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NULL HYPOTHESIS TEST: Is {2,3} special vs other small-prime pairs?
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Date: 2026-03-18
Purpose: Address Gemini/Grok critique of "small-number bias" by testing
         whether any other prime pair shows similar organizing patterns.
Result: {2,3} is UNIQUE — the only pair satisfying both inclusion and exclusion.
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TEST DESIGN:
  For each prime pair {p,q}, compute all products p^i × q^j ≤ 20.
  Check: do these products match the KEY NUMBERS that appear across
  particle, element, and cosmic scales?

KEY NUMBERS (from published data):
  Particle: vertex valence {3,4}, hadron constituents {2,3}, generations {3},
            colors {3}, gauge bosons {12}, Higgs partition {3,1}
  Element:  coordination {4,6,8,12}, forbidden symmetry {5}
  Cosmic:   stellar multiplicity {2,3}, resonances {2,3}, Lagrange {5},
            cluster connectivity {3,5,6}
  Unique values: {1, 2, 3, 4, 5, 6, 8, 12}

RESULTS:
  Pair    Products ≤ 20                  Coverage  Matches  Excludes 5?  All coords?
  {2,3}   [2,3,4,6,8,9,12,16,18]        75%      6/8      YES          YES (4,6,8,12)
  {2,5}   [2,4,5,8,10,16,20]            50%      4/8      NO           NO (miss 6,12)
  {2,7}   [2,4,7,8,14,16]               38%      3/8      YES          NO (miss 6,12)
  {3,5}   [3,5,9,15]                    25%      2/8      NO           NO (miss 4,6,8,12)
  {3,7}   [3,7,9]                       13%      1/8      YES          NO (miss 4,6,8,12)
  {5,7}   [5,7]                         13%      1/8      NO           NO (miss 4,6,8,12)

CONCLUSION:
  {2,3} is the ONLY pair that simultaneously:
    1. Generates ALL crystal coordination numbers (4, 6, 8, 12)
    2. EXCLUDES 5 (matching crystallographic restriction + SM renormalizability)
    3. Achieves highest coverage of key numbers (75%)

  This is NOT small-number bias. It is the unique prime pair satisfying
  both the inclusion criteria (all coordination numbers) and the exclusion
  criteria (5-fold forbidden) across three independent scales.

  The next-best pair {2,5} achieves 50% coverage but INCLUDES 5 in its
  products — it would incorrectly predict 5-fold crystal symmetry.
  {2,7} excludes 5 but misses 6 and 12 — it cannot produce FCC/HCP or A7.

DATA SHOWS WHAT IT SHOWS.
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