Step 1. Pyramidal identity: ∏ₖ₌₁ⁿ k · ∏ₖ₌ₙ₋₁¹ k = n! · (n-1)! Step 2. For n = 7: 7! · 6! = 5040 · 720 = 3,628,800 = 10! Step 3. Day in seconds: 86,400 = 2⁷ · 3³ · 5² 10! = 2⁸ · 3⁴ · 5² · 7 Step 4. Quotient: 10! / 86,400 = 2 · 3 · 7 = 42 ∎
| Angle | Identity | Result |
|---|---|---|
| Algebraic | n!·(n−1)! at n=7 | = 10! |
| Number-theoretic | 10! / 86400 in primes | = 2·3·7 = 42 |
| Pyramid structural | apex 7 × second-from-apex 6 | = 42 |
| Calendar | 42 days = 6 weeks (lunar quarters) | = 6 × 7 |
| Bitcoin (paper §6.1) | 2016 blocks × 2.5° = 5040° = 7! | 7! is the apex |
Section 6.1 of the original paper establishes that one Bitcoin difficulty period sweeps 5040° = 7!. This addendum shows that 7! is not merely a computational endpoint — it is the apex of a pyramidal product whose total value, when read as seconds, equals 42 days exactly.
The sister identity (apex 6) yields exactly 1 day = 86,400 sec, grounding the chain in the unit of the solar day. Together: the pyramidal apex 6 encodes the day; the apex 7 encodes 42 days = 6 weeks; their ratio 42 = 6 × 7 is the lunar week (7) times its multiplier (6). The same 6 and 7 that govern π ≈ 22/7 (denominator 7) and the Metonic insertions (12 + 7 = 19) appear here as the apex pair of an exact-arithmetic pyramid.