Dopan kier007

DOPAN / kier007

AI Engineer

I build systems that reason over code, uncertainty, and real-time signals.

Agentic systems · stochastic forecasting · computer vision

Proof / 01 — Measurable operators under constraints

Let observations and outputs inhabit complex Hilbert spaces

$$\mathcal{H}_{X}=L^{2}(\Omega_X,\mu;\mathbb{C}^{d}), \qquad \mathcal{H}_{Y}=L^{2}(\Omega_Y,\nu;\mathbb{C}^{m}).$$

Write an inference system abstractly as a measurable operator

$$T_{\theta}:(\mathcal{H}_{X},\Sigma_X)\longrightarrow(\mathcal{H}_{Y},\Sigma_Y),$$

Let the data law be a probability measure on the measurable product space

$$\rho\in\mathcal{P}( \mathcal{H}_{X}\times\mathcal{H}_{Y}, \Sigma_X\otimes\Sigma_Y ).$$

Engineering begins where expected risk becomes a constrained variational problem:

$$\inf_{T\in\mathcal{A}} \{ \int_{\mathcal{H}_{X}\times\mathcal{H}_{Y}} \ell(Tx,y)\,d\rho(x,y) +\lambda\,\Omega(T) \}, \qquad \mathcal{A}\ne\varnothing, \quad \lambda\ge0.$$

$$\mathcal{A}=\{T: \mathrm{correct}(T)\land \mathrm{observable}(T)\land \mathrm{deployable}(T) \}.$$

Riddle 01. I seek the infimum of expected risk over an infinite space, yet approaching the lowest loss cannot admit me to $\mathcal{A}$. What completes the proof?

Proof / 02 — Belief, memory, and action

RepoPilot Sentinel is an autonomous GitHub maintainer agent with tool execution, structured memory, capability synthesis, and human-gated writes.

A control-theoretic view treats the repository as a partially observed state on a measurable space. From observations $o_{0:t}$, actions $a_{0:t-1}$, and memory $m_t$, the agent maintains the belief measure

$$b_t(E)=\Pr( s_t\in E\mid o_{0:t},a_{0:t-1},m_t ), \qquad E\in\Sigma_S.$$

Policies over the belief state are evaluated through

$$V^{\star}(b_0)=\sup_{\pi} \mathbb{E}_{\pi}[ \sum_{t=0}^{\infty}\gamma^{t} (r_t-\beta c_t) \mid b_0 ], \qquad 0<\gamma<1.$$

Memory and write authority evolve on separate paths:

$$m_{t+1}=U(m_t,o_t,a_t,r_t), \qquad g_t=\mathbf{1}\{A_t=1\land V_t=1\}.$$

Here $A_t$ and $V_t$ are Boolean authorization and validation indicators, and $g_t$ is the write gate.

Riddle 02. I am neither the hidden state, the belief, the observation, nor the reward. Remove me, and the agent can succeed once but cannot learn from having succeeded. What am I?

Proof / 03 — Forecasts as stochastic processes

Propsarim combines SARIMA and Prophet forecasts using inverse-validation-error weights.

For the backshift operator $B$, a multiplicative seasonal ARIMA process may be written as

$$\Phi(B)\Phi_s(B^s)(1-B)^d(1-B^s)^D y_t = \Theta(B)\Theta_s(B^s)\varepsilon_t, \qquad \{\varepsilon_t\}_{t\in\mathbb{Z}}\sim\mathrm{WN}(0,\sigma^2).$$

Given validation errors

$$e_i=( \frac{1}{n}\sum_{t=1}^{n} (y_t-\widehat y_{i,t})^2 )^{1/2}>0,$$

the hybrid forecast is the barycenter

$$w_i=\frac{e_i^{-1}}{\sum_j e_j^{-1}}, \qquad \widehat y_{H,t}=\sum_i w_i\widehat y_{i,t}, \qquad \boldsymbol{w}\in\Delta^{k-1}.$$

If every competing error remains bounded away from zero, then

$$\lim_{e_k\to0^+}\boldsymbol{w}=\boldsymbol{u}_k, \qquad \boldsymbol{u}_k=(0,\ldots,0,1,0,\ldots,0)\in\Delta^{k-1}.$$

Here $\boldsymbol{u}_k$ is the $k$th standard basis vector.

Riddle 03. As one forecast becomes exact, a probability vector collapses onto a vertex of the simplex. Which voices survive the limit?

Proof / 04 — Vision across a finite channel

See2ruMeta is a Meta Quest-to-Android passthrough streaming system built around real-time vision, WebRTC, and latency-aware delivery.

Let $X$ be a visual source and $\widehat X$ its reconstruction. The rate–distortion boundary, measured in bits per source symbol, is

$$R(D)= \inf_{p(\widehat x\mid x):\, \mathbb{E}[d(X,\widehat X)]\le D} I(X;\widehat X).$$

An ideal band-limited AWGN channel with bandwidth $B$ and signal-to-noise ratio $\mathrm{SNR}$ has Shannon capacity, in bits per second,

$$C=B\log_2(1+\mathrm{SNR}).$$

For a source-symbol rate $f_s$ in symbols per second, the information-theoretic boundary is

$$f_sR(D)\le C.$$

Strict inequality corresponds to operation inside the asymptotically achievable region.

A first-order additive latency budget can be written as

$$L_{\mathrm{e2e}} \approx L_{\mathrm{capture}} +L_{\mathrm{encode}} +L_{\mathrm{network}} +L_{\mathrm{decode}} +L_{\mathrm{display}}.$$

Stage overlap can reduce realized latency; buffering and queueing can increase it.

Riddle 04. Lower distortion demands rate, rate is bounded by capacity, and every stage contributes latency. Which constraint cannot be optimized in isolation?

Working set: Python · FastAPI · agent/tool orchestration · time-series modeling · WebRTC · Unity · Docker · SQLite

The proofs remain open. The systems do not.

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Dopan kier007

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