About Me

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Hi — I’m Yuta Agawa, an independent mathematical physicist in Japan. I look at physics through geometry — especially holonomy — and try to derive what we can straight from spacetime. Holonomy isn’t a handy tool to me; it’s the organizing principle. My program is foundational rather than interdisciplinary: start from geometric first principles and build outward into gauge theory and dynamics. (About toy models: they aren’t because I think reality is a toy — they’re there because clarity beats hand-waving.)

Two lines began for very human reasons. The reinforcement-learning work grew from living with OCD and looking for crisp math about compulsive behavior; when I didn’t find what I needed, I pushed analysis tricks from mathematical physics into a small, inspectable setting. The integrability project started while re-reading withdrawn notes; I spotted a clean geometric pattern in a class of flow equations and followed it. Both threads fit the same habit: turn an intuition into exact structure.

Selected works

1) Yang–Mills mass gap in SU(N) — constructive holonomy framework

Idea. Build the theory around Wilson holonomy and its induced non-local, gauge-covariant fields [1][2]. Start with the usual connection and curvature,

\[ \begin{aligned} A &= A_\mu\,dx^\mu \in \mathfrak{su}(N),\\ F &= dA + A\wedge A. \end{aligned} \]

and use loop holonomy around a family of based loops \(\gamma_x\) of size \(L\),

\[ U(\gamma_x,L)=\mathcal{P}\exp\!\Big(i\oint_{\gamma_x}A_\mu(y)\,dy^\mu\Big), \]

to define the holonomy-regularized field by conjugation,

\[ \begin{aligned} \hat A_\mu(x;L) &= U(\gamma_x,L)\,A_\mu(x)\,U^{-1}(\gamma_x,L),\\ \hat F_{\mu\nu}(x;L) &= U(\gamma_x,L)\,F_{\mu\nu}(x)\,U^{-1}(\gamma_x,L). \end{aligned} \]

One then writes a regulated action (schematically)

\[ \begin{aligned} S_L[A] &= \frac{1}{2g^2}\!\int d^4x\,\mathrm{tr}\,\hat F_{\mu\nu}(x;L)\hat F^{\mu\nu}(x;L)\\ &\quad + S_{\text{gf}} + S_{\text{gh}}. \end{aligned} \]

and develops a multiscale polymer/cluster expansion adapted to these holonomy vertices. The induced measure converges weakly to a continuum limit and correlations exhibit exponential clustering [1]. A representative gauge-fixing estimate squeezes the set of “bad” configurations,

\[ 0\;\le\;\int_{\mathcal A_{\mathrm{bad}}}\! d\mu[A]\;\le\;K\,e^{-\kappa N_M}\xrightarrow[M\to\infty]{}0, \]

which delivers almost-sure uniqueness (a Gribov-type resolution) and underpins the mass-gap conclusion. A loop-geometry consistency check fixes the non-local vertices and preserves quantum corrections, giving the expected one-loop sign of the beta function (asymptotic freedom) inside the same framework [2].

2) Emergent integrability in a Wegner-type flow

Result. For a class of continuous diagonalization flows, the dynamics close to an exact second-order ODE with a conserved energy and time-reversal symmetry [3]:

\[ \begin{aligned} \ddot x(s) &= -\nabla V\big(x(s)\big),\\ E(x,\dot x) &= \tfrac12\|\dot x\|^2 + V(x),\qquad \frac{dE}{ds}=0. \end{aligned} \]

The flow behaves like a simple mechanical system with potential \(V\), which makes convergence and stability transparent where the operator picture is opaque.

3) Pathological reinforcement in sign-gated dual-trace TD learning

Setting. Linear value function and TD error

\[ \begin{aligned} V_\theta(s_t) &= \langle\theta,\phi_t\rangle,\\ \delta_t &= r_{t+1}+\gamma V_\theta(s_{t+1})-V_\theta(s_t). \end{aligned} \]

with sign-gated updates and two traces of unequal decay \(n^\pm\in(0,1)\) [4]:

\[ \begin{aligned} \Delta\theta_t &= \alpha\,\delta_t\Big(e_t^{+}\mathbf{1}_{\{\delta_t>0\}}+e_t^{-}\mathbf{1}_{\{\delta_t<0\}}\Big),\\ e_{t+1}^{\pm} &= n^{\pm}e_t^{\pm}+\phi_t. \end{aligned} \]

In a single-spike feature with a delayed stochastic outcome (delay \(L\ge2\), reward \(+R\) w.p. \(p\), penalty \(-S\) w.p. \(1-p\)), the expected update along the spike is exactly \(\gamma\)-independent and flips sign at the sharp threshold [4]:

\[pR\,(n^+)^{L-2}>(1-p)S\,(n^-)^{L-2}.\]

Equivalently, writing an effective gain ratio \(\kappa=(n^+/n^-)^{L-2}\), the hazard is \(pR\,\kappa>(1-p)S\). The independence from \(\gamma\) follows from a cancellation between bootstrapping and geometric trace sums; robustness bounds show the effect persists under leakage and bootstrap noise [4].

4) In development — holonomy as first principles

Unifying framework. A 4D geometric construction with a single connection whose holonomy encodes both gravity and gauge interactions [5][6]. Formally, on a principal bundle with group \(G\times\mathrm{Spin}(1,3)\),

\[ \begin{aligned} \mathcal{A} &= \omega\oplus A,\\ \mathrm{Hol}_x(\nabla_{\mathcal A}) &\subseteq G\times\mathrm{Spin}(1,3). \end{aligned} \]

and the effective low-energy symmetry emerges as the stabilizer of the holonomy class,

\[H=\mathrm{Stab}\big([\mathrm{Hol}_x(\nabla_{\mathcal A})]\big).\]

Non-trivial \(\pi_1\) (global topology) supplies holonomy sectors that act as geometric boundary conditions; Wilson lines then shift phases in the Higgs potential and couplings while respecting gauge invariance and Elitzur’s theorem [5][6].

Flavor (sketch). A horizontal \(U(1)_F\) with Stueckelberg mass \(M_X\) controls hierarchies while suppressing FCNCs [7],

\[ \begin{aligned} \mathcal{L}_{\mathrm{eff}} &\supset \tfrac12 M_X^2(B_\mu-\partial_\mu\sigma)^2+g_F J_F^\mu B_\mu,\\ \mathcal{L}_{\mathrm{low}} &\supset \frac{g_F^2}{M_X^2}\,J_F^\mu J_{F\mu}. \end{aligned} \]

with anomaly constraints and mixing angles tied back to the holonomy geometry. (This unified/holonomy-breaking/flavor trio is actively under construction; here I’ve highlighted only the essential pieces.)

References

  1. Agawa, Y. (2025). [Under Review] A Rigorous Proof of the Mass Gap in SU(N) Yang-Mills Theory. Zenodo. DOI: 10.5281/zenodo.14975444
  2. Agawa, Y. (2025). [Under Review] Quantum Corrections and Finite Gribov Uniqueness in a Non-local Gauge Theory: An Essential Addendum to the Proof of the Yang-Mills Mass Gap. Zenodo. DOI: 10.5281/zenodo.15809222
  3. Agawa, Y. (2025). Emergent Integrability in a Wegner-type Flow Equation. Zenodo. DOI: 10.5281/zenodo.14359669
  4. Agawa, Y. (2025). Pathological Reinforcement in Sign-Gated Dual-Trace Temporal-Difference Learning: Gamma-Invariant Necessary and Sufficient Conditions and Robust Bounds. Zenodo. DOI: 10.5281/zenodo.16889669
  5. Agawa, Y. (2024). [Under Development] Holonomy-Induced Gauge Symmetry Breaking on Non-Trivial Spacetime Topologies. Zenodo. DOI: 10.5281/zenodo.14272227
  6. Agawa, Y. (2024). [Under Development] A Unified Theory of Elementary Particles as Intrinsic Structures of Four-Dimensional Spacetime. Zenodo. DOI: 10.5281/zenodo.14233888
  7. Agawa, Y. (2024). [Under Development] Gauge-Invariant Mechanism for Mass Generation and Flavor Mixing in Four-Dimensional Spacetime. Zenodo. DOI: 10.5281/zenodo.14362777
  8. Agawa, Y. (2025). Technical Brief and Public Comment Submitted in Response to the NIST AI Standards "Zero Drafts" Pilot Project. Zenodo. DOI: 10.5281/zenodo.16888077
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