This blog strings together the research journeys across all current topics. It is a place for reflection, hunches and the occasional equation rendered with MathJax.
Working through the spectral action again felt like returning to an old puzzle. Computing the heat-kernel coefficients on $S^4$ gave the comforting numbers $a_0=\tfrac16$ and $a_2=\tfrac13$. The see-saw mechanism produced a light neutrino mass around $0.1\,\text{eV}$, a result that still surprises me for its elegance, and the one-loop $SU(3)$ running reaffirmed asymptotic freedom.
Pushing to $a_4=1/10$ with a SymPy script made the spectral action feel richer, and computing $J\approx3\times10^{-5}$ was a crisp reminder of how small CP violation really is. The near meeting of gauge couplings keeps grand unification tantalisingly close.
The teleportation protocol worked in simulation with unit fidelity; seeing the state reappear elsewhere always feels a bit like magic. A quick computation showed the GHZ state's entropy is $1$ bit, and an entanglement swapping routine yielded almost perfect concurrence, hinting that the entanglement web can be rerouted at will.
The algebraic punch of the CHSH value $2\sqrt2$ never fades. Entropy of the $W$ state at $0.918$ bits shows how democratically it shares entanglement, and a modest negativity in a mixed state underscored its fragility.
Exploring Wilson loops with tension $\sigma=0.2\,\text{GeV}^2$ suggested an area law, and fitting a propagator hinted at a mass around $0.48\,\text{GeV}$. The numbers are modest, yet each iteration sharpens intuition about how a gap could emerge.
Seeing an area law even on a $4\times4$ lattice was strangely satisfying, and the gentle slide of the $2+1$ dimensional coupling hints at confinement. Revisiting the $O(3)$ sigma model's $e^{-2\pi/g^2}$ gap felt like meeting an old friend.
Counting $30$ proper colorings of $C_5$ with three colors was a playful diversion. Partition growth at $p(100)=190\,569\,292$ came via Hardy–Ramanujan, and the periodic continued fraction of $\sqrt3$ reminded me how periodicity hides just beneath irrationality.
Confirming $C_5=42$ brought back combinatorics nostalgia. A tiny hyperbolic triangle enclosing $\pi/6$ area felt delightfully non-Euclidean, and summing prime reciprocals to $100$ reminded me how slowly they diverge.
I coaxed $a_0=\tfrac{2}{3}$ out of the spectral action and couldn't resist smirking when the CKM numbers lined up. Watching $g_1$ chase $g_2$ and $g_3$ without ever quite meeting them keeps my unification dreams stubbornly alive.
Teleportation at $F\approx0.93$ feels like magic with frayed edges. Concurrence peaking near $0.75$ and an entropy of $0.85$ bits told me the state was messy, but I still root for entanglement even when it misbehaves.
The Polyakov loop hovering close to one whispers deconfinement, yet the two-loop $g(10)\approx0.86$ slides toward weakness. A glueball mass near $0.68\,\text{GeV}$ smells plausible enough that I cling to the gap hypothesis.
Counting primes to $100$ gave the faithful $\pi(100)=25$ and the Catalans marching $1,1,2,5,14,42$ still look like a smug secret handshake. Watching Euler characteristic flip sign with genus reminds me topology can be cheeky.
A plain Hebbian run barely nudged the weight to $0.1$, while the STDP curve's $e^{-|\Delta t|/10}$ symmetry felt both biological and annoyingly fuzzy. Brains learn with spikes and timing, whether my analytical side likes it or not.