Mathematical Dimension
The ten CGB Mathematical Depositions, each drawn on three axes. The formulas and their plain-language readings are the same as on the Mathematical Animations page. Here every one of them turns in space. Free to view and open to everyone.
Each plot turns slowly on its own. Drag it to turn it yourself. The button stops and starts it.
Harmonic DecayLight
Add 1 + 1/4 + 1/9 + 1/16 + … forever and you get a piece of π-squared. Separately, watch a ringing bell fade as it oscillates — that fade is governed by the same ratio. Counting and dying waves, one structure.
The matter. A struck bell. The bronze stores the blow and gives it back a little weaker on every swing — that loss is the λ in the formula. Without something that rings and loses, there is no fade to measure, and the sum of fractions stays a sum of fractions.
In CRI-ONE: this ratio is the settle-time yardstick for our iterative solvers. The iteration schedule that reaches it is proprietary.
Entropic BridgeAbnormal
The disorder in a message, the irreversibility of heat, and the stiffness of free energy are the same inequality in different clothes. Shannon proved it for bits, Clausius for engines, Helmholtz for reactions — and they never met.
The matter. A gas in a cylinder, a wire carrying a message, a flask of reacting chemicals. Each is a crowd of small parts that can be arranged in many ways, and counting those arrangements is what entropy is. The same inequality holds for all three because all three are made of many pieces.
In CRI-ONE: every calculator that carries uncertainty from input to output is bounded by this inequality. The accounting rule that closes the bound is ours.
Voxel ResonanceUnique
A chip voxel vibrates at many frequencies at once; distance from its neighbours damps the signal like a bell curve. The seed’s matrix determinant scales everything — if det = 0 the voxel is dead, if large it screams.
The matter. One voxel of the chip — a small block of material with neighbours on every side. The d in the formula is a real distance to a real neighbour, and the damping is what the material in between does to the signal.
In CRI-ONE: our seed-matrix accelerator lives on top of Ψ(v). What we do with det(M) after that — the scoring gate, the prune order — is not public.
Photon Chromosome EncodingLight
A single photon carries data in three channels: colour, spin angle, and brightness. Count the bits in each and a good visible photon holds about 24 bits — the chromosome of light.
The matter. The instrument that reads the photon — a grating to split the colour, a polarizer to find the angle, a detector to weigh the brightness. The small deltas in the formula (δλ, δθ, Imin) are how finely those pieces of glass and silicon can tell two photons apart. Light carries the message; matter decides how much of it can be read.
In CRI-ONE: C is the channel-capacity ceiling every photon-transfer calculator respects. The encoding that saturates it is our IP.
Recursive Growth BoundCommons
Grow a chip like a tree and every branch splits into sub-branches; the total gate count at any depth follows a predictable ceiling. Too many wires vs gates is a losing design — the theorem tells you which regime you’re in.
The matter. Gates and wires. The recurrence only counts. It becomes a design limit because every gate takes up silicon and every wire takes up metal, and a die has only so much of either.
In CRI-ONE: we read T(n) at every recursion depth to sense wire-vs-gate imbalance in a layout. The balance rule that closes the design is proprietary.
Zero-Point FabricationAbnormal
Empty space vibrates at every frequency; the total is infinite. But 1+2+3+… secretly equals −1/12, and with that the infinite vacuum energy collapses to a tiny real force pulling metal plates together. Measured in the lab.
The matter. Two metal plates held a few nanometres apart. The plates decide which vibrations fit between them and which do not, and that difference is the force. With no plates there is no gap, and −1/12 stays a curiosity on paper instead of something a lab can measure.
In CRI-ONE: this is the vacuum-scale floor our fabrication calculators respect. The tolerance stack that sits on top of it is ours.
Golden Spiral ConvergenceUnique
The golden ratio shows up in sunflowers and galaxies — and in chip design: the ideal wiring fraction is 38.2%, exactly 1/φ². Deviate and you either starve for wires or waste silicon on empty metal.
The matter. Seeds on a sunflower head, and metal traces on a chip. Both must share a fixed amount of room without crowding each other. The ratio is only a number until something physical has to be packed.
In CRI-ONE: ropt = 0.382 is the target wiring fraction the routing calculators drive toward. The routing method that reaches it is proprietary.
Quantum Counting ParadoxAbnormal
A quantum chip holds a million answers at once, but you see only one when you look. The trick is rotating the hidden answer into view with exactly the right number of nudges — too few and it’s wrong, too many and you overshoot.
The matter. The qubits themselves — atoms, ions or tiny superconducting loops, held cold and still. Each nudge in the formula is a real pulse sent to that register, and looking at the answer is a real measurement that disturbs it. The count matters because matter can only be turned one step at a time.
In CRI-ONE: ⌊(π/4)√N⌋ is the rotation-pass ceiling our quantum-search estimators use. The pass schedule inside it is proprietary.
Thermal Noise FloorCommons
Physics says erasing one bit costs a minimum energy — about 3 billionths of a trillionth of a joule at room temperature. Divide your power budget by that and you have the absolute maximum computations per second any computer can ever reach.
The matter. Whatever holds the bit — the charge on a capacitor, the direction of a magnet, the position of a switch. The T in the formula is the temperature of that real object, and the energy is heat it must hand to its surroundings when the bit is erased. A bit stored in nothing would cost nothing. No such bit exists.
In CRI-ONE: every FLOPS number we report is clamped against the Landauer floor. No exceptions, no bypass.
Dimensional FoldUnique
A circle has area πr², a sphere volume 4/3πr³. Go higher and the volume grows, peaks at dimension 5, then shrinks to zero. A 100-dimensional ball is almost empty — everything hides at the surface.
The matter. A part with many measurements. Every dimension in the formula is one more quantity taken off a real object — a length, a voltage, a temperature. Take enough of them and the fold takes over: almost every part sits near the edge of its allowed range and almost none sits in the middle. The geometry is pure number. It starts to matter the moment there is a thing being measured.
In CRI-ONE: n* = 5 is the default projection dimension our under-constrained solvers fall back on. The basis matrix behind it is proprietary.
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