Appendix

Five-Term Functional and Robustness Scans

The free-energy functional that produces stable condensations, and the robustness scans that show the 3+e configuration is the minimum-energy outcome across parameter space.

Appendix

The Full Five-Term Functional and Robustness Scans

1. The Full Five-Term Functional

The Big Flare-Up Theory (BFUT) condensation energy functional has five terms. Each term encodes a distinct physical mechanism. Together they determine which configuration of substrate units is energetically preferred.

The full functional evaluated for a configuration of n units with k co-rotating:

E = coop + imb + geom + c3ph

1.1 Term by Term

Term 1 - Cooperation (coop)

coop = -J x pairs_sum(s)

pairs_sum(s) = sum of s_i x s_j over all distinct pairs i < j

Physical meaning: Co-rotating units attract each other by the Bernoulli mechanism. High substrate velocity at the shared interface between two co-rotating regions creates low pressure, drawing them together. The cooperation energy grows with the number of co-rotating pairs.

The jump from k=2 (one pair, -J) to k=3 (three pairs, -3J) is qualitative not gradual. This is why the 3-core is the first stable nucleus. Below k=3 the core cannot survive substrate fluctuations.

Term 2 - Imbalance Penalty (imb)

imb = λ x (sum(s))^2

Physical meaning: A net circulation asymmetry costs energy. If all units circulate in the same direction, sum(s) = n and the penalty is large. The balanced 2+2 configuration has sum(s) = 0 and zero penalty. The 3+1 configuration has sum(s) = 3-1 = 2, giving a moderate penalty λ x 4 = 2.4.

Term 3 - Geometric Cost (geom)

geom = (k-3)^2 + α x (n-k)

Physical meaning: Two independent geometric costs. First, (k-3)^2 penalises deviation of the primary group size from 3 - the three-sphere close-packing geometry. Second, α x (n-k) penalises each counter-circulating unit for the geometric asymmetry it introduces. When the expelled unit has mass fraction μ, this term scales as α x μ.

Term 4 - Circulation Phase Reward (c3ph)

c3ph = D_s x cos(3 x φ)

Physical meaning: The fifth term explicitly encodes the topology of the three-sphere packing into the energy functional. The phase φ measures the circulation configuration:

D_s must be positive. If D_s were negative, the 4+0 configuration would be the energy minimum and no stable charged matter would form. The fifth term gives E(4+0) = 6.10, with a 1D robustness of 97.56%, and explicitly encodes the three-sphere topology into the functional.

1.2 Parameters

2. The Per-Unit Energy Scan (Code 1)

Code 1 answers: for n co-rotating substrate units, which n minimises energy per unit E(n)/n? All units are at the primary phase φ = π/3, so cos(3φ) = -1 and c3ph = -D_s for all n.

Figure A. Total condensation energy E(n) for n=1 to 12. n=3 highlighted.

Figure B. Energy per unit E(n)/n for n=1 to 12. n=3 is the unambiguous minimum.

3. The Four-Unit Partition (Post 3-Core Formation)

The moment n=3 forms, the three-sphere packing geometry simultaneously creates the interstitial region. The four-unit bound system forms at E=1.400 model units. The three configurations and their full-functional energies:

Figure C. Four-unit partition energies. 3+1 is the clear minimum.

Figure D. Energy through the three stages of proton formation.

4. Three-Sphere Packing Geometry

Three substrate condensations of radius r_q in close-packed contact. The three centres form an equilateral triangle of side 2r_q. The outer radius of the assembly equals r_p, the measured proton charge radius. This is the only measured input.

r_outer = r_q x (1 + 2/sqrt(3)) = 2.1547 x r_q = r_p

r_q = r_p / (1 + 2/sqrt(3)) = 0.8414 / 2.1547 = 0.3905 fm

V_gap / V_q = (2*sqrt(3) - π) / (4*π/3) = 0.0770

Both results are universal geometric constants. No free parameters.

Figure E. Three-sphere packing geometry. Green arrows: co-rotating quarks. Red: interstitial unit (counter-rotates). Yellow dashed: outer radius = r_p.

5. Why the Interstitial Unit Counter-Rotates

The counter-rotation is mechanically imparted, not assumed. When the interstitial substrate exits through the gap between any two quarks, it encounters two co-rotating surfaces - one on each side. Both quarks rotate in the same direction. Each imparts a tangential force in the opposite direction to the passing substrate. Together they impart a net counter-clockwise torque.

This is the gear analogy: a gear placed between two co-rotating gears of the same handedness always rotates in the opposite direction. The result is the same regardless of which gap the substrate exits from, because all three quarks rotate in the same direction.

Counter-rotation in BFUT is the definition of opposite charge. The negative charge of the expelled unit is therefore not assigned or assumed. It is mechanically imparted during expulsion by the same co-rotation that defines the quarks as positively charged.

6. Proton Energy with Interstitial Unit

When the detached unit has mass fraction μ relative to a core unit, the P16 functional is evaluated with s = [+1, +1, +1, -μ]. The geometric asymmetry term scales with μ because the geometric displacement is proportional to the actual mass of the detached unit:

E(μ) = -J x pairs_sum([1,1,1,-μ])

+ λ x (3-μ)^2

+ (3-3)^2 + α x μ

+ D_s x (-1)

At μ=1.0 (standard 3+1): E = 1.4000 model units (baseline confirmed)

Minimum: E = 0.8958 model units at μ = 0.083

Driving force for expulsion: -0.504 model units

The result is robust across the full physical range μ = 0.077 to 0.23. Proton stability does not depend on fine-tuning the interstitial mass fraction.

Figure F. E(μ) vs μ across the full range 0 to 1. Minimum at μ=0.083. Green band: physical range.

Figure G. Proton energy across the physical range μ=0.077 to 0.230. Result is robust.

7. The Connecting Identity

P19 Section 21.6 establishes an algebraic identity connecting the interstitial volume fraction, the electron mass, and the compression energy. E_unit cancels, so the ratio is purely geometric:

E_gap = E_unit x V_gap/V_q = 298.661 x 0.0770 = 22.997 MeV

m_e_vss = E_unit / (6 x π^4) = 298.661 / 584.45 = 0.511009 MeV

E_gap / m_e_vss = 6 x π^4 x V_gap/V_q = 584.45 x 0.076996 = 44.9991, approximately 45.0

Physical meaning: The expelled substrate dissipates approximately 44/45 of the compression energy into the surrounding substrate during proton stabilisation. The remaining approximately 1/45 is retained as the stable counter-rotating condensate whose mass is m_e_vss. The chain from V_gap to E_gap to m_e_vss is one identity with E_unit as the common factor that cancels.

8. Modular Organisation: The Universal Structural Unit

For n = 3m co-rotating core units with m > 1, m independent three-core structures are always energetically preferred over a single large co-rotating condensate. The energy advantage grows with n. At n=24 the gap is 501.900 model units.

The comparison counts co-rotating core units only. The three-core unit is the preferred structural unit for matter at all scales, and its electron is generated subsequently by interstitial ejection.

Figure H. Total energy: single condensate vs modular units for n = 3m, from n = 3 to 24. The two coincide at n = 3.

Figure I. Energy advantage of modular organisation. ΔE = (n − 3)(1.1n − 2.5). Positive = modular wins. Gap grows with n.

9. Robustness: Parameter Space Analysis

The 3+1 selection is not a fragile result at a single parameter point. Scanning λ and α across [0.2, 1.2] with J=1.0 fixed:

Figure J. Parameter space map. Green: 3+1 is minimum energy. Red: other configuration wins. P16 working point marked.

10. All Key Results at a Glance