# RUN-024 — Round 21: the tail atlas rebuilds box by box, and every box closes with exact cores; the synthesis search values sit below the true minima

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-25
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_21_Lift_Tail_Synthesis_v0.1.md` (AMRAL-LUC-FC-R21, dated 2026-09-19) and its package, read from the archive:

- `TAIL_ATLAS.json`, `SYNTHESIS_SUMMARY.json`, `VALIDATION.json`, `State_Crystal`;
- the reference `B7_tail_boxes_depth26.json` (2,143 boxes);
- `round21_repro_entry.py`, read only after this tree's reconstruction, and never run.

**Tools:** [`run024_round21_lift_tail_synthesis.py`](../code/run024_round21_lift_tail_synthesis.py) — RUN-023's B7 lift with the pilot's sampled cores and with exact cores; this tree's exact hull; a phase-by-phase ridge scan; exact-area and bracketed-area placement searches for regular B7 and the smooth 7-fold witness
**Logs:** [`run024-round21-lift-tail-synthesis.json`](../data/gate-logs/run024-round21-lift-tail-synthesis.json)

**Verdict: PASS for the atlas as computed, the phase lock and the two lobes, the ridge's two channels, the finite-tail optimism theorem and the legal 7-fold family; PARTIAL for the atlas as a tail of B7 — with exact cores it is empty — and for every synthesis comparison, whose search values sit 3–6×10⁻⁶ below the true minima; UNMEASURED for the irregular and symmetry-breaking candidates, which the package does not record.**

- **The atlas rebuilds box by box.**
  - The depth-50 cell is the tracked branch's (RUN-022), τ = 3.82517×10⁻⁴.
  - B7's sampled tree gives 1,046, 1,397 and 2,143 open boxes at lift depths 22, 24 and
    26, with every node and CERT count.
  - All 2,143 depth-26 boxes of the package's file agree in path, box and bound to
    7×10⁻¹⁶.
  - The phase-lock and translation-radius quantiles agree to 5×10⁻¹⁶: medians 0.1886,
    0.1372, 0.1081; 95% quantiles 0.4785, 0.3330, 0.2446.
  - The two lobes are 1,000 and 1,143 boxes, with every circular centre, resultant and
    mean. One field is not what its name says: the atlas's `mean_phi` is the circular
    centre of 7φ divided by 7, not the mean of φ.
- **Every box of the atlas closes with exact cores.**
  - The same 2,143 depth-26 boxes, bounded with exact cores instead of the pilot's
    40-direction hull, all clear 0.835. The least clears it by 8.0×10⁻⁴.
  - B7's exact-core tree on this cell closes at lift depth 21, with 2,509 nodes, as
    RUN-023 found. Its last open boxes — 100, 39 and 35 at depths 18–20 — lie in both
    channels.
  - So the "structured adversarial atlas" is the map of where the pilot's sampling
    loses more than B7's margin. It is not where B7 fails.
- **The ridge has exactly two channels, but the package's ridge points are not their
  minima.**
  - Scanning the whole phase period in 64 steps, the least hull area over (x₇, y₇)
    has exactly two local minima: at 7φ ≈ 2.45 (0.8371407) and 7φ ≈ 5.89 (0.8371653).
    These are the two lobes' channels, as §6 says.
  - The package's two ridge points give their recorded areas to 10⁻¹⁵. But they lie
    8.0×10⁻⁶ and 1.1×10⁻⁵ above their channels' minima. Lobe A's point is not even a
    minimum at its own phase: a polish at that phase lowers it by 8.9×10⁻⁶.
- **The theorems are right.**
  - Finite-tail optimism (§10) is inclusion: a minimum over a subset of placements is
    at least the minimum over all, and can only fall as the subset grows. The
    cutting-plane record obeys it at every step.
  - The 7-fold family (§11) has width exactly 1. Its radius of curvature is at least
    λ/2 − 48|a|, so |a| ≤ λ/96 is sufficient. On 64 random draws it holds inside the
    bound and fails just outside it. The smooth trials' recorded minimum curvatures are
    1/2 − 48a.
- **The synthesis search values are below the true minima.**
  - Regular B7 at q† — the package's "baseline 0.83712806". At the package's own
    placement the exact hull area is 0.8371347. An exact-area search from 28 starts,
    the package's point among them, finds nothing below 0.8371340.
  - So 0.83712806 is not an area B7 attains. It sits 5.9×10⁻⁶ below B7's true
    minimum.
  - The smooth 7-fold witness h = ½ + a cos 7θ, searched again with a hull area
    bracketed at 80,000 directions: every one of the package's five minima lies 3.2 to
    5.9×10⁻⁶ below the lower end of the bracket.
  - The package's search area underestimates by a few 10⁻⁶: a sampled hull, it seems,
    though no code is shipped to say which.
  - The conclusions with a wide margin survive. B7's least exact value is above every
    smooth trial's upper bracket by at least 1.69×10⁻⁴. The irregular and k = 3
    candidates' recorded scores are 1.5×10⁻⁴ and 8.4×10⁻⁴ below B7's recorded one.
  - The one that does not survive is §14's "the final oracle returns to the regular B7
    baseline". 0.8371286 against 0.8371281 is a difference of 5.3×10⁻⁷, inside the
    area's own error.
- **What the package does not record.**
  - The shapes behind §14–17 — the cutting-plane iterates, the 30 irregular Reuleaux
    heptagons and the k = 3 candidates — appear only as scores. Nothing records a
    shape to test.
  - The only code shipped, `round21_repro_entry.py`, prints the package's own JSON
    files from an absolute path outside the package, and computes nothing.

Drill: 17 / 17 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
234 / 234 must-stay-green assertions held. Each of the twelve checks was blinded in
turn, and each was reported blind in exactly as many cases as plants name it.

---

## What Round 21 claims

1. **The tail** (§1–2). On the tracked depth-50 cell (τ ≈ 3.82517×10⁻⁴), B7's lift leaves
   1,046, 1,397 and 2,143 boxes open at depths 22, 24 and 26. The count may rise, but
   the volume falls: 2.494×10⁻⁴, 8.327×10⁻⁵, 3.193×10⁻⁵.
2. **Phase lock and lobes** (§3–7).
   - δ_lock = wrap(arg t₇ − 7φ₇). Its median |δ| falls 0.1886 → 0.1372 → 0.1081, and
     its 95% quantile 0.4785 → 0.3330 → 0.2446.
   - At depth 26 the boxes form two lobes: 1,000 near 7φ ≈ 2.5255, arg t ≈ 2.5536,
     |t| ≈ 0.01213; and 1,143 near 5.8819, 5.8989, 0.01248. Concentrations are
     0.98–0.99.
   - The area-minimising ridge at fixed φ lies in the same channels. The tail is a
     thickened ridge.
   - B23's counterexample (|t₂₃| ≈ 3.19×10⁻⁴) is a different, near-disc mechanism.
3. **Synthesis** (§8–10, §18–20).
   - Witness design is the max-min V* = max_K min_p F(K, p).
   - A cutting-plane loop alternates a shape step on known placements with a fresh
     placement oracle.
   - Finite-tail training is optimistic and can only get less so (v_{m+1} ≤ v_m).
   - A certificate-cost term belongs in the objective.
4. **Tests** (§11–17).
   - A legal 7-fold family (1 − λ)h_{B₇} + λ/2 + a cos(7θ + ψ), legal for |a| ≤ λ/96.
   - Regular B7 at q†: 0.83712806.
   - Smooth 7-fold at a = 0.0103: 0.8369528.
   - Tail-targeted training reaches 0.8377708, but a fresh oracle finds 0.8369560. Three
     more rounds return to 0.8371286.
   - The best irregular Reuleaux heptagon reaches 0.83697744, and a k = 3 pilot
     0.83628622.
   - No superior witness: B7 stays the primary deep-tail witness.
5. **Ledger and Round 22** (§21–23).

## A — sources

**A0.** 34 anchors. Round 21 has no reference list.

## W — rebuilt

| check | Round 21 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §1–2 | the cell; B7's sampled tree at depths 22–26 | PASS |
| W2 | §3–5, §7 | 2,143 boxes; quantiles; two lobes | PASS; `mean_phi` is misnamed |
| W3 | §6 | the ridge over the whole period; the package's ridge points | PASS for the channels; the points are not minima |
| W4 | — | the same boxes and tree with exact cores | every box closes; the tree closes at depth 21 |
| W5 | §8–11 | the theorem; the family; the record | PASS |
| W6 | §12–13 | B7 and the smooth family at q†, searched again | the package's values are below the true minima |
| W7 | §14–17 | what is recorded of the synthesised shapes | scores only |
| W8 | — | what is shipped | a script that prints the package's JSON |

**W1 — the volumes.**

- The atlas's base cell is the tracked branch's depth-50 cell, box and τ to 10⁻¹⁷.
- B7's tree is RUN-023's `lift` with the pilot's cores, run once to depth 26 and read
  at each depth:

  | lift depth | nodes | CERT | open | fraction |
  | ---: | ---: | ---: | ---: | ---: |
  | 22 | 11,133 | 4,521 | 1,046 | 2.494×10⁻⁴ |
  | 24 | 15,937 | 6,572 | 1,397 | 8.327×10⁻⁵ |
  | 26 | 21,883 | 8,799 | 2,143 | 3.193×10⁻⁵ |

  The box half-widths at each depth are the atlas's.

**W2 — the atlas.**

- Path by path, the 2,143 depth-26 boxes are the package's file: every centre,
  half-width and sampled bound to 7×10⁻¹⁶.
- At depths 22, 24 and 26, all ten quantiles of the translation radius and of
  |δ_lock| agree to 5×10⁻¹⁶.
- Splitting the depth-26 boxes by the nearer of the two centres of 7φ gives the
  atlas's lobes exactly:

  | lobe | boxes | 7φ centre | resultant | arg t centre | resultant | mean \|t\| |
  | --- | ---: | ---: | ---: | ---: | ---: | ---: |
  | A | 1,000 | 2.525491 | 0.9899 | 2.553558 | 0.9830 | 0.0121337 |
  | B | 1,143 | 5.881925 | 0.9877 | 5.898881 | 0.9778 | 0.0124807 |

  Every field matches to 10⁻¹² but `mean_phi`. The atlas's value is the circular centre
  of 7φ divided by 7 (0.3607844 and 0.8402750). The mean of φ is 0.3607748 and
  0.8402794.
- §7's |t₂₃| ≈ 3.19×10⁻⁴ is the B23 placement of Round 20 (RUN-023).

**W3 — the ridge.**

- m(φ) = min over (x₇, y₇) of the exact hull area with the base at the depth-50 cell's
  centre. It is computed at 64 phases across [0, 2π/7), each from three starts.
- It has exactly two local minima:

  | φ | 7φ | m(φ) |
  | ---: | ---: | ---: |
  | 0.35062 | 2.4544 | 0.8371407193 |
  | 0.84150 | 5.8905 | 0.8371653077 |

  Both lie in sec. 5's two channels.
- The package's two ridge points give their recorded areas to 10⁻¹⁵, but they are not
  the channels' minima:
  - φ = 0.35028, area 0.8371486776: 8.0×10⁻⁶ above the channel's minimum, and 8.9×10⁻⁶
    above a polish at its own phase;
  - φ = 0.85381, area 0.8371759985: 1.07×10⁻⁵ above its channel's minimum.

**W4 — the same tail with exact cores.**

- The 2,143 depth-26 boxes, each bounded by the exact hull of the disc and the three
  erosion cores (less the pilot's 10⁻⁹ pad): all ≥ 0.835. The least is 0.8358005576,
  and on every box the exact bound exceeds the sampled one by at least 1.20×10⁻³.
- That is the pilot's loss. RUN-022 measured 1.23×10⁻³ for the base alone at the depth-20
  centres. It is about half of B7's margin at these placements.
- B7's exact-core tree on the cell closes at lift depth 21, with 2,509 nodes. It has
  100, 39 and 35 open boxes at depths 18, 19 and 20. At depth 20, 22 lie in lobe A's
  channel and 13 in lobe B's.

**W5 — the theorems.**

- §10: for every K, min over P_m ≥ min over P ⊇ P_m, so v_m ≥ V*; and P_{m+1} ⊇ P_m gives
  v_{m+1} ≤ v_m. The record agrees:
  - training scores 0.8377708 → 0.8373826 → 0.8371980 → 0.8371877, never rising;
  - each fresh oracle value below its training score.
- §11:
  - the width is (1 − λ)·1 + λ + a[cos(7θ + ψ) + cos(7θ + 7π + ψ)] = 1;
  - the cosine term's radius of curvature is −48a cos(7θ + ψ), so ρ ≥ λ/2 − 48|a|.
  Numerically, 64 random (λ, ψ) with |a| ≤ λ/96 keep the width to 2×10⁻¹⁶ and ρ ≥ 0.
  With a = 1.02·λ/96, all 64 go negative somewhere.
- The five smooth trials' recorded minimum curvatures are 1/2 − 48a.

**W6 — the baselines.**

- Regular B7 at q†, the canonical image of Mishra's point, used by the package:

  | | value |
  | --- | ---: |
  | the package's "searched_min" | 0.8371280593 |
  | the exact hull area at the package's placement | 0.8371347054 |
  | a polish from that placement | 0.8371345412 |
  | the least of 28 exact-area searches | 0.8371339787 |

  The package's value is 5.9×10⁻⁶ below anything B7 was found to attain. Its placement
  is printed to eight digits, but the exact area near it never falls below 0.8371345 (the
  polish), and no exact search from 28 starts falls below 0.8371340. So the package's
  search area is not the exact hull area.
- The smooth 7-fold witness h = ½ + a cos 7θ, placed by rotation and translation:
  - the hull's area is bracketed by the contact-point polygon and the support-line
    polygon at N directions;
  - it is searched from 14 starts at N = 8,000 and polished at 40,000;
  - the bracket is taken at 80,000.

  | a | package | bracket | package below it by |
  | ---: | ---: | --- | ---: |
  | 0.006 | 0.8362521834 | [0.8362553546, 0.8362614196] | 3.2×10⁻⁶ |
  | 0.008 | 0.8366138051 | [0.8366196655, 0.8366262014] | 5.9×10⁻⁶ |
  | 0.009 | 0.8367717016 | [0.8367774562, 0.8367837740] | 5.8×10⁻⁶ |
  | 0.01 | 0.8369133231 | [0.8369189114, 0.8369255589] | 5.6×10⁻⁶ |
  | 0.0103 | 0.8369528003 | [0.8369581551, 0.8369648826] | 5.4×10⁻⁶ |

  Every smooth trial's upper bracket stays below B7's least exact value, by at least
  1.69×10⁻⁴.

**W7 — what is recorded.**

- For the cutting-plane iterates (§14), the 30 irregular candidates and 6 re-searched
  (§16), and the k = 3 candidates (§17), the package keeps scores only: no parameters, no
  support function, no vertices.
- The final oracle value, 0.8371286, is 5.3×10⁻⁷ from the B7 value it is compared with.
  The package's area was shown in W6 to sit 3–6×10⁻⁶ below true values.

**W8 — the code.** `round21_repro_entry.py` is nine lines. It reads the tail atlas and
the synthesis summary from an absolute path outside the package and prints four of
their fields. Nothing that computes the atlas, the ridge, the searches or the synthesis
is shipped.

## P — the package

**P1.**

- §4's medians and 95% quantiles are the atlas's, rounded to four digits.
- §5's lobes are the atlas's.
- §12–17's nine values are the synthesis summary's.
- The crystal carries the same numbers.

**P2.**

- `VALIDATION.json` describes the document and the crystal correctly: SHA-256, bytes,
  `$$` 122 and 16, both flags.
- Its digests of the atlas, the summary and the raw tail are the archive's.
- Its six key checks follow from the data. "Final oracle near regular" is true to 10⁻⁵.

## Round 21, read

| Round 21 | statement | verdict | note |
| --- | --- | --- | --- |
| §0 | summary; boxed verdicts | PARTIAL | the atlas is the sampled tail (W4); "not found" rests partly on unrecorded shapes (W7) |
| §1–2 | the tail's volumes | PASS | W1 |
| §3–5 | phase lock; two lobes | PASS | W2 |
| §6 | the ridge in the same channels | PASS | W3; the package's points are not the minima |
| §7 | two failure mechanisms | PASS as geometry | the B7 "failure" is the pilot's sampling (W4) |
| §8–10 | max-min; cutting plane; optimism theorem | PASS | W5 |
| §11 | the legal 7-fold family | PASS | W5 |
| §12–13 | B7 baseline; smooth family | PARTIAL | the ordering holds; the values sit 3–6×10⁻⁶ low (W6) |
| §14 | tail-targeted overfitting | UNMEASURED | shapes not recorded (W7); "returns to baseline" is inside the area's error |
| §15, §19 | interpretation; B7 remains primary | PASS | stronger than stated: exact cores close its tail (W4) |
| §16–17 | irregular heptagons; k = 3 | UNMEASURED | not recorded (W7) |
| §18, §20–23 | proof cost; next space; ledger; Round 22 | n/a | the ledger is reviewed in the registry |

## The status registry

Round 21's 13 judgments are reviewed, and its 2 first-used phrases defined: `NOT FOUND`
and `FORMALIZED + IMPLEMENTED`. Census v5 finds 10 of the 13: the ledger is written as
label lines, and the global verdict is a boxed set-off line.

| verdict | judgments |
| --- | ---: |
| PASS | 5 |
| PARTIAL | 5 |
| UNMEASURED | 1 |
| n/a | 2 |

The PARTIALs:

- "B7 LIFT-TAIL ATLAS: CLOSED" — built exactly, but the sampled tail;
- "NEW SUPERIOR WITNESS: NOT FOUND" and "superior 7-fold witness: NOT FOUND" — true of
  what can be rebuilt, silent on what is not recorded;
- "Cutting-plane witness synthesis: FORMALIZED + IMPLEMENTED" — the implementation is not
  shipped;
- "regular B7 local minimax interpretation: NUMERICAL-EVIDENCE" — its deciding difference
  is inside the area's error.

"nearby irregular Reuleaux7 improvement: NOT FOUND" is UNMEASURED.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's lobe count | 1 | A0 |
| the atlas's CERT count; the text's depth-24 fraction | 2 | W1 |
| a box's bound; a lobe's angle; `mean_phi` said to be the mean | 3 | W2 |
| a ridge minimum; the package's ridge area | 2 | W3 |
| the exact tree's closing depth | 1 | W4 |
| the cutting-plane record; a recorded curvature | 2 | W5 |
| the B7 baseline; a smooth bracket | 2 | W6 (the first also P1) |
| shape parameters said to be recorded | 1 | W7 |
| the shipped script said to compute | 1 | W8 |
| the atlas's recorded digest | 1 | P2 |
| a review dropped | 1 | R1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the tail boxes in reverse order;
- the atlas with its keys sorted.

**17 / 17 plants, 4 / 4 controls, 234 / 234 must-stay-green assertions; no check
without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 2 | 2 |
| W2 | 3 | 3 |
| W3 | 2 | 2 |
| W4 | 1 | 1 |
| W5 | 2 | 2 |
| W6 | 2 | 2 |
| W7 | 1 | 1 |
| W8 | 1 | 1 |
| P1 | 1 | 1 |
| P2 | 1 | 1 |
| R1 | 1 | 1 |

## What this round does not establish

- **Any bound.** The exact-core closures are floating-point values of this tree's own
  kernel, not certificates.
- **The global minimum of any witness.** Searches from many starts find what they find.
  The minima quoted here are the least found, and the ridge scan's resolution is one
  64th of the period.
- **The synthesised shapes of §14–17.** Not recorded, so not tested.