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.

u(t) = (T* - t)-beta, computed in your browser. Same curve, same tick marks; only the arithmetic of the axes changes.
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.

loading the 3D view
after the recordthe recordBurgers 1/deltabounded mimic1e-21e-31e-41e-51 - t, log scale, toward T* = 1
On the record the two differ by at most 2.33%; past the last sample the gap grows.
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.

Time view: the record, and two futures that fit it
after the recordtrue T*time t0.6200.6671510
Diagnostic view: elasticity d ln u / d ln(T* - t) at the fitted T*
true -b = -1.000-0.51-1.501e-21e-1T* - t, log scale, toward T*
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

What held, and what we killed

Approved is not governed. A claim counts here only when its artifact says so, and the losses get the same type size as the wins. Every figure below is rendered from a committed JSON file at build time.

Held up (2, and 0 with no gain)

  • Shu-Osher shock: log-derivative limiter sensor vs the best classical limiter (superbee)

    +6.30% / +6.39% / +23.58% L2

    Held-out runs (shift 10, shift 100, nx 800) at the threshold 5 tuned on the base case, where it scored +6.32%.

    cfd_nnc_comprehensive.json

  • Lane-Emden n=3 solved in log space

    7.7x fewer evals, 30x smaller error

    Part of the step reduction comes from a max_step cap on the classical run.

    tractability_benchmarks.json

Killed by our own runs (8)

  • Sod shock tube: the same limiter sensor

    -26.13% L2 vs superbee

    Best sensor setting L2 0.01180 against superbee 0.00935, after the exact solution gained its rarefaction fan. Reported, not hidden.

    cfd_nnc_comprehensive.json

  • Bessel J0 in log space is more accurate

    0.10x accuracy ratio

    Below 1: the log-space run was less accurate.

    tractability_benchmarks.json

  • Mixing schemes never lowers the spectral gap below the weakest one

    9 counterexamples in 88 valid cases

    Status in the artifact: REFUTED.

    spectral_gap_boundary_moo.json

  • CASCADE wins across many physics domains

    10 "1/r" cases, 2 distinct outcomes

    A synthetic toy benchmark: one 2-D problem relabelled per domain. Not citable.

    cascade_21_simulation_results.json

  • Every Pareto solution beats Van Leer on the Sod tube

    10 of 30 dominate

    Optimizer: pymoo NSGA-II. GlobalMOO was not connected for this run.

    shock_tube_dual_moo.json

  • Meta-derivative schemes preserve Lorenz chaos

    k = 0.3 collapses to a fixed point

    The attractor is destroyed, not preserved.

    chaos_corrected.json

  • A unitary region in the quantum phase scan

    0 of 1000 runs unitary

    Norm drift never reached the unitarity tolerance in the historical nonunitary blend model (revision e06aaa4).

    quantum_phase_corrected.json

  • Meta-BDF handles stiff Robertson kinetics

    4 of 5 Meta-BDF runs wrong

    Max error above 0.1 for every k > 0; only k = 0, the classical limit, and plain BDF/LSODA are right.

    robertson_stiff_ode.json

The CASCADE suite is a synthetic toy benchmark: one 2-D objective family relabelled per domain, scored by distance to the centre, which is not the singular point for every profile. The full table renders from its stored artifact. Each result names the optimizer that ran: mostly pymoo NSGA-II, with one historical GlobalMOO comparison (equal total budgets not verified) on the synergy page.

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.

loading the 3D view
e(s) = d ln theta / d ln alpha, s = ln alpha. Center of the mirror: (0, -1/2).
-3-2-101230.0-0.5-1.0s = ln alpha

triangular, square (grey), honeycomb, and the honeycomb reflected through (0, -1/2) (red dashes). A mirror-exact curve lands on its own reflection.

FCC vs BCC in 3D: (theta_FCC - theta_BCC) / theta_FCC on the grid s in [-3, 3]
-3-2-101231e-30-1e-3s = ln alpha
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.

candidate-envelope elasticity d ln I / d ln v, v <= 1/2
1e-41e-31e-21e-10.52/31/20enclosed volume v, log scale
heat-trace elasticity d ln Z / d ln t
1e-41e-31e-21e-11e01e1-3/2-1/20diffusion time t, log scale

Where to look next