# RUN-049 — UESFCM Cycles 15–17: the boundary-locking lemmas hold, both upper references and all three certificates reproduce, and all 115,511 area leaves hold in exact arithmetic; the references are loose to first order, Cycle 15's counterexample falls for only 1.6×10⁻⁷ along its ray, and "all other regions eliminated by finite certificates" includes 44 % of root 6217 that only a search closed **Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab **Date:** 2026-09-27 **Subject:** UESFCM Cycles 15–17 (all dated 2026-09-20), each package read from the archive: - Cycle 15 — corrected boundary locking. The package holds the document, the A_ref upper certificate and its verifier, the boundary-locking diagnostic, the state crystal and VALIDATION. - Cycle 16 — tight reference and local secant. The package holds the document, the tight A_ref certificate and its verifier, the tight-sublevel diagnostic, the state crystal and VALIDATION, and copies of Cycle 12's 76 remaining subroots and Cycle 14's closure summary. - Cycle 17 — tight-reference pruning and hierarchical competitive-core closure. The package holds the document, the five-cell certificate and its verifier, the prefix-tree certificates of cells 131 and 129 and their verifier, the 61 remaining cells, the competitive-core summary, the 68-cell search diagnostic, the state crystal and VALIDATION, and a copy of Cycle 10's verifier. They are evidence for UESFCM (RUN-043), not canonical documents. **Tools:** [`run049_uesfcm_cycles_15_17.py`](../code/run049_uesfcm_cycles_15_17.py) - RUN-047's replay: trees decoded, every leaf's witness bound rebuilt, every area leaf certified in exact arithmetic with this tree's exact eroded cores; - this tree's exact hull (RUN-009, with RUN-031's segments); - RUN-044's cancellation-free outer polygon, and the same polygon at 50 digits (mpmath); - golden-section search along rays. **Logs:** [`run049-uesfcm-cycles-15-17.json`](../data/gate-logs/run049-uesfcm-cycles-15-17.json) **Verdict: PASS. The lemmas hold. Both upper references reproduce, and all three certificates decode and pass leaf for leaf. All 115,511 area leaves also hold in exact arithmetic, each against its own threshold, the least by 2.66×10⁻⁵. Three things are noted:** - **The references are loose to first order.** Uniform directions miss the hull's 17 bridging-edge normals. With them added, 20,000 directions come within 3.1×10⁻⁹ of the exact hull and 500,000 within 4.9×10⁻¹². The tight reference 0.8349865 has 3.9×10⁻⁷ of room, not the stated 6.5×10⁻⁸. - **Cycle 16 §10 says "all other regions have already been eliminated by finite certificates".** But 44.2 % of root 6217's box lies outside the 200 subroots and was closed only in Cycle 11's search, whose trees are not shipped. Add the ρ₅ = 0 arc side that RUN-048 found uncovered. Cycle 11 calls this state packaging debt when it names 8001, 8015 and 8016. - **Cycle 15's crystal still carries "Gate 6217-A: CLOSED"**, which RUN-048 found not established. - **The lemmas (W1).** - Radial convexity (Cycle 15 §2) is right: only B3 moves, in one fixed direction, so Fáry–Rédei applies. - The long and local secant lemmas (Cycle 15 §6, Cycle 16 §6) are right: for a convex f the right derivative at s is at least any secant slope from the left. - The two-colour criterion (Cycle 15 §8) is right. Its proof uses one premise silently: every F3 point of the box has its ray's circle point inside the box. For root 6217 that holds, since the box is convex and the anchor lies on its face. - Tight-reference elimination and prefix-tree closure (Cycle 17 §1, §8) are right. - Numerically, the second differences of f along 200 random rays in the box are all positive (least 2.9). - **The counterexample to global monotonicity is genuine, but marginal.** At its stored configuration f(s) is 0.8360391144 (2×10⁻⁹ from the stored value). - The one-sided slopes at step 10⁻⁷ are both −0.04569, as stated. - But the ray turns 1.6×10⁻⁷ out: at step 10⁻⁵ the forward slope is +0.093. - So f falls for 1.6×10⁻⁷ and by 7.2×10⁻⁹, then rises. The ray's least value is 0.8360391072 (stored 0.8360391068), above A_ref. - **The upper references (W2).** - Cycle 15's 20,000-direction outer polygon comes to 8×10⁻¹⁶ of the stated area in double precision, and at 50 digits agrees with the stated 70-digit value to 43 digits. - Cycle 16's padded 500,000-direction polygon comes to 2×10⁻¹⁵; its final safety gives the stated upper. - The exact hull at the candidate is 0.8349861105592. So the true room is 1.389×10⁻⁵ under 0.835 and 3.894×10⁻⁷ under 0.8349865. - The polygons' excess over it is 7.8×10⁻⁶ at 20,000 directions and 2.7×10⁻⁷ at 500,000, plus 3.3×10⁻⁸ of padding. The hand-over normals remove almost all of it (above). - Cycle 15 stores the candidate's x5 rounded to 71 digits; Cycle 16 stores Cycle 12's string. - **The secants (W3).** - At the contact, f(s) − f(s − 10⁻⁴) = 3.27×10⁻⁶ (Cycle 16 §13's number) and f(s) − f(0) = 3.8×10⁻⁵. The outward slope is 0.213. - On Cycle 12's profile, 8 samples lie at or below 0.835. Their least f(s) − f(0) is 9.88424×10⁻⁶, against Cycle 15's 9.88746×10⁻⁶ (3×10⁻⁹ apart). - 2 of the 27 samples lie below the tight reference, at ρ = 0 and 0.00101, as Cycle 16 §9 says. - The searches behind Cycle 15 §12 (+0.0250) and Cycle 16 §5 and §8 store no points, so they are not reproduced. - **The five-cell certificate (W4).** Cells 28, 36, 68, 144 and 155. - 61,487 nodes, 30,467 area leaves and 279 NOT-F3 leaves; every stored count is this tree's. - Against 0.8349865 the witness bound passes on every leaf. The least slack is 3.5828319×10⁻⁹ in cell 155, and the per-tree minima agree to 3×10⁻¹⁶. - In exact arithmetic every area leaf clears 0.8349865, the least by 2.656×10⁻⁵. Every NOT-F3 box lies inside the ball. - **The prefix trees (W5).** The path bit 0 is the lower child. - Cell 131: 40,859 leaf paths, all distinct and none a prefix of another. The trie has 40,858 internal nodes and is complete. 38,893 leaves are area leaves and 1,966 NOT-F3. - Cell 129: 48,590 paths, 48,589 internal nodes, complete; 46,151 area and 2,439 NOT-F3. - Every NOT-F3 box lies inside the ball, and the witness bound passes on every area leaf. The least verified slacks are 1.2396455×10⁻⁹ and 6.1271319×10⁻⁹, 10⁻¹⁵ from the stated ones. - In exact arithmetic the least margins over A3+ are 3.022×10⁻⁵ and 3.183×10⁻⁵. - **The ledger (W6).** - The chain 76 − 8 = 68 − 7 = 61 holds row for row, and the 68 are exactly the diagnostic's cells. - The diagnostic's stored minimizers re-evaluate to 10⁻¹⁵, and all lie in their cells. Two lie just inside the F3 circle, cell 14 by 6.5×10⁻⁸ and cell 28 by 2.3×10⁻¹²: search tolerances. - Its ranking reproduces: 131 alone below 0.8349865; 131, 129, 125, 152 and 150 at or below 0.835. - **Root 6217's box:** the 200 subroots fill 55.85 % of it. The other 44.15 % was closed only in Cycle 11's search. - **The crystals (W7)** agree with the numbers rebuilt here. Cycle 15's still records Gate 6217-A as CLOSED. - **The verifiers (W8).** Read after the rebuild, never run. - Cycle 15's builds the outer polygon at 70 digits. It chooses each support piece by a float angle, which is harmless: at a switch both pieces give the same support, to second order. - Cycle 16's pads double-precision supports and sums the area in long double. - Cycle 17's two load Cycle 10's verifier from their own package (`dependencies/`, shipped). The prefix-tree verifier checks prefix conflicts, complete coverage, every NOT-F3 predicate and every area leaf. - **The package (P1).** Every VALIDATION digest matches the archive, and the replay counts are this tree's. - Cycle 17's VALIDATION records every file but the 68-cell diagnostic. - The dependency copies are byte-identical to their originals. Drill: 20 / 20 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 208 / 208 must-stay-green assertions held. Each of the ten checks was blinded in turn, and each was reported blind in exactly as many cases as plants name it. --- ## What the cycles claim 1. **Cycle 15.** - Global radial monotonicity out of the F3 circle is false; a numerical counterexample has outward slope −0.0457. - Radial convexity holds for fixed boundary parameters. An explicit feasible candidate has area below A_ref = 0.835, by a 20,000-direction outer polygon at 70 digits. - Boundary locking follows if the boundary parameters are covered by SECANT-LOCKED (f(s) ≥ f(0)) and AREF-PRUNED (the ray above A_ref) cells. 2. **Cycle 16.** - A padded 500,000-direction polygon tightens the reference to 0.8349865. - The long secant can fail where locking holds, so a local secant f(s) ≥ f(r₋) is added. - The responsibility is restricted to the 68 open cells, "all other regions" being eliminated already. 3. **Cycle 17.** - Five cells close against the tight reference with a 61,487-node certificate. - Cells 131 and 129 close against A3+ with hierarchical prefix-tree certificates, since their obstruction was scheduling, not geometry: 68 → 61. - Next: 125, 152, 150. ## W — rebuilt | check | cycle | rebuilt | result | | --- | --- | --- | --- | | W1 | 15–17 | the lemmas, read; convexity along rays; the counterexample | sound; the counterexample holds for 1.6×10⁻⁷ | | W2 | 15–16 | both outer polygons, at 50 digits and in double; the exact hull | to 43 digits and 2×10⁻¹⁵; loose to first order | | W3 | 15–16 | the secants at the contact and on the profile | agree to 3×10⁻⁹ | | W4 | 17 | the five-cell certificate, witness and exact | every leaf; least 2.66×10⁻⁵ exact | | W5 | 17 | the prefix trees: structure, witness, exact | complete, prefix-free; least 3.02×10⁻⁵ exact | | W6 | 15–17 | 76 → 68 → 61; the 68-cell diagnostic; the box's share | exact; 44 % of the box closed only in search | | W7 | 15–17 | the crystals | agree; Gate A still CLOSED in Cycle 15's | | W8 | 15–17 | the verifiers, read | sound as described | ## P — the package **P1.** Every VALIDATION digest matches the bytes hashed here and the archive's manifest. The replay counts are this tree's, and the dependency copies are byte-identical to Cycles 10, 12 and 14's files. Cycle 17 leaves its 68-cell diagnostic out of its VALIDATION. ## The cycles, read | cycle | statement | verdict | note | | --- | --- | --- | --- | | 15 | global radial monotonicity is false: outward slope −0.0457 | PASS | the ray turns 1.6×10⁻⁷ out (W1) | | 15 | radial convexity for fixed boundary parameters | PASS | positive second differences on 200 rays (W1) | | 15 | A_outer = 0.83499392544273890507… < 0.835 | PASS | to 43 digits (W2); loose to first order | | 15 | secant locking; the two-colour criterion | PASS | read; the circle point stays in 6217's box | | 15 | low-sublevel slope +0.0250; profile secant 9.887×10⁻⁶ | PASS as diagnostic | the second to 3×10⁻⁹; the first stores no point (W3) | | 16 | tight A_ref = 0.8349865 by a padded 500,000-direction polygon | PASS | to 2×10⁻¹⁵ (W2); true room 3.9×10⁻⁷ | | 16 | the long secant fails at a looser level; the local-secant lemma | PASS as diagnostic; PASS | no point stored; the lemma read | | 16 | f(s) − f(s − 10⁻⁴) ≈ 3.27×10⁻⁶ at the contact | PASS | W3 | | 16 | 2 / 27 profile samples below the tight reference | PASS | W3 | | 16 | all other regions are eliminated by finite certificates | FAIL | 44 % of the box only in search (W6); the ρ₅ = 0 arc side (RUN-048) | | 17 | tight-reference elimination; prefix-tree closure | PASS | read | | 17 | the five cells: 61,487 / 30,467 / 279, least 2.582831×10⁻⁹ | PASS | W4 | | 17 | cells 131 and 129 closed for A3+ | PASS | W5 | | 17 | 68 → 61; the competitive ranking | PASS | W6 | ## The status registry Not touched: the cycles are not canonical documents. ## The drill | kind | cases | named checks | | --- | ---: | --- | | Cycle 17's node count in the text | 1 | A0 | | the counterexample's stored slope; a ray's second difference | 2 | W1 | | a digit of Cycle 15's area; Cycle 16's padded area; the exact hull | 3 | W2 | | the profile secant; a profile sample's area | 2 | W3 | | a tree's stored leaf count; an exact margin below zero | 2 | W4 | | a prefix-tree leaf dropped | 1 | W5, P1 | | a NOT-F3 leaf typed as an area leaf; a witness failure | 2 | W5 | | a remaining cell dropped | 1 | W6, W7 | | a diagnostic minimum; the ranking | 2 | W6 | | a crystal's cell count | 1 | W7 | | the prefix-tree verifier's coverage test | 1 | W8 | | a VALIDATION digest; a dependency copy's bytes | 2 | P1 | Three controls: - the unmodified inputs; - the documents and verifiers with CRLF line endings; - the VALIDATION records with the keys sorted. **20 / 20 plants, 3 / 3 controls, 208 / 208 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 | 2 | 2 | | W5 | 3 | 3 | | W6 | 3 | 3 | | W7 | 2 | 2 | | W8 | 1 | 1 | | P1 | 3 | 3 | ## What this run does not establish - **Root 6217's closure.** 61 cells stay open, and the boundary-locking cover is not built in the cycles. Nor is the ρ₅ = 0 arc side covered (RUN-048), and 44 % of the box has only Cycle 11's search behind it. - **The searched diagnostics that store no point:** Cycle 15 §12, and Cycle 16 §5 and §8. - **Fáry–Rédei and Rogers–Shephard themselves**, cited as known; here they are only checked along rays. - **Root 3766 and F5**, untouched by these cycles.