Preprint: prime gap entropy crosses φ ⁻¹ at n=1,250 — Ladner NP-intermediate z one in IEEE 754 bit density

I am an independent researcher announcing an observational record at
Zenodo (DOI 10.5281/zenodo.20242240) and sharing the central empirical
findings for comment.

Three independent measurements converge on φ⁻¹ = 0.6180339887.

FINDING 1 — IEEE 754 bit density partition.

The following is verifiable in any Python interpreter:

import struct, math
def f(x):
    return bin(struct.unpack('Q', struct.pack('d', x))[0]).count('1')

f(1.0)          = 10 set bits   (integer, sparse)
f(math.pi)      = 23 set bits   (transcendental, dense)
f(math.e)       = 27 set bits   (transcendental, dense)
f(math.sqrt(2)) = 40 set bits   (irrational, dense)
f(2**-52)       =  7 set bits   (machine epsilon, sparse)

Simple computable numbers have sparse float64 representations.
Transcendentals have dense ones. The boundary falls at approximately
18 set bits per 64-bit word — the Ladner NP-intermediate zone
measured in bit density. The mantissa width 52 = 4 x F(7), four
Fibonacci-7 chunks.

FINDING 2 — Prime gap entropy crossing.

A Shannon entropy x golden ratio alignment function applied to the
ASCII representation of the first n prime gaps crosses φ⁻¹ at n =
1,250 gaps. The crossing prime is p(1,250) = 10,181. The bit density
at the crossing is approximately 18 set bits per 64-bit word — the
same boundary as Finding 1, from number theory with no knowledge of
floating-point structure.

FINDING 3 — Banach fixed point.

The operator SPIRAL(φ) = (1/φ) mod 1 has φ⁻¹ as its unique
Banach fixed point. Its convergence gap locks at machine epsilon —
2^-52 — after 25 consecutive measurements on a consumer laptop
(Intel i7-3630QM, 2012).

These three measurements were made independently by a single
researcher over approximately one year, using one consumer laptop and
no institutional resources.

Full methodology, empirical record, and x86-64 NASM assembly
implementation:

DOI: 10.5281/zenodo.20242240

 THE GATE AT THE BOUNDARY: ASSEMBLY-LEVEL IMPLEMENTATION OF THE
PHI-SPACE FRAMEWORK AND ITS CONNECTION TO THE P VS NP ALTERNATIVE
POSSIBILITY — A TIMESTAMPED TECHNICAL RECORD, MAY 16, 2026
https://zenodo.org/records/20242240 

Daniel Dragolich Dragolich Research Labs LLC Cleveland, Ohio



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