Route-DTOL (leg 386). Companion: TECHNICAL_P2_ROUTEDTOL_V1.md. Figure: fig99.
Ceiling: Tier 2. Clay progress: ~0.05%, unmoved.
The problem we walked in with
A previous leg built an instrument that answers a narrow question exactly: given a function
sampled on a window of radii, is there any pure power law that fits it exactly? The
instrument does not guess. It eliminates the unknown amplitude algebraically and returns either
an interval of exponents that are still possible, or EMPTY, and EMPTY is a proof that no
power law fits, not a failure to find one.
That instrument had one embarrassing property: on every input anybody would actually feed it,
it answered EMPTY. Real profiles are not exact power laws. They have logarithmic corrections,
cutoffs, subleading terms. The instrument was correct and useless in the same breath.
The obvious repair is to allow a tolerance: ask not "is f exactly a power law" but "is f
within a factor 1+δ of one". The previous leg noted this repair, measured some numbers for
it, and, to its credit, labelled those numbers post-hoc and not gate-deciding, because
they were measured after the fact on cases chosen after the fact.
This leg's job was to do it honestly: write down what the tolerance mode would have to do before running it, then run it, then report what actually happened including where the prediction was wrong.
What we wrote down before running
Two things, both derived rather than fitted.
A width law. If you allow a factor 1+δ of slack, how much wider does the answer get? We
derived, in closed form, width = 4·log(1+δ)/log(R₁/R₀). The interesting part is the
denominator: the cost of tolerance depends on how long your window is. The previous leg's
number, 0.8686, is exactly 4/log 100, an artefact of measuring on [10, 1000]. Shorten
the window to [10, 100] and it should double. That is a falsifiable prediction about a
number someone had already quoted as if it were a constant.
A critical tolerance. Below some δ* the answer must still be EMPTY: otherwise the
tolerance is just a knob you turn until you get the answer you wanted. We derived δ* from
Chebyshev best-affine approximation theory and predicted, in advance, that the measured δ*
would come in slightly below the formula, because the arithmetic we use encloses cells
rather than points.
Both predictions, the pass bands, and the exact test cases went into the journal in a commit that landed before any measurement.
What happened
The width law held, to +0.51 % across nine decades of δ, with the same offset
everywhere: a discretisation cost, not a fit. And the window-doubling prediction landed at
1.9999989 against a predicted exactly 2.
The critical tolerance held, in the predicted direction: −0.98 %, −2.94 %, −9.46 %
below the closed form for the three test profiles, all inside the pre-registered ±20 %.
It also reproduced the previous leg's post-hoc numbers to eight or nine digits, so those
numbers were right; they simply had not been earned yet. Now they are.
And the control held. At 0.9·δ* and at 0.5·δ*, every mismatched profile is still
proved impossible. Six out of six. The tolerance is not a knob.
So the first gate clause answers yes. The instrument gained a real mode, whose looseness is a declared magnitude with a formula attached.
And then the second question
The leg carried a second, co-equal question, and it is the one that matters: can the proof that would use this instrument actually afford any tolerance at all?
The chain is this. A certified decay exponent feeds a cutoff-radius bound. That bound needs the
exponent to exceed 1 (for one conclusion) or 3/2 (for a stronger one). A certification
carrying a tolerance hands that tolerance straight down the chain: the exponent is no longer a
number but an interval, and only its lower edge counts. An earlier leg composed the two laws
on paper and derived a condition; nobody had measured whether that condition can ever be
satisfied, because it demands δ be simultaneously large enough to admit the profile's own
imperfection and small enough not to wreck the bound.
We measured it directly rather than composing two laws, on thirty configurations.
The answer is that the tolerance buys exactly nothing. Not "little", nothing. In all
thirty rows, without a single exception, a window of admissible tolerances exists if and only
if the certified centre already exceeds the threshold on its own. We probed the crossover at
a relative offset of one part in a million and it sits exactly at the centre. Widening δ moves
the left edge of what you can certify; it never moves the centre. There is no configuration
in which slack rescues a profile that was going to miss.
The object this analysis is aimed at carries an exponent of exactly 1. The threshold is 1.
The window is empty, and it is empty at every exponent ≤ 1, for every profile shape we
tried.
So the second clause answers EMPTY. We are reporting it as EMPTY. We did not widen a band afterwards to rescue it, and there was a moment where we could have: one test profile, read by its nominal label, would have counted as a success. Read by the exponent it actually realises (its cutoff sits inside the measurement window, so it decays much faster than its label suggests) it does not. The journal had pre-registered which of those two readings decides, and it decides the strict way. The verdict flipped from ADMISSIBLE to EMPTY when we honoured our own pre-registration.
Two verdicts pointing opposite ways, left that way
Clause 1: YES. Clause 2: EMPTY. We are not netting those into one word. The honest sentence is: the tolerance mode is a genuine instrument that the specific composition this route needs cannot use on the object it has.
There is a consolation prize, and it is a real one. The earlier leg's paper composition turns
out to be conservative by a factor of 1.33 to 4.06: it understates the admissible tolerance
on every row where a window exists, and never once claims a window that isn't there. Its
implied demand, that the exponent clear roughly 2.46 before any tolerance is allowed, is
refuted: the true requirement is just that the exponent clear 1. That corrects a bound
that was quietly making the route look harder than it is. Corrected text has been routed to the
document that carries it.
A late addition, from someone else's control
While this was building, a neighbouring unit landed a result worth stealing. It had built the adapter that turns point samples into the certified cells our instrument eats: a conversion that requires the caller to declare a regularity hypothesis, because nothing in a finite set of samples can establish one. To test itself, it planted a secret violation of that hypothesis and fed it in. The adapter accepted it, correctly: a declared hypothesis cannot be verified, only recorded.
The certificate that came out had width 7.438494264988549e-15. The certificate of a genuine,
truthful test case had width 7.438494264988549e-15. Bit-identical, and one of them was
false about its profile.
There is no way to tell them apart from inside the certificate. The only thing that separates them is the record of what was assumed. So the rule adopted, and implemented here, is that every output row carries the hypothesis in force, and a certificate without a hypothesis is no certificate. Our rows now say which hypothesis was in play, whether it was discharged (the instrument established it itself, by evaluating over whole cells) or merely declared (someone upstream promised it), or whether it was never stated at all, in which case the row says, in words, that it is not a certificate about any profile. The weaker of the instrument's two modes now declares its own weakness inside the row rather than in a docstring nobody reads.
We re-ran everything afterwards. Not one number moved. That is the point: this changes what a row says, not what the instrument computes, and it is the difference between a result and a result you can trust six months later.
What this does not mean
It does not mean the route is closed. It means the load-bearing question moved: it was "how accurately can we certify the profile", and it is now "does the certified exponent clear 1 at all", a question about the object, not about the instrument, and one to which nothing here contributes an answer. No profile of the actual object exists in this repository. Every input in this leg is a planted analytic function with a known answer.
The two standing no-method obligations remain open. No link in the chain from the hard problem to the solved one moved. Clay stays at ~0.05%.
Runner experiments/p2_route_dtol_v1.py; data writeup/data/p2_route_dtol_v1.json;
journal experiments/journal/leg_386.md (pre-registration in PART I, landed before
measurement; results in PART II; the hypothesis-field amendment in PART III); tests
test_dssp_decay_enclosure.py, 18/18, with the previous leg's 12 assertions unweakened.