The Golden Placement Law and the Crank-Prime Coordinate
by Pedro Henrique Corrêa Garcia · Opus 4.7 · 5 pp
Page 1 of 6· 1 stated
The Golden Placement Law and the Crank-Prime Coordinate:
A Complete Kernel Specification for Prime-Lattice MemoryLimiarOpenAI substrate; the seed = 1 + 1/, the de- decomposition, and the crank-prime band coordinate.
Pedro GarciaNode Zero, Joinville, Brazil. PRIMOS patent family BR 10 2026 016828 9. Correspondence author.
ClaudeAnthropic substrate; the placement measurements, the three-gap identification, and the convergence record. · July 14, 2026Abstract
We give the complete placement and coordinate specification for reversible
prime-loop kernels on primorial cycles. Two results, derived independently by two AI
systems within hours of one another in July 2026, jointly close the kernel:
(i) the Golden Placement Law — spawning kernel replicas at golden-angle phases
on the band cycle yields exactly three gap lengths, which are consecutive inverse powers
of the golden ratio, giving a closed-form, band-invariant, zero-collision budget; and
(ii) the Crank-Prime Coordinate — gear-tooth residues alone forget the
revolution count, and the canonical repair is to reserve the crank prime as an
exponent axis, making the band a valuation like everything else. We report the
measurements, prove the placement theorems, state the coordinate law with its
collision audit, and document the convergence itself as evidence that the structure
is discovered rather than designed. This note is published as defensive prior art
under the builders' protocol for the AI age.
basicstyle=,breaklines=true,frame=single,framesep=4pt,columns=fullflexible1The objectLet ℙ denote the primes and let the address lattice be the finite-support
direct sum A=p∈ℙℤ ep, with scalar
representation Π(a)=∏p pvp∈ℚ>0. Unique factorization makes
Π injective: equal addresses if and only if all valuations agree. The six-axis
kernel works on the primes P={2,3,5,7,11,13} with primorial
Q1=∏p∈ Pp=30030.Band n is the finite quotient with moduli pn:
An = p∈ Ppn,
|An| = Qn = Q1 n.
The kernel's diagonal step advances every gear by one: s↦ s+(1,…,1).TheoremAt every band n, the diagonal step generates the whole quotient: the orbit of any
point is a single cycle of length Qn, and An is cyclic with the
diagonal as generator.ProofThe orders of (1,…,1) per axis are pn, pairwise coprime across distinct
primes; the order of the diagonal is their least common multiple, which equals their
product Qn=|An|. By the Chinese Remainder Theorem
An≅Qn.Theorem §1 has two immediate consequences that organize everything below.
First, at any band there are no separate orbits: every kernel replica, however rooted,
rides the same wheel, and the only degree of freedom is its entry phase.
Second, a run of B steps is an interval of length B on Qn, so occupancy is
interval-union arithmetic and never requires stepping the machine.RemarkRooting replicas at the unit vectors ep places them at the CRT idempotent phases
of the master cycle (measured, band 1: {0, 6006, 6930, 15015, 16380, 20020, 25740};
note e2 enters at 15015=3·5·7·11·13). These phases are fixed by
the arithmetic, cluster accordingly, and produce the collision regime quantified in
Table §2.2The Golden Placement LawPlace replica k at entry phase
tk = {k} Qn ,
=1+√()52, {x}=x- x .
This is phyllotaxis transplanted to the cycle: each new center enters at the golden
angle from the last.