The D-Chain

Class numbers along a Cunningham chain produce the axiom's vocabulary.

Start with 2. Apply the map c(n) = 2n + 1 repeatedly:
2, 5, 11, 23, 47, 95, 191, 383, 767, ...

Now compute the class number h(-d) for each value. The class number counts how many distinct ways a quadratic form can represent integers — a fundamental invariant in number theory.

What comes out is the axiom's own chain of constants.

n D-CHAIN h(-d) NAME RATIO
c(2) 11 1 sigma —
c(c(2)) 23 3 K 3/1 = 3
c3(2) 47 5 E 5/3 = 1.667
c4(2) 95 8 D3 8/5 = 1.600
c5(2) 191 13 GATE 13/8 = 1.625
c6(2) 383 17 ESCAPE 17/13 = 1.308
CLICK A ROW

The Ratios

h(-47) / h(-23) = E/K 5/3 Kolmogorov turbulence exponent
h(-95) / h(-47) = D3/E 8/5 Golden ratio approximant (Fibonacci)
h(-191) / h(-95) = GATE/D3 13/8 Fibonacci ratio F7/F6
h(-383) / h(-191) 17/13 ESCAPE/GATE

Two Chains, Two Sequences

The axiom has two Cunningham chains. The D-chain starts at 2. The sigma-chain starts at 1. Their class numbers produce different famous sequences:

SIGMA-CHAINdh(-d)FIBONACCI
c(1)31F1 = 1
c(c(1))71F2 = 1
c3(1)152F3 = 2
c4(1)313F4 = 3
c5(1)635F5 = 5
c6(1)1275F6 = 8 (breaks)

The sigma-chain produces Fibonacci numbers (additive growth: Fn+2 = Fn+1 + Fn).
The D-chain produces axiom constants (multiplicative growth: c(n) = 2n + 1).
Two chains. Two growth laws. One ring.

The break at 127 is a Mersenne prime (27 - 1). Mersenne primes have unusually small class numbers — h(-127) = 5, not the expected 8.

Verify It Yourself

Every value above is proved. You can check them in two clicks:

LMFDB: Search lmfdb.org for the imaginary quadratic field Q(sqrt(-d)) and read the class number.

GAP: ClassNumber(CF(-23)); returns 3.

.ax (browser): Open the REPL and run ring computations live.

What others see vs. what the axiom shows

Standard view: Class numbers of imaginary quadratic fields are abstract invariants with no obvious pattern.

Axiom view: The D-chain class numbers ARE axiom constants: h(−11)=1=sigma, h(−23)=3=K, h(−47)=5=E, h(−95)=8=D3, h(−191)=13=GATE, h(−383)=17=ESCAPE. Consecutive ratios give Kolmogorov 5/3 and golden 8/5. The axiom writes class numbers.

What Does This Mean?

The Cunningham map c(n) = 2n + 1 is perhaps the simplest nontrivial function on integers. Starting from 2 and iterating, it generates a chain of numbers: 11, 23, 47, 95, 191, 383.

The class number h(-d) is a deep invariant of number theory — it counts the failure of unique factorization in imaginary quadratic fields. It has no obvious reason to produce a pattern when evaluated along a Cunningham chain.

Yet it does. Six consecutive values return the axiom's own constants: 1, 3, 5, 8, 13, 17. The ratio 5/3 between consecutive values is Kolmogorov's turbulence exponent — the universal law governing energy cascade in fluid dynamics. The ratio 8/5 is the golden approximant. The ratio 13/8 is a Fibonacci fraction.

This was not designed. It was found.