# RUN-043 — UESFCM: seventeen Lebesgue cycles chain cleanly and never mark evidence as proof, but the target they anchor to is written down nowhere in the archive, the LinkBack types are used in one cycle, and the texts disagree on which states end the loop

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** UESFCM's package, read from the archive:

- `UESFCM_Unbounded_Expansion_Self_Referential_Finite_Closure_Methodology_v0.1.md` (EML-UESFCM-v0.1, dated
  2026-09-19);
- the Operational Protocol (EML-UESFCM-OP-v0.1, same date);
- `VALIDATION.json`.

Its one application is in the same folder, and is read here as evidence of how the method is used:

- seventeen Lebesgue cycles, 01–10 dated 2026-09-19 and 11–17 dated 2026-09-20;
- each cycle is a document and a state crystal.

Only their labels, chaining and ledger arithmetic are read here. Their mathematics is for the runs that follow.

**Tools:** [`run043_uesfcm.py`](../code/run043_uesfcm.py) — the documents parsed section by section; every leaf of the archive searched for the target and the first state; the crystals' arithmetic in exact decimals where the numbers are decimal
**Logs:** [`run043-uesfcm.json`](../data/gate-logs/run043-uesfcm.json)

**Verdict: PASS for fourteen of UESFCM's fifteen judgments, PARTIAL for "Problem Closure: DEFINED". The other twenty are labels.**

- **What UESFCM is.** A protocol text, and nothing in it is computed.
  - The protocol:
    1. fix a canonical target Q\*;
    2. expand without a preset bound;
    3. link every artifact back to Q\*;
    4. close each round into a finite state C_t;
    5. recentre on Q\* and what remains.
  - Its twelve DEFINED parts are defined, and its two NOT CLAIMED results are not claimed.
  - "Empirical benchmark validation: OPEN" is right.
- **Its vocabulary is consistent (W1).**
  - There are eleven LinkBack types, and the protocol lists the same eleven in the same order.
  - There are nine obligation states, fifteen artifact statuses, three terminal labels (§21) and five
    closed-for-now labels (§30). No label is listed twice.
  - The protocol's closure state C_t = (G, O, B, D, P) keeps five of §9's eight components.
- **Its stop rule is not consistent (W2).** Eight places say what ends the loop, and between them they give
  four different sets:
  - the abstract, §3 and the pseudocode of §50 give proof or counterexample closure only;
  - §21 adds TARGET-FAILURE ("三種主要終止狀態");
  - §61 and the protocol's runtime line add independence closure (§58) instead;
  - the protocol's §7 gives all of these and also SEARCH-STALLED.

  §49 treats SEARCH-STALLED as a trigger rather than a stop: "然後觸發" (then trigger) further representation,
  method, literature, independent-AI or DLMVC expansion. The protocol lists it under "Stop Conditions".

  The pseudocode is a `while True` loop with two returns and no break. It has no branch for a failed target,
  for a stall or for independence.
- **The seventeen cycles hold together as a chain (W3).**
  - Their codes run C01…C17, and each names its predecessor.
  - All 34 headers and crystals give one target ID.
  - Every cycle is `FINITE-CLOSED / PROBLEM-OPEN`, and every crystal has `problem_status: OPEN`.
  - PROBLEM-PROOF-CLOSED appears nowhere.
- **Their ledger holds (W3).** 39 identities recompute. They cover each number a crystal derives and each
  count one cycle hands the next. For example:
  - Cycle 04's certified lower = sample lower − orientation loss.
  - Cycle 10's verified slack = raw slack − the safe η.
  - Cycle 09's 649 near-ceiling roots − Cycle 10's 27 closed = Cycle 11's 622. And 622 = 617 verified
    + 3 in packaging debt + 2 persistent families.
  - Root 6217's 200 subroots − 124 closed = 76, then 76 − 8 = 68, then 68 − 5 − 2 = 61, across Cycles
    12, 14 and 17.
  - Cycle 12's contact x₃\* = Cycle 06's anchor x₃ − the safe margin 0.00882, exactly.
- **But the anchor is only a name (W3).**
  - Cycle 01 names its previous state `UESFCM-LUC-C00-v0.1` and its target `UESFCM-LUC-TARGET-v0.1`.
    Neither is in the archive, and all 1,459 leaves were searched:
    - the target ID occurs only as a reference, in the 34 cycle headers and crystals;
    - C00 occurs only in Cycle 01's header.
  - So the archive does not contain:
    - the Target Contract of §2, which I1 protects;
    - the obligations O₂–O₄ that Cycle 01 starts from.

    Cycle 01's master gap and public bounds do match Round 00's crystal.
  - No crystal carries a digest of its predecessor or of the target (§52 lists "target hashing"). One
    crystal, Cycle 05's, names a source.
- **Evidence is never marked as proof (W4, §45).**
  - The crystals file 15 strings under evidence keys, and the documents have 15 verdict lines whose subject is
    evidence. None of them carries PROVED, CLOSED, CERTIFIED or PASS.
  - One label joins a closure word to search: `SEARCH-CLOSED / CERTIFICATE-PACKAGING-DEBT`, for roots 8001,
    8015 and 8016 (Cycle 11 §13).
    - There it is kept apart from formal closure: "不得標 `INDEPENDENTLY VERIFIED PARENT ROOT`" (must not be
      labelled independently verified parent root).
    - The crystal carries those roots as packaging debt.
  - None of §40's banned phrases occurs.
- **The method's own labels are little used (W4).**
  - **LinkBack types.**
    - Two of the eleven occur in the cycles, `OPENS` and `REDUCES`, once each, in Cycle 01's LinkBack map.
    - That map's other four labels are not LinkBack types: EXTERNAL-DATA, CERTIFICATE-INTERFACE, SEARCH-ONLY
      and SEARCH-PRIOR.
    - Only Cycles 01 and 03 have a LinkBack section, although the protocol's Stage 3 makes LinkBack
      mandatory for every artifact.
  - **The method's labels as a whole.**
    - 29 of the method's 40 labels never occur. Among them:
      - all four closed-for-now labels of §30 (every cycle calls itself `FINITE-CLOSED / PROBLEM-OPEN`
        instead);
      - REOPENED, CERTIFIED-COMPUTATION and VERIFIED-THEOREM-APPLICATION.
    - The cycles use 84 labels of their own, mostly certificate leaf types: NOT-F3 24 times, AREA-PRUNED 13,
      ORBIT-COVERED 8, AREF-PRUNED 8.
  - **§39's round format.**
    - Every cycle recentres on entry and ends with a verdict.
    - 15 of 17 have a section for the next recentre (not 08 or 10), and all 17 crystals list it.
    - 10 have an obligation or responsibility ledger.
    - 2 have a finite-closure section.
- **Reopening (I7) is never exercised (W4).**
  - No closure is reopened in seventeen cycles. "Reopen" occurs only in prohibitions:
    - Cycle 16 §10, "Formal Responsibility Must Not Be Reopened";
    - Cycle 17, "Do not reopen".
  - What the cycles do take back are routes, and each is labelled:
    - Cycle 03's replacement family: `IMPLEMENTATION-TRAP / REJECTED-ROUTE`, `SEARCH-REJECTED`.
    - Cycle 07's global Hausdorff–Lipschitz bound: `REJECTED-AS-TOO-LOOSE`.
    - Cycle 15's global radial monotonicity: `REJECTED`. This was a stronger reading of Cycle 14's open gate.
      The counterexample that rejected it is itself labelled `NUMERICAL-DIAGNOSTIC`.
    - Cycle 16's global boundary restart: `REJECTED`.
- **Relations (W5).**
  - UESFCM relates itself to seven theories. Only one is in the archive: DLMVC.
  - The protocol's lagged line maps onto DLMVC. The line is old closure → blind reconstruct → audit →
    strengthen → hand back. DLMVC has a target rule (§4), Passes A, B and D (§7) and a return rule (§12).
  - DLMVC is of the same date and never names UESFCM. No DLMVC pass targets a cycle.
  - RCHM is named by 19 documents of the AMRAL and DLMVC series, but no RCHM document is in the archive.
  - UBE, Paper 08, UBGUL, SREG / PEC and RCIG are named only by UESFCM.
- **The package (P1).** VALIDATION describes both documents exactly:
  - digests and sizes;
  - line counts, where lines = newlines + 1;
  - no control characters;
  - `$$` and `$` paired;
  - no alternative delimiters.

Drill: 17 / 17 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 141 / 141
must-stay-green assertions held. Each of the eight checks was blinded in turn, and each was reported blind in
exactly as many cases as plants name it.

---

## What UESFCM claims

1. **Target.** Stage 0 builds a canonical target Q\* = (D, V, Q, P, E, S, C).
   - It is never silently mutated.
   - A correction makes a new version (§2).
2. **The cycle.** Q\* → Expand → LinkBack → FiniteClose → Recenter.
   - Every artifact gets a LinkBack type.
   - NULL never enters the main proof graph (§3–6).
3. **Closure state.** Each round is closed into a finite state C_t = (G, O, B, R, D, V, P, K).
   - History may grow; the working state is (Q\*, C_t) (§8–13).
4. **Termination.** The loop ends at proof or counterexample closure (§3, §21).
   - A failed target returns to Stage 0 (§21.3).
   - Independence closes relative to a theory (§58).
   - SEARCH-STALLED is not completion (§49).
5. **Invariants and failure modes.** Seven invariants (§41), seven failure modes (§42–48), and a status
   vocabulary (§40).
6. **Not claimed.** A convergence theorem, and universal solvability (§57).
7. **Open.** Benchmark validation (§55–56).

## A — sources

**A0.** 46 anchors: 40 in the methodology and 6 in the protocol.

The methodology's references are nine internal documents. W5 checks which of the theories they name are in
the archive.

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the LinkBack types in both documents; the obligation, artifact and closure labels; the two closure states | PASS: one vocabulary |
| W2 | the exits named in eight places; the pseudocode's control flow | four different sets; SEARCH-STALLED a stop in the protocol, a trigger in §49 |
| W3 | 17 headers and crystals; the archive searched for the target and C00; 39 ledger identities | the chain and the ledger hold; the target and C00 are not in the archive |
| W4 | §45 on 30 evidence labels; §40's banned phrases; the method's 40 labels counted; §39's format; reopening | no evidence marked as proof; 2 of 11 LinkBack types used, once each; 29 of 40 labels unused; nothing reopened |
| W5 | the seven related theories in the archive; DLMVC's sections | one of seven in the archive; the lagged line maps onto DLMVC |

## P — the package

**P1.** VALIDATION's record of each document agrees field by field with its bytes, hashed here, and with the
archive's manifest row.

## The judgments, read

| status | subject | verdict | note |
| --- | --- | --- | --- |
| DEFINED | Semantic Target Contract | PASS | §2; its one application's target is not in the archive (W3) |
| DEFINED | Unbounded Expansion | PASS | §5 |
| DEFINED | LinkBack Operator | PASS | §6; used in one cycle of seventeen (W4) |
| DEFINED | Finite Closure State | PASS | §8–10 |
| DEFINED | Self-Referential Recenter | PASS | §11 |
| DEFINED | Proof Obligation Ledger | PASS | §14–15 |
| DEFINED | Problem Closure | PARTIAL | each closure is defined; which ones end the loop differs from place to place (W2) |
| DEFINED | Anti-Drift Governance | PASS | §7, 24–26, 42 |
| DEFINED | UBE, RCHM, SREG / PEC relations | PASS | written down; the other side of each is not in the archive (W5) |
| DEFINED | DLMVC relation | PASS | the lagged line maps onto DLMVC (W5) |
| NOT CLAIMED | Formal convergence theorem | PASS | none stated; §55 Q3 asks it |
| NOT CLAIMED | Universal proof solvability | PASS | disclaimed in §57 |
| OPEN | Empirical benchmark validation | PASS | one application, no control, no DLMVC pass on a cycle |

## The status registry

UESFCM's 35 judgments are reviewed:

- 15 are judgments: 14 PASS and 1 PARTIAL;
- 20 are labels, reviewed n/a: the eleven LinkBack types, the three terminal labels, the four closed-for-now
  labels, PROBLEM-PROOF-CLOSED and SEARCH-STALLED.

It uses 21 phrases for the first time, and each is defined:

- NOT CLAIMED, from §62;
- the LinkBack types, in §6.1;
- the terminal labels, in §21;
- PROBLEM-PROOF-CLOSED, in §30;
- SEARCH-STALLED, in §49;
- the four closed-for-now labels, which §30 lists without defining. They are recorded as contextual.

| verdict | judgments |
| --- | ---: |
| PASS | 14 |
| PARTIAL | 1 |
| n/a | 20 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| a status line | 1 | A0 |
| a LinkBack type renamed in the protocol | 1 | W1 |
| a stall exit added to the pseudocode | 1 | W2 |
| §49's stall made a stop | 1 | A0, W2 |
| a remaining count; the target drifting; a target contract shipped | 3 | W3 |
| PROBLEM-PROOF-CLOSED in a verdict | 1 | W3, W4 |
| a diagnostic marked proved; a REOPENED label; a verdict heading renamed | 3 | W4 |
| a related theory in the archive; DLMVC naming UESFCM | 2 | W5 |
| a VALIDATION digest; the document's bytes | 2 | P1 |
| a review dropped; a definition dropped | 2 | R1 |

Three controls:

- the unmodified inputs;
- the methodology, the protocol, the cycles and DLMVC with CRLF line endings;
- VALIDATION with the keys sorted.

**17 / 17 plants, 3 / 3 controls, 141 / 141 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 2 | 2 |
| W1 | 1 | 1 |
| W2 | 2 | 2 |
| W3 | 4 | 4 |
| W4 | 4 | 4 |
| W5 | 2 | 2 |
| P1 | 2 | 2 |
| R1 | 2 | 2 |

## What this run does not establish

- **The cycles' mathematics.** Every certificate, bound and theorem of the seventeen cycles is for the runs
  that follow. W3's identities check only that the crystals agree with themselves and with each other.
- **The missing target.** `UESFCM-LUC-TARGET-v0.1` and Cycle 00 may exist outside the archive. Here they are
  references with nothing behind them.
- **UESFCM as a method.** One application, one problem, no control. §55's questions are not measured:
  - drift rate;
  - duplicate work;
  - convergence of the residual obligations;
  - kernel fidelity;
  - whether a lagged line finds more.

  W4's label counts describe how the method was used, not whether it works.