remembers nothing.
Confinement erases memory. Generational distance erases it faster.”