The Smooth Census

The Binomial Smooth Theorem

An integer is 11-smooth if its only prime factors come from {2, 3, 5, 7, 11}.

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. But at n = 13, the prime 13 enters 13! and divides C(13, k) for all 1 ≤ k ≤ 12 (by Lucas' theorem: 13 is prime, so 13 | C(13, k) whenever 0 < k < 13).

The smooth run length at depth k (counting values from n = k to n = 12) is:

As k varies from 1 to 12, this traverses the complete axiom vocabulary:

The run lengths {12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} = the full chain from D²*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. Above it, 13 contaminates everything.

Pascal's Triangle: The Smooth Boundary

Rows 1-14. Gold = 11-smooth. Red = contains prime > 11. Row 13 = the GATE wall.

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:

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.

The Non-Trivial Coincidences

Triangle numbers are binomial: T(n) = C(n+1, 2). Their smoothness follows from the Binomial Theorem. But these do NOT follow from binomials:

sigma(9) = 13 (sum of divisors)

sigma(9) = 1 + 3 + 9 = 13. The divisor sum of K² = 9 happens to EQUAL the gate prime. No binomial coefficient is involved. This is a number-theoretic coincidence: the divisors of the stop number sum to the gate.

F(7) = 13 (Fibonacci)

The b-th Fibonacci number is the GATE prime. The Fibonacci recurrence F(n) = F(n-1) + F(n-2) has no connection to 13 except that the cumulative additions hit 13 at exactly n = b = 7. Equivalently: F(b) = GATE.

B(5) = 52 = 4 * 13 (Bell numbers)

The E-th Bell number (counting partitions of a 5-element set) contains 13. Set partitions know about the GATE through the observer.

p(13) = 101 (partition function)

The GATE-th partition number is prime (101). The smooth run breaks at the GATE position itself. 101 is not a standard intruder, but 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 D²*K = 12. Each rung is an axiom constant, each gap is one step along the chain:

Gaps: 6-5=1=sigma. 8-6=2=D. 11-8=3=K. 12-11=1=sigma. The gaps themselves are axiom constants.

Combined with the Binomial Theorem (which fills ALL values 1..12), the axiom's smoothness vocabulary is COMPLETE. Every integer from 1 to 12 appears as a smooth run length, and every one names an axiom constant.

Why 13 Is the Wall

13 = GATE = D² + K² is the smallest prime larger than L = 11. Since 11-smooth means "primes ≤ 11 only," the first possible intruder IS 13. What's remarkable is not that 13 blocks — that's inevitable — but HOW it enters each sequence:

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

Smooth Run Explorer

Compute the initial 11-smooth run of a sequence:

Number Theory Thread

The Smooth Census connects to:

The Last Smooth Pair - (2400, 2401) = where consecutive smoothness ends. Same Stormer bound limits the shadow quartet.

The Partition Function - p(n) smooth zone: exactly D²*K = 12 values. GATE at 13.

The Universal Boundary - 8 intruder primes, all generated by 3 axiom maps. 13 = GATE is the first.

Shadow Evaluations - L = 11 smooth quartets of the shadow polynomial. Same L from a different function.

The Two Chains - Cunningham chains generate the axiom. shadow(13) = 6 = D*K (composite, chain stops).

The Bernoulli Connection - Bernoulli denominators: E = 5 smooth, 13 enters via von Staudt-Clausen.

The Fano-E8 Bridge - Eisenstein smoothness: D⁴ = 16 smooth divisor sums in E8 theta series.