Route-SCEL v1: samples → certified cell enclosures, under a named hypothesis

Leg 385, slot C, construction, difficulty HEAVY. Gate: YES. Module: solver/dssp_decay_samples.py. Tests: test_dssp_decay_samples.py, 17/17. Runner: experiments/p2_route_scel_v1.py (8.1 s). Data: writeup/data/p2_route_scel_v1.json. Figure: fig103. Novelty pass 00d8e4e, pre-registration 353cbf1: both committed before construction. Reads solver/dssp_decay_enclosure.py (leg 382, 104f5b3) and edits it nowhere. Companion: BLOG_P2_ROUTESCEL_V1.md.

Ceiling: TIER 2. CLAY_OBLIGATIONS.md §6 items 1 and 2 stay OPEN; item 1's admissible-cutoff half is entirely untouched by this leg. No L1 → L4 link moved. Clay ~0.05%.

1. The gap this closes, in leg 382's own words

solver/dssp_decay_enclosure.py:354:

"Supplying those is the caller's obligation and this routine cannot check it, if the caller has only pointwise samples, it must first convert them using a certified modulus of continuity, or a monotonicity hypothesis, and must state which."

certified_decay_from_cell_enclosures(r_lo, r_hi, f_lo, f_hi, ...) requires f_lo[i] ≤ f(r) ≤ f_hi[i] for every r in cell i. Any profile-producing unit holds point samples. The converter did not exist anywhere in the tracked tree (novelty pass §1: six modulus of continuity hits, none an implementation; zero sample→enclosure routines in solver/).

The negative fact that forces the design. From finitely many point evaluations alone, with no regularity hypothesis, no non-trivial enclosure is derivable. So the module's first behaviour is a refusal, not a computation.

2. The two paths

Let t = log r, g = log f, cells [r_i, r_{i+1}] with both endpoints sampled as certified intervals [f_lo_i, f_hi_i].

PATH A: MONOTONE. Caller declares f non-increasing (or non-decreasing) on the window. Extremes on a cell are attained at its endpoints, so f(r) ∈ [f_lo_{i+1}, f_hi_i]. Exact: no slack is added beyond the caller's own sample error bars.

PATH B: MODULUS. Caller declares ω(h) = L·h^α, α ∈ {1/2, 1}, in one of two kinds:

h/2 because every point of a cell is within half its width of one endpoint.

L may be a scalar or a per-sample array (cell i uses max(L_i, L_{i+1})).

No iexp in the substrate. solver/interval.py has ilog and no exponential, and leg 382 deliberately declined to add a transcendental whose remainder it would have to prove. The loglog kind is therefore closed with two elementary bounds using only + − × ÷:

e^{-x} >= 1 - x         (x >= 0)
e^{ x} <= 1/(1 - x)     (0 <= x < 1),   since 1/(1-x) = Σ xⁿ >= Σ xⁿ/n! = eˣ

applied outward. At ω ≈ 2.4e-3 the overestimate is O(ω²) ≈ 5e-6 relative: measured as max_relative_inflation = 2.4235738569e-3 against ω = 2.4177143477e-3, a 0.24% excess over the exact exponential. Inputs with ω(Δt/2) ≥ 1 are refused, not approximated.

Both may be declared, in which case the enclosures are intersected and the row records MONOTONE+MODULUS.

3. Refusal is a first-class output, and the conditionality is in the row

samples_to_cells returns INCAPACITY, never a guess, for: no hypothesis; a hypothesis the samples themselves contradict; non-increasing radii; any r ≤ 1; a non-positive sample bound; f_hi < f_lo; fewer than two samples; a bare number in place of a Modulus; α outside {1/2, 1}; ω(Δt/2) ≥ 1 in the loglog kind; and mutually inconsistent declarations. certified_decay_from_samples never calls leg 382's routine on a refusal (enclosure_called: false, p_lo: null): there is no code path producing a decay exponent without a named hypothesis attached to it.

Every accepted row carries hypothesis, hypothesis_detail, sample_exactness (enclosed vs declared-exact) and a conditional_on sentence, merged into leg 382's own output dict. §6 below is why that is not decoration.

Necessary conditions, never sufficient. A declared hypothesis implies checkable facts about the samples (monotone increments; |Δ| ≤ ω(h)). Violations that are certain after outward rounding refuse and name the cell. Passing proves nothing, §6.

4. PATH A reproduces leg 382 bit-identically

Window [10, 1000], N = 1000 cells, bracket [0,12]: leg 382's configuration, unmoved. Samples are certified enclosures produced by leg 382's own interval generator at point radii (a float64 evaluation of the planted formula is not its exact value, and laundering that away is the leak this module exists to plug).

Every difference is exactly zero, not "within tolerance", and the truth is inside every interval (K1: p ∈ [0.9999999999999963, 1.0000000000000038]). The baselines are re-computed in this run by calling certified_decay_interval, never transcribed.

Mechanism, and it is the same one leg 382 measured when its own width prediction failed (its §12a): for a monotone profile the cell enclosure's endpoints coincide with the pointwise values at the cell edges. Those edges are exactly the sample points. The conversion loses nothing. Prediction P1 CONFIRMED, in the strongest available form.

5. PATH B is sound, and cannot be made tight (predicted before it was measured

f = 3 r^-1, true log-log Lipschitz constant 1.0, declared ω(h) = 1.05·h (loglog)) a true, slightly conservative statement.

• Verdict INTERVAL, p ∈ [0.9989466218941611, 1.0010512713496271], truth inside.
• Width 2.1046494554660677e-3: pre-registered prediction ≈2.0e-3, band [7e-4, 6e-3]. P2 CONFIRMED.
• Ratio to PATH A: 2.829e11.

Log-log slopes: PATH A −0.0118 (rounding floor; leg 382 measured −0.0295 on its own ladder), PATH B −1.0068. P3 CONFIRMED. Extrapolating, PATH B reaches PATH A's 7.44e-15 at N ≈ 2.83e14 samples.

The modulus path cannot reproduce leg 382's exact-power widths at any feasible sampling density. It is sound, never tight. This was written into the pre-registration (P2, P3) before the module existed; it is not a post-hoc rationalisation of a disappointing number.

5a. A globally stated absolute modulus is useless on a multi-decade window

Same profile, declared ω(h) = 0.03·h where 0.03 = p·C·R₀^{−p−1} is the true global Lipschitz constant on [10, 1000].

• First cell whose lower enclosure is non-positive: r = 207.97 (pre-registered ≈208, band [100, 420]).
• Cells affected: 341 of 1000 (pre-registered ≥250).
• Downstream verdict: INCAPACITY ("profile enclosure touches or crosses zero; log is undefined") not a number. P4 CONFIRMED on all three components.

Mechanism: on a geometric grid h ∝ r, so ω(h/2) grows with r while f decays; the additive inflation overtakes the profile. The loglog kind, or per-sample local constants, are what such a window admits. Panel C of fig103.

5b. Declaring both

MONOTONE + loglog ω gives width 7.438494264988549e-15, difference from PATH A exactly 0.0. The intersection is dominated by the tighter statement. P12 CONFIRMED.

6. The controls: all five fired, none widened

P5–P9 all CONFIRMED. No threshold, amplitude, window or modulus was touched to make a control pass (the leg-361 rule).

6a. X3 is the leg's argument, stated precisely

The planted truth is f(r) = 3r^{-1}(1 + A·sin(2π(t − t₀)/Δt)) with Δt exactly the grid spacing in t, so the sine vanishes at every node: the samples are bit-identical to K1's. The adapter accepts (it provably cannot do otherwise) and the pipeline returns

• verdict INTERVAL, width 7.438494264988549e-15, containing p = 1,

which is numerically indistinguishable from K1's true certificate and is false about the true profile, which leaves the claimed enclosure in all 1000 cells by up to 5.14%.

Nothing is broken. The certificate is valid under the declared hypothesis; the hypothesis is false. The only thing separating the true certificate from the false one is the row's hypothesis / conditional_on field. That is why conditionality is recorded, and it is the one design decision in this module that is not a matter of taste.

X5 is the same trap on the modulus side: node-to-node increments satisfy L = 1.05 exactly (|Δg| = Δt ≤ 1.05Δt) while the within-cell excursion 0.05 dwarfs the inflation 2.4177e-3.

7. Where the pre-registration was WRONG, recorded, not amended

P11: REFUTED

Part I §5 predicted that X3's true profile, re-sampled on the half-cell-shifted grid, would be visibly non-monotone and refused. It is not: the adapter returns CELLS. The quarter-shifted grid also returns CELLS.

Mechanism. The wiggle has period exactly one cell in t, so a uniform shift by s cells multiplies every sample by the same constant (1 + A·sin(2πs)). A constant multiple of a monotone sequence is monotone, and its log-increments are unchanged, so both necessary conditions pass at every uniform shift; the half-shift is doubly invisible because sin(π) = 0. The pre-registration's reasoning was wrong about which grids can see the wiggle. The prediction stays on the record as wrong.

The correct statement, located post-hoc and labelled as such

NOT PRE-REGISTERED. Decides nothing about the gate, which is answered by X1–X5. Re-sampling the same true profile at other densities:

Detection is a commensurability phenomenon: ratios near an integer (500, 1000, 2000) or near 1 (997, 1001, 1010, the phase drifts too slowly) see nothing. A hypothesis-violating profile can hide from any fixed grid, which strengthens the leg's conclusion rather than weakening it: the hypothesis is doing the work, and the sampling never was.

8. Anti-tautology

1. The containment audit can report CLEAN, and does: worst signed relative excess −1.7067e-16 (S1), −2.4177e-3 (S2), −2.3079e-3 (S3), all strictly negative. P10 CONFIRMED. An audit that always fires would prove nothing.
2. No row returns the full search bracket.
3. The refusals are not universal: S1–S4 are accepted by the same code path that refuses X1, X2, X4.
4. Every accepted row carries a hypothesis.
5. Every leg-382 comparison number is re-computed in this run, not transcribed.

Ledger: 11 of 12 pre-registered predictions CONFIRMED, P11 REFUTED and recorded as refuted.

9. The gate, answered in its own wording

"Does the adapter convert a planted sampled profile with a stated true hypothesis into cell enclosures whose certification reproduces 382's exact-power result within its measured widths, AND does a planted hypothesis-VIOLATING input (samples secretly non-monotone / modulus understated) fire the refusal or a containment failure, controls able to fail, neither widened?"

YES, with the conjuncts stated separately because they are not equally strong:

• First conjunct: YES, via PATH A only. Monotonicity reproduces leg 382's four banked exact-power widths with difference exactly 0.0, truth inside every interval. PATH B does not and cannot (2.10e-3 at N = 1000, slope −1.0068, N ≈ 2.8e14 needed): a measured limitation, predicted in advance, and this leg does not claim a YES for both paths.
• Second conjunct: YES. X1, X2, X4 fire refusals; X3 and X5 are accepted (as they must be) and fire containment failures of +5.142e-2 relative and +4.772e-2 in log f. Nothing was widened; P11 failed and is recorded as failed.

Which hypothesis each result is conditional on: the reproduction rows and S4 on declared monotonicity (never verified); S2, S3, X5 on a caller-certified modulus. No row here is unconditional, and every row says so in its own conditional_on field.

10. Carried forward, unchanged by the YES

• CLAY_OBLIGATIONS.md §6 items 1 and 2 stay OPEN. Item 1's admissible-cutoff half is entirely untouched; this leg supplies the input contract of the certified-exponent third.
• No profile exists. Every input is a planted analytic known. The first real consumer remains whatever future unit produces a profile, and this leg claims nothing about one.
• Disclosed hole, inherited and only partly discharged. export.arxiv.org returned HTTP 429 to this leg exactly as it did to leg 382; the arXiv half of that leg's prior-art hole is still open. Crossref was reachable and searched weakly (novelty pass §3). No mathematical novelty is claimed, so only a citation could change.
• Left to a successor: leg 382's rel_tolerance δ mode and this adapter have not been composed, the δ a caller owes and the ω a caller owes are two separate statements of the same thing (accuracy), and whether they should be one input is unexamined here. A per-cell (rather than global) δ does not exist in leg 382's routine and would be its owner's change, not this leg's.
• No L1 → L4 link moved. Ceiling TIER 2. Clay ~0.05%.