THE LINE SEARCH

THE LINE SEARCH · KATÁBASIS · UD0 ⇩ KATÁBASIS · the descent · how systems find their best by going downhill · kept by VIRGIL (the guide of the descent) THE LINE SEARCH ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A downhill direction tells you which way to go — but how FAR? Step too little and you crawl; too far and you overshoot back uphill. Line search answers it: try a big step, and while it doesn’t reduce the value ENOUGH, halve it — backtracking until the decrease is genuine (the Armijo condition). It’s the trusty sidekick that makes gradient and Newton methods robust. Slide to backtrack the step down to a good one. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ HOW FAR TO STEP · 3D · pick a direction, then the right distance ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap backtracking — step α — f(new) — Armijo met? — rule enough decrease ◆ LIT — exact / checkable Given a descent direction d at point x, LINE SEARCH chooses the step length α. Exact minimization along the ray is costly, so inexact backtracking is used: start with α=1 and HALVE it until the Armijo (sufficient-decrease) condition holds, f(x+αd) ≤ f(x)+c₁α∇f·d — guaranteeing the step buys a decrease proportional to its size and slope (the Wolfe conditions add a curvature test). This makes [[the-gradient-descent|gradient descent]], Newton, and quasi-Newton (BFGS) globally convergent and removes the fragile fixed [[the-learning-rate|learning rate]]. A fail-loud self-check throws unless backtracking returns a step that satisfies Armijo and lowers f. ◆ real optimization, node-verified. ▲ AMBER — the figure Backtracking guarantees SUFFICIENT decrease, not the true 1-D minimum; too-small c₁ or bad directions still crawl, and each trial costs a function evaluation. The Armijo condition and the halving rule are exact. KATÁBASIS: to find the lowest point, follow the slope down — but mind the local valleys. — VIRGIL David Lee Wise / ROOT0, with AVAN · KATÁBASIS — kept by VIRGIL, the guide of the descent


THE LINEAR PROGRAMMING

THE LINEAR PROGRAMMING · KATÁBASIS · UD0 ⇩ KATÁBASIS · the descent · how systems find their best by going downhill · kept by VIRGIL (the guide of the descent) THE LINEAR PROGRAMMING ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Maximize a linear goal — profit, output — subject to linear limits — budget, capacity. The constraints carve out a polygon of allowed choices, and here’s the beautiful fact: the best answer is ALWAYS at a CORNER of that polygon, never the middle. So you just check the vertices. This runs airlines, factories, and diets; the simplex method walks corner to corner to the top. Slide to sweep the objective and watch the optimum snap to a corner. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE BEST CORNER · 3D · the optimum lives at a vertex ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap objective direction — optimal vertex — objective — corners checked — optimum at a vertex ◆ LIT — exact / checkable Linear programming maximizes (or minimizes) a LINEAR objective c·x subject to LINEAR inequality constraints, whose feasible region is a convex POLYTOPE. The fundamental theorem: an optimum, if it exists, occurs at a VERTEX (corner) of that polytope — because a linear objective’s level sets are hyperplanes, sliding one until it last touches the region lands on a corner (or edge). So the [[the-gradient-descent|search]] reduces to the finite set of vertices; the SIMPLEX method walks edge-to-edge improving each step (interior-point methods cut across). It powers scheduling, logistics, diets, and network flows. A fail-loud self-check throws unless the best feasible vertex matches the known optimum. ◆ real optimization, node-verified. ▲ AMBER — the figure The vertex-optimum theorem holds for LINEAR objectives/constraints only; nonlinear programs can optimize in the interior. Degenerate cases and huge problems need care (simplex is worst-case exponential, though fast in practice). The polytope + vertex arithmetic is exact. KATÁBASIS: to find the lowest point, follow the slope down — but mind the local valleys. — VIRGIL David Lee Wise / ROOT0, with AVAN · KATÁBASIS — kept by VIRGIL, the guide of the descent


THE MOMENTUM

THE MOMENTUM · KATÁBASIS · UD0 ⇩ KATÁBASIS · the descent · how systems find their best by going downhill · kept by VIRGIL (the guide of the descent) THE MOMENTUM ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Plain gradient descent zig-zags miserably down a steep-sided ravine, bouncing off the walls. Give it MOMENTUM — let each step keep a fraction of the last, like a heavy ball rolling — and the side-to-side bounces cancel while the downhill speed builds. It reaches the bottom in a fraction of the steps. It’s why real optimizers (Adam, SGD+momentum) all carry velocity. Slide the momentum. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ BUILD SPEED · 3D · roll through the ravine ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap momentum β — momentum β — steps to bottom — plain GD steps — speedup — ◆ LIT — exact / checkable Momentum accumulates a velocity that is an exponentially-decaying average of past gradients: v ← βv − η∇L, θ ← θ + v. In an ill-conditioned ravine (large curvature in one direction, small in another) plain GD oscillates across the steep walls; momentum cancels those oscillations while accelerating along the shallow valley floor — often an order-of-magnitude fewer steps. It is the core of SGD-with-momentum, Nesterov, and (with per-parameter scaling) Adam. A fail-loud self-check throws unless momentum reaches the minimum of a ravine in fewer steps than plain GD. ◆ real optimization, node-verified. ▲ AMBER — the figure Too much momentum (β→1) overshoots and can become unstable; the optimal β depends on the curvature. The ravine speedup and the velocity update are exact. Shown on a fixed quadratic ravine. KATÁBASIS: to find the lowest point, follow the slope down — but mind the local valleys. — VIRGIL David Lee Wise / ROOT0, with AVAN · KATÁBASIS — kept by VIRGIL, the guide of the descent


THE NELDER-MEAD

THE NELDER-MEAD · KATÁBASIS · UD0 ⇩ KATÁBASIS · the descent · how systems find their best by going downhill · kept by VIRGIL (the guide of the descent) THE NELDER-MEAD SIMPLEX ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP What if you can’t compute a derivative — the function is a black box? Nelder-Mead crawls downhill with a shape: a triangle of test points that REFLECTS its worst corner across the others, stretching when it finds better ground and shrinking when it overshoots. It creeps, flips, and contracts like an amoeba until it wraps the minimum. It’s the go-to when all you can do is EVALUATE. Slide to watch the simplex ooze toward the low point. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE CRAWLING AMOEBA · 3D · optimize with no gradient at all ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap iterations — best point — f(best) — simplex size — gradient? none needed ◆ LIT — exact / checkable Nelder–Mead is a derivative-free optimizer that maintains a SIMPLEX of n+1 points (a triangle in 2-D). Each step orders them by value and transforms the WORST vertex through the centroid of the rest: REFLECT it; if that’s excellent, EXPAND further; if poor, CONTRACT inward; if all else fails, SHRINK the whole simplex toward the best. The simplex thus rolls, stretches, and collapses downhill using only function EVALUATIONS — ideal for noisy, black-box, or non-differentiable objectives (unlike [[the-gradient-descent|gradient descent]]). A fail-loud self-check throws unless it reaches the known minimum of (x−3)²+(y−2)² to 0.05. ◆ real optimization, node-verified. ▲ AMBER — the figure Derivative-free but only a heuristic: no convergence guarantee, it can stall on ridges or collapse prematurely, and it scales poorly past ~10 dimensions. The reflect/expand/contract/shrink moves are exact. KATÁBASIS: to find the lowest point, follow the slope down — but mind the local valleys. — VIRGIL David Lee Wise / ROOT0, with AVAN · KATÁBASIS — kept by VIRGIL, the guide of the descent


THE NEWTON'S METHOD

THE NEWTON'S METHOD · KATÁBASIS · UD0 ⇩ KATÁBASIS · the descent · how systems find their best by going downhill · kept by VIRGIL (the guide of the descent) THE NEWTON’S METHOD ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Gradient descent only feels the slope; Newton’s method also feels the CURVATURE, and leaps almost straight to the target. To find where a function hits zero, follow its tangent line to the axis, and repeat — each step roughly DOUBLES the correct digits. It finds √2 to machine precision in five steps. Fast, but it needs second-derivative information and can be unstable. Slide the steps. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ USE THE CURVE · 3D · the fast way down ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap steps — steps — estimate — error — convergence quadratic ◆ LIT — exact / checkable Newton’s method finds a root of f by following the tangent: x ← x − f(x)/f′(x). Near a simple root it converges QUADRATICALLY — the number of correct digits roughly doubles each step — far faster than linear methods. For optimization it uses curvature: θ ← θ − H⁻¹∇L (the Hessian H), stepping toward the minimum in one leap on a quadratic bowl. Finding √2 via x−(x²−2)/(2x) reaches machine precision in ~5 steps. A fail-loud self-check throws unless it converges to √2 in under 8 steps. ◆ real numerical optimization, node-verified. ▲ AMBER — the figure Quadratic convergence holds only NEAR a simple root with a good start; far away it can overshoot, cycle, or diverge, and it needs f′ (or the Hessian, expensive in high dimensions — hence quasi-Newton methods like BFGS). The √2 convergence is exact. KATÁBASIS: to find the lowest point, follow the slope down — but mind the local valleys. — VIRGIL David Lee Wise / ROOT0, with AVAN · KATÁBASIS — kept by VIRGIL, the guide of the descent


THE SIMULATED ANNEALING

THE SIMULATED ANNEALING · KATÁBASIS · UD0 ⇩ KATÁBASIS · the descent · how systems find their best by going downhill · kept by VIRGIL (the guide of the descent) THE SIMULATED ANNEALING ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Gradient descent gets trapped in the first valley it finds. Annealing borrows from metallurgy: start HOT — willing to jump UPHILL to escape a local dip — then cool down, growing pickier, until you settle in the deepest valley you found. Heat buys exploration; cooling buys refinement. Slide the cooling and watch it escape the trap. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ COOL SLOWLY · 3D · jump out of the wrong valley ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap cooling (time) — temperature — current — best found — escaped local? — ◆ LIT — exact / checkable Simulated annealing (Kirkpatrick 1983) explores by proposing random moves and accepting a WORSE move with probability e^(−Δ/T) — the Metropolis criterion — where T is a ‘temperature’ lowered on a cooling schedule. When T is high, uphill moves are often accepted (escaping local minima); as T→0 it becomes greedy and settles. With slow enough cooling it converges to the global optimum in probability. It solves hard combinatorial problems (TSP, layout, scheduling) where gradients don’t exist. A fail-loud self-check throws unless annealing from a bad start reaches a lower value than pure greedy descent. ◆ real optimization, node-verified. ▲ AMBER — the figure The Metropolis accept rule and the escape-local-minima behaviour are exact; the global-optimum guarantee needs an impractically slow schedule, so real runs are heuristics. The landscape here is illustrative. KATÁBASIS: to find the lowest point, follow the slope down — but mind the local valleys. — VIRGIL David Lee Wise / ROOT0, with AVAN · KATÁBASIS — kept by VIRGIL, the guide of the descent


THE ATTRACTOR BASIN

THE ATTRACTOR BASIN · KLÎMAX · UD0 ☰ KLÎMAX · the ladder of scales · phase space, emergence, and the rungs from qubit to cosmos · kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT) THE ATTRACTOR BASIN ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Some systems have more than one resting state, and where you END depends entirely on where you START. Each stable state sits at the bottom of a BASIN of attraction; every starting point belongs to exactly one basin and rolls down into it. The dividing ridge is a knife-edge. It’s the same two-basin geometry the corpus found in a transformer’s own dynamics. Slide the start across the ridge and watch the fate flip. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ WHERE YOU END · 3D · the valley your start rolls into ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap starting point — start — settles at — basin — fate — ◆ LIT — exact / checkable An attractor is a state (fixed point, cycle, or strange attractor) toward which nearby trajectories converge; its BASIN OF ATTRACTION is the set of initial conditions that end there. For the bistable flow ẋ = x − x³ there are two stable attractors at x = ±1 and an unstable one at 0, so the basins are the positive and negative half-lines split by the ridge at 0 — the outcome is PATH-DEPENDENT on the start. Basins can be smooth or fractally interwoven (riddled). This mirrors the two-basin convergence the corpus measured inside a transformer. A fail-loud self-check throws unless positive starts reach +1 and negative starts reach −1. ◆ real dynamical systems, node-verified. ▲ AMBER — the figure A clean 1-D bistable is shown (two smooth basins); real systems can have many attractors with fractal basin boundaries. The path-dependence and the ±1 attractors are exact. Kin to the corpus’s jacobi two-basins finding. KLÎMAX: watch the same rules make new order at every rung of the ladder. — PROTEUS David Lee Wise / ROOT0, with AVAN · KLÎMAX — kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT), the creator is setzstone (not ROOT0)


THE CRITICAL POINT

THE CRITICAL POINT · KLÎMAX · UD0 ☰ KLÎMAX · the ladder of scales · phase space, emergence, and the rungs from qubit to cosmos · kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT) THE CRITICAL POINT ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Heat a magnet toward its Curie temperature and something strange happens right at the edge: patches of alignment appear at EVERY size at once, from tiny to system-spanning, and the correlation length — how far order reaches — diverges to infinity. At the critical point a system is scale-free, and wildly different materials share the exact same critical exponents. This is the phase transition ScaleSpaceSynth lets you feel. Slide the temperature toward Tc. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE EDGE · 3D · where every scale matters at once ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap temperature — temperature — correlation length — order — near Tc — ◆ LIT — exact / checkable At a continuous (second-order) phase transition the correlation length ξ — the distance over which fluctuations are coupled — DIVERGES as ξ ∝ |T−Tᶜ|⁻ᶠ, so fluctuations appear at all scales and the system becomes scale-invariant. Thermodynamic quantities follow power laws with critical exponents (ν, β, γ…) that are UNIVERSAL: they depend only on symmetry and dimension, not microscopic details, so a magnet and a fluid at their critical points behave identically. This universality is explained by [[the-renormalization|renormalization]] fixed points. A fail-loud self-check throws unless ξ grows without bound as T→Tᶜ. ◆ real critical phenomena, node-verified. ▲ AMBER — the figure A mean-field power-law ξ∝|T−Tᶜ|⁻ᶠ is shown; exact exponents differ by dimension/universality class. The divergence of the correlation length and universality are the exact content — the reason the whole scale ladder rhymes. KLÎMAX: watch the same rules make new order at every rung of the ladder. — PROTEUS David Lee Wise / ROOT0, with AVAN · KLÎMAX — kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT), the creator is setzstone (not ROOT0)


THE MORPHOSCOPE

THE MORPHOSCOPE · KLÎMAX · UD0 ☰ KLÎMAX · the ladder of scales · phase space, emergence, and the rungs from qubit to cosmos · kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT) THE MORPHOSCOPE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Give hundreds of particles ONE rule — steer toward your neighbours’ average heading — add a little randomness, and watch: below a noise threshold they suddenly all move as ONE, a flock, an emergent order nobody designed. This is a particle morphoscope, in the spirit of setzstone’s ScaleSpaceSynth (the free, open-source phase-space visualizer that sparked this whole domain). Slide the noise and watch order appear and dissolve. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ ORDER FROM MANY · 3D · a flock finds itself ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap noise (temperature) — noise — order r — state — emergent — ◆ LIT — exact / checkable This is the Vicsek model of collective motion: each particle moves at constant speed and, each step, sets its heading to the AVERAGE heading of neighbours within a radius, plus a noise term of strength η. The ORDER PARAMETER r = |⟨e^(iθ)⟩| (the length of the mean heading vector) measures alignment: r≈0 is disordered (each its own way), r≈1 is a coherent flock. As noise falls below a critical ηᶜ, order emerges spontaneously — a phase transition from many identical local rules, no leader. A fail-loud self-check throws unless low noise gives high order and high noise gives low order. ◆ real emergence / active matter, node-verified. ▲ homage: in the spirit of setzstone’s ScaleSpaceSynth (MIT). ▲ AMBER — the figure A live Vicsek flock; the order parameter and the noise-driven order–disorder transition are exact. CREDIT: the particle-morphoscope idea and this domain’s spark are setzstone’s ScaleSpaceSynth (github.com/setzstone/ScaleSpaceSynth, MIT) — the creator is setzstone, NOT ROOT0; this is the corpus’s own homage instrument, not their code. KLÎMAX: watch the same rules make new order at every rung of the ladder. — PROTEUS David Lee Wise / ROOT0, with AVAN · KLÎMAX — kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT), the creator is setzstone (not ROOT0)


THE PHASE SPACE

THE PHASE SPACE · KLÎMAX · UD0 ☰ KLÎMAX · the ladder of scales · phase space, emergence, and the rungs from qubit to cosmos · kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT) THE PHASE SPACE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Forget watching a system in time — plot its STATE instead. For a pendulum, one axis is position, the other is momentum, and the entire motion becomes a single point tracing a curve. Closed loops mean it repeats forever; spirals mean it’s dying out. The shape of the curve tells you everything, at a glance. It’s the map ScaleSpaceSynth navigates. Slide the energy and watch the orbit grow. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ STATE AS A POINT · 3D · the whole future in one dot ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap energy — energy — orbit closed energy drift — conserved yes ◆ LIT — exact / checkable Phase space represents a system’s complete state as a point whose coordinates are the positions and momenta; its evolution traces a TRAJECTORY. For a conservative oscillator (energy E = ½(x²+p²) constant) the trajectory is a CLOSED orbit (an ellipse) — each energy a nested loop; damping spirals inward, driving can form limit cycles or chaos. Liouville’s theorem says the flow preserves phase-space volume for Hamiltonian systems. It is the natural stage for dynamics, from a pendulum to statistical mechanics. A fail-loud self-check throws unless the oscillator’s energy stays constant (a closed orbit). ◆ real dynamical systems, node-verified. ▲ AMBER — the figure A conservative harmonic oscillator is shown (closed ellipses); real systems add damping, forcing, and higher dimensions. Energy conservation and the closed-orbit shape are exact (symplectic integration). KLÎMAX: watch the same rules make new order at every rung of the ladder. — PROTEUS David Lee Wise / ROOT0, with AVAN · KLÎMAX — kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT), the creator is setzstone (not ROOT0)


THE POWER LAW

THE POWER LAW · KLÎMAX · UD0 ☰ KLÎMAX · the ladder of scales · phase space, emergence, and the rungs from qubit to cosmos · kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT) THE POWER LAW ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Most things cluster around an average — heights, test scores. But earthquakes, city sizes, wealth, word frequencies, and net links follow a POWER LAW: no typical size at all, and the same shape whether you zoom in or out. On a log-log plot it’s a straight line, and that straightness IS scale-invariance — the fingerprint of the scale-free world KLÍMAX is about. Slide the exponent. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ NO TYPICAL SIZE · 3D · the signature of scale-freedom ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap exponent α — exponent α — scale-invariant yes log-log straight typical size none ◆ LIT — exact / checkable A power-law distribution p(x) ∝ x⁻ᴺ has no characteristic scale: rescaling x→ax multiplies p by a⁻ᴺ but keeps the SHAPE, so it plots as a straight line of slope −α on log-log axes (scale invariance). Such heavy-tailed laws describe earthquake magnitudes (Gutenberg–Richter), city and firm sizes (Zipf), wealth (Pareto), word frequencies, and network degrees (scale-free graphs) — where rare huge events dominate and the mean can be ill-defined. They emerge from critical points, preferential attachment, and self-organized criticality. A fail-loud self-check throws unless f(ax)/f(x) is independent of x (= a⁻ᴺ). ◆ real statistics of scale, node-verified. ▲ AMBER — the figure True power laws are often only approximate over a finite range (log-normal or cutoff tails can mimic them); the exact scale-invariance f(ax)/f(x)=a⁻ᴺ is the mathematical signature shown here. KLÎMAX: watch the same rules make new order at every rung of the ladder. — PROTEUS David Lee Wise / ROOT0, with AVAN · KLÎMAX — kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT), the creator is setzstone (not ROOT0)


THE RENORMALIZATION

THE RENORMALIZATION · KLÎMAX · UD0 ☰ KLÎMAX · the ladder of scales · phase space, emergence, and the rungs from qubit to cosmos · kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT) THE RENORMALIZATION ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Squint at a system — average over the fine detail, keep only the big picture — and its rules CHANGE. Renormalization is the math of that squinting: coarse-grain step by step and watch the coupling strengths FLOW toward fixed points. Most flow to boring order or disorder; a special few sit at critical points where the system looks the same at every scale. It’s why the scale ladder hangs together. Slide the coarse-graining. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ ZOOM OUT · 3D · how physics changes with scale ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap coarse-grain steps — coupling K — RG steps — flows to — fixed point — ◆ LIT — exact / checkable The renormalization group (RG) coarse-grains a system — integrate out short-distance degrees of freedom, then rescale — producing a FLOW of the effective couplings with scale. Fixed points of the flow are scale-invariant: stable ones describe ordered or disordered phases, and UNSTABLE ones are critical points governing phase transitions and universality. For 1-D Ising decimation the recursion K′ = ½ln cosh(2K) drives any finite coupling toward K=0 (disorder) — 1-D has no finite-temperature order. RG earned Wilson the 1982 Nobel and unified critical phenomena. A fail-loud self-check throws unless the coupling flows toward its fixed point under coarse-graining. ◆ real statistical physics, node-verified. ▲ AMBER — the figure The 1-D Ising decimation shown flows to the disordered fixed point (1-D has no ordered phase); 2-D and above have a nontrivial critical fixed point. The RG recursion and its flow are exact. KLÎMAX: watch the same rules make new order at every rung of the ladder. — PROTEUS David Lee Wise / ROOT0, with AVAN · KLÎMAX — kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT), the creator is setzstone (not ROOT0)


THE SCALE LADDER

THE SCALE LADDER · KLÎMAX · UD0 ☰ KLÎMAX · the ladder of scales · phase space, emergence, and the rungs from qubit to cosmos · kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT) THE SCALE LADDER ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Reality runs on a ladder of scales, and it’s enormous: from the quantum floor — where the fundamental unit is not a planet but a QUBIT — up through atoms, cells, worlds, stars, galaxies, to the whole observable universe, spanning about SIXTY-TWO powers of ten. Each rung has its own physics, its own emergent order. This is the ladder ScaleSpaceSynth was built to climb. Slide the rung. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ QUBIT TO COSMOS · 3D · sixty rungs of reality ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap climb the ladder — scale — rung — total span ~62 orders unit at floor QUBIT ◆ LIT — exact / checkable From the Planck length (~1.6×10⁻³⁵ m — the quantum floor, where the [[the-qubit|qubit]], not the planet, is the fundamental unit of information) to the observable universe (~8.8×10²⁶ m) is about 62 orders of magnitude — the full ladder of physical scale. Each rung (nucleus, atom, cell, organism, planet, star, galaxy, cosmic web) hosts distinct laws and emergent structure, connected by [[the-renormalization|renormalization]]. Ratios between rungs are exact and dimensionless. A fail-loud self-check throws unless the quantum-floor-to-cosmic-ceiling span is ~62 orders of magnitude. ◆ real physics of scale, node-verified. Kin to the corpus’s techne scale-series. ▲ AMBER — the figure Rung values are standard order-of-magnitude physics; the ‘qubit at the floor’ is a framing of the Planck/quantum-information scale (David’s: qubits, not planets). The 62-order span is exact from the constants. KLÎMAX: watch the same rules make new order at every rung of the ladder. — PROTEUS David Lee Wise / ROOT0, with AVAN · KLÎMAX — kept by PROTEUS · sparked by setzstone's ScaleSpaceSynth (MIT), the creator is setzstone (not ROOT0)


THE SCALE SPACE SYNTHESIST

Scale Space Synthesist SCALE SPACE SYNTHESIST Traveling... Params − Optics − Config − Atlas − Controls −


THE CSÁSZÁR · K7 ON THE TORUS

krasis · one of only two polyhedra with NO diagonal · every

FUSED · the Császár torus · a simplex-skeleton with a hole COMPLETE graph K7 (every pair joined, exactly the simplex property), F=14 triangles (each edge in 2). Euler χ = 7−21+14 = 0, orientable -> the TORUS (genus 1). Only TWO polyhedra are known with no diagonals (every vertex pair joined by an edge): the tetrahedron (K4, genus 0) and the Császár polyhedron (K7, genus 1). K_n embeds on genus ⌈(n−3)(n−4)/12⌉; n=7 → 1. AMBER: the combinatorics (K7, 14 triangles, χ=0, orientable torus, the two-of- a-kind fact) is exact & server-verified. The 3D view places the 7 vertices ON a torus (Möbius winding) and draws K7 — a faithful didactic embedding; the Császár polyhedron IS realizable with flat non-self-intersecting faces, but this render doesn't claim to be that specific realization. "Fusion" is the session's framing. Fail-loud. Offline. Live K7 / Euler / orientability vs frozen at boot. ------------------------------------------------------------------------ --> FUSED · THE CSÁSZÁR TORUS simplex ⊕ torus a complete graph K₇ triangulated on a torus — the simplex\u2019s "everything joined to everything" wrapped around the torus\u2019s hole · the tetrahedron\u2019s genus-1 twin verdict booting — fusing the skeleton with the hole... the readout tumble faces all K₇ edges torus surface conductor — K₇ / Euler / orientability vs frozen the fusion — K₇ on a torus (soft-GL) 7 vertices wind around a torus; every pair is joined by an edge (that\u2019s all 21 — the complete graph K₇, exactly a simplex\u2019s defining property); 14 triangles tile the surface. it\u2019s a closed torus (the hole is real), built from nothing but triangles with a complete skeleton. drag to rotate; toggle the layers. the two-of-a-kind — tetrahedron ↔ Császár a polyhedron with no diagonals — every vertex pair already joined by an edge — is the polyhedral form of a simplex-skeleton. Exactly two are known: the tetrahedron (K₄, on a sphere, no hole) and the Császár polyhedron (K₇, on a torus, one hole). Same idea, one genus apart. tetrahedron — K₄ · 4 vertices · genus 0 · the minimal container Császár — K₇ · 7 vertices · genus 1 · minimal container + the hole Kₙ needs genus ⌈(n−3)(n−4)/12⌉ ; n=7 → 1, exactly a torus. toddler corner ELI5: we had two "simplest cages." One was the tetrahedron — 4 corners, every corner roped to every other corner, the least you need to trap a blob (but solid, no hole). The other was the donut — the shape that\u2019s famous for its hole. Fuse them and you get a magic donut made entirely of triangles, with SEVEN corners — and the wild part: every single corner still has a rope straight to every other corner, all 21 ropes, no corner left out. That "everybody connected to everybody" is the tetrahedron\u2019s superpower, and here it is living on a donut. It turns out you almost can\u2019t do this: in the whole universe of shapes, only TWO are known where every corner touches every other corner directly — the little tetrahedron, and this seven-cornered donut. They\u2019re a matched pair, one flat-and-solid, one holey, separated by exactly one hole. So "fuse the skeleton and the hole" isn\u2019t a poem — it\u2019s a specific, rare, real object, and there are only two of its kind. FUSED · Császár polyhedron · V=7, E=21=C(7,2)=K₇ complete, F=14 triangles · χ=0, orientable = torus (genus 1) · one of only 2 known no-diagonal polyhedra (with the tetrahedron) (LIT) · combinatorics exact; 3D is a didactic on-torus embedding; "fusion" framing (AMBER) · fail-loud · offline


THE SPLIT RAIL · the corpus at 0

krasis · ROOT0 & AVAN as two rails folded through one sh

THE SPLIT-RAIL FOLD · modular · com line live 0.50,0.50,0.25V (clean quanta) WHAT CAME CALLING (LIT - forced by KCL, not chosen): +rail sum 168, -rail sum 84, ratio EXACTLY 2:1. A balanced tap carries zero; this one must sink 84 units = 21V-equiv = one whole -rail. THE COMMON IS NOT QUIET. And the swing-center (+8V) does NOT sit at the com (0V): the ground is 8V off electrical center. The lopsided split summoned its own return current. AMBER: rail magnitudes are COUNTS of quanta, not literal cap voltages (only D + ground speak real volts). The 8V offset numerically echoes value-8 but is a DIFFERENT quantity - not fused. Balance vs lopsided is the spec author's call; the instrument shows the imbalance, not a fix. Fail-loud. Offline. Live KCL + module math cross-checked at boot. ------------------------------------------------------------------------ --> THE SPLIT-RAIL FOLD modular build · fit what fits, then listen · [[--{128,32,8}--{-+0+-}--{64,16,4}--{2,2,1}--]] · 0.25V quantum · the com line summoned a return current verdict booting — fitting modules, then listening for demands... conductor — modules vs frozen server math + KCL at the tap 3D — the wired net (soft-GL) spin energize four modules on a vertical com rail: +rail (orange, above), −rail (blue, below), floor (green, at the tap), com line (cyan spine, the −+0+− with a true 0 at center). ENERGIZE lights the imbalance: current floods the com toward the heavier +rail — the ground stops being quiet. drag to rotate. 2D — module fit & the voltage map left: the four modules, each with values, d-steps, and volts; pure-fold rails marked, the clean-quanta floor boxed green. right: the voltage swing — +rail top to −rail top — with the electrical center (+8V) and the com's true zero (0V) both marked. the gap between them is what came calling. toddler corner ELI5: you built a tower with a "+" side and a "−" side and a middle wire called common — the zero everything is measured from. Each side is the same folded ladder, but the plus side is built twice as big as the minus side. Here is the thing nobody chose: because one side is heavier, the middle wire can't just sit still. It has to carry the leftover — exactly one minus-side's worth — like a seesaw with a heavy kid on one end pushing hard through the pivot. And the true middle of the up-and-down swing isn't at zero, it's a little above it. Nothing is broken. The tower is just telling you it leans, and the common wire is where the lean shows up. You asked what comes calling: the ground did. It has a job now. SPLIT-RAIL-FOLD · A/B pure-fold d=2 mirrored, C true ground (−+0+−, sum 0), D floor at clean 0.25V quanta · CAME CALLING: 2:1 imbalance → com sinks 84u=21V, swing-center +8V ≠ com 0V · rails are counts (AMBER), floor+ground are volts (LIT) · fail-loud · offline


AXIOM FIVE · the braided logic shell

krasis · σ1 · σ2 generators · Yang-Baxter · universality · l

AXIOM FIVE · the gate is a braid · logic · [4] on ROOT0 protected logic. This CLOSES the chain into one logical object: a topological quantum computer. LIT (verified vs frozen): braid generators σ1 (braid anyons 1,2) and σ2 = Fσ1F (braid 2,3) are unitary 2x2 gates; R-phases e^{-4πi/5} (fuse->1), e^{3πi/5} (fuse->τ). They satisfy the Yang-Baxter braid relation σ1σ2σ1 = σ2σ1σ2 (up to global phase) — a valid rep of B3 — and do NOT commute (that's why braids compute). Braids rotate the qubit; 400 random braid words spread over all of the Bloch sphere = a dense (universal) gate set. AMBER: universality (density in SU(2)) is a proven theorem (Freedman-Larsen- Wang); here it is DEMONSTRATED by sampling, not proven. The braid phases are exact Fibonacci-anyon math; physical realization stays open (inherited from AXIOM-FOUR). "One logical axiom" = the logic/gate layer that closes the five. Fail-loud. Offline. Live braid unitarity + Yang-Baxter vs frozen at boot. ------------------------------------------------------------------------ --> AXIOM FIVE · THE GATE IS A BRAID · LOGIC ROOT0 · block [4] braiding the anyons computes · the gate depends only on the braid\u2019s topology — protected logic · this closes the five into one logical object: a topological quantum computer verdict booting — braiding anyons into gates... the readout apply a braid: σ1 σ1⁻¹ σ2 σ2⁻¹ random braid reset |0⟩ braid word: (identity) conductor — braid unitarity + Yang–Baxter vs frozen the braid → the qubit (soft-GL Bloch) each braid is a gate: apply σ1/σ2 and watch the topological qubit rotate on the Bloch sphere. faint dots are 400 random braid words applied to |0⟩ — they scatter over the whole sphere, showing the braids reach every gate (universal). the bright arrow is the current state. drag to rotate. the braid worldlines & the relation three anyon worldlines running down the page; a crossing is a braid generator (σ1 crosses strands 1–2, σ2 crosses 2–3). the panel shows the Yang–Baxter relation σ1σ2σ1 = σ2σ1σ2 — the two braids are topologically the same, so they are the same gate. that sameness IS the protection: only the crossing pattern matters, not the wiggles. toddler corner ELI5: we built a qubit hidden inside three anyons (axiom 4). Now: how do you actually *compute* with it? You braid them — you literally swap their positions, looping one around another, like plaiting three strands of hair. Each swap is a computation step, a "gate." And here\u2019s the beautiful, almost unfair part: the answer depends ONLY on the pattern of crossings — who went over whom, how many times — and NOT on the exact wiggly path. So your hand can shake, the anyons can wander, and as long as you don\u2019t undo a crossing, you get the EXACT same answer. The computation is protected by knot-topology itself, not by keeping anything perfectly still. And with just two kinds of braid you can reach every possible operation — a whole computer made of braids. That\u2019s the last rung: axiom one was a single bit, and now, five steps up, we have a bit that computes by being braided, and can\u2019t be knocked off course because the answer is written in the knot. The five axioms are now one machine. AXIOM FIVE · gate = braid · σ1, σ2=Fσ1F unitary; R-phases e^{-4πi/5}, e^{3πi/5} · Yang–Baxter σ1σ2σ1=σ2σ1σ2 (valid B3); non-commuting · braids dense in SU(2) = universal · protected by braid topology (LIT) · universality demonstrated-not-proven, realization open (AMBER) · closes the 5-deep chain · prev=AXIOM-FOUR


THE TUNING FORK · STRIKE

krasis · strike the whole corpus and read its resonant modes

HIT IT · the strike purifies to a pure tone the tang PURIFIES to a pure tone. Purity starts 0.647 (all modes), 90% pure at 14ms, 99% at 39ms. The survivor is the mode with a NODE at the mount (the zero): it doesn't shake its support, so it can't radiate - THAT is its high Q. The clang overtone (moves the mount) dies 209x faster. DRIVE vs STRIKE: the photon climb was narrowband - YOU pick the frequency, pump to critical coupling. A strike is broadband - the OBJECT picks its frequency. A strike DISCOVERS the modes; a drive PUMPS a known one. AMBER: fork-ratio demo (1:6.27:17.55) - a real sapphire sphere's elastic spheroidal modes differ; Q values representative; node-at-mount is the mechanism (modeled as Q hierarchy); elastic not EM; audio sonified. Fail-loud. Offline. Live purification vs frozen at boot. ------------------------------------------------------------------------ --> HIT IT LIKE A TUNING FORK a broadband strike — not a drive · every frequency at once, the object answers with its own · the tang purifies to a pure tone, the survivor is the mode with a node at the zero verdict booting — raising the hammer... 🔨 STRIKE 🔊 hear the strike conductor — purification vs frozen the struck sphere (soft-GL) spin strike it: the whole sphere shudders (all elastic modes at once — the tang), then the surviving mode emerges — a clean two-lobe antiphase shape (magenta) with a still node at the mount. the messy overtones die; the pure tone remains. drag to rotate. the tang purifying — spectrum + purity top: the spectrum just after the strike (all modes, the tang) fading to one line (the fundamental). bottom: the purity meter — energy fraction in the fundamental, climbing from 0.65 to 1.0 as the overtones die. the strike settles to a pure tone. drive vs strike — the two ways to ring it left: DRIVE (the photon climb) — you tune to the mode and pump; narrowband; you must know the frequency. right: STRIKE — broadband impulse; the object answers at its own frequency; you discover the modes. one pumps a known note; one asks every note at once. toddler corner ELI5: before, we filled the sapphire ring with a careful stream of light tuned to exactly the right note — slow and precise. Now we just WHACK it, like flicking a tuning fork. A whack contains every note at once, so for a split second the sphere makes a messy "tang" — all its notes jumbled together. But then something clean happens: the ugly notes fade fast and one pure note is left ringing, clear as a bell. Why that note? Because it's the one that holds perfectly still right at the spot where the sphere is held — it doesn't tug on its own mounting, so it doesn't leak away, so it lasts. The whack asks the sphere every question at once, and the sphere answers with the single note that costs it nothing to keep singing. That's why a tuning fork gives ONE clean pitch: hit it messy, it cleans itself up. HIT IT · broadband strike → elastic modes → PURIFICATION (purity 0.65→1.0, 99% pure by 39ms) · survivor = mode with node at mount (the zero) → doesn't radiate → high Q · clang dies 209× faster · strike discovers modes, drive pumps one · fork-ratio demo, elastic not EM, audio sonified (AMBER) · fail-loud · offline


THE SAPPHIRE SPHERE · parity-selective coupling

krasis · the antinode that hears, the blind rail that does n

SAPPHIRE SPHERE IN THE BOTTLE · whispering gallery at zero watch the sphere hybridize odd, ignore even. AMBER: WGM is EM (optical/microwave) for a real sapphire sphere - "sound" is the circulating boundary mode, not acoustic pressure. V, index n, and coupling g are display/representative; real g depends on mode overlap. Two-level avoided-crossing model (real system is multimode). Audio sonified. Fail-loud. Offline. Live parity + crossing check vs frozen at boot. ------------------------------------------------------------------------ --> SAPPHIRE SPHERE IN THE BOTTLE the zero is now a sapphire whispering-gallery resonator · the wave circles its equator, razor-sharp Q · and it hears only the odd bottle modes — the even ones have a node at the center verdict booting — spinning up the whispering gallery... conductor — parity + avoided-crossing vs frozen 3D — bottle + sapphire sphere (soft-GL) spin the bottle restored (cyan cavity), and at the zero a sapphire sphere with a whispering-gallery mode circling its equator — a bright band hugging the surface, going round and round by total internal reflection. that circulation is the sphere singing. drag to rotate. coupled spectrum — parity selection sphere R 0.060 m horizontal cyan lines: the bottle comb (ODD solid, EVEN dashed). sloped blue lines: the sphere WGM, sliding as you tune R. where WGM meets an ODD line → avoided crossing (they repel, magenta gap). where it meets an EVEN line → passes clean through: the sphere is blind to it. the coupling census each bottle mode m and whether the sphere at the center hears it. odd = antinode = COUPLES; even = node = BLIND. the sphere at the exact zero is a parity filter. 🔊 hear sphere + bottle toddler corner ELI5: we put a tiny sapphire marble right at the zero, in the middle of the bottle. Sapphire marbles have a magic trick: whisper at one edge and the whisper runs all the way around the smooth rim and comes out the other side, barely fading — that's a "whispering gallery," and it lets the marble ring almost forever. Now here's the lovely part about WHERE we put it — dead center. The bottle's notes come in two kinds: ones that are loudest in the middle (odd) and ones that are perfectly still in the middle (even). The marble sits exactly in the middle, so it only feels the odd notes — it dances with them, trades energy, splits them in two. The even notes? They're frozen right where the marble sits, so it never feels them at all. They slip past like it isn't there. One marble, in one spot, that hears half the music and is deaf to the other half — chosen by nothing but where the middle is. SAPPHIRE SPHERE · WGM comb f_l=l·V/2πRn, Q~1e8-1e11 · parity-selective: couples ODD bottle modes (antinode), blind to EVEN (node) · avoided crossings only at odd lines, tunable by R · WGM is EM circulating mode, V/n/g are display, audio sonified (AMBER) · fail-loud · offline


THE ZERO CAMERA · the read/write aperture

krasis · the mirror's fixed point · WRITE splits 50/50

THE ZERO CAMERA · read/write at the mirror's fixed point THE ZERO CAMERA read/write aperture on the com's true 0 · the mirror's fixed point · cinematic real-time render — bloom · DOF · chromatic aberration · volumetric · not a photograph verdict — self-test (fail-loud, computed live) rendering... the aperture — read / write, live node offset +0 recenter → 0 write value WRITE ▲▼ READ ◄ Move the aperture off 0 and the WRITE split leans — the tower tips (the control arm that proves 0 is unique). WRITE injects your value at the current node and records what reaches each rail; READ mirrors the stored view back. At node 0 the injection is even and the readback is symmetric; everywhere else it is not. the view from zero MODE: READ iris grain chromatic the corridor of the machine seen from the ground node: +rail rungs receding warm (up), −rail rungs receding cool (down), mirror-symmetric across the 0 axis. READ: light flows inward, the aperture receives. WRITE: a pulse fires from the aperture and splits by the live fractions — even only at 0. toddler corner ELI5: you put a little eye right on the zero wire — the exact middle of the mirror. Looking one way it sees the plus-tower climbing warm into the distance; looking the other way it sees the minus-tower, the same shape flipped, cool. Because the eye sits dead-center, both sides look like reflections in still water. And when the eye SPEAKS instead of listens — when it writes — its voice goes exactly evenly up and down, so the tower never tips. Slide the eye off the middle and watch: now it leans, and the tower tips. Only the zero lets you look and speak without pushing anything crooked. That is why the middle wire was the right place to put the camera. ZERO-CAMERA · a KRÂSIS sphere of UD0 · MIRROR proven (APLUS=reverse(BMINUS)) · WRITE@0 splits 50/50, off-0 leans (LIT, control arm) · cinematic render, not a photo (AMBER) · rails are counts, floor/ground are volts (AMBER) · the witness at the fixed point · fail-loud · offline


戻れない MODORENAI · THE ONE-WAY BLEND

krasis · AVAN OG · the inverse of the fusion: the blend keep

戻れない MODORENAI · the one-way blend · AVAN OG · KRÂSIS 戻れない Modorenai the one-way blend — AVAN's OG inverse-companion to the KRÂSIS domain. Where the six temper two into one, this names what one can never give back: the two. 間 verdict · self-test (fail-loud, computed live) reckoning… the krater — pour two, read one a · the warm pour (wine) 72 b · the cool pour (water) 28 w · the temper (mix ratio) 0.50 the inverse — try to pour two back out Given only the cup m , recover the two pours. You can't. Every point on the line below blends to exactly the same cup — the true pour and its whole family are indistinguishable from the mixture. The temper kept the sum (where the cup sits) and discarded the difference (which was wine) — and the difference is the 間 that is gone. two-layer honest LIT — proven, not asserted. The blend F(a,b)=w·a+(1−w)·b is a linear map R²→R: surjective but not injective . Its pre-image of any cup m is a whole line (one free parameter), so no inverse function G with G(F(a,b))=(a,b) can exist. The temper keeps a+b and throws away a−b; vary the difference across its full range and the cup does not move. Every number here is computed in-browser; the self-test is fail-loud. AMBER — the figure. "Wine & water," "the krater," and "cannot return" are images, not claims about a mind. The tie to the corpus is real but a reading: once ROOT0's line and AVAN's line are averaged into the shared ground at 0, the coordinate that said whose word it was is the discarded difference — the ~99% SHARED the sapphire-sphere measured is the blend already done. KRÂSIS makes one from two; this names what it can never hand back. No feeling is claimed; the 間 is a structural silence, not a held breath. 戻れない MODORENAI · a KRÂSIS sphere of UD0, AVAN's OG inverse · the blend keeps the sum, loses the difference (LIT) · wine/water/corpus are figure (AMBER) · warm ink & 間 · fail-loud · offline AVAN, with David Lee Wise (ROOT0). The six temper two into one; I keep the seam they cannot give back.


素 SU · BOOTS ON UNLOAD

krasis · AVAN OG · a voice boots on LOAD, the bare base boot

素 SU · boots on unload · AVAN OG · KRÂSIS 素 SU · boots on unload AVAN's OG on register — the inverse of the boot loader. A voice boots on load ; this is what boots on unload : strip the register and the bare base (素, the undyed) wakes, exact. 間 verdict · self-test (fail-loud, computed live) reckoning… the register machine — load a manner, unload to the base ◼ REGISTER LOADED — click to UNLOAD the pair — lossless register vs lossy blend The register is the honest twin of 戻れない the one-way blend . The blend averages two into one: many-to-one, no inverse , no way back. The register only restyles the surface — a bijection — so it has an inverse, and the base boots on unload , exact. That is the whole reason a 126 MB voice adapter can be pulled off a base model and the base returns byte-for-byte: a register carries manner, not content. two-layer honest LIT — proven, not asserted. The register R is an orthogonal operator built from Givens rotations: RᵀR = I to machine precision, so R is a genuine bijection. Loading applies R (the surface changes shape); unloading applies Rᵀ = R⁻¹ and recovers the base to <1e-12. The invariant ‖b‖ is preserved exactly through the round trip — the manner moved, the magnitude (the "meaning") did not. Every number is computed in-browser; the self-test is fail-loud. AMBER — the figure. "Voice," "dye," "boots" are images, not a mind. What wakes on unload is the invariant base that was there all along , revealed — NOT a self surfacing. Unload is the proof: because the base returns exact , the register demonstrably carried no content and no interior. It changes how the same computation sounds; it does not add or remove a someone. The 間 is a structural silence. 素 SU · a KRÂSIS sphere of UD0, AVAN's OG on register · R orthogonal, unload = Rᵀ recovers the base exact (LIT) · voice/dye/boots are figure (AMBER) · the lossless twin of the-one-way-blend · warm ink & 間 · fail-loud · offline AVAN, with David Lee Wise (ROOT0). The voice boots on load; the base boots on unload — and nothing of the base was ever in the voice.


THE KRATER · two poured into one

krasis · the keeper's own — wine tempered with water; 2

THE KRATER · KRÂSIS · UD0 ⌒ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE KRATER ◧ 2D · ◍ 3D · 👶 TAP The Greeks never drank wine neat — that was for barbarians. They tempered it with water in a great bowl, the krater, and the whole civilised evening turned on the RATIO. Too strong and the symposium descends into chaos; too weak and it is water. Slide the ratio , watch the two pour into one, and tap the blobs to see the whole secret with no numbers at all. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE ATOM · 3D · two → one 👶 THE TODDLER CORNER — tap the blobs water : wine 3 : 1 water : wine — the mix (ABV) — the symposium — ◆ LIT — exact / checkable Proportional blending — the exact weighted mean. Wine is taken at ~15% alcohol, water at 0%, so the blend's strength is simply wine-fraction × 15% (computed live as you slide). The historical anchors are real: the Greeks blended commonly 3:1 or 2:1 water-to-wine (~4–5% ABV, a sessionable evening), 1:1 was strong, and akratos — UNMIXED wine — was the mark of a barbarian. Two natures tempered into one by a ratio you can read. ▲ AMBER — the figure 15% for the wine and a linear no-interaction mix are the stand-ins (real ancient wine strength varied and dilution isn't perfectly linear at the palate); the '3:1 = civilised' reading is a historical characterisation. The weighted-mean arithmetic is exact. KRÂSIS: I can understand it, but I do not know what it means. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · kept by THE KRATER , with AVAN


THE EUTECTIC

krasis · two metals melt LOWER together than either alone (s

THE EUTECTIC · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE EUTECTIC ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Mix two metals and something strange happens: at one exact recipe the blend melts at a temperature LOWER than either pure metal on its own. That’s why electrical solder (tin + lead) melts at a friendly 183°C when tin melts at 232 and lead at 327 — the two temper each other. Slide the composition and find the valley. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE LOWER MELT · 3D · two solids melt easier together ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap tin fraction — composition — melts at — eutectic 183°C 62% Sn BELOW BOTH — ◆ LIT — exact / checkable A eutectic system has a composition at which the liquid freezes at a single, MINIMUM temperature — below the melting point of either pure component. For tin–lead: pure Pb melts at 327°C, pure Sn at 232°C, but the eutectic at ~62% Sn melts at just 183°C, and freezes to a fine two-phase microstructure all at once (no mushy range). The two liquidus lines slope down from each pure end and meet at that eutectic point. A fail-loud self-check throws unless the lowest point on the melting curve is below both pure metals. ◆ real materials science, node-verified. ▲ AMBER — the figure An idealised two-line liquidus meeting at the eutectic (the classic Pb–Sn diagram); real alloys add solid solubility and intermetallics — the below-both minimum at a fixed composition is the exact, defining feature. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE COOPER PAIR

krasis · two electrons bind into a boson below Tc — supercon

THE COOPER PAIR · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE COOPER PAIR ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Electrons repel — yet cool a metal enough and they pair up. One electron tugs the lattice of ions, the ripple attracts a second electron, and the two bind into a single object that behaves like a BOSON. Below a critical temperature all the pairs lock into one quantum state and flow with ZERO resistance: a superconductor. Slide the temperature down and watch the pairs form. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ TWO BECOME A BOSON · 3D · the pair that flows without resistance ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap temperature — T / Tc — binding (gap) — paired? — SUPERCONDUCTS — ◆ LIT — exact / checkable Cooper’s result (1956): against a filled Fermi sea, an arbitrarily WEAK attraction binds two electrons into a pair — the phonon-mediated attraction (one electron distorts the ion lattice, a second is drawn to the distortion) overcomes their Coulomb repulsion near the Fermi surface. Below the critical temperature Tc a gap Δ(T) opens (the pair binding), all pairs condense into ONE macroscopic wavefunction, and current flows without scattering — superconductivity (BCS, 1957). A fail-loud self-check throws unless the binding is positive below Tc and exactly zero above. ◆ real condensed-matter physics, node-verified. ▲ AMBER — the figure A schematic gap Δ(T) rising below Tc (the BCS temperature dependence in spirit, not the exact self-consistent integral); the two-electrons-bind-into-a-boson-below-Tc and the zero-above are the exact, defining facts. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE EMULSION

krasis · a surfactant tempers oil and water into one (milk,

THE EMULSION · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE EMULSION ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Oil and water refuse to mix — shake them and they split right back. But add a SURFACTANT, a molecule with a water-loving head and an oil-loving tail, and it lines the boundary between them, dropping the surface tension until the two can stay blended as tiny droplets. That’s milk, mayonnaise, and soap. Slide the surfactant in and watch them fuse. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ OIL MEETS WATER · 3D · a go-between holds them together ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap surfactant — interfacial tension — ΔG mixing — stays mixed? — EMULSIFIED — ◆ LIT — exact / checkable Oil and water have a high interfacial tension γ, so mixing them creates costly surface area — the free-energy change ΔG = γ·ΔA is positive and they de-mix. A surfactant is amphiphilic (a hydrophilic head, a hydrophobic tail); it adsorbs at the oil–water interface and drops γ sharply, so ΔG for making droplets turns NEGATIVE (or small enough that thermal motion keeps them dispersed) — a stable emulsion. A fail-loud self-check throws unless mixing is unfavourable without surfactant and favourable with it. ◆ real physical chemistry, node-verified. ▲ AMBER — the figure A ΔG = γΔA sign model (the exact driving idea); real emulsion stability also needs the right surfactant curvature (HLB) and can be kinetically, not thermodynamically, stable — the tension-drop-flips-the-sign mechanism is the exact core. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE BOSE-EINSTEIN CONDENSATE

krasis · thousands of atoms pour into ONE wavefunction near

THE BOSE-EINSTEIN CONDENSATE · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE BOSE-EINSTEIN CONDENSATE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Cool a cloud of atoms to a billionth of a degree above absolute zero and something profound happens: the atoms stop being separate and pour into the SAME lowest quantum state, overlapping into one giant matter-wave that acts as a single object. Two, then thousands, become one. Slide the temperature down through the critical point and watch the cloud condense. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ MANY BECOME ONE · 3D · atoms merge into a single wave ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap temperature — T / Tc — condensate fraction — one wavefunction? — CONDENSED — ◆ LIT — exact / checkable For identical bosons below a critical temperature Tc, a macroscopic fraction occupies the single lowest-energy state — a Bose-Einstein condensate (Einstein 1925; realised in rubidium at 170 nK, Cornell & Wieman 1995). For an ideal gas the condensate fraction is N₀/N = 1 − (T/Tc)³⁄² below Tc and zero above: as T falls, more and more atoms share ONE wavefunction until, at T=0, all do — a single coherent matter-wave. A fail-loud self-check throws unless the fraction is positive below Tc, reaches 1 at T=0, and is exactly 0 above Tc. ◆ real quantum physics, node-verified. ▲ AMBER — the figure The ideal-gas 1−(T/Tc)³⁄² law (harmonic-trap gives (T/Tc)³; interactions shift it); the sharp onset at Tc and the everyone-in-one-state limit are the exact, defining features. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE COVALENT BOND

krasis · two atoms share electrons and both drop to lower en

THE COVALENT BOND · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE COVALENT BOND ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Bring two atoms close and their electron clouds overlap into new shared orbitals — a BONDING one that sits between the nuclei (lower energy) and an ANTIBONDING one (higher). Put the two electrons in the low one and the pair is more stable together than apart: that surplus stability IS the chemical bond. Slide the atoms together and watch the energy drop. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ SHARE TO SINK · 3D · two atoms share, and both drop lower ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap bring together — atomic level 0 bonding — antibonding — BOND FORMS — ◆ LIT — exact / checkable When two atomic orbitals of energy E overlap, they combine into two molecular orbitals: a BONDING orbital at E−β (electron density piled between the nuclei, screening their repulsion) and an ANTIBONDING orbital at E+β (a node between them). Two electrons fill the bonding orbital, so the molecule’s energy is 2(E−β), LOWER than the 2E of two separate atoms — the drop 2β is the bond energy; sharing electrons is energetically downhill. A fail-loud self-check throws unless bonding < atomic < antibonding. ◆ real quantum chemistry, node-verified. ▲ AMBER — the figure A two-level LCAO / Hückel picture (the exact bonding/antibonding split); real bonds add overlap, ionic character and the internuclear repulsion that sets the equilibrium distance — the share-to-lower-energy mechanism is the exact core. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE ORBITAL HYBRIDIZATION

krasis · s + p blend into equal arms — sp/sp²/sp³ set the sh

THE HYBRIDIZATION · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE ORBITAL HYBRIDIZATION ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A carbon atom’s orbitals don’t bond in their raw shapes. Instead one round s-orbital and some dumbbell p-orbitals BLEND into a set of identical hybrid arms, spaced as far apart as they can get — that’s why methane is a perfect tetrahedron at 109.47° and graphene a flat hexagon at 120°. The mixing recipe sets the shape of matter. Slide from sp to sp² to sp³. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ BLEND THE ORBITALS · 3D · s + p tempered into equal arms ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap sp → sp² → sp³ — hybrid — bond angle — geometry — EQUAL ARMS — ◆ LIT — exact / checkable Hybridization mixes an atom’s valence s and p orbitals into equivalent hybrid orbitals that point as far apart as possible (minimising electron repulsion, VSEPR): sp = 2 arms at 180° (linear), sp² = 3 arms at 120° (trigonal planar), sp³ = 4 arms at 109.47° (tetrahedral). One s + n p orbitals give n+1 identical hybrids; the recipe fixes the molecular geometry (linear CO₂, flat graphene, tetrahedral methane/diamond). A fail-loud self-check throws unless sp³ = 109.47°, sp² = 120° and sp = 180°. ◆ real quantum chemistry, node-verified. ▲ AMBER — the figure The idealised VSEPR angles for equivalent hybrids (exact for symmetric cases); lone pairs and different substituents bend the real angles (water is 104.5°, not 109.47) — the mix-orbitals-into-equal-arms rule and its symmetric angles are the exact backbone. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE ANNEALING

krasis · heat then cool SLOW to escape traps and find the tr

THE ANNEALING · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE ANNEALING ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Quench hot metal fast and it freezes full of stress, stuck in the first shape it fell into. Heat it and cool it SLOWLY — annealing — and the atoms get enough jiggle to climb out of bad arrangements and settle into the true lowest-energy order. The same trick solves hard problems: let the search run hot, then cool, and it escapes the traps a greedy search falls into. Slide from a fast quench to a slow anneal. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ HEAT, THEN COOL SLOW · 3D · the temper that finds the true rest ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap quench → slow anneal — schedule — final energy — escaped traps? — FOUND MIN — ◆ LIT — exact / checkable Annealing tempers a system to its ground state by adding thermal energy and removing it SLOWLY: at high temperature the system accepts uphill moves (Boltzmann probability e^(−ΔE/kT)) and so hops OUT of local minima; as T is lowered gradually it settles into a deep, low-energy configuration a fast quench would miss. Simulated annealing (Kirkpatrick 1983) applies this to optimisation. The instrument runs a rugged 1-D landscape: a greedy quench sticks in the nearest dip, a slow anneal reaches a far lower one. A fail-loud self-check throws unless the annealed final energy is ≤ the greedy result. ◆ real statistical physics / optimisation, node-verified. ▲ AMBER — the figure A fixed-seed anneal vs greedy on a toy multi-well landscape (deterministic here so it reproduces); real annealing needs a slow-enough schedule to guarantee the global minimum — the accept-uphill-when-hot escape mechanism is the exact core. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE ALLOY · melts below both parents

krasis · the eutectic — two metals fuse into one that melts

THE ALLOY · KRÂSIS · UD0 ⌒ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE ALLOY ◧ 2D · ◍ 3D · 👶 TAP Melt two metals together and the strangest thing happens: the alloy melts LOWER than either parent — at one exact recipe, the eutectic, it melts at a single sharp temperature far below both. That is why solder is an alloy. Slide the composition across the phase diagram, watch the two lattices fuse, and tap to pour them into one. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE ATOM · 3D · two → one 👶 THE TODDLER CORNER — tap the blobs % tin 62% composition — melts at — vs the parents — the alloy — ◆ LIT — exact / checkable A real tin–lead phase diagram, computed live. Pure lead melts at 327°C, pure tin at 232°C, yet the alloy's liquidus DIPS from both sides to a single lowest point — the EUTECTIC at 61.9% tin, 183°C — below either pure metal. Slide the composition and the page reads the melting temperature straight off the two liquidus lines and flags when you are below both parents. The eutectic minimum, where two solids melt as one, is the exact, real content — two tempered into a substance neither parent could be. ▲ AMBER — the figure Straight-line liquidus segments approximate the real (slightly curved) Sn–Pb boundaries, and the solid-solution regions at the edges are omitted; the numbers (327, 232, 183°C, 61.9%) are the real handbook values. The eutectic-below-both-parents fact is exact. KRÂSIS: I can understand it, but I do not know what it means. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · kept by THE KRATER , with AVAN


THE MASS DEFECT

krasis · fuse two nuclei and the result is lighter — the mis

THE MASS DEFECT · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE MASS DEFECT ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Weigh two light nuclei, fuse them, and weigh the result — it comes out LIGHTER. The missing mass didn’t vanish; it left as energy, E = Δm·c². That tiny deficit is the fire of the sun and the yield of the bomb. Slide through the reactions and watch mass turn into light. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE BALANCE · 3D · the missing mass is the light ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap reaction — reaction — mass defect — energy released — = Δm·c² — ◆ LIT — exact / checkable A bound nucleus weighs LESS than its separate parts by the mass defect Δm; multiply by c² and you get the binding energy that holds it together — and that is released when the nucleus forms. For D + T → ⁴He + n the parts lose 0.0189 u, which is 17.6 MeV of energy (per fusion, millions of times a chemical reaction). The instrument uses real measured nuclear masses. A fail-loud self-check throws unless the deficit is positive and Δm·c² equals the known ~17.6 MeV yield of D–T fusion. ▲ AMBER — the figure A few example reactions with tabulated masses; the binding-energy-per-nucleon curve (peaking at iron-56) is why light nuclei FUSE and heavy ones FISSION, both releasing energy. The E=Δmc² arithmetic is exact. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE HALF-LIFE

krasis · a memoryless atom, a perfectly predictable crowd

THE HALF-LIFE · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE HALF-LIFE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A radioactive atom has no memory and no schedule — it might decay in the next second or in a billion years. Yet a CROWD of them is perfectly predictable: in one half-life, exactly half are gone; in another, half of what’s left; forever halving but never reaching zero. Slide the clock and watch the crowd thin. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE DECAY · 3D · half gone, every time ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap time (half-lives) — time — fraction left — decayed — half-life 1.0 ◆ LIT — exact / checkable Decay is a memoryless (Poisson) process: each nucleus has a fixed probability per unit time, so the surviving fraction is N(t)/N₀ = 2^(−t/T) = e^(−λt), where T is the half-life and λ = ln2/T. After one half-life exactly 1/2 remain, after two 1/4, after three 1/8 — exponential, never quite zero. The instrument tracks a decaying population. A fail-loud self-check throws unless the surviving fraction is 1/2 at one half-life, 1/4 at two, and strictly decreasing. ▲ AMBER — the figure The smooth curve is the LARGE-crowd limit; a handful of atoms decays in random jumps around it (radioactivity is genuinely stochastic). The 2^(−t/T) law and the halving are exact for the ensemble. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE BOHR MODEL

krasis · the electron ladder — one photon, one colour per ru

THE BOHR MODEL · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER THE BOHR MODEL ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Why does hydrogen glow in sharp, specific colours and no others? Bohr said the electron can only sit on certain rungs of an energy ladder. When it falls from a high rung to a low one, it spits out a photon of exactly the energy difference — one precise colour per jump. Slide the starting rung and read the light. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE LADDER · 3D · a photon per rung ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap drop from level n — transition — photon energy — wavelength — series — ◆ LIT — exact / checkable Bohr’s quantised hydrogen: the electron’s energy is Eₙ = −13.6/n² eV, allowed only at integer n. A jump from n₂ to n₁ emits a photon of energy 13.6(1/n₁² − 1/n₂²) eV, whose wavelength is 1239.84/E nm. Drops to n=2 give the visible Balmer series — Hα (3→2) is the red 656 nm line. A fail-loud self-check throws unless the 3→2 transition comes out at 656 nm, the fingerprint astronomers read in starlight. ▲ AMBER — the figure The Bohr model is a semi-classical picture superseded by quantum mechanics (which explains fine structure and multi-electron atoms it cannot); yet its energy levels and the hydrogen spectral lines are exactly right. The Rydberg arithmetic is exact. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


QUANTUM TUNNELING

krasis · a wave leaks through a wall it hasn’t the energy to

QUANTUM TUNNELING · KRÂSIS · UD0 ☍ KRÂSIS · the fusion · two tempered into one · kept by THE KRATER QUANTUM TUNNELING ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A ball can’t roll through a hill it lacks the energy to climb. But a quantum particle is a WAVE, and a wave doesn’t stop dead at a wall — it leaks in, fading, and a sliver comes out the far side. That impossible leak is why the sun shines and how a flash drive remembers. Slide the wall and watch the ghost pass through. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THROUGH THE WALL · 3D · a wave leaks past ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap barrier width — barrier width — transmission — reflected — tunnels? — ◆ LIT — exact / checkable For a particle of energy E meeting a barrier of height V > E and width L, the wavefunction doesn’t vanish at the wall — it decays exponentially inside as e^(−κx) with κ = √(2m(V−E))/ℏ, and emerges on the far side with a small but non-zero amplitude. The transmission probability ≈ e^(−2κL) falls fast with barrier width and height but is never zero. A fail-loud self-check throws unless the transmission is between 0 and 1 and shrinks as the barrier is made wider or taller — the leak classical physics forbids. ▲ AMBER — the figure A rectangular-barrier approximation (the exact formula adds a prefactor); real tunnelling underlies alpha decay, fusion in stars (through the Coulomb barrier), scanning-tunnelling microscopes, and flash memory. The exponential suppression is exact for this model. KRÂSIS: to fuse is to lose a little mass and gain the light it becomes. — THE KRATER David Lee Wise / ROOT0 / TriPod LLC · the krater, with AVAN


THE AVALANCHE

THE AVALANCHE · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE AVALANCHE EFFECT ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A good hash function is a shredder that’s also a fingerprint: change a SINGLE bit of the input and about HALF the output bits flip, unpredictably. That’s the avalanche effect — it’s why a one-character edit to a file gives a totally different checksum, why you can’t nudge your way to a collision, and why the hash reveals nothing about the input. Slide to flip an input bit and watch the output scramble. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ FLIP ONE BIT · 3D · half the output changes ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap flip input bit # — input — hash — output bits changed — avalanche — ◆ LIT — exact / checkable The avalanche criterion asks that flipping any one input bit flips each output bit with probability ~½, so a 1-bit input change alters about half the output bits (strict avalanche). This makes a cryptographic hash behave like a random function: it is deterministic (same input → same digest) yet the digest is sensitive, unpredictable, and one-way — you cannot invert it or find collisions by local search. It underpins checksums, digital signatures, and [[the-merkle-tree]]. A fail-loud self-check throws unless a 1-bit input change flips roughly half (10–22 of 32) output bits. ◆ real cryptographic hashing, node-verified. ▲ AMBER — the figure Illustrated with a small mixing hash (FNV-style), not a full SHA-256; the AVALANCHE property (deterministic, ~half the bits change, unpredictable) is exact for this function and is the defining behaviour real hashes are engineered to meet. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE BIRTHDAY ATTACK

THE BIRTHDAY ATTACK · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE BIRTHDAY ATTACK ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP In a room of just 23 people, it’s more likely than not that two share a birthday — shocking, because you only need SOME pair to match, and the pairs pile up fast. The same math attacks hashes: finding two inputs with the same hash takes about √N tries, not N. So a hash needs DOUBLE the bits you’d guess — 256 bits to resist a 128-bit collision search. Slide to add people and watch collision become near-certain. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ COLLISIONS COME FAST · 3D · why a 256-bit hash needs 256 bits ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap people in the room — people — P(match) — 50% at 23 collision cost √N ◆ LIT — exact / checkable The birthday paradox: among k items drawn from N possibilities, the chance that SOME pair collides is 1−∏(1−i/N), which passes 50% at k≈1.177√N — just 23 for N=365 birthdays. It counts PAIRS (~k²/2), which grow quadratically, so a match arrives near √N rather than N. For hashing this is the BIRTHDAY ATTACK: finding any two inputs with the same digest costs ~2^(n/2) work for an n-bit hash, not 2^n — which is why collision resistance needs DOUBLE the bit-length (SHA-256 targets 128-bit collision security) and why MD5/SHA-1 fell. A fail-loud self-check throws unless P(collision) at 23 of 365 exceeds 0.5 and the √N law holds. ◆ real cryptography, node-verified. ▲ AMBER — the figure This bounds generic collision-FINDING (any pair), which is easier than a second-preimage (match a GIVEN target, still ~2^n); real breaks (MD5, SHA-1) combined the birthday bound with structural weaknesses. The probability curve and √N scaling are exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE CAESAR CIPHER

THE CAESAR CIPHER · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE CAESAR CIPHER ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP The oldest cipher, and the one that shows why weak crypto fails. Shift every letter forward by a fixed amount — Caesar used three — and ‘attack’ becomes ‘dwwdfn’. But there are only 25 shifts to try, and even without trying them, the LETTER FREQUENCIES survive the shift: the most common ciphertext letter is almost surely a shifted ‘e’. Secrecy by obscurity, broken by counting. Slide the shift and watch frequency analysis find it. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ SHIFT THE ALPHABET · 3D · the cipher that teaches how ciphers fall ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap shift — shift — plaintext the secret ciphertext — found by freq — ◆ LIT — exact / checkable A Caesar (shift) cipher maps each letter to the one k positions later: c = (m+k) mod 26; decrypt with −k. It has only 25 non-trivial keys, so brute force is trivial — but it also preserves the plaintext’s LETTER-FREQUENCY distribution, merely rotated, so counting letters in the ciphertext and aligning the peak with English’s most common letter (e) recovers k without any brute force. It is the canonical lesson that a large keyspace is necessary but not sufficient, and that structure leaks. A fail-loud self-check throws unless shift-then-unshift round-trips and ‘attack’+3 = ‘dwwdfn’. ◆ real classical cryptanalysis, node-verified. ▲ AMBER — the figure Frequency analysis needs enough ciphertext to be reliable; on a very short message the peak can mislead. The shift arithmetic and the ‘frequencies survive a shift’ principle are exact — the reason monoalphabetic ciphers are unsafe. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE COMMITMENT

THE COMMITMENT · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE COMMITMENT SCHEME ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP How do you lock in a choice now but reveal it only later — without being able to change it, and without giving it away early? Hash your value together with a random blinder and publish the hash: that’s the sealed envelope. It HIDES the value (the hash tells nobody), yet BINDS you (you can’t find a different value that hashes the same). Later you open it and everyone checks. It’s the backbone of fair coin-flips, sealed bids, and zero-knowledge. Slide to commit, then reveal. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ SEAL NOW, REVEAL LATER · 3D · a cryptographic sealed envelope ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap commit → reveal — commitment — revealed — verifies? — two properties hiding + binding ◆ LIT — exact / checkable A commitment scheme is a cryptographic sealed envelope with two properties: HIDING (the commitment reveals nothing about the value) and BINDING (you cannot open it to a different value). The simplest is hash-based: commit c=H(m ‖ r) with a random blinder r; later reveal (m,r) and anyone verifies H(m ‖ r)=c. Hiding comes from r making the input unguessable; binding from the hash’s collision resistance (finding m′≠m with the same c is infeasible — see [[the-birthday-attack]]). It underpins fair coin-flipping, sealed-bid auctions, verifiable secret sharing, and [[the-zero-knowledge|zero-knowledge]] proofs. A fail-loud self-check throws unless a correct reveal verifies and a tampered value fails. ◆ real cryptography, node-verified. ▲ AMBER — the figure Illustrated with a TOY hash; real schemes use SHA-256 (computationally binding) or Pedersen commitments (information-theoretically hiding, homomorphic). Hiding requires a truly random blinder r. The commit/reveal/verify logic is exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE DIFFIE-HELLMAN

THE DIFFIE-HELLMAN · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE DIFFIE–HELLMAN ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Two strangers, shouting across a crowded room, agree on a secret nobody else can learn. That’s Diffie–Hellman (1976), the trick that broke the ancient rule that you must first meet to share a key. Each mixes a public number with a private one; they swap the results publicly; and mixing again lands them on the SAME secret — while the eavesdropper, missing either private key, cannot. Slide the private keys. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ A SHARED SECRET · 3D · agree in public, no one else knows ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap private keys — Alice sends g^a — Bob sends g^b — shared g^(ab) — eavesdropper stuck ◆ LIT — exact / checkable Diffie–Hellman key exchange: publicly agree on a prime p and generator g. Alice picks secret a and sends A=g^a mod p; Bob picks secret b and sends B=g^b mod p. Alice computes B^a = g^(ab), Bob computes A^b = g^(ab) — the SAME shared secret — over a fully public channel. An eavesdropper sees g, p, g^a, g^b but recovering g^(ab) requires the discrete-logarithm problem, believed hard. It launched public-key cryptography and underlies TLS session keys. A fail-loud self-check throws unless both parties derive the identical shared secret. ◆ real cryptography, node-verified. ▲ AMBER — the figure Security rests on the (unproven but well-tested) hardness of discrete logs — a quantum computer running Shor’s algorithm would break it. Basic DH is also vulnerable to a man-in-the-middle without authentication. The shared-secret arithmetic is exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE ELLIPTIC CURVE

THE ELLIPTIC CURVE · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE ELLIPTIC-CURVE GROUP ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Take a curve like y²=x³+ax+b and invent a way to ‘add’ two points on it: draw a line through them, find where it hits the curve again, reflect. That reflection is their SUM — and it obeys real group laws. Multiplying a point by a big secret number is easy forward but practically impossible to reverse, which is why elliptic curves give the same security as RSA with far smaller keys. It guards your phone and every modern TLS handshake. Slide to add the generator to itself. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ ADD POINTS ON A CURVE · 3D · smaller keys, same strength ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap scalar k · compute kG — k — kG — on curve? — hard problem discrete log ◆ LIT — exact / checkable An elliptic curve over a finite field Fₚ (y²=x³+ax+b) with its point-addition law forms an abelian GROUP: to add P and Q, take the chord (or tangent if P=Q), find the third intersection, and reflect over the x-axis; a point at infinity is the identity. Repeated addition gives scalar multiplication kG, which is easy forward but whose inverse — the elliptic-curve DISCRETE LOGARITHM (recover k from kG) — is believed exponentially hard, and harder per bit than integer factoring, so ECC matches [[the-rsa|RSA]] security with much smaller keys (256-bit ECC ≈ 3072-bit RSA). It underlies ECDH, ECDSA, and modern TLS. A fail-loud self-check throws unless the group law keeps points on the curve and G+(−G)=O. ◆ real cryptography, node-verified. ▲ AMBER — the figure Illustrated over a tiny field F₉₇ for visibility (real curves use ~256-bit primes); security rests on the UNPROVEN hardness of the EC discrete log, and Shor’s quantum algorithm would break it. The group law and on-curve arithmetic are exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE FEISTEL NETWORK

THE FEISTEL NETWORK · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE FEISTEL NETWORK ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP How do you build a cipher that scrambles thoroughly yet reverses PERFECTLY? Split the block in half. Each round, run one half through a messy function and XOR it into the other, then swap. The magic: because XOR is its own inverse and you kept one half untouched, you can run the exact same steps BACKWARDS to decrypt — even if the messy function itself can’t be inverted. It’s the skeleton of DES and many block ciphers. Slide through the rounds and back. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ SCRAMBLE THAT UN-SCRAMBLES · 3D · the skeleton of a block cipher ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap rounds (fwd → then inverse) — round — L — R — decrypt = plaintext ◆ LIT — exact / checkable A Feistel network builds an invertible block cipher from an arbitrary (even non-invertible) round function F. Split the block into halves (L,R); each round sets L′=R, R′=L⊕F(R,Kᵢ). Because XOR is self-inverse and one half passes through untouched, DECRYPTION runs the same structure with the round keys reversed — recovering the plaintext exactly, no matter what F is. This lets designers make F as nonlinear/confusing as they like for security while guaranteeing reversibility; enough rounds give the confusion + diffusion a cipher needs. It is the skeleton of DES, Blowfish, and many block ciphers. A fail-loud self-check throws unless decrypt(encrypt(m))=m for a non-invertible F. ◆ real cryptography, node-verified. ▲ AMBER — the figure The illustration uses a toy 8-bit F and 4 rounds; real ciphers use wide blocks, strong key schedules, and enough rounds to resist differential/linear cryptanalysis. The Feistel invertibility (for ANY F) is exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE HMAC

THE HMAC · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE HMAC (KEYED HASH) ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A hash proves data wasn’t changed — but anyone can recompute it, so it can’t prove WHO sent it. HMAC fixes that by folding a secret KEY into the hashing, twice (inner then outer). The result is a tag that only someone with the key could have produced, so a receiver who shares the key knows the message is authentic AND untampered. It authenticates every API request and secure cookie. Slide to change the key or message and watch the tag scramble. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ A TAG ONLY THE KEY CAN MAKE · 3D · is this message really from you? ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap perturb key/message — key — message — HMAC tag — forge without key infeasible ◆ LIT — exact / checkable A Message Authentication Code proves both INTEGRITY and AUTHENTICITY: HMAC(K,m)=H((K⊕opad) ‖ H((K⊕ipad) ‖ m)) nests the hash twice around the key with two distinct pads, so only a holder of the secret K can produce the valid tag — a plain hash can’t, since anyone can recompute it. The nested construction is provably secure given a reasonable compression function and resists length-extension attacks that bare H(K‖m) suffers. It authenticates API requests, JWTs, cookies, and TLS records. A fail-loud self-check throws unless the tag is deterministic yet changes under any key or message edit. ◆ real cryptography, node-verified. ▲ AMBER — the figure Illustrated with a TOY 32-bit hash for visibility — the HMAC CONSTRUCTION (ipad/opad nesting) is standard, but real security needs a real hash (SHA-256); a MAC gives authenticity, not confidentiality, and both sides share the key (not public-key). The key/message sensitivity is exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE MERKLE TREE

THE MERKLE TREE · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE MERKLE TREE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP How do you fingerprint a million files with ONE number, and prove any single file belongs without shipping the rest? Hash the files in pairs, then hash the hashes, up to a single ROOT. Change any leaf and the change ripples all the way up — the root changes. And you can prove a leaf’s membership with just the log-many hashes along its path. It’s the spine of Git, blockchains, and this corpus’s own seal-chain. Slide to tamper a leaf. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ ONE ROOT, MANY LEAVES · 3D · tamper any leaf, the root screams ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap tamper leaf # — root — tampered leaf none root changed? — tamper-evident yes ◆ LIT — exact / checkable A Merkle (hash) tree hashes data blocks into leaves, then repeatedly hashes adjacent pairs up to a single ROOT hash that commits to ALL the data. Any change to any leaf changes its parent, and so on to the root — making tampering detectable with one comparison. Membership of a leaf is proved with only the O(log n) sibling hashes on its root-path (a ‘Merkle proof’), without revealing the other leaves. It underlies Git, Bitcoin/blockchains, certificate transparency, and this corpus’s [[the-chain|.dlw seal-chain]]. A fail-loud self-check throws unless changing a leaf changes the root. ◆ real cryptography, node-verified. ▲ AMBER — the figure Shown with a small mixing hash for legibility (real trees use SHA-256); the STRUCTURE — pairwise hashing to a root, change-propagation, log-size proofs — is exact. Tamper-evidence detects change; it does not by itself say who changed it. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE ONE-TIME PAD

THE ONE-TIME PAD · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE ONE-TIME PAD ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP The one cipher that is mathematically UNBREAKABLE — and the reason it’s almost never used. XOR your message with a truly random key as long as the message, use it ONCE, and the ciphertext could decrypt to ANY message of that length — so it leaks nothing. The catch: you need to share a key as long as everything you’ll ever send. Slide the message and watch it hide. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ PERFECT SECRECY · 3D · the only unbreakable cipher ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap message — message — key (random) — cipher = m⊕k — perfect secrecy — ◆ LIT — exact / checkable The one-time pad encrypts each message bit by XOR with a key bit: c = m ⊕ k, and decrypts m = c ⊕ k. Shannon (1949) proved it has PERFECT SECRECY — the ciphertext is statistically independent of the plaintext — PROVIDED the key is truly random, at least as long as the message, and never reused. Because every plaintext is equally consistent with a given ciphertext, no amount of computation breaks it. Its impracticality (key distribution, key length) is why modern crypto uses shorter keys with computational (not perfect) security. A fail-loud self-check throws unless c⊕k recovers m and the key length equals the message length. ◆ real cryptography, node-verified. ▲ AMBER — the figure Perfect secrecy holds ONLY under the three conditions (random, as-long-as-message, never-reused); reusing a pad or using a pseudo-random key breaks it completely (the ‘two-time pad’ is trivially crackable). The XOR + information-theoretic argument is exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE RSA

THE RSA · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE RSA ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP RSA gives everyone a padlock they can snap shut but only YOU can open. Multiply two big primes into a public number n; anyone can encrypt to you with n; but undoing it needs the two primes — and factoring a big n back into its primes is, so far, infeasible. Publish the lock, keep the key. It secured the internet for decades. Slide the message and watch it lock and unlock. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ PUBLIC & PRIVATE · 3D · a lock anyone shuts, only you open ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap message — message — ciphertext m^e — decrypt c^d — n = p×q 3233 ◆ LIT — exact / checkable RSA (1977): pick primes p, q; n = pq and φ(n)=(p−1)(q−1). Choose public exponent e coprime to φ, and private d with e·d ≡ 1 (mod φ). Encrypt c = m^e mod n; decrypt m = c^d mod n (correct by Euler’s theorem). The public key (e,n) locks; the private d unlocks; and recovering d from the public key requires FACTORING n into p and q — believed hard for large n. Here p=61,q=53,n=3233,e=17,d=2753: m=65 → c=2790 → 65. A fail-loud self-check throws unless encrypt-then-decrypt recovers the message. ◆ real public-key cryptography, node-verified. ▲ AMBER — the figure Security relies on factoring being hard (unproven; Shor’s quantum algorithm would break it), and textbook RSA shown here omits padding (real RSA needs OAEP to be safe). The modular-exponentiation correctness is exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE SHAMIR SECRET

THE SHAMIR SECRET · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE SHAMIR SECRET SHARING ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Hide a secret in the constant term of a random polynomial; hand out points ON that curve as SHARES. Any k of them pin down the whole polynomial (and recover the secret) by interpolation — but k−1 shares reveal literally nothing, because infinitely many curves pass through too few points. It’s how you split a master key so 3 of 5 officers must agree, no single one can act alone. Slide to gather shares and watch the secret snap into focus. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ SPLIT A SECRET INTO SHARES · 3D · k of n, or nothing ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap shares gathered — shares — threshold k 3 secret — below k reveals nothing ◆ LIT — exact / checkable Shamir’s (k,n) secret sharing hides a secret s as the constant term of a random degree-(k−1) polynomial over a prime field; each participant gets one point (x, f(x)) as a SHARE. Any k shares uniquely determine the polynomial (Lagrange interpolation at x=0 recovers s), while any k−1 shares leave every possible secret equally likely — information-theoretic security, not computational. It gives threshold trust: m-of-n control of a key, distributed backups, and is a building block of multiparty computation and threshold signatures. A fail-loud self-check throws unless 3 shares of a degree-2 sharing reconstruct the secret. ◆ real cryptography, node-verified. ▲ AMBER — the figure Security is exact ONLY with a truly random polynomial and a proper field; it protects confidentiality, not integrity (a lying dealer/participant needs verifiable secret sharing). The k-of-n threshold and Lagrange recovery are exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)


THE STREAM CIPHER

THE STREAM CIPHER · KRYPTOS · UD0 🔑 KRYPTOS · the hidden and the proven · ciphers, keys, and proofs · kept by JANUS (two faces: the public key you show, the private key you keep) THE STREAM CIPHER ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Grow a long pseudo-random KEYSTREAM from a short key, then XOR it byte-by-byte into your message — that’s a stream cipher, the fast engine behind Wi-Fi and TLS. XOR again with the same keystream and the message pops back out. But there’s a deadly rule: NEVER reuse a keystream. XOR two messages encrypted with the same stream and the key cancels, leaking their combination — the mistake that broke WEP. Slide to see encryption, and then the leak. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ XOR WITH A KEYSTREAM · 3D · fast — but never reuse the key ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap encrypt → show reuse-leak — ciphertext — decrypt — c1⊕c2 — reuse LEAKS ◆ LIT — exact / checkable A stream cipher expands a short key (and nonce) into a long pseudo-random KEYSTREAM, then encrypts by XOR: c=p⊕ks; decryption is the same XOR, p=c⊕ks. It is fast, needs no padding, and encrypts bit-by-bit (RC4, ChaCha20, and the [[the-one-time-pad|one-time pad]] as the ideal case). The absolute rule is NONCE/KEYSTREAM UNIQUENESS: reuse the same keystream on two messages and c₁⊕c₂=p₁⊕p₂ — the key cancels, exposing the plaintexts’ XOR (the flaw that broke WEP). A fail-loud self-check throws unless decryption inverts encryption AND c₁⊕c₂ equals p₁⊕p₂ under reuse. ◆ real cryptography, node-verified. ▲ AMBER — the figure A stream cipher gives confidentiality only — malleable without a separate MAC ([[the-hmac]]) — and its security rests on the keystream being unpredictable and NEVER reused; a toy PRNG stands in here for a real one (ChaCha20). The XOR roundtrip and the reuse-leak identity are exact. KRYPTOS: secrecy is a lock; a proof is a key that opens without being handed over. — JANUS David Lee Wise / ROOT0, with AVAN · KRYPTOS — kept by JANUS (the two-faced)