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F-theory encodes physics as the shape of a hidden geometry: attach an elliptic curve to every point of a base space, and the places where that curve degenerates are where symmetries and matter appear. F-Theory Lab makes that dictionary something you can push around with sliders. Every readout in it is a toy — the module says so permanently and on purpose — but the structure it teaches is real, and the discipline of separating the two is the point of the module as much as the geometry is.
Before you start — Comfort with polynomials in two variables helps and nothing else is assumed. The Kodaira type names look forbidding and are just labels for a short list of ways a shape can pinch.
The surface is written in Weierstrass form, y² = x³ + f·x + g, with f and g polynomials over the base. The discriminant Δ = 4f³ + 27g² is what matters: where it vanishes, the fibre pinches. Start with a configuration where it never vanishes.
Do — Open the smooth fibration. Fibration validity is at maximum and everything downstream of it — classification, gauge group, matter — reads zero. A perfectly regular geometry is exactly as interesting as it sounds.
Because Δ = 4f³ + 27g², a zero needs f to go negative — the two terms have to cancel. That makes a curve in the base where the fibre degenerates, and Kodaira classified the ways it can do so into a short finite list. The module assigns those names by magnitude tests rather than true vanishing orders, which is the specific thing "Toy Model" is warning you about.
Do — Open the single locus and read the classification gauge. One region, one type, and the anomaly balance sits in its band. Now drag the f coefficients back toward positive and watch the region vanish.
The dictionary from fibre type to gauge group — III → SU(2), IV → SU(3), I*₀ → SO(8), IV* → E₆, III* → E₇, II* → E₈ — is a genuine F-theory result and is not the toy part. Where two regions carrying different groups come close, the model suggests matter living on the seam between them. That mechanism is standard; the way this module detects it, by grid adjacency, is not.
Do — Open the two-stack configuration. Several regions, more than one symmetry, and a toy matter pair. Try to reach three distinct groups at once — distinct groups are worth far more than repeating one, and a deep pocket of the right size is what buys the exceptional groups.
This is the step the module exists for. Six readouts feed the total; a seventh, anomaly balance, is displayed and never scored, because the band it rewards is an analogy to a K3 surface's fixed budget of degenerations rather than anything derived — and a number nobody can defend should not be worth points. The status badge reads "toy grading — not scientific validation" every single time it renders, for the same reason.
Do — Open any configuration and read the boundary badge on each explanation. They are not all the same: the gauge dictionary is a mathematical model, the matter spectrum is a speculative proxy, and nothing in the module claims established physics. Then get the same gauge group using one fewer defect — restraint is the only thing minimality rewards.
Before any of the geometry can mean anything, the numbers have to be numbers. This checks that your shape evaluates to a finite value at every point of the grid, and that you kept the coefficients inside the range the model is defined on.
Try — This is the one readout you should never be optimising. If it is not at maximum, read the status badge first — the configuration is a failed model regardless of what the rest of the panel says.
Picture a small doughnut attached to every point of a flat sheet. Where the discriminant vanishes that doughnut pinches shut — and pinches are where the interesting physics lives, but they cannot happen everywhere at once. This scores how much of your sheet keeps a healthy, unpinched fibre.
Try — Drive every coefficient toward zero and watch this collapse while the singularity readouts climb. The two are opposed on purpose — a geometry that is singular everywhere is not a geometry.
Where the fibre pinches, it pinches in one of a small number of standard ways. This scores what fraction of your pinched patches are clean enough to be given one of those standard names.
Try — Place a defect and shrink the constant terms of both f and g toward zero. As the representative cell drops below 0.01 the region jumps to the deep tier — and a large deep region is what buys E₇ or E₈ on the gauge readout.
In this picture the kind of pinch decides which force lives there. Different pinch types carry different symmetries, so this scores how many genuinely different ones your configuration manages to produce at once.
Try — Aim for three different symmetry groups rather than one very deep one. A distinct group is worth 25; a sixth region is worth nothing.
A model that needs a long list of hand-placed adjustments to reach its result is a weaker model than one that gets there with fewer. This scores how restrained your configuration is.
Try — Reach the same gauge group with one fewer defect. That is worth 8 points here and frequently costs nothing anywhere else on the panel.
Where two regions carrying different symmetries touch, the model suggests a new kind of particle living on the seam between them. This counts how many different seams your configuration has.
Try — Move two regions with different symmetries toward each other until a pair label appears. The score moves in steps of 40, so the question worth asking is which pairs you can create at all — not how close you can get them.
Shown, but never scored. It asks whether your configuration has a plausible amount of degeneration overall — not too little, not too much — by analogy with a surface that has a fixed budget of pinch points to spend.
Try — Watch this while you tune coefficients toward a deep singularity. Falling off the top of the band means you are degenerating the whole base rather than a locus in it — the same failure fibration validity reports, read from the other side.
Surfacing sources you can verify is a deliberate anti-pseudoscience measure, not a bibliography. Nothing on this page asks to be taken on trust.
Glossary — every term used above, defined once.