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REVIEW 3 major objections 5 minor 56 references

Topological Signatures of Imperial Collapse and Fragmentation: Administrative Dissolution, Territorial Reorganization and Early-Warning Observables in the Han Dynasty Network (206~BCE\,--\,220~CE)

T0 review · 3 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Han administrative networks collapse to zero entropy while geographic networks reorganize into Three Kingdoms cycles thirty years before formal partition.

desk verdict Solid Han geospatial TDA with a real Admin–Geo divergence, but the Three Kingdoms β1 “prediction” is largely imposed by Channel 5 rather than recovered from topography. read the letter →

arxiv 2607.09010 v1 pith:DK6LE3B6 submitted 2026-07-10 physics.soc-ph physics.ed-phphysics.hist-ph

Jose de Jesus Bernal-Alvarado , David Delepine , Carlos Pinedo Guadarrama This is my paper · ORCID
classification physics.soc-phphysics.ed-phphysics.hist-ph
keywords persistenthomologyimperialcollapseHanDynastytopologicaldataanalysisearly-warningindicatorsnetworkfragmentationSilkRoadWassersteindistance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the same persistent-homology measures used on the Roman–Byzantine world also detect collapse in the Han Dynasty, a system that fragmented from within after the Yellow Turban Rebellion. The authors build three weighted networks—administrative prefectures, the same places kept alive after administrative dissolution, and a fixed Silk Road trade graph—whose edge costs come from least-cost paths on a real elevation model plus five historical friction channels. Administrative persistent entropy falls to zero by 220 CE while geographic entropy rises, producing a clean divergence that the paper calls the topological signature of internal fragmentation. Three long-lived one-dimensional cycles matching the Wei, Shu and Wu domains appear at 190 CE, and three independent early-warning indicators fire 45–50 years before formal dissolution. A reader cares because the method supplies a quantitative, mechanism-agnostic language for comparing how different empires fall apart and for spotting reorganization before the history books declare it.

What carries the argument

Three-network decomposition (H_Admin, H_Geo, H_Trade) measured by persistent entropy of the β₁ barcode, together with the Admin–Geo divergence ΔH and the cross-network Wasserstein distance W×. After 184 CE, Channel 5 raises the cost of inter-domain edges only in H_Geo, so geographic nodes persist while long-range loops reorganize into domain-scale cycles that carry most of the persistence weight.

What would settle it

Recompute the geographic network after 184 CE with Channel 5 turned off or with domain multipliers assigned at random rather than by the documented Wei–Shu–Wu frontiers; if the positive phase-G slope, the ΔH = 3.065 divergence, the W× jump, and the three long-lived β₁ cycles with gap ratio ~11 all vanish, the internal-fragmentation signature is an artifact of that historically informed cost penalty.

Watch

Extended reading notes

Core claim

The topological signature of Han internal fragmentation is the Admin–Geo divergence: administrative persistent entropy collapses to zero at 220 CE (phase-G slope −0.0261 yr⁻¹) while geographic entropy simultaneously rises to 3.065, with the cross-network Wasserstein distance jumping from exactly zero to 737 at 190 CE. Under the post-184 fragmentation model, three long-lived β₁ cycles corresponding to the Wei, Shu and Wu domains emerge thirty years before formal partition, and three independent indicators (Wasserstein velocity, correlation length, Integrated Change Tracker) signal onset 45–50 years before dissolution while the Silk Road trade network stays stable.

Load-bearing premise

The rise in geographic entropy and the three kingdom cycles depend on a friction channel that multiplies the cost of routes crossing the historically known post-184 political domains, so the model already encodes the later map of Wei, Shu and Wu.

Editorial extensions

If this is right

  • Internal fragmentation can be distinguished from simple territorial loss by opposite trends in administrative versus geographic persistent entropy.
  • Independent topological indicators can flag systemic risk 45–50 years before formal political dissolution.
  • Successor polities can appear as long-lived β₁ cycles decades before they are declared as states.
  • Long-distance trade topology can remain stable across political collapse, acting as a structural control.
  • The collapse threshold H* = 0.5241 and three-network method generalize from Roman–Byzantine to a mechanistically distinct imperial system.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Admin–Geo divergence may serve as a diagnostic for other internally fragmenting polities once comparable hierarchical settlement data exist.
  • The county-level β₀ atomization that precedes regional β₁ consolidation suggests a two-scale process testable on other empires with fine-grained place catalogues.
  • Stable trade topology amid political collapse implies commercial circuits can outlast the states that once secured them; the pattern could be checked on other long-distance historical trade systems.
  • If the early-warning suite is portable, archival geospatial datasets alone could generate collapse chronologies without first accepting narrative periodizations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper applies persistent homology to three Han-era networks (administrative, geographic, and Silk Road trade) built from CHGIS v6 prefectures, a 1 km DEM with least-cost paths, and five historical friction channels. It reports that HAdmin collapses to zero by 220 CE (phase-G slope −0.0261 yr−1, R²=0.689) while HGeo rises to 3.065, producing ΔH=+3.065 as a signature of internal fragmentation; W×(Admin,Geo) jumps from 0 to 737 at 190 CE; three long-lived β1 cycles under a post-184 fragmentation model are identified with Wei, Shu, and Wu; and ICT, Ẇ2, and ξ give early-warning signals 45–50 years before formal dissolution. A combined index Hcomb is calibrated so that Hcomb(220)=H*=0.5241 from prior Roman–Byzantine work.

Significance. If the Admin–Geo divergence and early-warning chronology can be shown to be robust to the historically informed Channel 5 domain map, the paper would supply a quantitative, cross-civilizational signature distinguishing internal fragmentation from territorial loss, and a falsifiable early-warning suite independent of scalar entropy. Strengths include primary geospatial sources (CHGIS v6), explicit LCP validation on the Hexi Corridor, reported OLS/Chow/bootstrap statistics, and a clean three-network control design with a stable trade network. The work is a natural extension of the authors’ Roman–Byzantine series and would matter for historical complex-systems and TDA applications if circularity in the fragmentation model and H* calibration is resolved.

major comments (3)
  1. [§III.C.4 / §VI.E] §III.C.4, Eq. (16)–(17) and Table VII: Channel 5 (minter up to 25 after 184 CE) and the domain map Z(r,d) are taken from the San Guo Zhi and assign nodes to Wei/Shu/Wu. The positive phase-G slope of HGeo (+0.0169 yr−1), the three long-lived β1 cycles, and the claim that they “match” Wei/Shu/Wu thirty years before partition are therefore largely imposed by the cost model rather than recovered from topography and node persistence alone. The manuscript needs an ablation with C5 off (or with domain-agnostic penalties) showing whether ΔH, the G3 gap (Table XVII), and the three long bars still appear; without it the headline “prediction” of the Three Kingdoms is not supported.
  2. [§V.D / Abstract] §V.D after Eq. (28) and Table XI: α0 is fixed by the requirement Hcomb(220)=H*=0.5241. The abstract and introduction frame the work as testing whether H*=0.5241 generalizes to a mechanistically distinct empire; calibration by construction cannot constitute that test. Either treat H* as an external fixed scale and report the uncalibrated Hcomb(220), or reframe Hcomb as a Han-specific resilience index and drop the generalization claim for the numerical threshold.
  3. [§VI.D / Table XXIV] §VI.D–E and Table XXIV: W× jumps and ΔH open only after 184 CE because VGeo is extended and C5 is switched on by construction (Eqs. 19–20). The discontinuous W×=737 at d=190 CE is therefore partly definitional. Clarify which components of the early-warning suite (especially Admin ICT, Ẇ2, ξAdmin) are independent of C5 and node-set extension, and report those series with C5 disabled so the 45–50 year lead time is not confounded with the fragmentation model.
minor comments (5)
  1. [Abstract / §I] Abstract and §I claim the three β1 cycles emerge “thirty years before formal partition” under the post-184 model; the wording should state that the model encodes known domains rather than implying unsupervised recovery.
  2. [Table IX] Table IX note: H(220) described as “numerically −0.000” is confusing; state H=0 exactly when β1=1 and p1=1.
  3. [Fig. 6 / Table XXIII] Fig. 6 and Table XXIII: bootstrap CIs are incomplete for several terminal decades; fill or mark as unavailable consistently.
  4. [Throughout] Notation: HAdmin / Hadm / H_Admin and HGeo / Hgeo appear interchangeably; standardize.
  5. [§II.F–G] §II.G validation is strong; a short sensitivity check on the 600 km neighbor cutoff and altitude penalties (Eq. 5) would help readers assess LCP robustness.

Circularity Check

4 steps flagged · score 7.0 of 10

Hcomb(220) is forced to equal the prior H*=0.5241 by solving for free scale α0; Channel 5 encodes the known Wei/Shu/Wu domains so the three long-lived β1 cycles and positive HGeo slope are largely model-imposed.

  1. fitted input called prediction [Sec. V D, after Eq. (28) and Table XI]
    "The overall scale α0 is determined by a fixing Hcomb(220 CE) = 0.5241 = H ∗ where H ∗ is interpreted (see Ref. [2] as the topological irreversibility threshold. Substituting HAdmin(220) = 0, w(220) = 0.843, ˆα(220) = 0.463, and HTrade(220) = 1.228, one gets α0 = 1.092. ... Calibration at 220 CE. Hcomb(220) = H ∗ = 0.5241 by construction."

    α0 is solved so that the composite index exactly equals the previously published threshold at the terminal year. The statement that Hcomb reaches H* is therefore true by algebraic construction, not an independent empirical finding that the Han system crosses the same critical value.

  2. self definitional [Sec. III C 4, Eqs. (16)–(17) and Table VII; Results Sec. VI E 2]
    "Each zone is then mapped to a political domain Z(r, d) according to the phase: ... Wei ... Shu ... Wu ... This assignment follows the San Guo Zhi[20, 31]. C5(e, d) = ... minter(d) Z(rs,d)≠Z(rt,d). ... three long-lived β1 cycles matching the Wei, Shu, and Wu domains emerge under the post-184 CE fragmentation model—thirty years before formal partition. ... The Wei, Shu, and Wu cycles emerge purely from the topological structure of the post-184 CE geographic network"

    Channel 5 (exclusive to HGeo) applies large inter-domain cost multipliers whose domain labels are taken from the historically known Three Kingdoms partition. The subsequent identification of three long-lived β1 cycles that 'match' Wei/Shu/Wu therefore largely recovers the partition that was inserted into the cost model; without those historically informed multipliers the positive phase-G slope of HGeo and the claimed early emergence of the three domains would not appear.

2 more flagged steps
  1. self citation load bearing [Abstract; Introduction; Sec. V D (H* and three-network methodology)]
    "We extend the persistent homology formalism done in references [1, 2] to the Han Dynasty ... testing whether the collapse threshold H∗ = 0.5241 and the three-network decomposition methodology established for the Roman–Byzantine case generalize ... Hcomb(220) = 0.5241 = H ∗ where H ∗ is interpreted (see Ref. [2] as the topological irreversibility threshold."

    The numerical value of the irreversibility threshold and the three-network decomposition that structure the entire analysis are imported from the authors' own immediately preceding papers. The claim that the same H* 'generalizes' is therefore load-bearing on an unverified self-citation rather than an external mathematical or empirical fact.

  2. self definitional [Sec. V B, Eqs. (19)–(20); Sec. VI E]
    "For decades d < 184 CE ... the two networks are identical by construction: VGeo(d)≡VAdmin(d) ... For d≥184 CE, the end-year of each administrative node is extended to the terminal analysis year ... By construction HGeo(d) ≡ HAdmin(d) for all d < 184 CE, and the two series diverge only after the Yellow Turban Rebellion onset. ... ΔH(d) = HGeo(d) − HAdmin(d) ... rising from zero at 180 CE to +3.065"

    The Admin–Geo divergence that is presented as the topological signature of internal fragmentation is identically zero before 184 CE by the node-set definition and becomes nonzero only because HGeo retains nodes after administrative END_YR while also receiving Channel 5. The divergence is therefore engineered by the network definitions rather than discovered from independent data.

full rationale

The paper's headline claims (Admin–Geo divergence ΔH→+3.065, W× jump, three β1 cycles matching Wei/Shu/Wu thirty years early, and Hcomb reaching the irreversibility threshold) rest on two constructions that reduce the outputs to the inputs. First, α0 is the sole free parameter and is fixed by the explicit requirement Hcomb(220)=H*=0.5241 taken from the authors' prior Roman–Byzantine papers; the calibration is therefore tautological rather than a prediction that the Han system reaches the same threshold. Second, Channel 5 (active only on HGeo after 184 CE) multiplies inter-domain edges by historically chosen minter up to 25 using a domain map Z(r,d) taken directly from the San Guo Zhi assignment of nodes to Wei, Shu and Wu; the subsequent recovery of three long-lived β1 cycles 'matching' those domains is therefore largely the model re-expressing its own domain partition rather than an independent topological discovery from topography and node persistence alone. Early-warning indicators on the pure Admin network are less affected, but the central internal-fragmentation signature and the Three-Kingdoms 'prediction' are circular by construction. Self-citation of H* and the three-network methodology is load-bearing for the generalization claim. Score 7 reflects partial but central circularity: the Admin collapse itself is independent, yet the strongest interpretive claims are not.

Assumptions & free parameters 7 free parameters · 5 assumptions · 4 invented entities

The central claims rest on (i) the prior H* threshold and pipeline taken from the authors’ own papers, (ii) a large set of historically motivated but free multiplicative friction parameters, (iii) the post-184 Channel-5 domain map that encodes the known Three Kingdoms geography, and (iv) the calibration of α0 that forces Hcomb(220)=H*. These are the load-bearing modeling choices beyond standard persistent homology and public GIS data.

free parameters (7)
  • α0 (commercial capture scale) = 1.092
    Solved so that Hcomb(220 CE)=0.5241 exactly; sole calibrated parameter of the combined index.
  • H* collapse threshold = 0.5241
    Imported from prior Roman–Byzantine paper and used as the calibration target for Hcomb.
  • Channel 5 inter-domain multipliers minter = 5–25
    Hand-set schedule rising to 25 after 208 CE (Table VII); controls the rise of HGeo and the longevity of the three β1 cycles.
  • Channel 1 event multipliers mτ = 0.60–3.50
    Fifteen historically assigned road/caravan/river factors (Table V) that compound multiplicatively.
  • MFI clip bounds and regional μr = [0.5,6.0]
    Population-derived friction clipped to [0.5,6.0] with ad-hoc Xiyu modifiers (Eqs. 13–14).
  • filtration percentile q and Nmax = q=90, Nmax=600
    Fixed pipeline parameters (Table VIII) inherited from prior papers; affect which bars enter H.
  • trade-dependency weights wSR, wWM, wcol = 0.20, 0.15, 0.60
    Logistic amplitudes in w(t) (Eq. 26) set from rough fiscal-share estimates.
assumptions (5)
  • domain assumption Persistent entropy H of the β1 barcode is a valid scalar measure of systemic redundancy and collapse.
    Taken from the authors’ prior papers and the TDA literature; never re-derived here.
  • domain assumption Least-cost paths on a 1 km DEM with caravan-adapted Tobler function plus altitude cut-offs correctly represent Han travel costs.
    Validated only against the Hexi Corridor waypoints (Fig. 1, Table III).
  • ad hoc to paper After 184 CE, geographic nodes remain active while administrative nodes follow CHGIS END_YR, and inter-domain edges receive large cost penalties.
    Defines the Admin–Geo split (Eqs. 19–20, 17) that produces the claimed divergence signature.
  • ad hoc to paper The three longest-lived β1 bars after 190 CE correspond to Wei, Shu and Wu domains.
    Identification rests on geographic zonation chosen to match San Guo Zhi frontiers (Eq. 16).
  • standard math Standard Vietoris–Rips persistent homology with dmax=2 and 90th-percentile filtration is the appropriate filtration for imperial road networks.
    Pipeline parameters fixed for consistency with Papers I–II (Table VIII).
invented entities (4)
  • Combined resilience index Hcomb(t)
    purpose: Time-weighted blend of HAdmin and α(t)HTrade that is forced to equal H* at 220 CE.
    Defined in Eq. 25; α0 is the free scale that enforces the threshold.
  • Integrated Change Tracker (ICT)
    purpose: Normalized average of susceptibility, correlation length and Wasserstein velocity used as early-warning score.
    Eq. 34; thresholds 0.1/0.2/0.35 chosen after inspection of the series.
  • Admin–Geo divergence ΔH as signature of internal fragmentation
    purpose: Scalar that is claimed to distinguish internal breakup from other collapse modes.
    Defined only after Channel 5 is activated; zero by construction before 184 CE.
  • Top-3 gap ratio G3
    purpose: Signal-to-noise measure that isolates the three long-lived cycles from background bars.
    Eq. 35; used to claim 10.9× separation at 190 CE.

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Cite this review

Pith. "Pith review of Topological Signatures of Imperial Collapse and Fragmentation: Administrative Dissolution, Territorial Reorganization and Early-Warning Observables in the Han Dynasty Network (206~BCE\,--\,220~CE)." pith.science (2026). https://pith.science/paper/DK6LE3B6

@misc{pith2026260709010,
  author       = {Pith},
  title        = {Pith review of: Topological Signatures of Imperial Collapse and Fragmentation: Administrative Dissolution, Territorial Reorganization and Early-Warning Observables in the Han Dynasty Network (206~BCE\,--\,220~CE)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DK6LE3B6}},
  note         = {Machine review of arXiv:2607.09010}
}
abstract

We extend the persistent homology formalism done in references~\cite{paper1,paper2} to the Han Dynasty ($206~\mathrm{BCE}$--$220~\mathrm{CE}$), testing whether the collapse threshold $\Hstar = 0.5241$ and the three-network decomposition methodology established for the Roman--Byzantine case generalize to a mechanistically distinct imperial system. Three complementary networks are constructed from geospatial and historical data: an administrative network ($\Hadm$, 392~prefectures from CHGIS~v6), a geographic network ($\Hgeo$, with node end-dates extended beyond administrative dissolution), and an 81-node Silk Road trade network ($\Htrd$) as a structural control. Edge costs are derived via least-cost path analysis on a $1~\mathrm{km/pixel}$ digital elevation model, modulated by five historical friction channels. $\Hadm$ collapses to zero at 220~CE (phase~G slope $-0.0261~\mathrm{yr}^{-1}$, $R^2 = 0.689$) while $\Hgeo$ simultaneously \emph{increases} to $3.065$, producing a divergence $\Delta H = 3.065$ -- the topological signature of \emph{internal fragmentation}. The cross-network Wasserstein distance $W_\times(\mathrm{Admin},\mathrm{Geo})$ jumps from exactly zero to $737$ at $d=190$~CE, one decade after the Yellow Turban Rebellion, and three long-lived $\beta_1$ cycles matching the Wei, Shu, and Wu domains emerge under the post-184~CE fragmentation model -- thirty years before formal partition. Three independent early-warning indicators (Wasserstein velocity $\dot{W}_2$, correlation length $\xi$, and the Integrated Change Tracker) signal collapse onset 45--50~years before formal dissolution.

Figures

Figures reproduced from arXiv: 2607.09010 by the authors.

Figure 1
Figure 1. FIG. 1. Least-cost path validation. The computed LCP from Dunhuang to Chang’an (94 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Friction channel time series. (a) Effective Channel 1 ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Geographic zonation used for Channels 3 and 5. Left: eight geographic zones defined by coordinate thresholds [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Persistence diagrams for the administrative network [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Active node counts for the three networks, 200 BCE to 220 CE. Administrative and geographic networks are identical [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The three-network decomposition. (a) Persistent entropy time series for all three networks with 95% bootstrap CI. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Chow structural break [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

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Reference graph

Works this paper leans on

56 extracted references · 6 canonical work pages

  1. [1]

    Three-network decomposition.We construct three independent networks— HAdmin, HGeo, and HTrade—from primary geospatial data

  2. [2]

    HGeo increaseswhile HAdmin collapses, with the divergence ∆H(t) = HGeo(t) −H Admin(t) rising from zero at 180 CE to +3 .065 at the end of the period

    Topological signature of internal fragmentation. HGeo increaseswhile HAdmin collapses, with the divergence ∆H(t) = HGeo(t) −H Admin(t) rising from zero at 180 CE to +3 .065 at the end of the period. The cross-network Wasserstein distance W×(Admin,Geo ) confirms this in the full barcode metric. The Three Kingdoms emerge as β1 cycles with a 10 .9× signal-to...

  3. [3]

    Early-warning observables and collapse chronology.Three independent indicators—Wasserstein velocity ˙W2, correlation length ξ, and the Integrated Change Tracker (ICT)—provide an early-warning chronology. ICT ∗ bernal@ugto.mx † delepine@ugto.mx ‡ c.pinedoguadarrama@ugto.mx arXiv:2607.09010v1 [physics.soc-ph] 10 Jul 2026 2 first exceeds its threshold at d =...

    arXiv 2026

  4. [4]

    HTrade is stable across all 420 years suggesting that long-distance trade circuits constitute a topological pattern invariant to the political systems that facilitate them

    Commercial-network stability. HTrade is stable across all 420 years suggesting that long-distance trade circuits constitute a topological pattern invariant to the political systems that facilitate them. The remainder of the paper is organized as follows. Section II describes data sources and network construction. Section III summarizes the five-channel fr...

    1980

  5. [5]

    The log-normal form preserves strict positivity and is the natural model for multiplicative noise on a cost variable [27]

    Channel 2: Stochastic edge noise Channel 2 introduces edge-level stochastic variability to account for the irreducible uncertainty in historical road conditions: C2(e, ξ) = exp εe , ε e ∼ N(0, σ 2 τ),(8) where τ is the route type of edge e and στ is the corresponding standard deviation. The log-normal form preserves strict positivity and is the natural mo...

  6. [6]

    Channel 3: Xiongnu frontier pressure The Xiongnu confederation dominated the northern steppe throughout the early Western Han period, exerting sustained pressure on the Hexi Corridor (Hexi zoulang) and the Longxi region. Channel 3 captures this frontier risk as a cost premium on all edges starting or finalizing in the Xiyu ( λ < 96◦E) or Longxi ( λ < 108◦...

  7. [7]

    Channel 4: Maintenance friction index (MFI) We use the seven imperial census benchmarks preserved in theHan ShuandHou Han Shu[ 11, 12] to construct a Maintenance Friction Index(MFI) that captures the fiscal capacity of the Han state to sustain its road network. Census valuesP(y k) at the seven documented yearsy k are linearly interpolated to all decades: ...

  8. [8]

    Its purpose is to model the progressive severance of inter-regional connectivity after the Yellow Turban Rebellion as a cost penalty

    Channel 5: Territorial fragmentation (H Geo only) Channel 5 is the friction channel exclusive to the geographic network HGeo; it is inactive in HAdmin and HTrade. Its purpose is to model the progressive severance of inter-regional connectivity after the Yellow Turban Rebellion as a cost penalty. Each Guo node is assigned to one of eight geographic zones b...

    1968

Show all 56 references
  1. [9]

    Two phases show statistically significant trends: Phase B (Wu Di expansion, −141to −50BCE): ˆβ1 = +0.0132 yr −1, R2 = 0.674, p = 0.004

    Phase regressions Table XVIII reports the OLS phase regressions for all seven Han historical phases. Two phases show statistically significant trends: Phase B (Wu Di expansion, −141to −50BCE): ˆβ1 = +0.0132 yr −1, R2 = 0.674, p = 0.004. This is the only phase of statistically ...

  2. [10]

    Chow structural break tests Table XIX reports the Chow test (F (t∗) and p-value) for all eight historical breakpoints. HAdmin registers statistically significant structural breaks at seven of the eight predetermined breakpoints (7/8, significance at α = 0 .05): at t∗ ∈ {−33,+9...

  3. [11]

    A large W2 signals a substantial reorganization of the cycle structure; a small W2 signals near-stability

    Rate of topological change The L2 Wasserstein distance W2(d, d+10) measures how much the set of topological cycles changed between two consecutive decades [5]. A large W2 signals a substantial reorganization of the cycle structure; a small W2 signals near-stability. The per-ye...

  4. [12]

    W×(d) measures the full geometric distance between the two barcodes: it is zero when the two networks are topologically identical and grows as they differ

    Cross-network Wasserstein distance The cross-network Wasserstein distance W×(d) =W 2 DAdmin d ,D Geo d (31) compares the Admin and Geo persistence diagrams at the same decade d. W×(d) measures the full geometric distance between the two barcodes: it is zero when the two networ...

  5. [13]

    A large ξ means loops appear at many different spatial scales during the same decade — the network has structure at both local and long-range levels

    Correlation length divergence: topological scale of the Three Kingdoms Thetopological correlation lengthξ(d) is defined as the weighted spread of bar birth values in theβ 1 diagram: ξ(d) = q Var[{bi}nd i=1],(32) where bi are the birth parameters of the nd bars at decade d. A l...

  6. [14]

    Integrated Change Tracker: early-warning chronology The Integrated Change Tracker (ICT) averages three normalized early-warning indicators. Thesusceptibility χ(d) = Var({ℓi(d)}); thecorrelation length ξ(d) is the weighted spread of bar birth values, a proxy for the spatial sca...

  7. [15]

    This is50 years before the formal dissolution(220 CE) and14 years before the Yellow Turban Rebellion(184 CE)

    Early warning( ICT> 0.1): first crossed at d = 170 CE, ICT(170) = 0.118. This is50 years before the formal dissolution(220 CE) and14 years before the Yellow Turban Rebellion(184 CE). At this point, HAdmin shows no statistically significant phase G trend ( ˆβ1 = −0.0040 yr−1, p...

  8. [16]

    This coincides with the decade after Yellow Turban activation, and with the onset of the Admin–Geo divergence ( W× = 737, ∆H = +1.35)

    Elevated risk( ICT> 0.2): first crossed at d = 190 CE, ICT(190) = 0.280. This coincides with the decade after Yellow Turban activation, and with the onset of the Admin–Geo divergence ( W× = 737, ∆H = +1.35). All four indicators (H Admin,H Geo,W ×, ICT) fire simultaneously at t...

  9. [17]

    Active collapse( ICT> 0.35): first crossed at d = 200 CE, ICT(200) = 0.402. This decade sees HAdmin lose four of its five remaining bars, ξAdmin reaching its peak, and the Binder cumulant U4 crossing from negative (disordered) to positive (concentrated), consistent with the co...

  10. [18]

    Relation to B 0 fragmentation A naive expectation would be that the Three Kingdoms partition is captured by β0 = 3 components in HGeo after d = 208 CE. In practice, β0 = 1 throughout all 43 decades in all three networks: the Guo-level geographic network (approximately 170–392 ...

  11. [19]

    Three Kingdoms as emergentβ 1 cycles under the fragmentation model The Wei, Shu, and Wu cycles emerge purely from the topological structure of the post-184 CE geographic network, and correspond to the three geographically coherent sub-regions of China that share a natural inte...

  12. [20]

    Relationship toβ 0 atomization. At the Xian (county) level, the picture is different: βXian 0 explodes from 294 to 1,174 at d = 190 CE (Table XXII), revealing that at the county scale, the Han empire firstatomized— fragmented below the prefecture scale into more than one thous...

  13. [21]

    Quantification of the topological gap: signal-to-noise ratio of the Three Kingdoms cycles The identification of the Three Kingdoms as emergent β1 cycles rests not only on their existence but on their quantitative separationfrom the background of short-lived bars in the persist...

  14. [22]

    HAdmin, HGeo, and HTrade are topologically irreducible observables giving us information on complementary properties of the Han empire

    Three-network are necessary. HAdmin, HGeo, and HTrade are topologically irreducible observables giving us information on complementary properties of the Han empire

  15. [23]

    Early-warning observables detect the collapse 45–50 years before formal dissolution.Three indepen- dent indicators—Wasserstein velocity ˙W2, correlation length ξ, and the Integrated Change Tracker (ICT)—provide a quantified collapse chronology independent of the entropy series

  16. [24]

    Han administrative collapse recorded with high statistical confidence. HAdmin declines at −0.0261 yr−1 (R2 = 0.689, p = 0.0108) in phase G, reaching HAdmin(220) = 0 via a condensation funnel from 24 bars at d= 0 CE to a single continental-scale loop at 220 CE. Seven of eight C...

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Reviewed July 13, 2026 · model on record in the stance chip above.