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Research / Quantum information

A GHZ-mixture witness against BCWW

A four-qubit GHZ mixture violates the same proposed inequality by 0.113302 bits.

Evidence assessed 2026-10-02

A mixture of a four-qubit GHZ state and two basis states violates the BCWW candidate inequality [1] by 0.113302…0.113302\ldots bits. The relevant marginal entropies are classical, so this construction does not establish a quantum advantage.

State and marginal entropies

Define

∣GHZ4⟩=∣0000⟩+∣1111⟩2,ρ=34∣GHZ4⟩ ⁣⟨GHZ4∣+18∣1010⟩ ⁣⟨1010∣+18∣1001⟩ ⁣⟨1001∣.|\mathrm{GHZ}_4\rangle=\frac{|0000\rangle+|1111\rangle}{\sqrt2}, \qquad \rho=\frac34|\mathrm{GHZ}_4\rangle\!\langle\mathrm{GHZ}_4| +\frac18|1010\rangle\!\langle1010| +\frac18|1001\rangle\!\langle1001|.

The BCWW functional uses only proper subsystems. Tracing out a nonempty complement removes the GHZ off-diagonal term. Every required entropy therefore equals the Shannon entropy of the corresponding marginal of

P(0000)=P(1111)=38,P(1010)=P(1001)=18.P(0000)=P(1111)=\frac38,\qquad P(1010)=P(1001)=\frac18.

With logarithms in base two, substitution gives

F=254−98log⁡23−158log⁡25=0.113302008774…>0.F=\frac{25}{4}-\frac98\log_2 3-\frac{15}{8}\log_2 5 =0.113302008774\ldots>0.

The exact sign certificate is

250=1125899906842624>600677490234375=39515.2^{50}=1125899906842624>600677490234375=3^9 5^{15}.

The retained standard-library verifier reconstructs each marginal and checks the closed form [2]. Its October 2 replay succeeded.

The interactive calculation uses ordinary floating point. The exact certificate in the text establishes the sign at the stated witness.

The slider assigns weight pp to the GHZ projector and (1−p)/2(1-p)/2 to each basis-state projector. The integer certificate applies at p=3/4p=3/4; other plotted values use floating point.

Attribution and retained attempt

The provenance note credits the fifth arm of an operator-led verification fleet dated August 10, 2026, and records five verification approaches, including blind arithmetic and fresh code. The complete original fleet transcripts were not recovered, so this attribution cannot be reconstructed from a complete discovery trace.

The retained pursuit [3] began on August 30 and revisited the witness and surrounding claim. Its two child attempts ended down after rate-limit and incomplete-stream failures. The archive contains journal events and the witness artifact, but no native session JSONL. The viewer shows this later repair attempt, not the August 10 fleet; it does not certify complete capture of the original messages, children, or source bytes. This is a second witness to the same BCWW refutation.

References

  1. N. Bao, C. Cao, M. Walter, and Z. Wang. Holographic entropy inequalities and gapped phases of matter . J. High Energy Phys. 09 (2015), 203. arXiv:1507.05650v2, Eq. (4.6). DOI: 10.1007/JHEP09(2015)203.
  2. GHZ-mixture entropy verifier . Exact retained Python source. Standard-library reconstruction and closed-form check; replayed October 2, 2026.
  3. BCWW repair pursuit . Retained event and finding projection, August 30, 2026. A later attempt, distinct from the August 10 verification fleet. Source hashes identify the inspected files.

Agent record

Find or exclude repairs to the already-refuted BCWW candidate within a bounded grammar.

Recorded run

3 recorded agents · 0 native sessions · 241 / 241 events · capture incomplete

1× compresses the full recorded span into 30 seconds.

Activity over time
0 ms14.0 min27.9 min41.9 min55.9 min69.8 min83.8 minrootrootgate: prior-art sweep for published BCWW 4.6 repairsgate: prior-art swe…gate retry: prior-art sweep (crash-safe, out/ only)gate retry: prior-a…Recorded findingsRecorded findingsbegin · 2026-08-30 23:29:28 UTCspawned · 2026-08-30 23:38:47 UTCprogress · 2026-08-30 23:39:27 UTCprogress · 2026-08-30 23:40:03 UTCprogress · 2026-08-30 23:41:03 UTCprogress · 2026-08-30 23:41:44 UTCprogress · 2026-08-30 23:42:08 UTCprogress · 2026-08-30 23:42:42 UTCprogress · 2026-08-30 23:43:24 UTCprogress · 2026-08-30 23:44:06 UTCprogress · 2026-08-30 23:44:42 UTCprogress · 2026-08-30 23:45:22 UTCprogress · 2026-08-30 23:46:05 UTCprogress · 2026-08-30 23:46:42 UTCprogress · 2026-08-30 23:47:47 UTCprogress · 2026-08-30 23:48:35 UTCprogress · 2026-08-30 23:49:34 UTCprogress · 2026-08-30 23:50:00 UTCmaterialized · 2026-08-30 23:50:13 UTCprogress · 2026-08-30 23:50:48 UTCsettled · 2026-08-30 23:52:06 UTCR1's GHZ4-mixture second witness pinned exactly: rho = 3/4 GHZ4 + 1/8|(A,C)> + 1/8|(A,D)>, c46 = -0.113302009 bits, classical-reducible · 2026-08-30 23:52:40 UTCspawned · 2026-08-30 23:54:45 UTCprogress · 2026-08-31 00:00:14 UTCsettled · 2026-08-31 00:02:17 UTCmetered · 2026-08-31 00:06:19 UTCmetered · 2026-08-31 00:07:44 UTCmetered · 2026-08-31 00:09:19 UTCmetered · 2026-08-31 00:10:54 UTCmetered · 2026-08-31 00:12:45 UTCmetered · 2026-08-31 00:14:47 UTCmetered · 2026-08-31 00:17:28 UTCmetered · 2026-08-31 00:21:32 UTCmetered · 2026-08-31 00:27:56 UTCmetered · 2026-08-31 00:34:20 UTCmetered · 2026-08-31 00:40:43 UTCmetered · 2026-08-31 00:46:55 UTCmetered · 2026-08-31 00:53:16 UTC
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Agents3

No session traces supplied.

root

Director · opencode/zai-coding-plan/glm-5.3 (configured; served identity unknown)

0 entries · 0 tool calls at this time

No retained conversation content at the selected time. Source events remain inspectable.

1 / 241
Original assignment
objective
Find or exclude repairs to the already-refuted BCWW candidate within a bounded grammar.
supplied Knowledge
The corrected classical witness, GHZ witness, and earlier no-go findings were supplied.
deliverables
Check prior art first; then exact witness screening, a copy-lemma proof attempt, or a bounded conjecture with a falsifier battery.
constraints
The task forbade re-verifying the settled BCWW refutation. It named an OpenCode director, four roles, a $25 budget, and small storage. These are intended conditions, not measured expenditure.
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Sources, redactions & capture gaps

Reviewed assignments and recorded findings are published where available. Native message bodies outside the August 5 review remain unpublished. Original source hashes and physical line numbers are retained.

Tool-name and command keyword rules; these are annotations, not measured research value or evidence of independence.

Unavailable: zero-valued legacy counters are not treated as a bill.

3 agents · 0 native sessions · 0 attributed sessions · 4 source files.

Recorded terminal: no-winner · all-children-down

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