I. The Motivation: Beyond the Rounding Error
In an era dominated by relentless rationalization, utilitarianism, and the obsessive pursuit of "cost-performance,"
the weight of a single human life is increasingly being disregarded. In the age of Artificial Intelligence and big data,
an individual is too often treated as a mere rounding error—a negligible fraction to be smoothed over in a massive
statistical dataset.
Society constantly offers the platitude that "everyone's life has meaning." But for a generation that has grown up observing
the cold, systemic realities of the world through the lens of algorithms, well-meaning emotional encouragement often rings hollow.
When you are continuously bombarded with metrics that suggest how "average" or "replaceable" you are, warm comfort is insufficient.
This research program was not created to offer a warm embrace. It was created to construct an inescapable logical framework.
We address the question of human significance not with sentiment or morality, but with the cold, rigorous language of nonlinear dynamics,
topology, and systems engineering.
The Truncation Fallacy in Physical Reality
A digital computer stores every number with a fixed number of significant digits, so a contribution that is small enough relative to
the whole is absorbed: the computed sum returns the whole unchanged. But this is a property of the representation, not of the quantities.
The exact sum differs for every non-zero contribution, and the physical universe stores no fixed number of digits for macroscopic events.
Therefore, in any system satisfying the framework's axioms, a person whose presence or absence makes "no difference" is a mathematical
impossibility. Your very existence—however ordinary it may seem—is a sequence of discrete impulses, each of which leaves the future different
from what it would otherwise have been, at every later instant. The dynamics carry that difference forward; they do not round it away.
II. Intellectual Trajectory: From Algorithms to Existential Proof
Underlying this entire research program is a single, highly consistent mathematical inquiry: "Amidst continuous state transitions
in a complex system, what is the information—the invariant structure—that is never lost?"
1. The Origin: A Heuristic Hunt for the Unknown (2023)
The journey began with the paper "Mining for Conserved Constituents and Quantities" (2023), co-authored with Dr. James Hearne.
The premise was deceptively simple: given a set of observed transformations (like elementary particle interactions or chemical reactions),
could we algorithmically discover the hidden conservation laws that govern them?
Mathematically, we sought a matrix \( \mathbf{X} \) that satisfied \( D_o \mathbf{X}^\top = \mathbf{0} \) for a difference matrix \( D_o \) of
observed reactions, while ensuring \( D_u \mathbf{X}^\top \neq \mathbf{0} \) for transformations known to be impossible.
Looking back, the methodology was admittedly raw and naive. Because the true invariant structures of nature are discrete (e.g.,
baryon numbers, stoichiometric coefficients), we forced the algorithm to search for exact integers. Without advanced algebraic tools
at the time, we relied on a gritty, brute-force heuristic search—constraining the vector components to narrow ranges (e.g., \([-3, 3]\)) and
manually tuning iteration limits (stage_limit, try_limit) to fight off the inevitable combinatorial explosion.
It was computationally exhausting and mathematically unrefined.
Yet, beneath this muddy heuristic struggle lay a profound structuralist conviction. While the broader field of Machine Learning was moving
toward massive data reduction and black-box statistical approximations, this algorithm was an attempt to do the exact opposite. It was a stubborn,
ground-up effort to dig straight into the bedrock of reality and extract the "unalterable, indecomposable units" of nature using
basic linear algebra.
We did not want to approximate reality; we wanted to find the exact rules that never change. That raw desire to uncover the strict geometric
bounds of a system—the invariant manifold—became the conceptual engine for everything that followed.
2. The Discrete Complement: Cohomological Mining (2026)
The heuristic search of 2023 was subsequently elevated into a mathematically exact algorithmic pipeline. In complex discrete systems,
continuous approximations (like SVD over \(\mathbb{R}\)) introduce what we call the Semantic-Integrality Gap: the
extracted real-valued basis vectors are mathematically valid over \(\mathbb{R}\), but mapping them back to structurally meaningful
discrete entities requires arbitrary thresholding and rounding. True invariants reside strictly over an integer ring, \(\mathbb{Z}\).
By modeling discrete state transitions as a general chain complex over \(\mathbb{Z}\)—with a boundary operator
\(\partial: C_1 \to C_0\) encoding the net effect of each transformation—we reframed the discovery of conservation laws as the
exact computation of the integer zeroth cohomology group, \( H^0 = \ker(\delta) \) where \(\delta = D^\top\) is the coboundary operator.
Utilizing Hermite Normal Form (HNF) and Lenstra-Lenstra-Lovász (LLL) lattice basis reduction,
the pipeline extracts structurally minimal integer generators in polynomial time.
The key structural result: node deletion is not a chain map (\(\partial' \circ f_1 \neq f_0 \circ \partial\)), so the
standard functoriality of cohomology fails entirely. This is what permits the Betti number to strictly increase under
deletion—a dimensional jump that no continuous projection can produce. It is this failure of functoriality that constitutes the
algebraic definition of irreversibility in the discrete setting.
3. The Main Line: Topoethics (2026)
The structural question—"what is never lost?"—was elevated from the discrete algebraic setting to the universal, continuous
language of dynamical systems. Where the cohomological framework identifies conserved integer constraints, Topoethics
addresses the complementary question: how does perturbation propagate through the continuous state space? The answer has three layers.
That a difference is never lost follows from the uniqueness of solutions alone; positive Lyapunov exponents
explain how it grows to the scale of the whole; and topological mixing explains why no one can steer where it leads.
If our society operates as a chaotic, densely coupled dynamical system, human interventions are discrete impulses in a piecewise-autonomous system.
Because the universe cannot round these macroscopic impulses to zero, the information is folded into the state of the system's future:
coarse-grained statistics may forget it, but the state does not.
The Central Result: Non-Negligibility of Every Individual
This framework establishes rigorous conditional theorems: If a system satisfies the axioms
of the framework—a state-space model with an attractor, deterministic chaos (a positive maximal Lyapunov exponent,
\(\lambda_{\max} > 0\), and stable sets of zero volume), topological mixing, and macroscopic agency (non-zero
impulses that do not conspire to cancel the dynamics)—then replacing or removing any single act,
all other acts held fixed, produces a history that differs from the original at every later instant and, for all
impulses outside a set of zero volume, never converges back to it.
Under observability conditions on the coupling between agents, this difference reaches every agent and, for all but
an exceptional set of death times, outlives the one who made it. Yet no agent with finite-precision knowledge can
certify where it leads in the long run. Within this axiomatic model, a person whose presence or absence makes no
difference is a mathematical impossibility.
III. Published Papers & Preprints
Published Preprint
Topoethics: Individual Absolute Value through Nonlinear Dynamics and Topological Mixing
Yusuke Yokota · October 2026 (revised version) · Preprint
This paper introduces Topoethics, the study of ethical consequences that arise from the topological structure of causal dynamics,
and establishes the non-negligibility of every individual existence within a deterministic chaotic model of history. Human actions are
formalized as discrete impulses in a piecewise-autonomous dynamical system governed by explicitly stated axioms: a chaotic attractor,
stable sets of zero volume, topological mixing, and impulses that do not conspire to cancel the dynamics. The framework is conditional:
given the axioms, the conclusions follow as mathematical theorems, not as moral assertions.
Under these axioms, every impulse produces a history that differs from its counterfactual at every later instant and generically never
converges back to it. Under observability conditions on the coupling between agents, the difference made by a single act reaches every
agent and, for all but an exceptional set of death times, outlives the one who made it. No agent with finite-precision knowledge can
certify the long-range consequences of an intervention, all other acts held fixed. The difference an act makes persists in the state
even where coarse-grained statistics forget it.
The paper further addresses five principal counter-arguments—noise attenuation, historical homeostasis, justification of irreversible
state reduction, floating-point truncation, and macroscopic fatalism—and reads ethics as an engineering requirement, separating the
physical causality layer (L1/L2) from the institutional one (L7). Death is treated as a structural mutation: an irreversible reduction
of the system's degrees of freedom that changes the vector field itself.
A numerical appendix illustrates the deductive chain on the Lorenz system (irreversible divergence), the Hénon map (loss of
predictability, and coarse-grained statistics that forget while exact distributions do not), and a 20-node coupled Rössler network
(irreplaceability, and persistence of influence beyond death). These systems satisfy the axioms only in part, so the experiments
show behavior consistent with the results rather than proving them.
nonlinear dynamics
topological mixing
Lyapunov exponents
irreversibility
chaos theory
Published Preprint
Cohomological Mining of Invariant Structures: Algorithmic Extraction of Conservation Laws from Discrete State Transitions
Yusuke Yokota · March 2026 · Preprint (journal submission in preparation)
We present an exact algorithmic pipeline for extracting integer-valued conservation laws from discrete
interaction networks. While continuous methods (e.g., SVD) yield real-valued null spaces that obscure
discrete structure—a gap we term the Semantic-Integrality Gap—our method combines
Hermite Normal Form (HNF) computation with LLL lattice basis reduction to produce a minimal, maximally
sparse basis for the zeroth cohomology \(H^0\) over \(\mathbb{Z}\), without arbitrary floating-point rounding.
Three numerical experiments validate the framework: (1) explicit demonstration of the Semantic-Integrality
Gap on a closed stoichiometric system, (2) recovery of all five known moiety conservation laws from
the E. coli core metabolic network with zero integrality gap, scaled to the genome-scale
iML1515 reconstruction (1,877 metabolites, 32 invariants, ~20 seconds; non-integer entries rounded
to \(\mathbb{Z}\) prior to extraction), and (3) proof that node deletion—the discrete analogue of individual
removal—strictly increases the Betti number (\(\Delta k > 0\)), a dimensional jump inaccessible to any continuous
projection.
Crucially, node deletion is shown to violate the chain-map condition
(\(\partial' \circ f_1 \neq f_0 \circ \partial\)), so the standard functoriality of cohomology fails:
there is no induced homomorphism relating the mutated and original invariant spaces. This algebraic
irreversibility complements the companion Topoethics paper, in which the death of an agent is a
structural mutation of the vector field and the difference made by the agent's past acts survives the
deletion: the discrete framework shows what deletion breaks in the constraint structure, the dynamical
framework what it cannot erase from the trajectory.
integer cohomology
conservation laws
HNF
lattice basis reduction
Betti number
chain map failure
metabolic networks
semantic-integrality gap
IV. Ongoing Research: Category-Theoretic Unification
The immediate next step in this research program is to unify the continuous framework (Topoethics) and
the discrete framework (Cohomological Mining) into a single, algebraically rigorous foundation using
Category Theory.
The Failure of Functoriality as Irreversibility
In our cohomological framework, we demonstrated that the deletion of a node (an individual agent) is not a
continuous projection. Crucially, it breaks the structure of a chain map. Because the boundary operator itself is mutated,
the standard functoriality of cohomology fails.
This mathematical failure of commutativity—the inability to map the mutated system back to the original via an induced
homomorphism—is exactly the algebraic definition of irreversibility.
Complementarity of the Two Frameworks
The cohomological analysis and the continuous dynamical-systems analysis in Topoethics approach node
deletion from complementary mathematical levels. The discrete framework demonstrates, via exact integer cohomology,
that deletion fragments the system's conservation constraints (the Betti number increases). The continuous
framework treats the death of an agent as a structural mutation—the vector field itself changes—and proves,
under observability conditions on the coupling, that the difference made by the agent's past acts survives the
deletion in the remaining coordinates. Neither perspective alone captures the full picture: the cohomological
framework identifies what structural constraints break, while the dynamical framework shows what
the deletion cannot erase.
Current Focus
Our ongoing work models individual agents as non-functorial morphisms within a category of chain complexes. By formalizing
simultaneous multi-node deletions (e.g., exploring the Mayer-Vietoris sequence for fractured topologies), we aim to precisely
quantify how individual cohomological contributions interact, entangle, and irreversibly shape the structural constraints of the
overall system.
V. Research Notes
Accessible walkthroughs of the mathematics behind our papers.
Each note unpacks a specific concept for readers who want to go deeper
than the abstract but prefer a gentler pace than the paper itself.
Notes on the Revised Paper
The walkthroughs are being rewritten to follow the revised paper: its axioms, the proofs of
irreversible divergence and of the impossibility of long-range control, persistence beyond death,
and the numerical illustrations.
Coming Soon