TECHNICAL, Route-DTOL v1 (leg 386): the pre-registered tolerance mode, and the tolerance the §4 composition can afford

Companion: BLOG_P2_ROUTEDTOL_V1.md. Figure fig99 (writeup/figures/fig99_route_dtol_v1_delta_window.png, rebuilt from JSON alone by experiments/p2_route_dtol_v1_evidence.py). Runner experiments/p2_route_dtol_v1.py; data writeup/data/p2_route_dtol_v1.json; journal experiments/journal/leg_386.md (PART I = pre-registration, committed caf48e8 before any measurement; PART II = results); novelty log writeup/novelty/leg_386.md (36d01c7).

CEILING: TIER 2 in every branch. CLAY_OBLIGATIONS.md §6 items 1 and 2 remain OPEN. No L1 → L4 link moved. Clay stays ~0.05%.

0. The two gate clauses, answered in the gate's own wording

Clause 1. At the freshly-measured δ*: do the three non-power planted cases certify with contained truth and reported widths, does the exact power law still certify at δ = 0 with 382's widths reproduced, AND does a sub-δ* control remain EMPTY (the mode must not turn the instrument into a fit with extra steps)?

YES. 1a ✔ (width law), 1b ✔ (three cases certify at δ*, truth contained), 1c ✔ (δ = 0 reproduces 382's widths to a factor 1.0002), 1d ✔ (6/6 EMPTY below δ*).

Clause 2. Does the admissible-cutoff analysis tolerate δ > 0 AT ALL?

EMPTY at the α value in play. At α_centre = 1, the α the banked Type-I object carries, the composed window D = { δ ≥ 0 : verdict = INTERVAL and p_lo > α_threshold } is empty, and it is empty at every α_centre ≤ threshold. 0 of 30 measured rows are admissible with a realised α_centre ≤ 1.

They are not netted. The mode is a real instrument (clause 1) that the specific composition §4 needs cannot use on the object §4 has (clause 2).

1. Definitions

Leg 382's instrument certifies P_cert(f, R₀, R₁) = { p ≥ 0 : ∃ C > 0, C·r^{−p} = f(r) ∀ r ∈ [R₀, R₁] } by exact Fourier–Motzkin elimination of log C from an outward-rounded interval enclosure of the log–log tube. Verdicts INTERVAL / EMPTY (a proof of a negative) / INCAPACITY.

The mode this leg pre-registers and lands:

P_cert^δ(f, R₀, R₁) = { p ≥ 0 : ∃ C > 0,  f(r)/(1+δ) ≤ C·r^{−p} ≤ f(r)·(1+δ)  ∀ r ∈ [R₀,R₁] }

δ = 0 recovers P_cert exactly and remains the default. A nonempty interval is still not proof of a power law: it is the statement that no exponent outside it is consistent within δ.

2. The width law (clause 1a): derived, then measured

In the log–log plane with t = log r, y = log f(r), a tolerance δ inflates the tube by h = log(1+δ) on each side. Fourier–Motzkin over a window [t₀, t₁] gives half-width 2h per edge pair, hence

width(δ) = 4·log(1+δ) / log(R₁/R₀)

pre-registered in PART I §2.1 with pass band [0.95, 1.05] on the ratio measured/predicted.

PASS. The uniform +0.51 % is the cell-enclosure overhead, constant over nine decades of δ, which is what makes this a law rather than a fit.

Disagreement with leg 382's lead 0.8686, reported as a disagreement: measured 0.87284, i.e. +0.49 % above the lead. The substantive correction is not the digit. It is that 0.8686 is 4/log 100, a property of the WINDOW, and it doubles to 1.7457 on [10, 100]. Any downstream threshold quoted at "0.434 per unit δ" (half-width) is quoted at a window, and leg 381's composition inherits that dependence silently. Panel (a) of fig99.

3. The critical tolerance (clause 1b + the fresh δ*)

Write f(r) = C·r^{−p₀}·φ(r). In the log–log plane P_cert^δ becomes nonempty exactly when the tube of half-width h admits an affine function through log φ, i.e. when h ≥ E∞(log φ), the Chebyshev best-affine approximation error. Hence

δ* = exp(E∞(log φ)) − 1, with transition exponent p_c = p₀ − b* (b* the optimal affine slope).

PART I §2.2 predicted in advance that the measured δ* would land below this value, because the cell enclosure is slightly wider than the exact tube, with band ±20 %.

Two findings in opposite directions, both reported. (i) No disagreement with leg 382: the leads reproduce to 5e-9/5e-8/4e-6 relative; they were correct, they were simply not yet earned, and they are now measured under a pre-registration. (ii) A systematic −0.98 %, −2.94 %, −9.46 % against this leg's own closed form, in the predicted direction and inside the band; the deficit grows as δ* shrinks, consistent with a fixed discretisation width being a larger share of a smaller threshold. The continuum formula is an upper bound on the measured δ*, not an equality: PART I did not say so and should have.

Certification at the freshly-measured δ* (N = 1000, mode cells, bracket [0, 12]):

Each interval lies inside the profile's own exact local-slope range, the instrument certifies no exponent the profile never exhibits. As pre-registered, none contains the nominal p₀ = 2, and that is recorded rather than repaired: a profile that is not a power law has no true exponent, and the transition exponents match the closed forms to 0.0021 / 0.0147 / 0.0010.

4. δ = 0 unchanged (1c) and the sub-δ* control (1d)

Truth contained in all four; a factor 10 was pre-registered as sufficient and a factor 1.0002 was achieved. The mode changed nothing at δ = 0.

Sub-δ* control: at 0.9·δ* and 0.5·δ*, 6/6 EMPTY (C1 0.284127/0.157848, C2 3.017581/1.676434, C3 0.062765/0.034870). Below its threshold every mismatch is still proved impossible: the mode is not a knob, and leg 382's pinned "EMPTY at every ε down to 1e-12" regression test is untouched and still passing.

5. Clause 2, the mechanism, measured rather than composed

§4's composition is: α_lo = α_centre − 0.434·δ must exceed 1 (fixed-ball energy, critical L³) or 3/2 (global L²). Leg 381 read this as H-381: half-width = 0.434·δ, vanishing at δ = 0, giving δ < (α_centre − 1)/0.434. PART I §4 registered the alternative H-shift: half-width = 0.434·(δ − δ*), vanishing at δ*. The discriminant, at δ = 1.1·δ*:

H-shift, by the pre-registered rule (H-381 required ±10 %; all three are below a seventh of it). Stated precisely: the intercept is confirmed (the half-width vanishes at δ*, not at 0) while the near-transition slope coefficient is not 0.434 and is pinned only to within a factor 0.51–1.88.

What this refutes. H-381 implies a profile with δ* = 3.35 needs α_centre > 1 + 0.434·3.35 = 2.456 before any tolerance is admissible. Refuted. Because the half-width vanishes at δ*, the requirement is α_centre > 1 and nothing more. Measured δ_max against leg 381's law at threshold 1:

Leg 381's law understates the admissible δ_max by 1.33×–4.06× and never once claims a window that is not there. It is conservative, not wrong-signed: nothing built on it is unsafe; it is simply tighter than the measurement requires.

6. Clause 2: the answer, its window table, and its α-sensitivity

Measured directly by cutoff_admissible_delta_window, 3 shapes × p₀ ∈ {1, 1.5, 2, 2.5, 3} × thresholds {1, 3/2} = 30 rows.

The window is non-empty if and only if the realised certified centre exceeds the threshold. 30 rows, 0 exceptions. Six crossover probes at relative offset 1e-6 place the crossover exactly at the realised centre (admissible just below, EMPTY just above). δ buys ZERO headroom on the threshold: widening δ moves the LEFT edge of what can be certified and never moves the centre.

(Representative rows; all 30 in the JSON. δ_max = 10 rows are capped at the search ceiling and flagged capped. Panel (b) of fig99.)

α-sensitivity, stated as §4's consumers need it: the dependence is A STEP, NOT A SLOPE. The window is empty for every α_centre ≤ threshold and non-empty for every α_centre > threshold, uniformly across all three mismatch shapes, and once open it is wide (0.735 to ≥ 9.93 in δ). At α_centre = 1 it is EMPTY; at α_centre = 1.0001 it is non-empty. So the answer does not vary shape-to-shape at fixed realised α: it varies only across the threshold, and the question §4 must ask is not "how accurate is the profile" but "does the certified centre clear 1 (or 3/2) at all".

The nominal-vs-realised trap, disclosed in full. One row, the rational cutoff at nominal p₀ = 1, is admissible, and a nominal reading of the α ladder would have returned ADMISSIBLE for clause 2 on the strength of it. It is not admissible at α = 1: the measurement window [10, 1000] contains its cutoff at r = 300, so it realises α_centre = 2.0328. PART I §4 fixed the decision at the realised α before any measurement ("the clause is decided at the α actually realised and not at a nominal one"), precisely so this could not be counted. The JSON records both (nominal_p0_reading: 1 admissible shape; realised_alpha_reading_OPERATIVE: 0). The verdict flipped ADMISSIBLE → EMPTY when the pre-registration was honoured. That is a faithfulness fix in the stricter direction, not a post-hoc rescue; no band was widened, and no rescue is reported as gate-deciding.

7. TEXT ROUTED TO INTEGRATION: two files this leg may not edit

Both are integration-owned. Neither has been touched by this leg. The exact proposed text follows.

7.1 CLAY_OBLIGATIONS.md §4: proposed replacement for the paragraph beginning "§4 IS NOT DISCHARGED BY A δ = 0 CERTIFICATION (user ruling, 2026-08-11)."

§4 IS NOT DISCHARGED BY A δ = 0 CERTIFICATION (user ruling, 2026-08-11); THE δ QUESTION IS NOW ANSWERED, AND THE ANSWER IS EMPTY AT THE α IN PLAY (leg 386, DTOL). The admissible cutoff radius is a function of the certified exponent, so a certification carrying a tolerance δ passes that tolerance into the cutoff bound. Leg 386 pre-registered the δ mode before running and measured the composition directly rather than composing two legs' laws. Three corrections to what was written here. (i) The width law is width = 4·log(1+δ)/log(R₁/R₀); leg 382's 0.8686 is 4/log 100, a property of the window, doubling to 1.7457 on [10,100]: any half-width quoted at 0.434 per unit δ is quoted at a window. (ii) Leg 381's δ < (α_centre − 1)/0.434 is conservative by 1.33×–4.06× on every measured row and never optimistic; its implied demand α_centre > 1 + 0.434·δ* (≈ 2.456 at δ* = 3.35) is REFUTED: the certified half-width vanishes at δ*, not at δ = 0, so the requirement is just α_centre > 1. (iii) The tolerance buys ZERO headroom on the threshold. Over 30 measured configurations (3 mismatch shapes × 5 exponents × thresholds {1, 3/2}) the admissible window D = { δ ≥ 0 : INTERVAL and p_lo > α_threshold } is non-empty if and only if the realised certified centre already exceeds the threshold: 0 exceptions, crossover located at the centre to 1e-6. The dependence on α_centre is a step, not a slope; once open the window is wide (0.735 to ≥ 9.93 in δ). The banked Type-I object carries α = 1, so its composed δ window is EMPTY, and would be at any α_centre ≤ 1. The load-bearing question for §4 is therefore not the certification tolerance but whether a certified α_centre exceeding 1 (or 3/2) can be produced at all; no profile of route 4's object exists in this repository, and leg 386 contributes nothing to that question. §4's δ sub-question is CLOSED (answer: EMPTY at α = 1). §4 itself remains OPEN, on the admissible-cutoff half and on the missing profile. Method status: NO KNOWN METHOD IN THIS REPOSITORY. Named in §6.

(Integration note: the standing "Until DTOL lands, §4 stays OPEN in every route-4 gate" clause is satisfied on its own terms, DTOL has landed with a pre-registered δ mode, but §4 does not thereby close, and PROG-R4 should keep it open on the two grounds named in the last sentence, not on leg 386's account. §6 items 1 and 2 stay OPEN in every branch.)

7.2 capabilities.py (APPLIED IN THIS LEG'S COMMIT, not routed

The coordinator's mid-leg amendment directed that a validated-text update belonging to this leg's own module ship in the same commit as the module change) the standing rule adopted after a mid-rebase HEAD published a solver module without its capability row and reddened test_every_solver_module_is_indexed for every agent. The solver/dssp_decay_enclosure.py entry's holds and validated texts were therefore edited here (test count 12/12 → 18/18, the fresh width-law and δ* numbers, the zero-headroom limit, the refuted α demand, and the hypothesis-field contract). Its CEILING sentence stands unchanged and still applies.

7bis. THE HYPOTHESIS FIELD, the standing rule, implemented here

A mid-leg amendment adopted a standing rule on every consumer of the enclosure chain: every output row carries the hypothesis in force; a certificate-without-hypothesis is no certificate. It is earned rather than hygienic. Leg 385's control X3 planted a secret monotonicity violation and was accepted (as it must be, the hypothesis being caller-declared and unverifiable from samples) and the certificate it produced had width 7.438494264988549e-15, bit-indistinguishable from the true certificate of the genuine planted known K1. Same width, false about its profile. Only the recorded hypothesis separates them.

What this leg shipped. hypothesis / hypothesis_detail / conditional_on are now present on every return path of every function in solver/dssp_decay_enclosure.py, bound in the same breath as δ and before any early return, so no path can emit a row whose tolerance or whose hypothesis is implicit. Specifically:

The dependency deliberately points one way (adapter → enclosure, never back), so the three shared strings are duplicated rather than imported and test_hypothesis_strings_match_the_adapter fails loudly if leg 385's module renames one.

No measurement changed. The runner was re-run in full after the amendment; every number in this document, in fig99 and in the JSON is bit-for-bit what it was before (δ* to all 17 digits, ratios 1.005051–1.005074, both coefficients, both gate answers). The field is an addition to what each row says, not to what the instrument computes.

A related result cited rather than rediscovered (leg 385, banked): prediction P11 was REFUTED and recorded as refuted, a shifted grid does not detect a node-aligned wiggle, because a uniform shift multiplies every sample by one constant; detection is a commensurability effect (N = 1100, 1500 refuse; 500, 997, 1001, 1010, 2000 do not), so a violating profile can hide from any fixed grid. Also banked: a globally-stated absolute modulus is unusable across three decades (341/1000 cells non-positive from r = 207.97) and must be stated in log–log. Neither bears on how δ* was measured here (this leg's inputs are interval-evaluated closed forms, never samples, which is exactly the case that records EXACT-INTERVAL-EVALUATION) but both bear on any caller who arrives with samples, and they are why the UNDECLARED row says what it says.

8. Anti-tautology checks (9/9) and reproduction

1. No planted known returns the full [0, 12] bracket at any δ ✔
2. No exact power law is ever certified EMPTY (4 exponents × 4 tolerances) ✔
3. Every mismatch is EMPTY at δ = 0 ✔
4. δ recorded on every row ✔: 4b, on the INCAPACITY paths specifically ✔; this closed a real gap in leg 382's module, whose zero-crossing early return dropped rel_tolerance
5. Widths non-decreasing in δ; EMPTY downward-closed ✔
6. Hypothesis recorded in every row ✔, 6b: the hypothesis separates what the width cannot ✔ (a cells row and a nodes row of the same profile carry different hypotheses, and an undeclared caller comes back UNDECLARED rather than inheriting one it never declared); 6c: the composed window carries the hypothesis it was composed from ✔

.venv/bin/python experiments/p2_route_dtol_v1.py            # ~61 s -> writeup/data/p2_route_dtol_v1.json
.venv/bin/python experiments/p2_route_dtol_v1_evidence.py   # JSON only -> fig99
.venv/bin/python test_dssp_decay_enclosure.py               # 18/18

9. Deviations, disclosed

1. p₀ = 1.25 dropped from the pre-registered clause-2 ladder: the planted generators go through ipow_half_integer, which accepts only half-integers by leg 382's deliberate design. A limit of the planted profile, not of the enclosure (which searches the continuum). Nothing in either clause turns on it: the step is located to 1e-6 by the crossover probes.
2. Two files beyond the literal territory list, both under this route's own name: experiments/p2_route_dtol_v1_evidence.py (required convention for a registered figure) and this BLOG/TECHNICAL pair.
3. CLAY_OBLIGATIONS.md NOT edited (forbidden): exact proposed text routed in §7.1 above. capabilities.py WAS edited, in the same commit as the module change, under the coordinator's mid-leg amendment; this is a departure from the dispatch brief's literal territory list and is disclosed as one (§7.2). 3b. The hypothesis field was added after the pre-registration was written, under the same amendment. It is an addition to the interface, not to either gate clause, and the full re-run confirmed it changed no measured value. Neither clause was re-decided.
4. External novelty endpoints refused (arXiv and Semantic Scholar, both HTTP 429, banked verbatim in writeup/novelty/leg_386.md §3). No external prior-art result was obtained; no methodological novelty is claimed, so nothing rests on it, but the hole is real and open.

10. Interface for leg 389 (CT2C), which consumes this mode

Argument rel_tolerance (name deliberately unchanged from leg 382: it is capability-indexed and has a consumer), default 0.0, float, ValueError on < 0, validated and bound before any early return. Arguments hypothesis and hypothesis_detail on certified_decay_from_cell_enclosures, both defaulting to None and both recorded (as UNDECLARED) rather than dropped. Recorded on every row: rel_tolerance, tolerance_mode ∈ {"exact", "relative"}, predicted_width_exact_power_law, hypothesis, hypothesis_detail, conditional_on. Read hypothesis before width: §7bis is the reason. Helpers exported: predicted_width(delta, r0, r1) (float reference, never a certified bound) and cutoff_admissible_delta_window(...) -> {admissible, delta_min, delta_max, width, alpha_threshold, alpha_centre, p_lo_at_delta_min, delta_star_lo/hi, capped, reason, config}.

Still owed, and not by this leg: the samples→cells conversion (a caller with pointwise samples needs a certified modulus of continuity or a stated monotonicity hypothesis); a sharp near-transition half-width law (the H-shift intercept is confirmed at δ*, its slope coefficient is pinned only to a factor 0.51–1.88); and an α_centre for the actual object.