# RUN-002 — Round 00: the baseline it measures itself against, and the interface it fixes

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-24
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_00_Methodology_v0.2.md` (AMRAL-LUC-FC-R00, dated 2026-09-18), read from the archive; byte-identical to the curated copy by [RUN-001](./RUN-001-CORPUS-IDENTITY.md)
**Tools:** [`run002_round00_baseline_interface.py`](../code/run002_round00_baseline_interface.py) (sympy 1.14 for the symbolic identities; exact rationals elsewhere)
**Logs:** [`run002-round00-baseline-interface.json`](../data/gate-logs/run002-round00-baseline-interface.json), arXiv snapshot [`arxiv-round00-citations.json`](../data/external/arxiv-round00-citations.json) (retrieved 2026-09-24)

**Verdict: PASS**, with one `OPEN` observation and one `PARTIAL`.

- The published interval Round 00 adopts, `0.8344 ≤ a_Leb ≤ 0.8440935944`, is reported
  correctly: every title, author, first-version date, quoted number and quoted
  statement matches its arXiv record, and the stated gap `0.0096935944` is exact.
- Its support-function transformation law is a true identity for rotations and
  translations; its containment interface is a true theorem, with both of its
  hypotheses load-bearing and both stated.
- `OPEN`: Round 00 writes the placement group as `E(2)` but parametrises and
  describes only rotations and translations. Immaterial for mirror-symmetric test
  bodies; material for chiral ones. Handed to Round 01 and every round whose
  witness bodies lack a mirror symmetry.
- `PARTIAL`: no arXiv paper since the baseline improves either bound — measured on
  arXiv only.

Drill: 20 / 20 plants caught by exactly the named checks, 8 / 8 controls
undisturbed, 146 / 146 must-stay-green assertions held; each check blinded in turn
is reported blind in exactly as many cases as plants name it.

---

## What Round 00 claims, and what it does not

Round 00 is a methodology freeze. Its own status block says no new bound is
claimed and no finite closure is proved, and nothing in it contradicts that. It
still fixes three things every later round inherits:

1. the problem statement — `a_Leb` as the infimum of `Area(U)` over convex `U`
   containing a rigid image of every compact planar set of diameter at most 1;
2. the external baseline it measures itself against;
3. the support-function interface of its §9: the transformation law
   `h_{gK}(θ) = h_K(θ − φ) + t·u_θ` for `g = (R_φ, t)`, the slack field
   `δ = h_U − h_{gK}`, and containment as `δ ≥ 0` on the whole circle.

This round checks (2) and (3). The values are extracted from Round 00's own bytes by
anchored patterns, never retyped — so a drill that edits the document is seen by the
checks.

## A — the baseline

| check | what | result |
| --- | --- | --- |
| A1 | every baseline value found in Round 00 exactly once; the interval, the gap line and the two bound statements agree with each other | PASS |
| A2 | each citation's title, authors and first-version date match arXiv; the quoted numbers and statements appear in the abstract | PASS — 4 / 4 |
| A3 | stated gap = upper − lower, exactly | PASS — `0.8440935944 − 0.8344 = 0.0096935944` |

What A2 compared, per citation:

| arXiv | Round 00 says | the record says |
| --- | --- | --- |
| 1810.10089 | Gibbs, 2018, upper bound `0.8440935944` | v1 2018-10-23; abstract states `0.8440935944` |
| 2608.30538 | Mishra, **2026-08-31**, rigorous lower bound `0.8344`; constant-width test sets; exhaustive subdivision certified | v1 **2026-08-31** (a v2 followed on 2026-09-01); abstract states `0.8344`, built on the disc and the constant-width curves on the equilateral triangle and regular pentagon, an exhaustive subdivision of a five-dimensional space, a 486,799,600-node certificate and an independent verifier |
| 2609.01284 | Zeng, 2026-09-01, `Λ_1 ≤ Λ_2 ≤ ⋯ ≤ a_Leb`, `a_Leb = lim Λ_M`, `0 ≤ a_Leb − Λ_M ≤ CM^{-2}` | v1 2026-09-01; abstract states the monotone hierarchy, the limit and the `C M^{-2}` rate verbatim, and a certified `0.834` |
| 2606.14625 | Wichiramala & Panraksa, 2026 | v1 2026-06-12; about Wetzel's 30-60-90 triangle and unit arcs — listed as background only, and Round 00 uses it for nothing |

The date check compares against the **first** version, not the current one: the
drill's M3 moves Mishra's date to the v2 date and requires A2 to trip.

**What A2 does not establish.** It shows Round 00 *reports* these results
correctly. It does not show the results are right. Anything later in the line that
rests on Mishra's bound or Zeng's hierarchy rests on them at `PASS by cited theorem`
at most, and this tree will say so each time.

### Currency of the baseline — PARTIAL

Round 00 fixes its baseline "as of 2026-09-18". If a better bound had appeared since,
the line's target would mean something else. An arXiv search on 2026-09-24 (three
queries: Lebesgue universal cover; Lebesgue covering problem; universal covering
constant width), newest first, finds nothing after 2026-09-01. It does find three
related papers Round 00 does not cite; none supersedes the baseline:

| arXiv | subject | bearing on the planar interval |
| --- | --- | --- |
| 2606.04458 (Xie, v4 2026-07-08) | certified planar lower bound `0.833` | superseded — it is the 2026 preprint Mishra's abstract says it improves on |
| 2607.27227 (Chen, v2 2026-08-06) | universal covers in ℝ³; polytopes in ℝᵈ | none |
| 2512.04023 (Arman, Bondarenko, Prymak, Radchenko) | the asymptotic, high-dimensional problem | none |

Not measured: journals, other preprint servers, anything arXiv's search did not
surface for those queries. Hence `PARTIAL`, not `PASS`.

## B — the interface

| check | what | result |
| --- | --- | --- |
| B1 | for `g = (R_φ, t)`: `(R_φx + t)·u_θ = x·u_{θ−φ} + t·u_θ` as a symbolic identity in `x, t, φ, θ` | PASS |
| B2 | evidence for the observation below: the improper-motion law, and a body for which it matters | verified |
| B3 | the containment interface needs `U` closed and convex; each hypothesis dropped admits an exact counterexample | PASS by cited theorem |

**B1.** The identity is pointwise; the support function follows exactly, since `x ↦
R_φx + t` is a bijection from `K` onto `gK`, so the supremum of the left side over
`gK` is the supremum of the right side over `K`: `h_{gK}(θ) = h_K(θ − φ) + t·u_θ`.
Nothing is sampled.

**B3.** For a closed convex `U` and compact `K`: `gK ⊆ U` gives `h_{gK} ≤ h_U`
trivially; conversely, a point of `gK` outside `U` is strictly separated from `U` by
some direction `u`, where `h_{gK}(u) > h_U(u)`. That is the strict separation theorem
for a point and a closed convex set — cited, not re-proved, hence `PASS by cited
theorem`. Both hypotheses were instantiated and both are needed:

- *drop closed:* `U` the open unit disc, `K` the closed unit disc. `h_K = h_U = 1` in
  every direction (for `U` as a supremum), yet `(1, 0) ∈ K \ U`.
- *drop convex:* `U` the annulus `1/4 ≤ |x|² ≤ 1`, `K = {0}`. `h_U = 1 ≥ 0 = h_K`, yet
  `0 ∉ U`.

Round 00 §9 states the interface "對凸閉集" — for closed convex sets. Both hypotheses
are there.

One step is left implicit: §1 takes the infimum over convex `U`, §9 works with
closed ones. The gap is harmless and closes in a line — the closure of a convex set
is convex, covers whatever the set covers, and has the same area, since the boundary
of a planar convex set is Lebesgue-null — but it is a step, and Round 01, which
Round 00 tasks with fixing the target domain, is where it belongs.

## The observation — `OPEN`: which group moves the test bodies

Round 00 writes the placement group as `E(2)` — conventionally the full isometry
group, reflections included. Everywhere it describes or parametrises that group, it
means rotations and translations only: §1 calls the motions 剛體運動 (rigid
motions), §8.3 says the freedom of `g ∈ E(2)` "包括 rotation 與 translation", and §9
writes `g = (R_φ, t) ∈ E(2)`. Its law is the law of the orientation-preserving half.

The other half obeys a different law. For `g = (R_φ ∘ σ, t)`, `σ` a reflection:

`h_{gK}(θ) = h_K(φ − θ) + t·u_θ`

— verified as a symbolic identity (B2), and Round 00's law verified *not* to hold
there.

**When it matters.** For a test body with a mirror symmetry, a reflected placement
is a rotated one, and the two conventions cannot differ. The disc and the regular
Reuleaux polygons — every test body Mishra and Zeng use — are mirror-symmetric. A
chiral body is not: the triangle `(0,0), (3,0), (1,2)` has three distinct sides, and
listing its vertices by increasing opposite side gives orientation `−1`; its mirror
image gives `+1`. Proper motions preserve both lengths and orientation, so no
rotation and translation carries the triangle onto its mirror image (exact
arithmetic, B2).

The consequence for a lower-bound certificate: if a test body is chiral and the
certificate enumerates only rotated and translated placements, it proves a bound
for covers that must hold a *properly* congruent copy — not, without a further
argument, for covers allowed to hold a reflected one. Round 00's own risk register
(Risk E) plans an asymmetric adversarial branch; that is where this becomes live.

**Handed on.** Round 01 is tasked by Round 00 with "the `E(2)` action". This tree
will check there which convention the line adopts, and in every later round whose
witness bodies lack a mirror symmetry, whether the placements enumerated match it.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| a bound edited everywhere in the document | 1 | A2 + A3 |
| the gap line's result edited | 1 | A3 |
| Mishra's date moved to the v2 date | 1 | A2 |
| a cited title changed by one character | 1 | A2 |
| a cited arXiv id changed | 1 | A1 + A2 |
| the hierarchy's rate changed to `CM^{-1}` | 1 | A1 |
| a conflicting second lower-bound statement inserted | 1 | A1 |
| the interval's upper value edited, the Gibbs line not | 1 | A1 |
| the arXiv snapshot's number, statement or author order altered | 3 | A2 |
| the transformation law with the sign of `φ` flipped, the translation negated, the translation rotated, the rotation dropped, or the improper law substituted | 5 | B1 |
| the chirality witness replaced by an isosceles triangle | 1 | B2 |
| the open disc taken closed; the annulus hole closed up; the witness point moved inside the open disc | 3 | B3 |

Controls, nothing may trip: the unmodified inputs; the document with CRLF line
endings; a title carrying an HTML entity for its apostrophe; abstract numbers written
with a trailing zero; the law rewritten with `θ − φ + 2π`; the law written as
`x·(R_{−φ}u_θ)`; a second scalene body; a second counterexample point inside the
annulus hole.

**20 / 20 plants, 8 / 8 controls, 146 / 146 must-stay-green assertions.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A1 | 4 | 4 |
| A2 | 7 | 7 |
| A3 | 2 | 2 |
| B1 | 5 | 5 |
| B2 | 1 | 1 |
| B3 | 3 | 3 |

## What this round does not establish

- That Mishra's `0.8344` or Zeng's hierarchy is correct. Cited, not verified.
- Anything about Round 00's governance — its claim-status vocabulary (§21.6) and its
  canonical-source rules (§25) are measurable properties of the whole corpus, and
  are the next round.
- Anything about Rounds 01–37.