A blowup, read as a straight line.
u(t) = (T* - t)-beta races to infinity at T*. Measure it in multiplicative arithmetic, the bigeometric calculus, and the same curve is a line whose slope is the exponent. For a power law the readout is an identity: D_BG[xn] = en on x > 0.
- time t
- -
- u(t)
- -
- du/dt (classical)
- -
- elasticity
- -
- D_BG = e^elasticity
- -
- note
- elasticity = d ln u / d ln(T* - t). du/dt climbs without bound; the elasticity sits on -beta.
This is a diagnostic. u still reaches infinity at T*; the readout measures the exponent of the law and leaves the singularity where it is.
Reading a blowup law
Hand the lens a record that stops short of T*. It fits log f against log(T* - t) with T* left free and reports what the record is consistent with: a power b on this window. Below, the Burgers analyticity-strip width, an exact law with b = 1.5 at T* = 1, sits next to a smooth mimic that stays bounded. Drag the record closer to T* and watch both fits agree.
Every exponent at once
The hero curve, for every beta from 0.25 to 3, as one sheet. Slide it from the classical chart to the log chart and, on the same vertices, every slice straightens into a line of slope beta. Then step to the data horizon.
- Burgers 1/delta(t), exact strip-width law
- fit T* = 1.00001, fit b = 1.509
- consistent with power b = 1.509 on this window
- Truth: delta reaches 0 at t = 1 (b = 1.5).
- mimic (s^2 + eps^2)^(-3/4), eps = 1.00e-4
- fit T* = 1.00001, fit b = 1.501
- consistent with power b = 1.501 on this window
- Truth: smooth everywhere; its maximum 1.06e6 sits at t = 1.
A cascade, and a law you can tune
Sample u' = up up to a cutoff, optionally with a log-periodic ripple from a geometric cascade (ratio 2 gives omega = 2 pi / ln 2, about 9.07). The fitted T* and b come from the log-log fit; the elasticity panel is a diagnostic view only. With 1% multiplicative Gaussian noise on 160 samples (p = 2, cutoff 99%, 50 seeded trials), the fit reads b with RMS error 0.0012, while the elasticity curve read point by point is off by 0.24 RMS. The ripple spacing in the elasticity returns the cascade ratio.
- true T*
- 0.6667
- fit T*
- 0.6664 (-0.04%)
- true b, 1/(p-1)
- 1.0000
- fit b
- 0.9951
- ripple ratio set
- 2.00
- ripple ratio from elasticity
- 2.01
- samples
- 160
- bounded future sup
- 193.0
The mirror
Take the theta function of a lattice, theta(alpha) = sum of exp(-pi alpha |x|2) over every lattice point, origin included, and read its elasticity in the log chart. For any isodual lattice in d dimensions, e(s) + e(-s) = -d/2 exactly, so in 2D the curve is point-symmetric about (0, -1/2). This is classical Jacobi/Poisson duality, seen through the lens. Computed at build time by tested code.
triangular, square (grey), honeycomb, and the honeycomb reflected through (0, -1/2) (red dashes). A mirror-exact curve lands on its own reflection.
- triangular: max |e(s) + e(-s) + 1|
- 5.8e-15
- mirror-exact
- square: max |e(s) + e(-s) + 1|
- 4.2e-15
- mirror-exact
- honeycomb: max |e(s) + e(-s) + 1|
- 0.068
- not a lattice: the mirror breaks (dashed = its reflection)
- FCC vs BCC at s = 0
- 2e-15 apart
- tie at s = 0, order swaps across it (grid only)
Two dimension meters on one torus
On the unit cubic 3-torus, two elasticities read off a dimension. The ball, cylinder and slab candidate envelope for the isoperimetric profile, drawn for enclosed volumes up to one half (beyond that the complement of each region is the better candidate), steps 2/3, 1/2, 0. The heat trace slides from -3/2 to 0 with no plateau between. Same space: one meter steps, the other slides.
Where to look next
Demos
Browser models near hard regions: crack tips, vortices, orbits. Illustrative, each labelled.
Tools
Invariance checker, simulators and benchmarks that run the arithmetic on your inputs.
Results
Stored runs with their controls. Above: Shu-Osher L2 error by limiter on the base case, from the artifact; the sensor threshold was tuned on this same case.
Evidence
What held, what we killed, and every null result. Above: the spectral-gap cases, failures in red.