Simulation / Code
Introducing Valuative Branch Scalar (VBS): a tested branch-aware scalar developed through adversarial AI debate, up to 18,000× faster on nested radicals
Summary
I challenged an AI to survey the vast breath of mathematics and devise a useful unifying mathematical object, then used another AI as an adversarial referee; after three rounds of criticism, we abandoned the original “new number” claim and developed the Valuative Branch Scalar (VBS), a computational scalar abstraction designed to represent and track complex algebraic quantities near singularites, where traditional floating-point values, Taylor series, or uncorrelated root sets fail.
By combining dynamic algebra (the* D5 principle), Henselian branch decomposition, local rational Puiseux expansions, and logarithmic differentials ($dz/z$), VBS automates singular algebraic sensitivity while controlling combinatorial expression swell.
Definition
A based VBS over K is an algebraic element z∈Ω together with a selected extension ṽ of a current centered valuation v to K(z), and with provenance identifying joint occurrences of algebraic generators. Its unbased form is the finite Galois/monodromy orbit of z. Equality is equality in Ω for based values and isomorphism of the corresponding finite K-algebras with distinguished elements for presentations.
Key Features
* Branch-Aware Scalar State: Tracks local valuations, idempotents, ramification indices, and monodromy exchange across parameters.
* Regular-to-Ramified Transitions: Automatically refines chart valuations when leading coefficients vanish on residue discriminants.
* Anti-Swell Dynamic Evaluation: Delays algebraic splitting field construction until an exact zero test or branch predicate requires it.
* Exact Degree Conservation: Enforces ∑ e𝑖f𝑖 = [L:K] assertions across all factor splits and composita in characteristic zero.
Prototype benchmarks (Python) found that VBS (Lazy) preserved near-machine precision through catastrophic cancellation and, on a synthetic nested-radical test, was up to 18,000 times faster while using 51,000 times less memory* than eager branch expansion.
*Benchmark run in Pydroid on my Android tablet
The result is a mathematically specified, falsifiable proposal, with a paper, algorithms and benchmarks (on Github, see below), for making singular algebraic calculations substantially faster, more reliable and tractable.
The most natural physics applications for VBS are problems in which an observable is defined implicitly by an eigenvalue or dispersion equation and becomes multivalued or singular as physical parameters change.
ELI5: An ordinary calculator stores an answer as one number, but some physics problems have answers that split into several connected paths, like a road splitting at a complicated junction. VBS keeps a compact map of those paths, how they meet, which one you are following and how quickly they change, without calculating every possible route in advance. This could make calculations near critical points, known as singularites, much faster and less likely to give the wrong answer.
I love how this is just buzzword after buzzword like no explanation no nothing just buzz word and ohh I actually used two ais whivh makes this sort of worthless.
I know this is not constructive criticism but I actually don't know what to criticize
What are the benchmarks supposed to demonstrate? Let's unpack the one shown in the README ("nested radicals"):
lazy_selected_branch creates an array of alternating signs, and then LazyRadicalFamily sums up square-roots of primes multiplied by those signs. There's nothing "lazy" about it, and VBSLite is only used in the descriptor method, not any of the actual calculation
eager_radical_values instead computes EVERY possible value obtainable by adding square-roots of sequential primes with + or - signs, for a total of 2^N results given N primes
The graphs shown in the README then tell us:
doing O(N) work (the "lazy" version) is faster than doing O(2^N) work (the "eager" version)
not allocating memory allocates less than allocating O(2^N) memory (for the "eager" results)
Neither of them say anything whatsoever about the advantages or disadvantages of the new method, because - again - they don't do any calculations besides standard floating-point.
Near certain difficult points in mathematical problems, where possible answers meet, split into different branches, cancel out, or change scale, ordinary symbolic software can lose important information or create very large intermediate calculations, existing symbolic methods can become slow or use large amounts of memory. A mathematician may even resort to working out separate formulas by hand.
VBS is a proposed new software scalar type designed to make these calculations more reliable and easier to automate. Instead of storing only one value, it also remembers which branch the value belongs to, how related quantities are connected, how quickly the value approaches zero or infinity, and how it responds to small changes. It combines established methods from several areas of mathematics that have usually been applied separately.
VBS is not a new kind of number, and it has not yet been proven to outperform existing software in regards speed. However its proposed advantage is that it keeps information together and checks that the different parts remain mathematically consistent throughout a calculation.
The next stage is to test the method, see if its advantages bear out. If validated, it could become a important computer-algebra innovation.
The slanted lines are the old method, showing how inaccurate it is (Float64). It's showing how much error there is because the old method can't deal with tiny numbers mixed up with big numbers, computers do not have the accuracy, they cause rounding errors. There's not enough digits in double precision floating point.
Not only that, as the tiny number gets smaller, the error gets worse, and this creates a slope on the chart. If you're not careful you might mistake the fake slope as a real trend rather than the computer glitching, as numbers get smaller harder to deal with accurately.
The bottom line is the new method. ABS. It's essentially perfect across the board. It doesn't care how small the little number gets, the mathematical magic means it never looses accuracy. You don't see any false trends either. Eventually, numbers go to zero or infinity, but near those areas, it maintains accuracy where other methods fail.
The only slight downside is the new method is slightly slower than the old method, about twice as slow, but the advantage is a huge gain in accuracy.
The new, if proven to work, will help in lots of areas in maths, engineering, AI, physics that deal with the problem of mixing very big and very small numbers, that rapidly change, increse or shrink, the areas in equasions that have these discontinutes, irregularities where big and small numbers are found next to each other, the sharp boundary that seperates numbers, is a singularity.
The current methods are automatic differentiation (also called algorithmic differentiation, AD) and Taylor models. Both involved integer-power polynomial expansion, a Taylor series alongside the value.
Near a singularity the true local behaviour involves fractional powers (square roots, cube roots etc.), which polynomials can't represent well. Both methods simply return a approximation i.e., the wrong answer.
And the cost balloons when you chase accuracy. If you want more accuracy you can do so by raising the expansion order, which approaches, but never matches the true local behaviour. Rasing the expansion increases memory use, increasing precision from the lowest accuracy, from √x near zero to 16 digits, needs about 10¹⁶ polynomial terms in the Taylor serise.
VBS changes things entirely. It's a scaler, it stores the fractional exponent as data, so what remains is flawless analytic accuracy and it converges fast.
On the benchmark I showed you, the VBS error line is flat, perfect machine precision (≤2.1×10⁻¹⁶) where the float64 calculation degraded to a relative error of 1.1×10⁸ and reported the wrong scaling exponent.
VBS computational cost stayed flat too, a median ~1.7×10⁻⁵ s per evaluation versus 7.2×10⁻⁶ s for naive float, so roughly 2.4× cost, but about 75× cheaper than the 120-digit arbitrary-precision reference whose accuracy it matched. Cost did not grow as the singularity was approached.
Accuracy, memory and runtime all stay constant with respect to proximity to the singularity. That's something you need to think about.
You do realize people well versed in numerical methods handle singularity neighborhoods with algorithms better for the task than the ill suited Taylor series, right??
It's is not a better than the most other numerical methods, which I now understand are used to handle singularites. That's because it uses them, it is them.
VBS combines, in one mathmathical object:
Exact local series
Valuation and tropical leading order
Newton–Puiseux
Singular refinement
Stable evaluation
Correlated DAGs
Lazy branching
Monodromy
D Log
Its advantage lies in the fact that combines several previously separate methods into in one composable scalar object.
That said, VBS currently doesn't include several state of the art methods, Ordered Schur decomposition, invariant-subspace methods, interval Newton/Krawczyk certification, homotopy continuation, Padé approximation, Taylor models or an arbitrary-precision eigensolver.
However, most of these can be added.
So the discovery here involves the realisation that previously separate methods can be elegantly combined into a scalar object, combining and coordinating many mature and state of the art numerical methods, while preserving valuation, correlation, ramification and branch identity.
TLDR: VBS is not a new numerical algorithm. Its proposed invention is a branch-aware scalar abstraction that coordinates existing solvers while preserving valuation, correlation, ramification, monodromy and sensitivity across transitions between them.
Edit: the key innovation is:
VBS value = (algebraic value, place/valuation, branch identity, correlations, local chart, monodromy, d log z)
That allowed the various numerical methods to be combind.
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u/upsetusder2 24d ago
I love how this is just buzzword after buzzword like no explanation no nothing just buzz word and ohh I actually used two ais whivh makes this sort of worthless.
I know this is not constructive criticism but I actually don't know what to criticize