Paper Citation Record · LEDGER

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code

As of 9 August 2026, this Paper Citation Record lists 37 of 37 outbound references and 1 inbound Pith citation observation for arXiv:2607.00134.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2607.00134 v2

Coverage vector

measured 37 of 37 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-07-12T09:43:07.405489Z

measured 38 of 38 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-09T06:31:02.800959+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-08T19:23:26.023659Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: pith, observed 2026-08-08T19:23:27.868781Z

Reference resolution

37 of 37 outbound references displayed

verified exact0
verified fuzzy0
unresolved37
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 65e735ac-6edb-4a32-b582-45123812909a · outbound

This paper cites (B3) is a statement about theuniversalpart ofS (n).

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code (B3) is a statement about theuniversalpart ofS (n)

Reference 1

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8ef98c54-6026-44cd-90b7-68911b34dfd7 · outbound

This paper cites (B3) was proved in Ref.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code (B3) was proved in Ref

Reference 2

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3e7dc000-0ed0-43e2-a8d6-0719c0438b7b · outbound

This paper cites [69, 70] is aground-statestate- ment.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code [69, 70] is aground-statestate- ment

Reference 3

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9e05d382-f584-4b94-8d58-f779a3f50255 · outbound

This paper cites A single four-region combi- nation [Eq.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code A single four-region combi- nation [Eq

Reference 4

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9ead8930-8b48-4c31-a431-dc1d944ba47d · outbound

This paper cites (3) at (h x, hz) = (0,0) (couplingsJ e,J m kept general; the main text setsJ e = Jm = 1) and the symmetric eight-bipartition topolog- ical combination of CC [their Eq.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code (3) at (h x, hz) = (0,0) (couplingsJ e,J m kept general; the main text setsJ e = Jm = 1) and the symmetric eight-bipartition topolog- ical combination of CC [their Eq

Reference 5

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ddfbb9e6-bf27-4c43-b66d-89867f155404 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 6

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f963d951-663b-4536-8b1c-5fd86176ca57 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 7

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a1dd6ec7-8d55-4ec1-ae4b-86474c4764aa · outbound

This paper cites This is verified empirically, with the crossover scale supplied by Eq.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code This is verified empirically, with the crossover scale supplied by Eq

Reference 8

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation af19020b-c123-4a73-8498-fca4bbb22410 · outbound

This paper cites (D20) for the bipartitions that host a collective mem- brane operation, and Eq.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code (D20) for the bipartitions that host a collective mem- brane operation, and Eq

Reference 9

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 96f2464f-ee69-4e8c-b5c1-f11b80aeae4c · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 10

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b58f510e-a298-4b53-b211-f2c4d991261f · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 11

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c3979fd7-2903-4863-a2fd-c5d3e1870271 · outbound

This paper cites For each slicesletD(s) ={p:b p(s) =−1}be the flux-defect set; under lattice dualityDis a set of dual links, aZ 2 1- chain ˜D.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code For each slicesletD(s) ={p:b p(s) =−1}be the flux-defect set; under lattice dualityDis a set of dual links, aZ 2 1- chain ˜D

Reference 12

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ee8dcb4a-9267-4632-9f37-18b80d0b4cf7 · outbound

This paper cites (C6) Here TrH⊗H runs over the doubled space, and Tr A and Tr ¯A over the factors of the originalH=H A ⊗ H ¯A.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code (C6) Here TrH⊗H runs over the doubled space, and Tr A and Tr ¯A over the factors of the originalH=H A ⊗ H ¯A

Reference 13

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9ab612db-daec-4b91-9154-f496459eb040 · outbound

This paper cites Connected.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Connected

Reference 14

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ec0e6554-1cdb-40a9-a800-cbe554a4efbb · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 15

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c636f6de-aca7-46da-acdc-0453b55bca66 · outbound

This paper cites III A)—so the combination is the conditional mu- tual informationI(b:d|ac) =−S(A 1) +S(A2) +S(A3)− S(A4), which fixes the sign vectorσ= (−,+,+,−).

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code III A)—so the combination is the conditional mu- tual informationI(b:d|ac) =−S(A 1) +S(A2) +S(A3)− S(A4), which fixes the sign vectorσ= (−,+,+,−)

Reference 16

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b275c239-dc84-4603-b81b-c64047bcf1c3 · outbound

This paper cites For its boundary-local dressing to cancel in the matched-boundary combination [Eq.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code For its boundary-local dressing to cancel in the matched-boundary combination [Eq

Reference 17

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c3400fd3-96fc-4061-86a3-a49c67ffc622 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 18

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 37ebeea0-2254-4956-b8fe-b9aa7b27e601 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 19

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 711eeea1-6ad1-47f5-b256-47c6e5bb58ae · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 20

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ecb0da4d-bbf7-40f9-8ed6-52d3ead754c0 · outbound

This paper cites Proposition 1 at generic fields is an all-orders, term-by-term result resting on asinglephys- ical input—the finite-ξclustering bound of Eq.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Proposition 1 at generic fields is an all-orders, term-by-term result resting on asinglephys- ical input—the finite-ξclustering bound of Eq

Reference 21

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a05e3791-b373-4d6a-8d55-f70255780714 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 22

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5a3ee16b-9ba3-4b01-b0c0-321f23dccfa9 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 23

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation cdfaeec8-102a-4929-b055-bba155a3a7bf · outbound

This paper cites 6(a)] is supplemented by a companion FSS campaign at the plateau temperatureT= 0.5, withL∈ {10,12,14}, using the identical chain-trick pipeline and production parameters of Sec.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code 6(a)] is supplemented by a companion FSS campaign at the plateau temperatureT= 0.5, withL∈ {10,12,14}, using the identical chain-trick pipeline and production parameters of Sec

Reference 24

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7f72f38d-d2d3-43ee-89f2-17f979504ea0 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 25

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d6c2dc33-a1f7-421e-8c77-0ba24f8ae59b · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 26

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2e72d9fe-ece1-43f4-948f-b2df0c5df982 · outbound

This paper cites The bare topo- logical entanglement entropy is thereforenotan FDLU invariant—not even constant along one FDLU orbit, let alone under the broader quasi-local channels of Sec.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code The bare topo- logical entanglement entropy is thereforenotan FDLU invariant—not even constant along one FDLU orbit, let alone under the broader quasi-local channels of Sec

Reference 27

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 28cb1b23-dd08-4a19-aaaf-870659c4c699 · outbound

This paper cites [37]):β g (the classical gauge coupling, not the inverse temperature) tracks the magnetic couplings (Jm, hx, T) andKthe electric fieldh z.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code [37]):β g (the classical gauge coupling, not the inverse temperature) tracks the magnetic couplings (Jm, hx, T) andKthe electric fieldh z

Reference 28

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9f2dddd0-4d83-453a-b776-697d7bcae05b · outbound

This paper cites II B), which likewise van- ishes]: either way the bare holonomy bit is destroyed as L→ ∞.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code II B), which likewise van- ishes]: either way the bare holonomy bit is destroyed as L→ ∞

Reference 29

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a0da32b6-4a0c-4066-b4e5-10d971bdebbc · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 30

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e7383cb9-e8b6-43de-ad69-c67321543bef · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 31

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 15f36024-7ec3-4743-a803-e2f539bc3dac · outbound

This paper cites (F3) is the thermal-ensemble analog of the random- plaquette/accuracy-threshold problem [91, 106].

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code (F3) is the thermal-ensemble analog of the random- plaquette/accuracy-threshold problem [91, 106]

Reference 32

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a154c46e-29bf-4abc-9cca-37f2773b4f63 · outbound

This paper cites Its value on the trivial class is fixedmodel- independentlyby the light-cone (channel-invariance) ar- gument of Sec.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Its value on the trivial class is fixedmodel- independentlyby the light-cone (channel-invariance) ar- gument of Sec

Reference 33

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ccea5249-98a5-4dd4-b3af-abc593758372 · outbound

This paper cites Confined/Higgs.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Confined/Higgs

Reference 34

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f7f7f84a-2061-4756-8d40-763aa6aaae28 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 35

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c7c0f1ca-70a4-47e0-a87a-0836a3ed7672 · outbound

This paper cites an unresolved cited work.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code Unresolved cited work

Reference 36

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 313f81e5-f168-4808-a044-1b82a40f44fa · outbound

This paper cites That is a misla- bel here.

Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code That is a misla- bel here

Reference 37

Resolution: unresolved; no resolver link, observed 2026-07-12T09:43:07.405489Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

Observation e007d205-0667-43fa-b952-7376ef5fd455 · inbound

Quantized topological invariant of symmetry-projected Gibbs states cites this paper.

Quantized topological invariant of symmetry-projected Gibbs states Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code

Reference 28

Resolution: verified exact; local_arxiv, observed 2026-08-08T19:23:27.874194Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.