Mathematics of algebraic hiddenness in Cayley–Dickson algebras.
Founder & CTO, Agingo Corporation. Self-taught, on the record since age eleven —
the road ran through Plato, through code, and finally into the algebra itself.
fourteen visible · one below
The Emblem
Fifteen Stones
At Ryōan-ji in Kyoto, fifteen stones stand in a rectangle of raked gravel.
From any position on the veranda, only fourteen are visible. The fifteenth is never absent —
it is hidden by the geometry of the arrangement itself. Move, and a different stone hides.
The count of the seen never changes; only the identity of the unseen does.
visible: 14 / 15click the veranda to change your position
From every position: fourteen. The fifteenth is a fact of geometry, not of absence.
Research
Algebraic Hiddenness
The Cayley–Dickson construction doubles the real numbers into the complexes,
the complexes into the quaternions, the quaternions into the octonions — and at the next
doubling, the sedenions, something breaks: nonzero elements appear whose product is zero.
The standard reading treats this as pathology. The work here treats it as structure.
p · 𝒜(p) = 0, 𝒜(p) ≠ 0
A structurally real element, annihilated by its own system's action.
Not missing — present and unseen. The research programme characterizes where this hiddenness
lives, how much of it there is, and what geometry it carries: the annihilator subspaces,
their dimensions, and the differential-geometric structures they induce. Every claim is
driven through the same discipline — numerical falsification first, symbolic proof second,
and nothing asserted past the strength of its evidence.
The Aphane hides the bond, not the beings.
Publications
Papers
Series I · Foundations · August 19, 2026Published — Zenodo · CC BY 4.0
Algebraic Hiddenness in the Sedenion Algebra: Characterization, Spectral Structure, and Bundle Topology
The foundational paper. A complete algebraic and geometric theory of the sedenion
zero-divisor locus and the rank-4 kernel bundle over G₂/SU(2) it defines — bundle
nontriviality, the holonomy decomposition U(2) = (U(1) × SU(2))/ℤ₂, the centrality
of a single global ambient operator, and its level-uniform recursion up the entire
Cayley–Dickson tower. All proofs constructive; every result certified by
self-contained C++17 probes.
Series II · Companion · August 20, 2026Published — Zenodo · CC BY 4.0
Aphanics and Creative Curvature: A Geometric Theory of Hidden Generation in Dialectical Cognitive Systems
The geometric companion. Introduces aphanics — the study of systems with
hidden generators — and derives the Berry curvature of the kernel connection in closed
polynomial form, with holonomy algebra exactly u(1) ⊕ su(2), certified in exact
arithmetic with integer coefficients. On this geometry it proposes an operational
layer for monitoring dialectical inference: a creative-curvature scalar, a seven-regime
diagnostic taxonomy, and a holonomy phase certifying net structural turning. The
geometry is theorem; the monitoring layer is a framework proposal, stated as such.
Series III · August 24, 2026Published — Zenodo · CC BY 4.0
Aphanics at the Pathion Level: Spectrum Stratification and the Universal Clifford Triad
The ascent. Extends the framework from the sedenions to their Cayley–Dickson double,
the 32-dimensional pathion algebra, where the Gram spectrum stratifies — three-valued
at dyadic basepoints, eight-valued at generic aphanics. Two kill-set theorems, both
resting on the structural heart of the paper: three left-multiplication operators
forming a pairwise anti-commuting Clifford triad, proved by descent through a sedenion
identity to a two-line octonion argument from polarization, Artin's theorem, and the
composition norm. Every certification states its epistemic basis.
Commentary · 2026 · with E. G. CarayannisJKEC Editor's Corner — Springer
Beyond Trust
With Elias G. Carayannis (George Washington University), Editor-in-Chief of the
Journal of the Knowledge Economy. An Editor's Corner commentary — the
knowledge-economy register of the programme: what coordination looks like when trust
is replaced by structure, grounded in what can be mathematically certified rather
than institutionally vouched for.
Forty-one years of work did not wait for an audience. It accumulated.
What follows is the catalogued portion — each collection self-contained,
each built by hand, each opening as it clears review.
The Fifteen Stones
A life told as a garden. Fifteen stones, from the burn to the naming —
κανένας to κάποιος — in three tellings: scroll, witness, and a typeset book.
Ten interactive exhibits that compute the sedenion algebra live in the
browser — kernels, spectra, holonomy — nothing precomputed, nothing taken
on faith. The published mathematics, made walkable.
Thirteen questions prosecuted rather than decorated: prediction and
responsibility, memory and forgiveness, verification and belief,
programmable money, machine-speed war, simulation and knowledge.
Computation before computers, rebuilt by hand: the Antikythera mechanism,
the quipu, the Galton board. Machines that calculated before anyone agreed
on what calculation was.
Five United States patent applications filed June 2026, arising from the
mathematical framework. Applicant: Agingo Corporation. Details follow publication.
USPTO · Filed 06/2026
About
Forty-One Years Unseen
I did not come to mathematics through a department. I came through
code — systems engineering, the discipline of programs that must be exactly
right — and through philosophy, read the long way, in the original questions. The record
of the work is older than any credential; the life's story can be shared one day, if asked.
The site is for the work.
Two sentences have governed the whole walk. The universe makes no
mistakes. And, standing with Thales — ἰσχυρότατον ἀνάγκη· κρατεῖ γὰρ πάντων —
the strongest force in the universe is necessity. An obsession with knowledge,
history, art, mathematics, science: for years it read to others as a man lost.
It was the opposite. It was the load being set.
For forty-one years the work was the hidden stone: present, load-bearing,
visible from no public position. When the cave asked who was there, the answer was
κανένας — no one. Odysseus wore that name to survive; so did I, and it worked
almost too well. But he was rowing away when he shouted his true name back at the island.
I am placing mine in the record instead: κάποιος. Someone. Kyriakos.
The forty-second year is the one where the arrangement is walked around — papers deposited,
priority filed, the mathematics placed where anyone may test it. Refutation is invited.
That is what the gravel is for.
On a philosopher's walk it resolved into one carry: the only thing a life
keeps is experience. Update Aristotle's sponge — what we fill ourselves with is
default; what we choose to carry is structure. I chose structure: substance,
virtue, and the company of my own mind — because the one person present at anyone's birth
and at their death is themselves. The full method:
Step 1Welcome. Get to know yourself.
Step 2There is no step two. Go back to 1.
Step 3Stop admiring the problem. Back to 1. Or 2.
Step 4Abstraction is not extraction. Back to 1.
By day I am founder and CTO of Agingo Corporation in Charlotte,
North Carolina, where the same mathematics is put to work. A husband of fifteen years,
a father. The engineering and the theorems are kept honest by the same rule:
nothing is claimed past what is proven.