============================================================================== MVPS -- Three-Layer Mathematical Structure A technical reference for the coherence framework of the MVPS bundle Leonardo Melegassi Catellix Research draft-melegassi-ippm-mvps-bundle author Version: 1.1 Date: 2026-05-20 (v1.0) / 2026-05-20 (v1.1, errata) Status: reference document, accompanies the I-D and the toolkit ============================================================================== ============================================================================== Errata and revisions -- v1.1 ============================================================================== This revision keeps the same path, scope, and section numbering as v1.0 (so the link distributed alongside the I-D continues to resolve), and corrects eight specific imprecisions (E1-E8 below) that surfaced under technical review. No code, no test, and no operational threshold was changed; the reference implementation in app/mvps_layer{1,2,3}.py is unchanged and the 70 unit tests still pass. The revisions are descriptive, not mechanical: they bring the prose of v1.0 into line with what the code actually computes, and they re-label as conjectures / hypotheses what v1.0 framed as theorems. E1. Phi_D calibration constant (Sec. 4.1). The text in v1.0 stated "k = 6.25 calibrates Phi_D = 0.5 at D^2 = chi^2(3, 0.95) = 7.81". This is *arithmetically false*: exp(-7.81 / 6.25) ~= 0.287, not 0.5. The constant in the code (k = 6.25) is correct as the operational decay rate that the toolkit ships with, but the *characterisation* of what k = 6.25 does is corrected in Section 4.1: the half-value Phi_D = 0.5 is attained at D^2 = k * ln 2 ~= 4.33 (i.e. the chi^2(3, ~0.77) quantile). The WATCH (D^2 = 7.81) and ALARM (D^2 = 11.34) thresholds map under k = 6.25 to Phi_D ~= 0.287 and Phi_D ~= 0.163 respectively. Recalibrating k to 11.27 would be required to obtain Phi_D = 0.5 at the chi^2(3, 0.95) quantile; this is *not* shipped, and the choice of k is logged as open question E2 (centroid+temperature recalibration on labelled data). E2. JSD normalisation upper bound (Sec. 2.2.a). v1.0 stated "JSD_norm = JSD / log 2 in [0, 1]". For two distributions, log 2 is the upper bound; for N >= 2 distributions weighted uniformly, the upper bound is log N. The reference implementation already uses the correct general normaliser (jsd / log_2(min(N, |alphabet|)) in mvps_layer1.jsd_normalized); v1.1 reflects this in Sec. 2.2.a. E3. Naming of the K block and the H functional (Secs. 2.4, 3.2). The K block names (Kirchhoff, Clausius, Hamilton) and the H functional are presented in v1.1 as *physics-inspired operational observables*, not as derivations from Noether's theorem or from a Lagrangian. Specifically: - K_1 (Kirchhoff): genuine application of the current-conservation law to a packet-forwarding hop. Kept verbatim. - K_2 (Clausius-like): an operational observable on the *signed change* of Shannon entropy of the path distribution between two snapshots, normalised by the alphabet capacity. It is not the second law of thermodynamics; the second law concerns the monotone direction of entropy in closed systems and does not apply here. - K_3 (Hamilton-like): the coefficient of variation of the H functional over a trailing window. There is no Lagrangian, no canonical coordinates, and no stationary-action principle involved; the analogy is descriptive. - H = -log(C_1 C_2 C_3) is an *operational energy / Boltzmann- weight* construction, not the Hamiltonian of classical mechanics. It is kept because it is mathematically convenient (non-negative, additive across axes, exponential of a probability-like product). E4. Boguna-Krioukov connection (Sec. 2.5.2). The link from Delta < 0 to negative curvature, and thence to hyperbolic embedding of the Internet graph, is *suggestive*, not a theorem. Boguna-Krioukov address graph-topology embeddability; Delta addresses metric-triangle violations on RTT. v1.1 marks the connection as a research hypothesis (E13 in the catalogue). E5. T_1 SRLG fallback (Sec. 3.3). When SRLG metadata is absent, T_1 falls back to a max-IP-share proxy. v1.1 makes explicit that this proxy measures path-overlap, not anti-SPOF, and that the metric label "anti-SPOF" applies *only* in the SRLG-aware regime. E6. Inter-layer relations R1-R5 (Sec. 6). v1.1 labels each of R1-R5 with its evidential status (empirical, conjectured, hypothesised) instead of presenting them uniformly as "formal relations". E7. Canonical scenarios (Sec. 5). The four-row tables (alpha, beta, gamma, delta) are produced by the reference implementation on synthetic bundles constructed for the test harness. They validate the implementation against its specification; they do *not* yet validate either against real Internet measurement at scale. v1.1 raises this caveat from Sec. 8 to the abstract and to a framed warning at the start of Sec. 5. E8. det(Sigma) invariance (Sec. 2.5.5). Phrased in v1.0 as "the conjecture is that det(Sigma) is near- invariant"; in v1.1 the conjectural status is marked in-line (boxed) so a fast reader does not mistake it for an established property. ============================================================================== Abstract. This document specifies, in self-contained mathematical form, the three-layer observability framework that has emerged from the MVPS (Multi-Vantage Path Snapshot) bundle: an instantaneous spatial layer (Layer 1), a temporal-structural layer (Layer 2), and a phase-and- transition layer (Layer 3). Each layer is a finite collection of dimensionless scalars (and, in Layer 3, one categorical variable) whose values lie in [0, 1] or in the unit simplex, are computable from a stream of MVPS bundles using only stdlib arithmetic, and have a rigorous interpretation grounded in classical results: Einstein's causal bound, Shannon entropy, Jensen-Shannon divergence, the Jaccard index, Fiedler's algebraic connectivity, Menger's theorem, the Birkhoff ergodic theorem, the Heisenberg-Gabor uncertainty principle, the Ornstein-Uhlenbeck process, and the critical-slowing- down precursor of Scheffer et al. The intent is not to introduce new mathematics; it is to organise existing mathematics into a framework that is *empirically falsifiable on Internet path data*. The document is structured for a reader with graduate-level training in either applied mathematics, statistical physics, or empirical Internet measurement. No prior exposure to the MVPS bundle is required beyond the definitions in Section 1. Three honest framings used throughout this document. (a) Names like "Hamilton", "Kirchhoff", "Clausius" appear in the second-layer K-block. They are *physics-inspired observables*: descriptive analogues that borrow notation and intuition from classical mechanics, electrical-circuit theory, and statistical thermodynamics. With the exception of K_1, which is a verbatim application of Kirchhoff's current law to a packet-forwarding hop, these names do *not* denote derivations from Noether's theorem, the second law of thermodynamics, or Hamilton's principle. The H functional H = -log(C_1 C_2 C_3) is likewise an operational "energy / Boltzmann-weight" construction, not the Hamiltonian of classical mechanics. Section 8.9 of this v1.1 makes this explicit. (b) The four canonical scenarios (alpha, beta, gamma, delta) of Section 5 are produced by the reference implementation on synthetic input bundles constructed for the test harness. They validate the *implementation against its specification*; they do not yet validate either against real Internet measurement at scale. Empirical validation on RIPE Atlas / CAIDA Ark / a transit operator's telemetry is open work (catalogue items E2, E12, M16, M24). (c) Several relations in this document are conjectural -- det(Sigma) near-invariance under equilibrium (Sec. 2.5.5), the boundedness of inter-layer relations R2/R4/R5 (Sec. 6), and the formal reduction K_3 <-> tau_CSD (Sec. 6, R3). They are presented as conjectures, not theorems, and are flagged in-line as such in v1.1. Companion artefacts: - draft-melegassi-ippm-mvps-bundle (the data structure) - app/mvps_layer{1,2,3}.py (reference implementation) - tests/test_mvps_layer{1,2,3}.py (70 unit tests, all green) - thesis-kit/PHD-CONTINUATIONS.txt (academic-pipeline catalogue, 115 entry points) ============================================================================== 1. Notation and the MVPS bundle ============================================================================== Throughout, t denotes a discrete tick (one MVPS bundle per tick). B(t) : MVPS bundle at tick t. Formally a JSON object with a vantage list V = {V_1, V_2, ..., V_N}, N >= 2. V_i : vantage i. Has an ordered hop list H_i = (h_i^1, h_i^2, ..., h_i^{L_i}). Each hop carries an IP, an RTT in milliseconds, a geographic anchor in RFC 7946 GeoJSON Point form (when available), an ASN (when available), and optional metadata (timestamps, SRLG memberships, flow counters, MPLS LSE stack). E_i : the edge set of vantage i, E_i = { (h_i^k, h_i^{k+1}) : 1 <= k < L_i }. E_i is the directed sequence of consecutive-hop pairs. d_ab : great-circle distance in metres between two geographic anchors a and b (Haversine formula). c_f : speed of light in the relevant medium. Default c_f = 2.0e8 m/s for terrestrial fibre (refractive index ~ 1.5). Per-region calibration is an open question (see thesis-kit M4). RTT(V_i): one-way round-trip time observed by vantage i. The Einstein causal bound below uses pairwise RTT sums. x_+ : max(x, 0). 1[P] : indicator of predicate P. log : natural logarithm. log_2 in entropic expressions when explicitly stated; otherwise log = ln. All layer scalars are dimensionless and lie in [0, 1] unless explicitly stated. The single non-scalar output of the entire framework is Layer-3's phase label Phi_K, which lies in a finite set of operational regime names. ============================================================================== 2. Layer 1 -- instantaneous spatial coherence (C_1, C_2, C_3) ============================================================================== Layer 1 quantifies the agreement, at one instant, between the N >= 2 independent observers that produced B(t). It has three primary axes (C_1, C_2, C_3), a Hamiltonian H built from them, and six hidden structural observables that live *between* the three axes. --------------------------------------------------------------------------- 2.1 C_1 -- causal coherence (Einstein + entropy) --------------------------------------------------------------------------- Definition. C_1 = min(C_1^Einstein, C_1^tau), where: C_1^Einstein = 1 - (1/M) * sum_{(a,b) : a < b} 1[ RTT(V_a) + RTT(V_b) < 2 * d_ab / c_f ] C_1^tau = exp( - H_v ), H_v = -sum_p p_log p with M the number of ordered vantage pairs (a, b) with a < b considered, and H_v the Shannon entropy of the empirical distribution of *path fingerprints* observed in a moving window. Interpretation. C_1^Einstein is one minus the fraction of vantage pairs whose sum of RTTs falls below 2 d_ab / c_f, the *minimum physically possible* round-trip time given the geographic separation. A violation of this bound is not "noise"; it is a violation of special relativity in the assumed medium (most likely indicating a wrong geographic anchor, a wrong c_f for the link in question, or an instrumentation error). C_1^tau is the exponential of the negative entropy of the fingerprint distribution: when the same fingerprint dominates the window (H_v -> 0), C_1^tau -> 1; as the network produces increasingly diverse fingerprints, H_v grows and C_1^tau collapses. The min(...) is deliberate: a violation of either the physical bound *or* the temporal-stability bound deserves to depress C_1. Range. C_1 in [0, 1]. C_1 = 1 iff (i) no pair violates 2 d_ab / c_f and (ii) one fingerprint is observed deterministically in the window. --------------------------------------------------------------------------- 2.2 C_2 -- informational coherence (Jensen-Shannon / match-rate) --------------------------------------------------------------------------- Two production regimes: 2.2.a Continuous-distribution regime. Let p_v be the empirical hop-distribution of vantage v (a probability mass function over hop IPs visited in the bundle). Define M = (1/N) sum_v p_v (the centroid distribution). JSD( {p_v} ) = (1/N) sum_v KL( p_v || M ) where KL is Kullback-Leibler divergence. JSD is non-negative and symmetric (Lin 1991, IEEE Trans. Info. Theory, Theorem 3) and is uniformly bounded *above* in two distinct ways depending on N: (i) for N = 2 distributions: JSD <= log 2 ; (ii) for N >= 2 distributions weighted uniformly: JSD <= log N ; (iii) when distributions are supported on a finite alphabet A: JSD <= log( min(N, |A|) ), which is the tightest of the three for typical bundles where |A| (distinct hop IPs observed) is small. The reference implementation uses normaliser (iii) (function ``jsd_normalized_from_distributions`` in app/mvps_layer1.py: ``jsd / log_2(min(N, |alphabet|))``). In particular, C_2 is computed as C_2 = 1 - JSD_norm( {p_v} ), JSD_norm = JSD / log_2( min(N, |A|) ) in [0, 1]. (Errata note for v1.0 readers: v1.0 stated ``JSD / log 2`` in this section, which is correct only for N = 2 vantages. The code already used the general normaliser; only the prose in v1.0 was wrong. See E2 in the errata list at the head of this document.) 2.2.b Point-wise canonical-fingerprint regime (F001 lab). When the workload is a controlled corpus of canonical fingerprint comparisons (as in the differential-testing harness), C_2 reduces to the match rate: C_2 = (1 / C(N,2)) * sum_{i < j} 1[ fp(V_i) = fp(V_j) ] The two regimes coincide at C_2 = 1 when all vantages agree point-wise. Range. C_2 in [0, 1]. --------------------------------------------------------------------------- 2.3 C_3 -- topological coherence (Jaccard on edge sets) --------------------------------------------------------------------------- Definition. C_3 = (1 / C(N,2)) * sum_{i < j} | E_i intersect E_j | / | E_i union E_j | Interpretation. C_3 is the mean pairwise Jaccard index of the *edge sets* of the N observed paths. C_3 = 1 iff all vantages traversed the same directed sequence of router-router edges. C_3 close to 0 indicates either heavy ECMP / multipath, anycast convergence on a distant node, or a topological event (BGP re-route). Why edges and not nodes. The Jaccard on nodes is degenerate when all paths share a destination (they always do): the destination hop forces a non-trivial overlap. Edges respect the directionality of traversal. Range. C_3 in [0, 1]. --------------------------------------------------------------------------- 2.4 Hamiltonian (operational) and residuals --------------------------------------------------------------------------- Operational H functional. H(t) = -log C_1(t) - log C_2(t) - log C_3(t) = -log( C_1(t) * C_2(t) * C_3(t) ) H is non-negative; H = 0 iff all three coherences are saturated at 1. H increases as any coherence collapses. Interpreting C_i = exp(-E_i) as a Boltzmann factor with "energy" E_i = -log C_i, the product C_1 * C_2 * C_3 = exp(-H) becomes a Boltzmann weight of the joint state. The framework then admits a partition-function- like form Z(beta) = sum_states exp(-beta * H) over a finite catalogue of reference states (the seven phases of Layer 3); this is exploited in Section 4. Naming caveat (v1.1). H here is *not* the Hamiltonian function of classical mechanics. There is no Lagrangian, no canonical coordinates, and no stationary-action principle behind the present definition. H is an operational energy / Boltzmann-weight construct chosen for three reasons: (a) H >= 0, with H = 0 only at the joint ideal state; (b) H is additive across axes, so collapses on any single axis aggregate without cancellation; (c) the exponential of -H is dimensionless and lies in (0, 1], which fits the partition- function machinery of Section 4. Calling H the "Hamiltonian of the system" is an analogy, not a derivation. Residuals. Predictor: C_hat_i(t) = alpha * C_i(t-1) + (1 - alpha) * C_hat_i(t-1) Residuals: S_i(t) = C_i(t) - C_hat_i(t) EWMA with default alpha = 0.3. S_i(t) is the one-step forecast error; its first two moments are the operational inputs to the six hidden observables and to Layer 2 / Layer 3. --------------------------------------------------------------------------- 2.5 Six hidden Layer-1 observables --------------------------------------------------------------------------- These quantities lie *between* the three axes; they are not functions of any single C_i. 2.5.1 rho -- gauge fidelity. rho = C_2 / C_3. Interpretation. C_2 measures fingerprint agreement; C_3 measures topological agreement. Their ratio quantifies how faithful the fingerprint *function* is to the underlying topology. rho = 1 indicates a bijective gauge; rho = 0 indicates a "gauge gap" (e.g. F001: two distinct implementations produce different fingerprints for the *same* observed topology). rho >> 1 would indicate over-canonicalisation (impossible under correct C_2, used as a sanity check). 2.5.2 Delta -- triangle defect (informational curvature). For three vantages a, b, c: Delta_abc = ( RTT_ab + RTT_bc - RTT_ac ) / RTT_ac Delta is the relative violation of the metric triangle inequality. In Euclidean geometry Delta >= 0 is identically true; in real networks Delta < 0 occurs *systematically* (BGP can route a->c more cheaply than a->b->c). The systematic deviation is the operational analogue of *negative curvature*. Research hypothesis (not theorem). The hyperbolic-embedding literature for the Internet graph (Boguna-Krioukov, Nature Comm. 2010 [Boguna-Krioukov-2010]) treats *graph topology*, not RTT-metric triangle violations. Whether Delta < 0 measured on RTT correlates monotonically with the negative curvature of the hyperbolic embedding is a research hypothesis (catalogue entry E13), not a derivation. The connection drawn here is suggestive and is offered as motivation for the choice of observable, not as a proof. 2.5.3 R -- reciprocity violation. For an ordered pair of vantages (a, b): R_ab = | OWD(a -> b) - OWD(b -> a) | / ( OWD(a -> b) + OWD(b -> a) ) where OWD is one-way delay. R measures the asymmetry of the forward and reverse paths, which is the dominant source of systematic error in classical traceroute interpretation. R = 0 iff the network is reciprocity-symmetric (rare in BGP-routed systems). R is directly observable on OWAMP / TWAMP-instrumented vantages (RFC 4656, RFC 5357). 2.5.4 Sigma -- residual covariance tensor. Sigma_{ij}(t) = Cov( S_i, S_j )_{tau in window} , i, j in {1, 2, 3}. Sigma is the 3x3 symmetric covariance matrix of the EWMA residuals over a trailing window. It is accumulated online via Welford's algorithm (Welford 1962, Technometrics) so the memory cost is O(1) and the numerical stability is provable. 2.5.5 det(Sigma) -- phase volume of the residual. The determinant of Sigma is the squared volume of the uncertainty ellipsoid of (S_1, S_2, S_3). +-------------------------------------------------------------+ | CONJECTURE T1 (not theorem; not yet empirically supported | | at scale). | | | | det(Sigma) is conjectured to be near-invariant under | | network equilibrium, and to jump discontinuously at | | structural phase transitions. Empirical falsification of | | this conjecture, on either RIPE Atlas or a transit | | operator's telemetry, is doctoral-scale work (catalogue | | entry D8). The toolkit *measures* det(Sigma) honestly and | | exposes it; it does *not* claim that the conjectured | | invariance has been demonstrated. | +-------------------------------------------------------------+ 2.5.6 sigma_t * sigma_f >= 1 / (4*pi) -- Heisenberg-Gabor. Let sigma_t be the RMS temporal dispersion of an observation window and sigma_f the RMS spectral dispersion of its Fourier transform. Gabor (1946) proved that for any L^2 signal: sigma_t * sigma_f >= 1 / (4 * pi) This is the *physical* floor on joint time-frequency observability. For MVPS, it bounds the precision with which a "tick rate G" (the equilibrium sampling cadence) and a "regime change frequency" can simultaneously be resolved. The bound is independent of instrumentation cost; it is a mathematical fact about the L^2 inner product. The six observables together turn Layer 1 from a 3-vector into a fully-equipped geometric structure: three coordinates, a Boltzmann weight, a metric tensor on the noise (Sigma), and a fundamental observability bound. ============================================================================== 3. Layer 2 -- temporal and structural coherence (D, K, T) ============================================================================== Layer 2 is the natural derivative of Layer 1 along *time* and *graph structure*. It is exactly 9 scalars, organised as 3 x 3. There is intentionally no scalar aggregate ("MVCI v2"); the canonical object is the 9-vector. Combining axes with different units into a single weighted average was the structural design flaw of the legacy MVCI v1.x (where four of the seven first axes did not even compute the quantity their public glossary claimed). --------------------------------------------------------------------------- 3.1 Block D -- dynamics (regularity prerequisites) --------------------------------------------------------------------------- D_1 -- Lipschitz regularity. L_hat(t) = sup_{i < j} || C(t_j) - C(t_i) ||_2 / (t_j - t_i) over the supplied trajectory window. With saturation sat: D_1 = 1 - min( L_hat / sat , 1 ) Theoretical anchor: Arzela-Ascoli (Rudin 1976, Theorem 7.25) -- a family of continuous functions is relatively compact in the uniform topology iff it is pointwise bounded and equicontinuous. Equicontinuity = uniform Lipschitz. D_1 -> 0 therefore signals that the C-trajectory leaves the function space C^0_Lip and enters a regime where derivatives and OLS slopes are statistically meaningless. D_1 is a *gatekeeper*: without D_1 sufficiently high, neither K nor T is statistically meaningful. D_2 -- temporal causality (Lorentz hop order). For each vantage v and consecutive hops (h_v^k, h_v^{k+1}): D_2 = 1 - (1/M) * sum 1[ arrival_ts(h_v^{k+1}) < arrival_ts(h_v^k) ] where M is the count of consecutive-hop pairs with valid timestamps. A violation is a clock-skew, retransmission, or reverse-path observation. D_2 = 1 means all observed arrivals respect their light-cone order. D_3 -- ergodicity (Birkhoff). D_3 = exp( - || _t - _v ||_2 ) where _t is the time-average of the Layer-1 vector over a trailing window from a single vantage, and _v is the space-average over vantages at the current instant. The Birkhoff ergodic theorem (1931) states that for an ergodic measure-preserving system, time-averages converge almost- surely to space-averages. D_3 = 1 therefore signals operational ergodicity (multi-vantage observation is exchangeable with prolonged single-vantage observation). D_3 << 1 reveals systematic non-ergodicity, which is itself a regime, not a measurement defect. --------------------------------------------------------------------------- 3.2 Block K -- physics-inspired observables (Kirchhoff, Clausius- like, Hamilton-like) --------------------------------------------------------------------------- K is the *physics-inspired* block: three observables whose names borrow from electrical-circuit theory, statistical thermodynamics, and classical mechanics. Of the three, *only K_1 is a verbatim application of a physical conservation law* (Kirchhoff's current law to a packet-forwarding hop). K_2 and K_3 are descriptive analogues; they are useful operational observables, but they are not derivations from Noether's theorem, the second law of thermodynamics, or Hamilton's principle. v1.1 of this document makes that distinction explicit (see also Sec. 8.9). K_1 -- Kirchhoff (flow conservation). For each hop h with byte counters bytes_in(h) and bytes_out(h): K_1 = exp( - (1/H) * sum_h |bytes_in(h) - bytes_out(h)| / max(bytes_in(h), bytes_out(h)) ) where H is the number of hops with valid flow metadata. Justification. The Kirchhoff current law (1845) applied to a packet-forwarding hop: in the absence of drops, generation, or observation gaps, bytes_in = bytes_out. K_1 = 1 iff every observed hop balances; K_1 << 1 quantifies the mean relative imbalance. K_1 = None (unmeasurable) when no hop carries flow counters. K_2 -- Clausius-like (delta-entropy of the path distribution). With H_paths(t) the Shannon entropy of the empirical distribution of paths in B(t): K_2 = exp( - | H_paths(t) - H_paths(t-1) | / log N_paths_max ) where N_paths_max = max distinct paths over the two snapshots. Operational interpretation. K_2 is a *normalised absolute change* of Shannon entropy between two consecutive snapshots, scaled by the alphabet capacity. K_2 ~ 1 when the network's path distribution is approximately stationary across the two snapshots (low entropy production); K_2 << 1 when the distribution is reorganising (BGP UPDATE storm, route flap, anycast re-anchoring). Naming caveat (v1.1). The label "Clausius" is descriptive: it evokes the original 1865 statement that entropy change in a closed system is non-negative. K_2 is *not* a statement about monotone direction in a closed system, and an Internet path distribution is not a closed thermodynamic system. K_2 is a symmetric (absolute-value) fluctuation observable; it equally fires on entropy increases and entropy decreases. We retain "Clausius" only as a memorable mnemonic. K_3 -- Hamilton-like (coefficient of variation of H). Over a trailing window of size W: K_3 = exp( - std(H) / mean(H) )_{t-W+1..t} Operational interpretation. K_3 is a monotone transform of the coefficient of variation (std / mean) of the H functional over a trailing window. K_3 -> 1 when std(H) << mean(H) (the H trajectory is approximately constant, i.e. the joint coherence state is stable); K_3 -> 0 when std(H) is comparable to or exceeds mean(H) (high relative dispersion of H). By construction, K_3 responds to the same regime shifts that fire tau_CSD in Layer 3, but via a different mathematical object (variance of H vs. slopes of variance and autocorrelation of the individual axes). The two are not identical; their formal relation is open question T8. Naming caveat (v1.1). The label "Hamilton" is, again, a mnemonic. There is no Lagrangian L, no action integral S = integral L dt, and no canonical pair (q, p) anywhere in K_3. Hamilton's principle (1834) characterises classical motion as the stationary point of an action integral, which is a variational statement about *trajectories of canonical coordinates*; K_3 is none of those things. We retain "Hamilton" only because the H functional it operates on is itself *named* (operationally) Hamiltonian; see Sec. 2.4 caveat. --------------------------------------------------------------------------- 3.3 Block T -- structural topology (invariants of the graph) --------------------------------------------------------------------------- Construct the undirected aggregate graph G(B) = (V_G, E_G) whose vertices are hop IPs observed across all vantages and whose edges are the union of consecutive-hop pairs. T_1 -- anti-SPOF (max SRLG share) / T_1' -- max-IP-share fallback. Let f(s) = | { v in vantages : path(v) traverses SRLG s } | / | vantages |. T_1 = 1 - max_s f(s) A Shared Risk Link Group (SRLG) is a set of links whose simultaneous failure is correlated (a shared underlying duct, a shared power plane, a shared submarine cable). T_1 quantifies how much of the topology is exposed to a single correlated failure. The label "anti-SPOF" applies to T_1 *only when SRLG metadata is present*. Fallback (v1.1 honest framing). When SRLG membership is unavailable, the implementation falls back to T_1' = 1 - max_h (n_vantages_through_h / N), the maximum IP-share across vantages. T_1' measures *path overlap* on a single hop IP, not correlated-failure exposure: two vantages may share an IP without sharing the underlying physical risk group, and two vantages may share an SRLG without sharing any IP. The metric label "anti-SPOF" therefore does *not* apply to the fallback; the toolkit reports T_1' alongside an explicit ``mode = "ip_share_fallback"`` flag and v1.1 of this document recommends downstream consumers gate on that flag before interpreting T_1 as anti-SPOF. T_2 -- algebraic connectivity (Fiedler ratio). Let L = D - A be the graph Laplacian, with eigenvalues 0 = lambda_1 <= lambda_2 <= ... <= lambda_n. Define: T_2 = lambda_2 / lambda_max. Theoretical anchor. Fiedler (1973, Czech. Math. J.) proved that lambda_2 > 0 iff G is connected, and that lambda_2 is a measure of "how connected" G is. Cheeger's inequality (1970, Princeton) yields lambda_2 / 2 <= h(G), where h(G) is the isoperimetric (conductance) constant: the *minimum fraction* of edges that must be cut to bisect the graph. Therefore T_2 is a lower bound on the difficulty of cutting the network into disjoint pieces. T_3 -- path diversity (Menger). For destination pairs (s, d) in G: T_3 = (1 / P) * sum_{(s,d)} mincut_edge_disjoint(s, d) / target_max_cut where P is the number of pairs evaluated and target_max_cut is an operational benchmark (e.g. 4 for Tier-1 transit). Theoretical anchor. Menger's theorem (1927, Fund. Math.): the maximum number of edge-disjoint s-d paths equals the minimum edge cut separating s from d. T_3 is a normalised mean of edge-disjoint cuts; high T_3 means many parallel paths between any pair of points; low T_3 means a single cut can isolate flow. The 9-vector (D_1, D_2, D_3, K_1, K_2, K_3, T_1, T_2, T_3) is the canonical Layer-2 object. No scalar aggregate is published; the discrimination lives in the *signature* across the nine axes (see Section 5). ============================================================================== 4. Layer 3 -- phase and transition (Phi, tau, omega) ============================================================================== Layer 3 projects the continuous Layer-1 state into a *discrete* phase Phi and supplies two real-valued statistics that quantify transition precursors (tau) and classification coherence (omega). The letter Phi follows the statistical-mechanics tradition: a phase is a region of state space where qualitative properties are homogeneous, whose boundary is a singularity of a partition function, and whose transitions are preceded by *critical slowing down* (Scheffer et al., Nature 461:53, 2009). --------------------------------------------------------------------------- 4.1 Block Phi -- detection (where are we now?) --------------------------------------------------------------------------- Reference centroids: a finite catalogue {mu_k : k = 1..K} of phases in the [0, 1]^3 cube. The default K = 7, with engineering priors: BASELINE (0.95, 0.65, 0.95) PHYSICAL_DEGRADATION (0.55, 0.55, 0.80) BGP_ROUTING_INSTABILITY (0.90, 0.85, 0.35) CONGESTION (0.70, 0.45, 0.92) CAUSAL_BREACH (0.40, 0.60, 0.75) SILENT_FAILURE (0.88, 0.55, 0.88) HOSTILE_COORDINATED (0.50, 0.80, 0.50) Operational K_* labels: K_BASELINE, K_DRIFT, K_INFORMATIONAL, K_STRUCTURAL, K_CAUSAL_BREACH, K_SILENT_FAILURE, K_HOSTILE. Phi_D -- Mahalanobis distance to the BASELINE centroid. D^2(t) = (C(t) - mu_BASELINE)^T Sigma_BASELINE^{-1} (C(t) - mu_BASELINE) Phi_D = exp( - D^2 / k ), default k = 6.25. What k = 6.25 actually does (v1.1, corrected). The shipped default is k = 6.25. Under this default: Phi_D = 0.5 when D^2 = k * ln 2 ~= 4.331 which is the chi^2(3) quantile at p ~ 0.7723. At D^2 = 7.81 (chi^2(3, 0.95), the operational WATCH threshold): Phi_D = exp(-7.81 / 6.25) ~= 0.287. At D^2 = 11.34 (chi^2(3, 0.99), the operational ALARM threshold): Phi_D = exp(-11.34 / 6.25) ~= 0.163. Under the null hypothesis H_0 (network is in BASELINE and Sigma_BASELINE is correctly specified), D^2 ~ chi^2(3); the false-alarm rates of the WATCH (D^2 >= 7.81) and ALARM (D^2 >= 11.34) thresholds remain 0.05 and 0.01 respectively *because the thresholds are stated on D^2*, not on Phi_D. What v1.0 of this document said (incorrectly). v1.0 stated "k is calibrated so that Phi_D = 0.5 at D^2 = chi^2(3, 0.95) = 7.81". That sentence is arithmetically false: Phi_D = 0.5 at D^2 = 7.81 would require k = 7.81 / ln 2 ~= 11.27, not 6.25. The reference implementation (app/mvps_layer3.phi_d_mahalanobis) uses k = 6.25 and is internally consistent; only the *characterisation* of what k = 6.25 does was wrong in v1.0. Operational implication. k = 6.25 is the shipped default, *not* a derived constant. It produces an aggressive (steep) decay of Phi_D as D^2 grows: Phi_D ~ 0.29 at the WATCH boundary and Phi_D ~ 0.16 at the ALARM boundary, which is operationally usable when Phi_D is interpreted as a "soft confidence in BASELINE" rather than as a calibrated probability. Recalibrating k to obtain Phi_D = 0.5 exactly at the chi^2(3, 0.95) quantile, or to a different chi^2 quantile, is part of open question E2 (centroid + temperature recalibration on labelled production data). Phi_P -- Bayesian posterior over phases. P(phi = k | C) = pi_k * N(C; mu_k, Sigma_k) / sum_j pi_j * N(C; mu_j, Sigma_j) where pi_k is the prior on phase k (uniform unless overridden) and Sigma_k is the per-phase covariance (default isotropic). Implementation uses a numerically stable log-sum-exp softmax. Phi_P = max_k P(phi = k | C). Phi_K -- operational label. Phi_K = argmax_k P(phi = k | C), mapped through the class -> K_* table. Honest caveat. The centroid table mu_k was calibrated on *legacy* (v1.x) definitions of (C_1, C_2, C_3). Layer 1 v1 replaced those definitions with the Einstein / JSD / Jaccard triple defined in Section 2. Recalibration on labelled data produced under Layer 1 v1 is open question E2 (six months of labelled bundles; one Master's thesis). --------------------------------------------------------------------------- 4.2 Block tau -- precursor of transition (are we about to switch?) --------------------------------------------------------------------------- tau_CSD -- Critical Slowing Down (Scheffer). For each axis C_i in the window: beta_var = OLS slope of rolling variance of C_i beta_rho1 = OLS slope of rolling lag-1 autocorrelation of C_i CSD = 0.5 * sigmoid(k * beta_var) + 0.5 * sigmoid(k * beta_rho1) with k = 30 (Scheffer 2009). For CSD to exceed 0.5, *both* sigmoids must be > 0.5, which requires *both* slopes to be positive. This AND-gate is what makes CSD falsifiable rather than triggered by single-axis noise. tau_CSD = 1 - CSD. Theoretical anchor. In a dynamical system approaching a bifurcation, the leading eigenvalue of the linearised flow approaches zero from below: the system loses its restoring force. Two consequences (Wissel 1984; Held & Kleinen 2004; Scheffer 2009) are: rising variance (less restoring) and rising lag-1 autocorrelation (slower relaxation). Both occur *before* the bifurcation itself. tau_CSD is the operationalisation of this universal precursor. tau_dphi -- distance to the Voronoi boundary. With D^2_phi the Mahalanobis distance to the winning centroid and D^2_phi' to the runner-up: tau_dphi = | D^2_phi - D^2_phi' | / ( D^2_phi + D^2_phi' ) tau_dphi = 1 means the point is well inside one Voronoi cell; tau_dphi = 0 means the point lies on the boundary between two cells (any perturbation flips the Phi_K label). tau_OU -- Ornstein-Uhlenbeck decorrelation time. With rho_1 the lag-1 autocorrelation of an axis series: T_OU = - delta_t / log(rho_1) tau_OU = exp( - T_OU / T_ref ) where delta_t is the tick spacing and T_ref is a reference operational scale (default T_ref = 4 * G; G is the equilibrium sampling cadence at which the chi^2(3) FAR is calibrated -- empirically G ~ 8.4 min today). Theoretical anchor. An Ornstein-Uhlenbeck process with mean- reversion rate theta has lag-tau autocorrelation exp(-tau * theta); inverting yields T_OU = 1/theta. Networks near a transition have decreasing theta (the loss of restoring force noted above), so T_OU grows and tau_OU collapses. --------------------------------------------------------------------------- 4.3 Block omega -- coherence of the classification (do we trust Phi?) --------------------------------------------------------------------------- omega_H -- 1 - normalised entropy of the posterior. omega_H = 1 - H(P) / log K, H(P) = -sum_k P_k log P_k. omega_H = 1 iff posterior concentrated on one phase; omega_H = 0 iff posterior uniform (classifier indecisive). omega_M -- top-1 minus top-2 margin. omega_M = P_top1 - P_top2 omega_M < 0.05 should trigger manual review (the classifier is effectively in a tie). omega_C -- rolling agreement with the last N labels. omega_C = (1/N) sum_{i=t-N+1..t} 1[ Phi_K(i) = Phi_K(t) ] omega_C = 1 iff the regime has been stable for the last N ticks; omega_C oscillating iff the regime is flapping. The Layer-3 output is the triple (Phi_K, (Phi_D, Phi_P), (tau_CSD, tau_dphi, tau_OU), (omega_H, omega_M, omega_C)). ============================================================================== 5. Canonical scenarios and discrimination ============================================================================== +---------------------------------------------------------------------------+ | WARNING -- nature of validation in this section. | | | | The four scenarios (alpha, beta, gamma, delta) below are synthetic | | bundles constructed for the test harness. The numbers tabulated in | | Sections 5.1, 5.2, 5.3 are produced by the reference implementation | | on those synthetic bundles. They validate the *implementation against | | its specification* (and they do, the 70 unit tests pass), but they do | | *not* validate the framework against real Internet measurement at | | scale. | | | | Empirical validation against RIPE Atlas, CAIDA Ark, or a transit | | operator's telemetry is ongoing work, and is logged as catalogue | | items E2, E12, M16, and M24. A reader who needs *empirical* | | evidence (as opposed to specification-correctness evidence) should | | treat Section 5 as illustrative and look at those catalogue items | | for the data sets, methodology, and timelines. | +---------------------------------------------------------------------------+ Each layer admits four canonical synthetic scenarios that exercise its sensitivities orthogonally. These are reproducible with the reference implementation (Section 7). alpha -- ideal operation beta -- isolated structural failure (local outage, SPOF) gamma -- precursor (rising variance and autocorrelation; Phi_K has not yet flipped) delta -- multi-axis failure (DDoS / flapping) The discrimination matrix below is produced by the reference implementation; values to three decimal places. --------------------------------------------------------------------------- 5.1 Layer 1 -- canonical scenarios --------------------------------------------------------------------------- Scenario | C_1 | C_2 | C_3 | H | C_total ------------------+---------+---------+---------+----------+--------- alpha (ideal) | ~ 1.000 | ~ 1.000 | ~ 1.000 | ~ 0.000 | ~ 1.000 beta (canon gap) | ~ 1.000 | ~ 0.000 | ~ 1.000 | infty | ~ 0.000 gamma (syn shift) | ~ 0.500 | ~ 0.500 | ~ 0.500 | ~ 2.079 | ~ 0.125 delta (desync) | ~ 0.700 | ~ 0.700 | ~ 0.700 | ~ 1.069 | ~ 0.343 Each scenario produces a distinct signature on (C_1, C_2, C_3, H). The "canonicalisation gap" of beta is the F001 lab regime: same topology, different fingerprints -- a rho = 0 / gauge_gap state. --------------------------------------------------------------------------- 5.2 Layer 2 -- canonical scenarios --------------------------------------------------------------------------- Scenario | D_1 | D_2 | K_1 | K_2 | K_3 | T_1 | T_2 | T_3 ---------+-------+-------+-------+-------+-------+-------+-------+------ alpha | 1.000 | 1.000 | 1.000 | 1.000 | - | 0.500 | 0.106 | 0.250 beta | 1.000 | - | 1.000 | 1.000 | - | 0.000 | - | - gamma | 0.293 | 1.000 | 1.000 | 1.000 | 0.407 | 0.500 | 0.106 | 0.250 delta | 1.000 | 1.000 | 0.435 | 0.368 | 1.000 | 0.500 | 0.000 | - Each scenario produces a distinct signature on the 9-vector. Note the *orthogonal* coverage: beta is the only scenario with T_1 = 0; gamma is the only scenario with D_1 << 1 AND K_3 << 1; delta is the only scenario with K_1 << 1, K_2 << 1, AND T_2 = 0. --------------------------------------------------------------------------- 5.3 Layer 3 -- canonical scenarios --------------------------------------------------------------------------- Scenario | Phi_K | Phi_P | Phi_D | tau_CSD | tau_dphi | omega_M | omega_C ----------+--------------+--------+--------+---------+----------+---------+-------- alpha | K_BASELINE | 0.726 | 1.000 | 0.500 | 1.000 | 0.456 | 1.000 beta | K_STRUCTURAL | 1.000 | 0.000 | 0.500 | 0.658 | 1.000 | 1.000 gamma | K_BASELINE | 0.544 | 0.819 | 0.502 | 0.345 | 0.099 | 1.000 delta | K_STRUCTURAL | 0.864 | 0.000 | 0.200 | 1.000 | 0.748 | 0.600 The most operationally consequential cell is gamma's Phi_K = K_BASELINE coupled with omega_M = 0.099 and tau_dphi = 0.345: the classifier *has not yet flipped* but the posterior margin is collapsing and the operating point is approaching the Voronoi boundary. This is precisely the cell where preventive action is still possible. The delta cell's omega_C = 0.600 quantifies regime flapping. ============================================================================== 6. Inter-layer composition ============================================================================== Layer 1 C(t) in [0,1]^3 + H = -log(C_1 C_2 C_3) + six hidden observables | | (time series of L1 vectors) v Layer 2 (D_1, D_2, D_3, K_1, K_2, K_3, T_1, T_2, T_3) in [0,1]^9 | | (D_1 gates K and T -- without | temporal regularity, conservation | axes are statistically meaningless) v Layer 3 Phi(t) discrete + (tau_CSD, tau_dphi, tau_OU) + (omega_H, omega_M, omega_C) Cross-layer relations. v1.1 labels each relation with its evidential status. The five relations below are *not* uniformly theorems; they range from empirically supported to merely conjectured. The honest reader should treat them according to their label. R1 (gating) -- STATUS: empirical, threshold open. D_1 below a threshold L_critical implies that the OLS slopes inside tau_CSD lose statistical meaning (variance and lag-1 autocorrelation slopes are no longer well-defined estimators of restoring force). The qualitative direction is empirically robust on synthetic CSD scenarios (Sec. 5.3 gamma row); the *precise* threshold L_critical is open question T6. R2 (Sigma to D_1) -- STATUS: conjecture (not proven). It is conjectured that the principal eigenvalue of Sigma lower-bounds (L_hat * dt)^2 / 4 in any window of duration dt. A proof *sketch* (not a proof) is in the app/mvps_layer1.py module docstring; a rigorous proof at thesis level is part of D4. Until then, R2 is a conjecture. R3 (K_3 and tau_CSD) -- STATUS: empirical, formal reduction open. Both react to a regime shift, by different mathematical objects: K_3 is the coefficient of variation of H; tau_CSD is the AND of OLS slope sigmoids on (variance, autocorrelation) of individual axes. They are correlated by construction on the synthetic gamma scenario (Sec. 5.3). The *formal* reduction (or formal independence) is open question T8. R4 (T_2 and accessible transitions) -- STATUS: conjecture (not yet tested empirically). Conjecture: with T_2 below a critical value the Phi-trajectory in Layer 3 is restricted to Voronoi-adjacent transitions (no "jumps" across the diagram); above the critical value, the transition graph on Phi is fully connected. Tests over real RIPE Atlas data are open question E12. Until those tests are run, R4 is speculative. R5 (rho and gauge equivalence) -- STATUS: structural claim, proof in progress. rho = C_2 / C_3 detects the gauge gap (rho -> 0) that characterises F001 and similar canonicalisation failures. The structural claim is that Profile v1 ensures rho = 1 modulo Profile-v1-equivalent bundles. The full formalisation (and proof of the equivalence-class statement) is in progress under PHD-CONTINUATIONS.txt D1; the empirical direction (rho collapses on F001-class bundles) is robust on the test harness. ============================================================================== 7. Implementation and reproducibility ============================================================================== Reference implementation is Python 3.10+, stdlib-only, available under permissive licensing alongside the I-D. app/mvps_layer1.py Layer 1 + six hidden observables app/mvps_layer2.py Layer 2 (D / K / T) app/mvps_layer3.py Layer 3 (Phi / tau / omega) tests/test_mvps_layer{1,2,3}.py 70 unit tests (all green) Each module exposes a self-validation entry point that runs the four canonical scenarios and emits the tables of Section 5. No external numerical dependencies: dense symmetric-3x3 eigendecomposition uses a closed-form 3-degree characteristic polynomial inversion; Welford's covariance update is implemented inline; Ford-Fulkerson on unit-capacity graphs is implemented inline for T_3. --------------------------------------------------------------------------- 7.A Visual evidence package (v1.1, synthetic only) --------------------------------------------------------------------------- A reproducible visual companion to this document is hosted at: https://www.catellix.com/v11-evidence.html The page renders nine figures: five clinical cases (BASELINE, BGP_ROUTING_INSTABILITY, CONGESTION, CAUSAL_BREACH, F001_GAUGE_GAP) and four conjecture tests (T1 det(Sigma), tau_CSD AUC precursor, Phi_D FAR under autocorrelation, mutual information of (C_i, C_j)). Each figure has a side-panel of three buttons -- [Theory], [Inputs], [Repro] -- so a reviewer can read the per-figure derivation, the generation parameters (seed, rho, sigma, T), and the reproduction command. Every figure is watermarked SYNTHETIC DATA and the page carries a top-level warning to that effect. Reproducibility receipt. The script ``scripts/render_v11_evidence.py`` is deterministic (fixed seed, stdlib + matplotlib only) and emits a manifest at ``frontend/static/data/v11-evidence.json`` with SHA-256 of every figure. A reviewer who re-runs the script can bit-compare the hashes; any mismatch indicates a regression. Status. The visual package validates the *implementation* against its specification on a synthetic regime where each axis is excited orthogonally. It is not, and is not claimed to be, empirical validation against real Internet measurement; that is open work (E2, E12, M16, M24). ============================================================================== 8. Limitations and honest caveats ============================================================================== 8.1 Centroid calibration. The Phi centroids mu_k are *engineering priors* from the legacy MVCI v1.x catalogue. They were not calibrated on bundles produced under the Layer 1 v1 definitions (Einstein / JSD / Jaccard). Empirical recalibration over six months of labelled production bundles is required before Phi_K is operationally trustworthy. This is open question E2. 8.2 c_f calibration. The default c_f = 2e8 m/s is appropriate for terrestrial fibre; submarine and intra-datacentre links require per-region calibration. Open question M4. 8.3 K_1 / K_2 observability. K_1 and K_2 require byte-level flow metadata (sFlow / NetFlow / IPFIX) per hop, which is not present in most public traceroute datasets. Validation on a real operator's telemetry is open question M16. 8.4 G (equilibrium sampling cadence). The chi^2(3) FAR of Phi_D is calibrated only when consecutive ticks are approximately independent. G makes the regime explicit (today G ~ 8.4 min). For cadences finer than G a HAC / batch-means correction is required and is not yet shipped. 8.5 T_2 / T_3 scale. The Laplacian eigendecomposition is dense and caps at n = 128 nodes. Beyond this an external eigensolver is required (LAPACK ssyevd, ARPACK Lanczos for the leading eigenpair). 8.6 Sigma estimator robustness. The Welford accumulator is the maximum-likelihood Gaussian estimator and breaks down under outliers. A robust replacement (MCD, Tyler M-estimator) is the subject of M10. 8.7 Conjecture T1. det(Sigma) is *conjectured* to be a near- invariant under network equilibrium. The conjecture is not proven; empirical evidence at scale is open work. 8.8 Number of phases K. The choice K = 7 is the legacy engineering prior. Justification via BIC / AIC / silhouette on real data is open question M24. 8.9 Physics-inspired naming (K block and H functional). With the sole exception of K_1 (verbatim Kirchhoff's current law on a packet-forwarding hop), the names "Kirchhoff", "Clausius", "Hamilton" used in the K block, and the label "Hamiltonian" used for H = -log(C_1 C_2 C_3), are physics-inspired analogues, not derivations from Noether's theorem, the second law of thermodynamics, or Hamilton's principle. They were chosen for mnemonic value and operational descriptiveness. A reader expecting a Lagrangian, an action integral, or a strict statement of entropy monotonicity in a closed system will *not* find one in this framework, and should not expect to. 8.10 Phi_D temperature constant k. The shipped default k = 6.25 is *not* a calibrated constant in the rigorous sense; it is an engineering prior. Under k = 6.25, Phi_D = 0.5 occurs at D^2 ~= 4.33 (chi^2(3, ~0.77)), and at the WATCH/ALARM thresholds (D^2 = 7.81 / 11.34) Phi_D evaluates to ~0.287 / ~0.163. v1.0 of this document mis-stated this calibration; v1.1 corrects it (Sec. 4.1, errata E1). Whether to recalibrate k to 11.27 (so that Phi_D = 0.5 at the WATCH boundary) or to leave k = 6.25 as a conservative aggressive-decay prior is part of open question E2. 8.11 Empirical validation. Sections 5.1, 5.2, 5.3 are produced on *synthetic* bundles. They demonstrate specification correctness of the implementation, not empirical validity of the framework against real Internet measurement. Validation against RIPE Atlas / CAIDA Ark / a transit operator's telemetry is open work (catalogue items E2, E12, M16, M24). v1.0 stated this in Sec. 8; v1.1 raises it to the abstract and to a framed warning at the head of Sec. 5. 8.12 T_1 anti-SPOF semantics. The metric label "anti-SPOF" applies to T_1 only when SRLG metadata is available (Sec. 3.3). The max-IP-share fallback T_1' measures path overlap on a single hop IP, which is *not* the same observable; consumers of T_1 should gate on the toolkit's ``mode = "ip_share_fallback"`` flag before interpreting T_1 as anti-SPOF. ============================================================================== 9. References ============================================================================== [Boguna-Krioukov-2010] Boguna, M., Papadopoulos, F., Krioukov, D. "Sustaining the Internet with hyperbolic mapping". Nature Communications 1:62 (2010). [Birkhoff-1931] Birkhoff, G. D. "Proof of the Ergodic Theorem". Proc. Nat. Acad. Sci. 17:656-660 (1931). [Cheeger-1970] Cheeger, J. "A lower bound for the smallest eigenvalue of the Laplacian". In Problems in Analysis, Princeton University Press, pp. 195-199 (1970). [Fiedler-1973] Fiedler, M. "Algebraic connectivity of graphs". Czech. Math. J. 23:298-305 (1973). [Gabor-1946] Gabor, D. "Theory of communication". J. Inst. Elec. Eng. 93:429-457 (1946). [Held-Kleinen-2004] Held, H., Kleinen, T. "Detection of climate system bifurcations by degenerate fingerprinting". Geophys. Res. Lett. 31:L23207 (2004). [Lin-1991] Lin, J. "Divergence measures based on the Shannon entropy". IEEE Trans. Info. Theory 37:145-151 (1991). [Menger-1927] Menger, K. "Zur allgemeinen Kurventheorie". Fund. Math. 10:96-115 (1927). [Poincare-1908] Poincare, H. "Science et Methode". Flammarion, Paris (1908). See Livre I, Chapitre II. [RFC-4656] Shalunov, S., Teitelbaum, B., Karp, A., Boote, J., Zekauskas, M. "A One-way Active Measurement Protocol (OWAMP)". RFC 4656 (2006). [RFC-5357] Hedayat, K., Krzanowski, R., Morton, A., Yum, K., Babiarz, J. "A Two-Way Active Measurement Protocol (TWAMP)". RFC 5357 (2008). [RFC-5952] Kawamura, S., Kawashima, M. "A Recommendation for IPv6 Address Text Representation". RFC 5952 (2010). [RFC-7946] Butler, H., Daly, M., Doyle, A., Gillies, S., Hagen, S., Schaub, T. "The GeoJSON Format". RFC 7946 (2016). [RFC-8785] Rundgren, A., Jordan, B., Erdtman, S. "JSON Canonicalization Scheme (JCS)". RFC 8785 (2020). [Rudin-1976] Rudin, W. "Principles of Mathematical Analysis", 3rd ed. McGraw-Hill (1976). [Scheffer-2009] Scheffer, M. et al. "Early-warning signals for critical transitions". Nature 461:53-59 (2009). [Welford-1962] Welford, B. P. "Note on a method for calculating corrected sums of squares and products". Technometrics 4:419-420 (1962). [Wissel-1984] Wissel, C. "A universal law of the characteristic return time near thresholds". Oecologia 65:101-107 (1984). ============================================================================== End of document. v1.0 2026-05-20 Andradina, SP, Brazil. (initial release) v1.1 2026-05-20 Andradina, SP, Brazil. (errata: Phi_D calibration prose, JSD upper bound, K-block naming framing, R1-R5 evidential status, synthetic-scenario warning, T_1 fallback framing, det(Sigma) conjecture box.) This document is the technical reference companion to draft-melegassi-ippm-mvps-bundle. It is intended for reviewers with graduate-level training in either applied mathematics, statistical physics, or empirical Internet measurement. Source of truth: app/mvps_layer{1,2,3}.py in the toolkit. "La mathematique est l'art de donner le meme nom a des choses differentes." -- Henri Poincare ==============================================================================