The number that always answers, and the number that knows when to refuse

Route-DEXC v1, leg 382. Figure: fig100. Data: writeup/data/p2_route_dexc_v1.json.

A confident number for a profile that has no number

Give this repository's screen a profile shaped like r^(-2) + 0.01·r^(-1), and ask it for the far-field decay exponent. It answers 1.500000.

That profile does not have a decay exponent. On the window it is asked about, [10, 1000], it starts out looking like r^(-2) and ends up looking like r^(-1); the crossover sits right in the middle, at r = 100. There is no exponent p for which the profile is C·r^(-p). The answer 1.500000 is the least-squares compromise between two different behaviours, and nothing in the output says so.

That is not a bug in solver/dssp_screen.py. Its routine is called fitted_far_field_decay_exponent and it does exactly what the name says: np.polyfit of log|V| against log r over twelve points. A fit always returns a number. That is what fits are for.

The trouble is what the number is for. CLAY_OBLIGATIONS.md §4 spells it out: to cut a self-similar profile off and get finite-energy data (which the Clay statement requires, and which this object does not currently have) you need to choose a cutoff radius and bound the perturbation the cutoff introduces, and both of those are functions of the decay exponent. A compromise slope with no error bar cannot carry that weight. §4 says so in one sentence: fitted is not sufficient here. §6 then lists it as one of two obligations in this programme with no known method in this repository.

Leg 382 built the method. This is what it can and cannot do.

The idea, which is not clever

Write t = log r and g = log f. Then "the profile is a power law with exponent p" is just "g is the straight line c − p·t", where c = log C is the amplitude nobody cares about.

Now instead of a best-fit line, enclose the profile. Chop [10, 1000] into a thousand cells and, using the interval arithmetic this repository already has, compute for each cell a rigorous box that the profile provably lies inside across that whole cell, not at sample points, across the entire cell, so there is no gap for the profile to wiggle through between samples.

Then ask a different question. Not which line fits best?, but which lines fit at all? Every cell says the line has to pass through its box. Each of those is a linear inequality in the two unknowns (c, p). Eliminate the amplitude c (in two variables that elimination (Fourier–Motzkin) is exact, no approximation) and what is left is precisely the set of exponents that are still possible.

Three things can come out:

• an interval (no exponent outside these bounds is possible;
• EMPTY) no exponent is possible, at all. The profile is provably not a power law here;
• INCAPACITY: the machinery could not say, and says so.

None of this is new mathematics, and leg 382's novelty pass says so in as many words. What is new is that this repository can now do it: the pass swept the tree and found that every log-log exponent anywhere in it is a least-squares fit, without exception.

What it does on things whose answer is known

Four planted profiles with exact exponents: 1, 2, 2.5, 3 (the 2.5 deliberately not an integer, to catch an instrument that quietly snaps to round numbers). All four exponents land inside the certified interval, and the intervals are narrow: widths of 7.4e-15, 1.6e-14, 2.0e-14, 1.6e-14. That is the floating-point rounding floor. The certified answer and the fitted answer agree to twelve decimal places, which is exactly what should happen when the profile really is a power law.

Then the mismatches. The two-power blend above: EMPTY. A profile with a far cutoff, r^(-2)/(1 + (r/300)^4): EMPTY. A logarithmic correction r^(-2)·log r, whose effective slope only wanders between about 1.57 and 1.86, subtle enough that a fit would never blink: EMPTY.

The fitted column, on those same three profiles, returns 1.500000, 2.808892 and 1.768252: three confident numbers for three profiles that have no exponent. Panel A of fig100 is just those two columns side by side.

This mattered enough to be locked down in advance. This repository has been burned before by an instrument that could not fail (leg 340) and by the temptation to widen a window until something passes (leg 361), so the window, the planted knowns, the controls and even the predicted numbers were all written down and committed before the module existed. The commit order is in the git log and can be checked.

The part that went wrong, which is the interesting part

The pre-registration made a prediction about how the certified interval would widen with coarser cells. It was wrong: the interval turned out to sit at the rounding floor regardless. Fine; the instrument is better than expected.

The second prediction was wrong in a way that actually matters.

Take an exact r^(-2) profile and perturb it by one part in a trillion. Ask the instrument for the exponent. It answers EMPTY.

And it is right. A profile perturbed at the 1e-12 level is not exactly C·r^(-p) for any p, so the set of exponents that fit it exactly really is empty. The pre-registration had called "the truth stays inside" a soundness requirement whose violation would be a defect. It was not a defect. It was a mis-statement of what had been defined.

But follow it through: no numerical profile is ever exactly a power law. So the instrument as first specified would answer EMPTY for every real input it would ever be handed. It would be rigorous and completely useless.

What it costs to be useful

The fix is the obvious one, and its honesty depends entirely on where the extra number comes from. Ask for exponents consistent with the profile to within a stated relative accuracy δ: where δ is not a dial to turn until something passes, but a bound the caller already owes about their own profile.

With that, the instrument reports (panel B):

certified exponent half-width ≈ 0.434 × δ

Know your profile to a part in a million, get the exponent to about four parts in ten million. Clean, linear, and it tells a future consumer exactly what accuracy to aim for before running anything.

And the price is measurable too (panel C). Each mismatch has a critical tolerance δ*, the sloppiness at which it stops being caught:

An exact power law, by contrast, is never excluded at any tolerance) so the instrument is not merely fail-happy. It discriminates, and the table says by how much.

What this does not mean

It would be easy to over-read this, so, plainly:

There is no profile. Route 4 has not produced one. This instrument has been validated on planted analytic knowns and on nothing else. Its first real consumer is a future unit that does not exist, and this leg claims nothing whatsoever about one.

§4 is not discharged. §4 wants three things: the certified exponent, the admissible cutoff radius, and the size of the perturbation the cutoff introduces. This is the first. The other two are untouched. Both of CLAY_OBLIGATIONS.md §6's no-method items stay open: item 1 because only half of it was attempted, item 2 because it was not attempted at all and, per leg 314, is not the kind of thing a computation can supply.

The fitted column is still there. Nothing was replaced. solver/dssp_screen.py was read and edited nowhere. The certified column is recorded alongside the fitted one, in the same row of the same JSON, which is the only arrangement in which the comparison in panel A can be made at all.

And the ceiling has not moved: Tier 2, no link of the L1 → L4 chain touched, Clay odds unchanged at ~0.05%.

What did change is small and real. An obligation that read "no known method" now has a named instrument, a test suite, and a measured price list. That is one line of a specification turning into something you can run.

Technical companion: TECHNICAL_P2_ROUTEDEXC_V1.md. Module: solver/dssp_decay_enclosure.py. Tests: test_dssp_decay_enclosure.py (12/12). Runner: experiments/p2_route_dexc_v1.py. Novelty pass: writeup/novelty/leg_382.md. Pre-registration and results: experiments/journal/leg_382.md.