Math at a Glance

This is the single-page cheat sheet for the intelligence layer. Every technique that appears in FrankenTUI’s runtime decisions is listed here with its consuming subsystem, canonical formula, and the performance property it buys. Use this table as a navigation hub — each row links to the full explanation.

For the motivation behind the whole layer, see intelligence/overview.

Bayesian inference

Technique	Where used	Core formula	Performance impact
Bayes factors	Command-palette scoring	$\dfrac{P(R\mid E)}{P(\neg R\mid E)} = \dfrac{P(R)}{P(\neg R)} \prod_i \mathrm{BF}_i$	Better ranking with fewer re-sorts
Evidence ledger	Explanations for probabilistic decisions	$\log \dfrac{P(R\mid E)}{P(\neg R\mid E)} = \log \dfrac{P(R)}{P(\neg R)} + \sum_i \log \mathrm{BF}_i$	Debuggable, auditable scoring
Log-BF capability probe	Terminal-capability detection	$\log \mathrm{BF} = \log \dfrac{P(D \mid H)}{P(D \mid \neg H)}$	Robust detection from noisy probes
Log10-BF coalescer	Resize-scheduler evidence ledger	$\mathrm{LBF} = \log_{10} \dfrac{P(E \mid \text{apply})}{P(E \mid \text{coalesce})}$	Explainable, stable resize decisions
Bayesian hint ranking	Keybinding-hint ordering	$V_i = E[U_i] + w_{\mathrm{voi}} \sqrt{\operatorname{Var}(U_i)} - \lambda C_i$	Stable, utility-aware hints
Beta-Binomial	Diff-strategy selection	$p \sim \operatorname{Beta}(\alpha, \beta)$ with binomial updates	Avoids slow strategies as workload shifts
Bayesian height predictor	Virtualized-list preallocation	$\mu_n = \dfrac{\kappa_0 \mu_0 + n \bar{x}}{\kappa_0 + n}$ + conformal $q_{1-\alpha}$	Fewer scroll jumps

Change detection and sequential testing

Technique	Where used	Core formula	Performance impact
BOCPD	Resize coalescing	Run-length posterior + hazard $H(r)$	Fewer redundant renders during drags
Run-length posterior	BOCPD core	$P(r_t = r \mid x_{1:t})$ recursion	Fast regime switches without thresholds
E-process	Budget alerts, throttle	$W_t = W_{t-1}(1 + \lambda_t (X_t - \mu_0))$	Safe early exits under continuous monitoring
GRAPA	Adaptive e-process	$\lambda_{t+1} = \lambda_t + \eta \nabla_\lambda \log W_t$	Self-tuning sensitivity
CUSUM	Budget change detection	$S_t = \max(0, S_{t-1} + X_t - \mu_0 - k)$	Fast drift detection
CUSUM hover stabilizer	Mouse hover jitter	$S_t = \max(0, S_{t-1} + d_t - k)$	Stable hover targets without lag
Alpha-investing	Sequential FDR	$E[V]/E[R] \le W_0$ (wealth process)	Bounded false-alert rate across monitors

Conformal prediction

Technique	Where used	Core formula	Performance impact
Vanilla conformal	Risk bounds	$q = \operatorname{Quantile}_{\lceil (1-\alpha)(n+1) \rceil}(R)$	Stable thresholds without tuning
Mondrian conformal	Frame-time risk gating	$\hat{y}^+ = \hat{y} + q_{1-\alpha}(\lvert r \rvert)$ per bucket	Safe budget gating with sparse data
Conformal rank confidence	Command-palette stability	$p_i = \dfrac{1}{n} \sum_j \mathbf{1}[g_j \le g_i]$ (gap-based p-value)	Deterministic tie-breaks, stable top-k

Value of information and sampling

Technique	Where used	Core formula	Performance impact
VOI sampling	Expensive measurements	$\mathrm{VOI} = \operatorname{Var}(p) - E[\operatorname{Var}(p \mid \text{sample})]$	Lower overhead in steady state

Control theory and scheduling

Technique	Where used	Core formula	Performance impact
PID / PI	Degradation control	$u_t = K_p e_t + K_i \sum e_s + K_d \Delta e_t$	Smooth frame-time stabilisation
MPC (evaluation)	Controller check	$\min_{u_{t:t+H}} \sum \lVert y_{t+k} - y^\star \rVert^2 + \rho \lVert u_{t+k} \rVert^2$	Confirms PI is sufficient
SOS barrier	Admissible-region proof	$B(x) = \sum_{i+j\le 6} c_{ij} x_1^i x_2^j > 0$	Constant-time safe/unsafe classification
Jain’s fairness	Input guard	$F = \dfrac{(\sum x_i)^2}{n \sum x_i^2}$	Prevents rendering from starving input
Smith’s rule + aging	Queueing scheduler	$\text{priority} = \dfrac{w}{r} + a \cdot \text{wait}$	Fair throughput under load

Data structures with statistical guarantees

Technique	Where used	Core formula	Performance impact
Count-Min Sketch	Width cache, timeline aggregation	$\hat{f}(x) = \min_j C_{j, h_j(x)}$	Fast approximate counts
W-TinyLFU admission	Width-cache admission	admit iff $\hat{f}(x) \ge \hat{f}(y)$ (doorkeeper → CMS)	Higher hit rate, fewer recomputes
PAC-Bayes	Sketch calibration	$\bar{e} + \sqrt{\operatorname{KL}(q \\| p)/(2n)}$	Tighter error bounds
Fenwick tree	Virtualized lists	Prefix sums with $i \pm (i \wedge -i)$	$O(\log n)$ scroll + height queries
Summed-area table	Tile-skip diff	$\mathrm{SAT}(x, y) = A(x, y) + \mathrm{SAT}(x-1, y) + \mathrm{SAT}(x, y-1) - \mathrm{SAT}(x-1, y-1)$	Skip empty tiles on large screens
S3-FIFO cache	Hot-key cache	Small (10%) + Main (90%) + Ghost FIFOs	Scan-resistant, better than LRU/ARC at lower overhead

Visual effects (deterministic math)

Effect	Core equation	What it produces
Metaballs	$F(x, y) = \sum_i \dfrac{r_i^2}{(x - x_i)^2 + (y - y_i)^2}$ , iso-surface $F \ge \tau$	Smooth, organic blob fields
Plasma	$v = \dfrac{1}{6} \sum_{k=1}^6 \sin(\phi_k(x, y, t))$	Psychedelic interference bands
Gray-Scott	$\partial_t u = D_u \nabla^2 u - u v^2 + F(1 - u);\; \partial_t v = D_v \nabla^2 v + u v^2 - (F + k) v$	Reaction-diffusion morphogenesis
Clifford attractor	$x_{t+1} = \sin(a y_t) + c \cos(a x_t);\; y_{t+1} = \sin(b x_t) + d \cos(b y_t)$	Strange-attractor filaments
Mandelbrot / Julia	$z_{n+1} = z_n^2 + c$ (escape-time colouring)	Fractal boundaries + deep zooms
Lissajous / harmonograph	$x = A \sin(a t + \delta);\; y = B \sin(b t + \phi)$	Phase-locked curves
Flow field	$\vec{v}(x, y) = (\cos 2\pi N,\; \sin 2\pi N);\; p_{t+1} = p_t + \vec{v} \Delta t$	Particle ribbons
Wave interference	$I(x, t) = \sum_i \sin(k_i \lVert x - s_i \rVert - \omega_i t)$	Multi-source ripples
Spiral galaxy	$r = a e^{b \theta};\; \theta(t) = \theta_0 + \omega t$	Logarithmic-spiral starfields
Spin lattice (LLG)	$\dfrac{d \vec{S}}{dt} = -\vec{S} \times \vec{H} - \alpha \vec{S} \times (\vec{S} \times \vec{H})$	Magnetic-domain dynamics

Animation and UI dynamics

Technique	Where used	Core formula	Performance impact
Damped spring	Animation transitions	$x'' + c x' + k (x - x^\star) = 0$	Natural motion without frame-rate artefacts
Easing curves	Fade/slide timing	$t^2$ , $1 - (1 - t)^2$ , cubic variants	Predictable velocity shaping
Staggered cascade	List animations	$\text{offset}_i = D \cdot \text{ease}(i / (n - 1))$	Coordinated, non-uniform entrances
Sine pulse	Attention pulses	$p(t) = \sin(\pi t)$	Smooth 0→1→0 emphasis
Perceived luminance	Dark/light probe	$Y = 0.299 R + 0.587 G + 0.114 B$	Reliable theme defaults

Cross-cutting summaries

Technique	Where used	Core formula	Performance impact
Rough-path signatures	Workload fingerprinting, regression detection	$S(X)^k_{i_1 \ldots i_k} = \int dX^{i_1} \cdots dX^{i_k}$	Reparameterisation-invariant trace comparison
Incremental view maintenance	Derived-state updates	Signed deltas + bilinear join rule + 50% fallback	Derived views updated in $O(\Delta)$

Using this table

Each row linking to an explanation page is a live link; hover to see the preview, click to read the full motivation, math, worked example, Rust interface, and debugging notes.
Rows without links describe techniques whose primary explanation lives elsewhere (data-structure pages under /reference, effect pages under /extras, animation under /style). They are included here for completeness so the full mathematical surface of FrankenTUI is visible in one place.
The “performance impact” column is deliberately qualitative. Hard numbers are in the individual pages’ worked examples and in the benchmark reports referenced by /testing/benchmark-gate.

If a decision in FrankenTUI surprises you, grep this table for the subsystem. The corresponding intelligence page will have the evidence-sink event name and a jq one-liner. That’s usually a one-minute path from “huh?” to “oh, I see exactly why.”

Cross-references

Intelligence overview — the narrative introduction to why this table exists.
/operations/frame-budget — the operator-facing view of the same machinery.
/reference/telemetry-events — the schema of the evidence sink’s event catalog.