============================================================================== MVPS TERRESTRIAL MOBILE / VEHICULAR PROFILE -- MATHEMATICAL PROOF (Coherence monitoring across vantages moving on land over cellular (5G/LTE) links with handovers. Third member of the space/sea/land trio. Companion to draft-melegassi-ippm-mvps-terrestrial-mobile-00.) "On land the sky is not jammed; the network moves under your feet. The enemy of coherence here is the handover, not the clock." ============================================================================== DEFENSIVE SCOPE. This profile is about DETECTING anomalies in the network / coherence telemetry of mobile terrestrial vantages (intrusion, comms tampering, rogue base stations). It contains NO navigation, targeting, or kinetic function and emits none. Mathematical assembly: Leonardo Melegassi (Catellix) Produced: 2026-05-28 Companion: docs/draft-melegassi-ippm-mvps-terrestrial-mobile-00.txt Inherits: docs/MVPS_ARCH_PROOF.txt (D-8 Architecture-Invariance Theorem) Core: docs/MVPS_MATHEMATICAL_EXISTENCE_PROOF_V4.txt (T1, T2, T9, T3') Siblings: docs/MVPS_MARITIME_EDGE_PROOF.txt (D-18, sea), draft-melegassi-ippm-mvps-orbital-coherence-00 (D-7, space) Method: docs/MVPS_METHODOLOGY_PROOF.txt (claim discipline M-1..M-9) Numerical receipt: scripts/validate_terrestrial_mobile.py (exit 0, 7/7) evidence/terrestrial_mobile_receipt.json ============================================================================== 0. THE SPACE / SEA / LAND TRIO ============================================================================== MVPS promotes its theorems to any surface satisfying axioms A1..A5 (D-8). Three domain profiles re-establish the one axiom each domain stresses -- A1, bounded joint clock skew -- and inherit the rest: SPACE (D-7, orbital) : propagation delay + Doppler over LEO links. SEA (D-18, maritime): DIL connectivity + GNSS-denied holdover. LAND (D-19, this) : cellular HANDOVER + RAN scheduling jitter. On land GNSS is normally AVAILABLE, so holdover drift (the maritime problem) is negligible. The terrestrial difficulty is mobility through the radio access network: a vantage hands over between base stations, its path and latency change, and slot-based scheduling adds jitter. ============================================================================== 0.1 IMPORTS ============================================================================== Core v4.0 basis I1..I12, in particular I12 (Minsker/Cohen, geometric-median max-bias) for A5. Standard cellular timing model: Q1. A GNSS-disciplined base station holds time to a residual eps_sync; the UE is aligned to the base station by Timing Advance to a residual eps_ta. Q2. Slot-based scheduling bounds delivery jitter by tau_jit (5G numerology mu: slot = 1ms / 2^mu; LTE ~ 1 ms). Q3. A handover induces a bounded residual mis-timing tau_ho; make-before-break (NR DAPS) gives tau_ho ~ 0, break-before-make gives tau_ho ~ interruption time. ============================================================================== 1. DEFINITIONS ============================================================================== F-T1. eps_sync : GNSS/PTP residual at the base station (s). F-T2. eps_ta : timing-advance residual of UE alignment (s). F-T3. tau_jit : RAN scheduling jitter (s). F-T4. tau_ho : residual mis-timing during/after a handover (s). F-T5. v : ground speed of the vantage (m/s). F-T6. EFFECTIVE JOINT SKEW under terrestrial mobility: skew_eff = 2*(eps_sync + eps_ta) + tau_jit + tau_ho . F-T7. T_tick : the deployment coherence tick (s). ============================================================================== 2. LEMMAS AND THEOREMS ============================================================================== ------------------------------------------------------------------------------ L-TER-1 [T]. (A1 under cellular mobility.) If skew_eff = 2*(eps_sync + eps_ta) + tau_jit + tau_ho < T_tick, then axiom A1 holds for the terrestrial-mobile deployment on tick T_tick. ------------------------------------------------------------------------------ Proof. Two vantages are each misaligned from base-station time by at most eps_sync + eps_ta (Q1), in opposite directions, contributing 2*(eps_sync + eps_ta) to pairwise skew. Scheduling adds at most tau_jit (Q2) and an in-flight handover adds at most tau_ho (Q3). Summing gives skew_eff; the hypothesis is exactly the A1 inequality on T_tick. [] Validator: L-TER-1 (5G-uRLLC 0.128 ms, LTE 33.0 ms, HSR 54.0 ms; all below their ticks). ------------------------------------------------------------------------------ L-TER-2 [T]. (Maximum tolerable handover interruption.) For fixed eps_sync, eps_ta, tau_jit, T_tick the largest handover residual preserving A1 is tau_ho_max = T_tick - tau_jit - 2*(eps_sync + eps_ta). ------------------------------------------------------------------------------ Proof. Set skew_eff = T_tick and solve for tau_ho; monotonicity (L-TER-3) makes the threshold one-sided. [] Validator: L-TER-2 (5G-uRLLC: tau_ho_max ~ 99.87 ms at a 100 ms tick; round-trips to exactly 0.100000 s). OBSERVE: with make-before-break handover (tau_ho ~ 0) even sub-second ticks have ample margin -- the binding term on land is the handover interruption, which modern NR mechanisms drive toward zero. ------------------------------------------------------------------------------ L-TER-3 [T]. (Monotonicity.) skew_eff is strictly increasing in each of eps_ta, tau_jit, tau_ho. ------------------------------------------------------------------------------ Proof. Partial derivatives 2, 1, 1 are all positive. [] Validator: L-TER-3. ------------------------------------------------------------------------------ L-TER-4 [T]. (Doppler is dominated.) The time uncertainty contributed by Doppler over one tick is (v/c)*T_tick, and at vehicular / high-speed- rail velocities it is below 1% of the RAN jitter tau_jit; hence it is absorbed and need not appear in F-T6. ------------------------------------------------------------------------------ Proof. For a fractional velocity v/c, the accumulated timing error over T_tick is at most (v/c)*T_tick. At v = 300 km/h = 83.33 m/s, v/c = 2.78e-7, so over T_tick = 0.1 s the term is 2.78e-8 s = 27.8 ns, versus tau_jit on the order of 1e-4 s. The ratio is ~3e-4 << 1e-2. [] Validator: L-TER-4 (5G 11.0 ns, LTE 83.4 ns, HSR 27.8 ns; each < 1% of its tau_jit). ------------------------------------------------------------------------------ T-TER-INHERIT [T]. (Inheritance.) If A1 holds on T_tick (L-TER-1) and the compromised-vantage fraction f < 1/2, then the core theorems T1, T2, T3', T9 hold on the terrestrial-mobile surface VERBATIM, by the D-8 Architecture-Invariance Theorem. ------------------------------------------------------------------------------ Proof. A2, A3, A4, A5 are inherited unchanged; A1 is re-established by L-TER-1. The premise set of D-8 is met; promotion is immediate. [] Validator: A-TER-INHERIT (promotes T1, T2, T9, T3' at f = 0.2). ------------------------------------------------------------------------------ B-TER-1 [T]. (Byzantine / spoofed vehicles.) With f < 1/2 of vantages compromised or spoofed, the geometric-median aggregator has finite max-bias b(f) = C*f/(1-2f) (I12), diverging only as f -> 1/2. ------------------------------------------------------------------------------ Proof. Direct from I12 on the vantage observation set; the surface changes neither aggregator nor bound. [] Validator: B-TER-1 (b(0.2)=0.333, b(0.4)=2.000). ============================================================================== 3. CONJECTURE (with falsification protocol -- methodology M-6) ============================================================================== C-TER-1. CONJECTURE: a coordinated rogue / false base-station cluster (an IMSI-catcher fleet) injects a rank-low, correlated timing/identity signature across mobile vantages that the multi-vantage D^2 detector flags BEFORE any single UE crosses its per-node threshold. (a) observable: cross-vantage correlated TA / cell-identity anomaly versus per-UE max-z, on the coherence tick; (b) data source: fleet RAN measurement reports plus a controlled false-base-station testbed; (c) test: one-sided detection-time gain, Wilson 95% lower bound > 0; falsified if the lower bound <= 0; (d) current blocker: licensed spectrum for a controlled rogue-gNB testbed; held as [C] until that receipt exists. This conjecture is NOT promoted to a theorem; the profile's guarantees (L-TER-1..B-TER-1) do not depend on it. ============================================================================== 4. WHAT THIS PROVES / DOES NOT PROVE ============================================================================== Proves: (i) A1 holds on land under explicit cellular-timing budgets (L-TER-1), with a closed-form handover tolerance (L-TER-2); (ii) Doppler is provably dominated at terrestrial speeds (L-TER-4); (iii) the core detection and Byzantine theorems inherit verbatim (T-TER-INHERIT, B-TER-1). Does NOT prove / explicitly disclaims: (a) any rogue-base-station detection guarantee (conjecture C-TER-1); (b) operation when tau_ho >= T_tick - tau_jit - 2*(eps_sync+eps_ta) without enlarging the tick or adopting make-before-break; (c) anything outside DEFENSIVE coherence monitoring -- no kinetic, targeting, or navigation claim exists or is implied. ============================================================================== 5. REFERENCES ============================================================================== [I-D.melegassi-ippm-mvps-terrestrial-mobile] this document's companion. [ARCH] draft-melegassi-iab-mvps-architecture-00 (D-8 invariance). [MVPS-V4] MVPS Mathematical Existence Proof v4.0 (T1, T2, T9, T3'). [MARITIME] draft-melegassi-ippm-mvps-maritime-edge-00 (D-18, sea sibling). [ORBITAL] draft-melegassi-ippm-mvps-orbital-coherence-00 (D-7, space). [I12] Minsker, S.; Cohen, M., geometric-median max-bias. ============================================================================== END OF MVPS TERRESTRIAL MOBILE / VEHICULAR PROFILE MATHEMATICAL PROOF ==============================================================================