The Smooth Census

run(k) = GATE - k = 13 - k

An integer is 11-smooth if its only prime factors come from {2,3,5,7,11} -- the five axiom primes. For n <= 12, every binomial coefficient C(n,k) is 11-smooth. At n = 13, the prime 13 enters every C(13,k) for 0 < k < 13. Seven independent classical sequences confirm the same boundary. The GATE is the universal wall.

The Binomial Smooth Theorem

For n <= 12, the factorial n! contains only primes <= 11. So C(n,k) = n!/(k!(n-k)!) is a ratio of 11-smooth numbers, hence 11-smooth. At n = 13, the prime 13 enters 13! and divides C(13,k) for all 0 < k < 13 (by Lucas theorem: 13 is prime, so 13 | C(13,k)).

Binomial Smooth Theorem (S716, PROVED)

C(n,k) is 11-smooth for all k <= n <= 12. At n = 13, every C(13,k) = 13 * (smooth number). The GATE is the sole intruder. The smooth run length at depth k is run(k) = GATE - k = 13 - k.

k	Run = 13-k	Axiom name	C(13,k)/13
1	12 = D^2*K	Trinity heart	1 = sigma
2	11 = L	Protector	6 = D*K
3	10 = D*E	Bridge*Observer	22 = D*L
4	9 = K^2	Spider web	55 = E*L
5	8 = D^3	Spider legs	99 = K^2*L
6	7 = b	Depth	132 = D^2KL
7	6 = D*K	Thorn	132 = D^2KL
8	5 = E	Observer	99 = K^2*L
9	4 = D^2	Bridge squared	55 = E*L
10	3 = K	Closure	22 = D*L
11	2 = D	Bridge	6 = D*K
12	1 = sigma	Ground	1 = sigma

Run lengths {12,11,10,...,1} = the full chain from D^2*K down to sigma. The row C(13,k)/13 is PALINDROMIC and every quotient is 11-smooth. Pascal's triangle is smooth below the GATE.

The Independent Smooth Census

Classical sequences whose smooth runs are NOT explained by binomial coefficients. Each has an initial 11-smooth run whose length is an axiom constant, blocked by 13 = GATE:

Sequence	Run	= Axiom	Blocker
Bernoulli denom B_2n	5	E	2730 = 235713
Fibonacci F(n)	6	D*K	F(7) = 13
Catalan C_n	6	D*K	429 = 31113
Sum of divisors sigma(n)	8	D^3	sigma(9) = 13
Triangle T(n)	11	L	78 = 2313
Partition p(n)	12	D^2*K	p(13) = 101 (prime)
Bell B(n)	5	E	52 = 4*13

In 6 of 7 sequences, 13 = GATE appears in the blocking VALUE. In partitions, 13 is the blocking POSITION (p(13) is the first non-smooth partition number). The GATE blocks either way.

How 13 Enters Each Sequence

Each entry mechanism is independent. Divisor sums, recurrences, combinatorial products, von Staudt-Clausen -- all different mathematics, same wall.

sigma(9) = 13

Sum of divisors

1 + 3 + 9 = 13. Divisors of K^2 sum to GATE. No binomial involved.

F(7) = 13

Fibonacci

The b-th Fibonacci number IS the gate. Cumulative addition: 1,1,2,3,5,8,13.

B(5) = 52

Bell numbers

4 * 13. E-th Bell number contains 13. Set partitions know the GATE through the observer.

p(13) = 101

Partitions

GATE-th partition number is prime. CRT(101) = (E, D, sigma, K, D): all axiom elements.

The Smooth Ladder

Ordering by run length, the sequences form a ladder from E = 5 to D^2*K = 12. Each rung is an axiom constant. Gaps: 6-5=1=sigma. 8-6=2=D. 11-8=3=K. 12-11=1=sigma. The gaps themselves are axiom constants.

Run = 5 = E

Bernoulli, Bell

Observer-depth. Two sequences share the rung.

Run = 6 = D*K

Fibonacci, Catalan

Bridge * closure. Thorn pair.

Run = 8 = D^3

sigma(n)

Spider legs. The divisor function.

Run = 11 = L

Triangle numbers

Protector. T(n) = C(n+1,2): binomial.

Run = 12 = D^2*K

Partition function

Trinity heart. GATE at position, not value.

Combined

All integers 1..12 appear

With the Binomial Theorem filling all values 1..12, the axiom's smoothness vocabulary is COMPLETE. Every run length names an axiom constant.

Contrast

Aspect	Standard view	Through the axiom
Smooth runs	Scattered counts: 5, 6, 8, 11, 12	Every count is an axiom constant: E, DK, D^3, L, D^2K
Blocker	13 happens to be the next prime after 11	13 = GATE = D^2 + K^2. Structural boundary
Entry pattern	Each sequence has its own mechanism	All mechanisms hit 13. Row 13 of Pascal spells the full chain
Ladder gaps	1, 2, 3, 1 -- no meaning	sigma, D, K, sigma. The gaps ARE the axiom

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