Many Nets, One Field

Research thesis, published 2026-09-27; revised 2026-10-02 (see the revision log). Research record: elementalcollision/physarum (private; binding decisions in research/VALIDATED.md and its amendments VALIDATED_R2.md to VALIDATED_R9.md). Nothing described here has been run.

The page that started it

Jenson's write-up describes Jones's model as two coupled layers: an agent layer (the “data map”, particles with a heading, a location and three sensors — front-left, front, front-right) and a continuum layer (the “trail map”, a 2D grid of intensities). Each affects the other: particles deposit onto the trail map and sense it to steer.

Jenson's write-up divides each tick into six sub-steps; Jones's 2011 text gives the same sequence as a sensory stage, a motor stage and a diffusion-and-damping pass: sense the trail field at the three offset sensors (Jones's 2010 paper, §2.1, states that strong local coupling, with a sensor offset of at least 3 px, is necessary for the complex behaviours to emerge; a 2017 supplement co-authored by Jones gives the same minimum); rotate toward the winning sensor; attempt to move forward one cell; if the target cell is free, deposit chemoattractant there, and if it is occupied, abandon the move and pick a fresh random heading instead — no deposit (a rule also stated formally by Adamatzky & Jones); then diffuse the whole trail map through a 3×3 mean filter, and decay it (Jones's later texts say the values “may be damped” or are “degraded”, while the 2010 paper names only a “decay factor”; Jenson's specific implementation decays multiplicatively). Jones supports both periodic and fixed absorbing boundaries, with qualitatively different long-run patterns.

Jones updates the particle population in a random serial order each step (why is discussed below). Jenson's implementations usually drop the one-particle-per-cell exclusion rule — “I usually ignored this step, preferring the patterns that arose without it” — while noting that it is “crucial for exact mimicry” of the organism because it “approximates a sort of conservation of matter”. Jenson separately adds a contagion behaviour: an infected particle lays an infected trail that spreads to other particles — the page's own illustration that a label can ride the shared medium.

Jones frames pattern formation as a tension between efficient foraging (searching a maximal area) and efficient nutrient transport (minimal transport distance, with fault tolerance).

The thesis

What appears open is not any single mechanism but their combination: one grid carrying a trail channel per net; agents attracted to their own net's channel and repelled by the others; and a cap on agents per cell that withholds reinforcement under congestion. The combination is applied to silicon assembly (interposer and redistribution-layer, RDL, routing) and evaluated against real baselines at real sizes. Five of the seven ingredients below have a published precedent. The other two — targeting interposer/RDL routing, and using chemorepulsion for spacing — are application choices for which our searches found none.

“Capacity” here means agent exclusion, at most one agent per lattice cell, which caps how fast the trail can be reinforced (L1 below). It is not a routing capacity: a global-routing cell carries many tracks, and an RDL layer's capacity is set by wire pitch and design rules. A router built this way would generalise exclusion to a per-cell cap c, so L1's bound becomes s = c·d plus any pin attractant, and it would still have to check real track capacity once routes are extracted from the field.

We found no assembly of the ingredients for chip or package routing. The ISPD 2007 and 2008 global-routing contests used 16 distinct industrial circuits (adaptec1–5, newblue1–7, bigblue1–4; up to about 1.6 million nets, grids up to 973 × 1256 tiles, 6–8 metal layers) and ranked routers on each circuit by total overflow — how far routed demand exceeds edge capacity — then maximum overflow, then wirelength. None of the Physarum routers we could read reports results on these suites or on the later ICCAD 2019 and ISPD 2024/2025 suites, and none reports overflow (ICCAD 2019 did not itself score overflow; see Limits).

Apart from one paper discussed below, the only one with whole-circuit numbers is Nath's Physarum–particle-swarm hybrid (PhD thesis, Jadavpur University, 2023, ch. 5; a related article by Nath, Majumdar & Sarkar in Res. J. Eng. Technol. 14(1), 2023, could not be reached in full). It builds one rectilinear Steiner tree per terminal set, is tested on random sets of 15, 30 and 200 terminals, and reports only total wirelength on five ISPD'98 IBM circuits, against figures apparently taken from a 2013 wirelength-estimation paper — no overflow, capacities or net counts. A 2025 IEEE conference paper (Gowtham et al., ICIH 2025) also describes a Physarum–PSO (particle-swarm optimisation) hybrid for VLSI routing. We read its full text. It builds one rectilinear Steiner tree per terminal set, reports wirelength, runtime and run-to-run standard deviation on 15- and 30-node instances and on one ISPD'98 circuit (IBM01), and reports no overflow, capacities or net counts. Most values in its tables are identical to values in Nath's thesis (for example 489.2, 2514, 2450, 2441, 62815 and 62742), so we do not count it as independent evidence.

An ant-colony router, the nearest relative, has already been compared with contest-era academic routers (Arora & Moses, 2009, against “several state of the art academic routers”; its references include MaizeRouter and NTHU-Route), so the evaluation gap we found is specific to Physarum-style field routers.

Physarum-inspired chip routing is not new either. PORA (Guo & Huang, 2020) builds single-net obstacle-avoiding Steiner trees and does not report the two largest cases of its benchmark suite. Sun (2019) runs a Physarum graph-Steiner solver on SteinLib VLSI instances of up to tens of thousands of vertices, against GA, DPSO and SPH rather than state-of-the-art solvers. Hsu, Schaposnik Massolo & Schaposnik (2021 preprint; 2022 journal) route one 131-pad net; their preprint, though not the published version, names multiple trees and multiple layers as future work. All of these are single-net. Outside silicon, Taneichi & Rinoie apply a Tero-type Physarum solver (growth exponent γ = 2 with a saturating sigmoid) with per-system conductivities and inter-system growth penalties to aircraft system routing (a hydraulic case study: A340-500-based airframe, three systems, 25 actuators; J. Aircraft 2025, the journal version of AIAA 2020, and JJSASS 2024; see mechanism 1). Capacity-aware Physarum routing is not new either: Tero's flux–conductance solver has been given a link-capacity term for communication-network routing (Sonoda et al., ICNC 2024), and a capacitated Physarum algorithm for multi-origin–destination traffic assignment appeared as a 2020 preprint and in 2022 as a journal paper (Huang et al.). Neither is label-per-net, and neither is chip or package routing. The nearest bio-inspired circuit-routing precedent with one label per net is not Physarum at all: one ant colony per trace, on five hand-built PCB layouts of at most six traces (de Azambuja, Mourelle & Nedjah, CILAMCE 2024).

In networking, “Physarum routing” names two other literatures: per-hop routing protocols for wireless sensor, IoT and ad hoc networks that score next hops with Physarum-inspired rules (e.g. Li et al., 2008–2011; P-iRP, 2013), and Physarum solvers for network-design problems such as multicast trees and relay-node placement (reviewed by Sun, arXiv 1712.02910, and Awad et al., Artif. Intell. Rev. 56, 2023). Neither routes wires under capacity limits; we cite them only to avoid confusion.

Two claims, tested differently. The novelty claim — that no such system has been built and evaluated for package routing — is falsified by a single citation, and the Limits section says where our search could have missed one. The efficacy hypothesis is what the experiments below test: on interposer/RDL instances of at least 10³ nets, a router built this way should reach 100% routability with zero shorts, total wirelength within 5% of the best available baseline on at least 80% of instances, and bit-identical output run to run. It fails if extracted routes short, if it routes fewer nets than the baseline on more than 20% of instances, or if per-net state does not fit the local machines at 10³ nets. Why it might win at all: every step is a local, data-parallel field update that maps directly onto a GPU; congestion is resolved by the medium rather than by rip-up and reroute; and diagonal moves come free on an 8-neighbour lattice. (These thresholds are proposals, to be fixed before any run, not results.)

Seven ingredients, one at a time

A summary first, then the detail. KNOWN means the mechanism and its use in an adjacent EDA or agent setting are both published; PARTIAL means the mechanism is published but not evaluated for chip or package routing; APPARENTLY OPEN means our searches found no precedent at all — bounded, as the limits section says, by what those searches could reach; a non-Physarum competitor is named where one exists.

1. Labels per net

Each ingredient is published; we found no combination of them for chip or package routing. A labelled Physarum router exists outside silicon. Taneichi & Rinoie (J. Aircraft 62(1), 2025, the journal version of AIAA 2020-0012; J. Japan Soc. Aeronaut. Space Sci. 72(3), 2024) extend Tero's flux–conductivity solver — a graph ODE, not an agent model — so that each aircraft system (hydraulic G, B, Y) has its own conductivity, pressure and flow on a shared graph. In both papers each system's growth term is multiplied by a factor between 0 and 1 that shrinks as other systems' flow on the same edge rises; the 2024 paper adds a like factor for flow through the edge's end nodes, plus a non-local factor that damps growth on edges likely to be hit by the same rotor burst as another system's edges. A system's own flow reinforces its own routes; other systems' flow withholds that reinforcement. There is no repulsive term and no capacity limit: sharing is discouraged, never forbidden. On three systems feeding 25 flight-control actuators in an A340-500-based model, the edge factor alone cut route overlap from about 400–760 ft, depending on a, to about 25 ft (2025), and the edge and node factors together cut it from about 590 ft to about 30 ft (2024), close to the overlap that paper marks as unavoidable (all read from the papers' charts). With the edge factor alone, total length rose by up to about 10% at a = 1 and fell at a = 4 and 9 (2025); with both factors it did not rise (2024). The only comparison is with the unmodified solver. Its response function is a growth exponent γ = 2 with a saturating sigmoid (a = 1, 4 or 9): superlinear at low flow; not the 0 < μ < 1 redundancy regime and not μ = 1, so the exact two-edge results in L5 do not apply to it. (The 2024 paper's µ is a rotor-burst weight, not Tero's exponent.)

Labelled Jones-type agents exist too. In Versluis (MSc thesis, Utrecht 2018) each vehicle has its own particle kind and attractant; kinds ignore each other's attractant, exclude only their own kind, and are displaced only when a claimed city re-emits its claimant's attractant (at most 3 vehicles and 20 cities; tours about 24–52% longer than optimal or best-known). The explicit attract-own / repel-others rule, with the other species' trail channel acting as a repellent, is published for Jones-type agents only in computer graphics and computational art, never for routing: McGraw & Ferdousi, SIGGRAPH Asia 2021 Posters (ACM); Maggioli, Marin, Melzi & Rodolà's MoMaS (Computer Graphics Forum 41(7), 2022), a peer-reviewed procedural-texturing method in which each agent senses its own species' trail channel minus the other species' channels, obstacles are fixed, non-diffusing repellent fields, agents never exclude one another, and pattern stability is judged only empirically; and the non-refereed Softology (2019) and Lague variants. Competing labelled Physarum with explicit mutual repulsion and hard one-Physarum-per-cell exclusion exist as a hexagonal cellular automaton with a shared food field and no per-label trail (Awad, Pang, Lusseau & Coghill, ALIFE 2019; Soft Computing 2023), used for foraging, sensor coverage and multi-objective search, not routing.

Multi-commodity Physarum dynamics share one conductivity per edge: in Bonifaci et al. (TCS 920, 2022) “sharing pays off”; in Lonardi, Facca, Putti & De Bacco (Phys. Rev. Research 2021; Sci. Rep. 2022) an exponent β tunes between consolidating flows (β > 1) and spreading them to penalise congestion (β < 1). Huang et al. (J. Comput. Sci. 2022) keep link flow under a capacity by re-regulating conductivity in the update (enforced iteratively, not guaranteed) and, for traffic assignment, keep one conductivity per origin with capacity split by flow share. Outside Physarum: ant colonies attracted by their own colony's pheromone and repelled by other colonies' pheromone for optical-network path routing and wavelength allocation (Navarro Varela & Sinclair, CEC 1999, in two of their three variants; the third is described under mechanism 2), multi-type ant-colony optimisation for edge-disjoint paths (Nowé, Verbeeck & Vrancx 2004; Blesa & Blum), Kawamura et al. 2000, and per-species floor fields (Burstedde et al. 2001). Open: per-net self-deposited fields, repulsion from other nets' fields and a hard capacity or exclusion limit in one agent system, evaluated on VLSI or package routing at contest scale.

2. Deposit only on a successful move

Jones states the rule directly (2010, §2.1) and credits it, together with forward-biased sensing, with avoiding static clustering. It is restated by Tsuda, Jones, Adamatzky & Mills, who ask whether the plasmodium's “bottom-up circuit routing method” can be harnessed as an alternative to template-based circuit design, and model it with Jones's particles. It is not new even within agent-based congestion control: Burstedde et al.'s floor-field cellular automaton (CA) adds a trace only when a particle actually jumps, and Ant-Based Control (Schoonderwoerd et al. 1997) delays ants at congested nodes specifically to withhold reinforcement. What must not be claimed is that forgetting congestion history is new or absent from routers: Independence (Sharma, Ebeling & Hauck, FPL 2005), a placer that runs a modified PathFinder router in its inner loop, already decays that router's history cost geometrically, and TritonRoute-WXL and OpenROAD's detailed router (Kahng, Wang & Xu) decay history or marker costs in production code today. Only the original PathFinder history term is monotone.

Keeping congestion history in a record separate from the attractive steering field is also known: in pieces in routers, and as a whole in one ant-colony router. PathFinder's history cost h_n (monotone in the original; decayed in Independence, TritonRoute-WXL and OpenROAD) and the averaged link flows of traffic assignment (method of successive averages) are occupancy histories that act only as a cost; there is no attractive field. AntNet (Di Caro & Dorigo, JAIR 1998) keeps an exponentially weighted trip-time model beside its pheromone routing table and uses it to score reinforcement, not to choose next hops; it records trip times per destination, not occupancy. The closest agent-based precedent is the global update/occupancy (GU/O) variant of Navarro Varela & Sinclair (CEC 1999), an ant-colony router for optical networks: each ant is attracted by its own colony's pheromone, while a separate count of the ants of every colony, its own included, crossing each link, evaporated each cycle like the pheromone, sets the weight of repulsion in every link choice. It is graph ant-colony optimisation, with counts per link rather than per cell and no Jones-style sensing or diffusion. Montgomery & Randall's anti-pheromone (ANTS 2002) keeps a second pheromone for poor solutions, but that field steers directly. MCPM, the Monte Carlo Physarum Machine (Elek et al.), keeps a non-steering, attenuated occupancy trace beside its steering deposit, but uses it only as output. In the pedestrian floor-field CA and in Jones's model the occupancy trace is itself the attractive field. We found no source that feeds a decaying occupancy field that is not itself followed into the move cost of a Jones/Physarum agent model alongside a separate attractive trail; given these precedents it is a known combination carried into a new model, not a new mechanism.

3. Soft-then-hard capacity

This is established practice, not a novel schedule: analytic global placement lets cells overlap and raises the density penalty gradually (ePlace, RePlAce, DREAMPlace), then legalises — Tetris-style row legalisation, then Abacus refinement. Jenson's collision-off vs. Jones's collision-on is the Physarum instance of exactly this two-phase pipeline, not a new one.

4. Deterministic parallel arbitration

Priority-based, at-most-one-agent-per-cell arbitration on the GPU is Lysenko & D'Souza's 2008 framework, where the higher-priority agent always wins a contested action. The cell-centric “pull”, or gather, form, in which each cell scans its interested agents and picks by priority, is FLAME GPU's method (Richmond, 2014). Yang et al. (SIGSIM-PADS 2018) systematise these methods, list determinism and freedom from bias as requirements, and warn that a fixed priority injects directional bias, which they remove with a fresh random permutation each cycle. Jones himself cited this literature when anticipating a parallel version of his model; see the correction below for what that anticipation actually said and what exclusion enables.

Fixed-point accumulation for determinism is AMBER's SPFP (single-precision, fixed-point) model (2013) and DREAMPlace's opt-in deterministic mode (fixed-point integer atomics, off by default since 2.1.0). EDA already treats run-to-run non-determinism as a defect: OpenROAD fixed a pin-access non-determinism that surfaced at global routing, and the ISPD 2018 and 2019 detailed-routing contests penalised non-deterministic output (3%). At least one global-routing contest (ICCAD 2019) applied a 3% score penalty to non-deterministic multi-threaded output. ISPD'24 contest slides report 14 of 18 final global-routing submissions deterministic, and all three top-ranked ones — a post-hoc observation in the results, not a scored metric, and only two of those three used a GPU at all. Peer-reviewed GPU/FPGA Physarum implementations already exist (Tsompanas & Sirakoulis 2012; Dourvas et al. 2015/2016) — but as synchronous cellular automata, deterministic by construction, not as Jones-style agents with exclusion. A later Physarum cellular-automaton agent model (Dourvas et al. 2019, on CUDA and on an FPGA design simulated in ModelSim) resolves contested moves by fresh per-step random priorities, a Physarum instance of random-priority gather arbitration. It reports that ModelSim FPGA results were the same as CUDA's (same success rates); determinism is neither claimed nor analysed, and the paper calls its model stochastic.

5. Silicon assembly first

No bio-inspired interposer or RDL router turned up in our searches; the literature we found uses ILP, A*, SAT and network-matching methods. Any-obtuse-angle RDL routing exists in some published routers (Chung, Chuang & Chang, DAC 2023, reporting 15.7% shorter wirelength than a traditional router limited to 90° and 135° turns), but octilinear routing remains the norm. A 2025 TCAD fan-out RDL router restricts segments to 90°, 45° and 135° directions and, like the DAC 2023 router, forbids acute turns for manufacturability. “RDL permits any-angle routing” is not a safe blanket statement. Redundancy at this scale is a live design concern already: Universal Chiplet Interconnect Express (UCIe)'s advanced-package interface reserves spare lanes (two redundant bumps per 32 data signals, at 25–55 µm pitch) and its standard-package interface instead degrades link width, both handled inside named repair states (RepairCLK, RepairVAL, RepairMB) of its link-training state machine. This is lane-level redundancy — spare wires and bumps switched in by repair logic — not redundant paths within a net. For signal nets a loop is a defect (see below); the redundancy a Physarum network offers transfers to power and ground, not to signals.

6. Tero for power delivery

Gupta & Kahng (VLSID 2006) already scale each power-mesh segment's conductance by 1 + |ΔV| / V_drop,max — a Tero-like flux-driven rule, without the name, reported to cut peak IR drop (the resistive voltage drop across a loaded mesh) up to 33% in one iteration, area-preserving. Slime-mould-inspired design of macro-scale electrical transmission networks is published (Lumbreras et al., Energies 2019); we found no work naming Physarum or Tero for on-chip power-delivery-network (PDN) sizing specifically, alongside nature-inspired PDN sizing that does exist under other names (river formation dynamics, particle swarm optimisation). The regime that keeps redundancy for a PDN is 0 < μ < 1, where every edge that carries flow survives (exact on the two-edge ring; on general graphs only the static statement L5′ holds, and convergence is open) — see the mathematics below, including a convex-optimisation reading of that same regime.

7. Chemorepulsion and self-organised spacing

Jones's chemorepulsion produces regular periodic spot and stripe arrays, distinct from the irregular patterns chemoattraction gives. As a design idea for dummy fill, decap or via/bump spacing this is apparently open, but it has an established competitor worth naming: density-repulsion spreading of TSVs in 3D-IC placement (Athikulwongse, Pathak & Lim, DAC 2012; journal version Athikulwongse, Ekpanyapong & Lim, IEEE TVLSI 2014), which targets thermal variation rather than spacing regularity as such. A second competitor uses Physarum directly (secondary; we could read only the abstract): in Awad, Pang & Coghill (2018), multiple Physarum, standing for mobile sensors, compete over interest points by chemo-attraction, with repulsion exerted by competing Physarum and by obstacles, for sensor deployment. “Turing patterns” is also, separately, a term already used for TSV copper-electrodeposition chemistry (Moffat & Josell, 2012) — a terminology collision, not prior art for the design idea.

A correction worth keeping

Right after saying he usually ignored Jones's collision step, Jenson writes: “Also (conveniently) this collision detection removes any sort of sequential dependence, allowing for increased computational parallelism.” Read literally, the sentence credits the collision step with the parallelism. Read in context, it almost certainly means that omitting the step does, and on that reading it is right: without exclusion no two particles contend for a cell, so every particle can update independently.

What needs correcting is the implication that exclusion forces serial processing. In his 2011 paper, Jones updates particles in a random serial order “to avoid any influence from sequential positional updates”; with exclusion, that order decides who wins a contested cell. In the discussion at the end of the same paper he anticipates GPU acceleration of “the environmental diffusion and possibly also the agent interactions, since the agent particles could be classed as SIMD operations”, citing the GPU agent-modelling work of Lysenko & D'Souza and Richmond et al. — a possibility he names, not a claim that it is straightforward. Exclusion can be kept and run in parallel, as below. The price is that the result is a different dynamics from random serial order, not a parallel implementation of it.

Gather arbitration is that different dynamics: every agent proposes a target cell from the frozen previous state, every free cell picks its highest-priority proposer, losers stay and redraw. This is what the GPU agent-based-modelling literature already uses (Lysenko & D'Souza; Richmond; Yang et al.'s bias-corrected priorities). It is a genuinely different dynamics from random serial order — not an equivalent parallel implementation of it — and the next section is exact about exactly how much it differs.

Fixed-point deposits close the loop: identical serialised additions of the same value are order-free even in float, but heterogeneous deposits (different weights, several labels summed into one field) are not, and IEEE binary32 addition is not associative at all — (2²⁴ + 1) + 1 = 16,777,216 but 2²⁴ + (1 + 1) = 16,777,218, an exact computation, not a measurement. With exclusion and gather arbitration, each cell has at most one writer per step, so deposits need no atomic operation at all.

The mathematics

Five propositions, plus a static statement (L5′) added in a second round, as stated in a separate line-by-line mathematics review, with their hypotheses. Each was proved on paper there and checked by exact rational arithmetic or a stated numerical run; none has been kernel-checked. Every numeric figure below is an exact computation or a numerical check of the mathematics (random trials of an inequality, or a small illustrative agent run), not a simulation of a router or a chip. All 31 checks in the review's script pass, and were re-run once before this page was drafted; a second, certificate script for L5′ (described below) passes all of its L5′ checks. Which tier each proposition could reach is the subject of the alignment section below.

L1 — capacity bound

Status: elementary; proved on paper in the review (not kernel-checked), with changes to the working sketch. Take the step order deposit → diffuse → decay, so T_t+1 = λM(T_t + D_t). Assume: the diffusion operator M has non-negative weights with row sums ≤ 1 (true of the periodic 3×3 mean, the zero-padded mean and the renormalised edge mean; sum-preservation is not needed); retention satisfies 0 < λ < 1; the initial trail satisfies 0 ≤ T₀ ≤ τ₀; and each cell receives between 0 and s of deposit per step. Then max T_t ≤ λ^tτ₀ + (1−λ^t)·c(s), with c(s) = λs/(1−λ) — in particular max T_t ≤ max(τ₀, c(s)), with ≤ rather than <. The proof is one line: the maximum obeys m_t+1 ≤ λ(m_t + s); unroll it. If deposits came after diffusion and decay, the constant would be s/(1−λ). Under exclusion, a successful mover deposits d, a blocked one deposits nothing, and no cell receives two arrivals in one step, so s = d. When pins or food sources also project attractant every step, s = d + max F, which is the case a router is in. The bound is sharp for the trail update on its own: a constant deposit on every cell meets it exactly. The agent rules cannot produce that pattern, because a full grid cannot move at all. In one constructed run (8×8 torus, one empty cell, 63 agents, λ = 0.9, 3,000 steps), a favourable serial order reached 99.2% of the bound, Jones's random serial order 5.3%, and gather arbitration 3.9%. Without exclusion, N agents moving together give exactly N times the single-agent field, so the worst case grows linearly in N; exclusion replaces that with a constant. Structurally this is the pheromone bound of MAX–MIN Ant System (Stützle & Hoos 2000, Proposition 4.1). The extra ingredient is the non-expansiveness of the mean filter, and the sub-step order changes the constant.

L2 — forgetting

Status: holds for a fixed deposit sequence; the coupled system is not a contraction. If the deposit field at each step is held fixed, the map is a λ-Lipschitz contraction in sup norm, so a deposit made k steps ago weighs at most λ^k today — the honest form of “decay is a forgetting rate”. It fails once deposits depend on the sensed field: sensing compares values by strict argmax, so the full map is not even continuous. A checked counterexample (the argument does not depend on the sensor offset): two runs whose trails differ by 2×10⁻⁶ at the start disagree on which sensor wins, send one agent straight and turn the other, and differ by 0.1 one step later — more than 50,000 times the 1.8×10⁻⁶ a contraction with λ = 0.9 would allow. There is no claim here that two runs from different starting trails converge to the same attractor; the non-contraction is a single constructed counterexample, not a broad empirical finding.

L3 — deterministic gather arbitration

Status: holds (determinism, exclusion, conservation); not equivalent to Jones's random serial order. Under gather arbitration — every agent proposes a target cell from the frozen previous state, every free cell picks its highest-priority proposer, losers stay and redraw — the next state is a pure function of the state, a seed and the step number, independent of thread schedule; no cell ever holds two agents; the agent count is conserved. The review's check was an ordinary program enumerating unlabelled configurations on a 4×4 king-move torus, k = 2 and k = 3 agents, all headings, all priority orders: C(16,2)·8² + C(16,3)·8³ = 294,400 configurations — not a kernel proof. (The kernel state counts in the alignment table below are for labelled agents: 15,360 for k = 2 and 1,720,320 for k = 3.) The lemma assumes lattice moves to one of the eight neighbours; with Jones's continuous headings, a move that stays inside the same cell needs its own rule. It is not the same distribution as random serial order: on a 1×3 ring with two agents facing the same way, serial order lets the rear agent move into a cell the front agent just vacated (P(both move) = 1/2); gather arbitration forbids this outright (P(both move) = 0) because the target was occupied when the step began. The two laws differ on 91,392 of the 294,400 checked configurations — about 31%.

L4 — fixed-point versus float deposits

Status: holds, refined. Integer sums are order-independent under any interleaving (the abelian group law on ℤ/2^w). Serialised float addition of identical addends is also order-free; what breaks determinism is heterogeneous addends (different weights, several labels summed into one cell) or re-association (a warp computing d+d before adding it). With exclusion and gather arbitration, each cell has one writer per step, so no summation happens at all — a plain store is deterministic even in float. With exclusion the trail never exceeds max(τ₀, λ(d+F)/(1−λ)) (L1), so a 32-bit fixed-point scale can be chosen once to rule out wrap-around; without exclusion the headroom must grow with the number of agents moving together. A fixed-point floor for diffusion and decay keeps the L1 bound exactly; what degrades is L2's forgetting, where two runs can stay up to ⌈1/(1−λ)⌉−1 least-significant bits apart.

L5 — Tero equilibria on the two-edge ring

Status: exact for two edges; known only for μ=1 on general graphs. On Tero's flux–conductance model, for two parallel edges of length ℓ₁ < ℓ₂: at μ = 1, the dynamics converge to the shorter edge alone from every positive start — the only regime for which Bonifaci, Mehlhorn & Varma (2012) prove, on any undirected graph with positive lengths and positive initial conductances, attraction to the minimum-cost unit flows, and convergence to the shortest path when it is unique. (For a graph with two edges between the same pair of nodes, that condition can fail: a zero initial conductance on the shortest edge stays zero, so the outcome then depends on where the dynamics starts, as Miyaji, Ohnishi, Tero & Nakagaki (2008) showed for a double-edge graph.) For 0 < μ < 1, both edges survive at a unique stable interior point with D₁/D₂ = (ℓ₂/ℓ₁)^μ/(1−μ) — not close together: at μ=0.9, ℓ₂/ℓ₁=1.1 gives a ratio of about 2.36. For μ > 1, exactly one edge survives, winner-take-all, and the rule is exact: the edge with the larger D_i(0)^μ−1/ℓ_i^μ at the start wins — not necessarily the shorter one.

For μ ≠ 1 we found no convergence theorem for Tero's ODE on general graphs (searched to September 2026). What is known: the two-edge ring above is solved exactly for every μ (the bifurcation at μ = 1 is Tero et al. 2007's, restated by Bonifaci 2017, §3.4). For μ > 1, global convergence to the shortest path fails: the ring has two stable single-edge equilibria, and the initial state decides which one survives (Tero et al. 2007; Bonifaci 2017, 2021); for 1 < μ < 2 the associated cost is non-convex, with a numerically observed link to branched transport (Facca, Cardin & Putti, JCP 2021). For 0 < μ < 1 the only full-support equilibrium is the unique minimiser of a strictly convex flux cost (L5′ below), but convergence to it is a conjecture in the continuum; on graphs we found no published proof (we have an unpublished derivation, not peer-reviewed). Bonifaci's revised model ẋ = x(g(|q|/x) − 1) converges on every graph to a fixed point for any increasing differentiable g with g(1) = 1, and to the optimum (the shortest path) when g(y) ≥ 1 + α(y − 1). That covers y^μ for μ ≥ 1 but not y^μ with μ < 1 or a saturating g (Karrenbauer, Kolev & Mehlhorn 2020, Thm 3). Whether that condition can be dropped is open (Karrenbauer, Kolev & Mehlhorn; Bonifaci 2021). The revised model's limit is always the single shortest path, so it has no parallel-path (redundancy) regime — the regime, 0 < μ < 1, that a power mesh would need.

L5′ — 0 < μ < 1 on general graphs: a static convex reading

Status: the static statement holds; partially sourced, and not a new result; convergence of the dynamics to it is open. Take a finite graph with lengths ℓ_e > 0, supplies summing to zero, and 0 < μ < 1. If (Q, D, p) is a Tero equilibrium (D_e = |Q_e|^μ) with nonzero flow on every edge, then Q is the unique minimiser of the strictly convex cost ∑_e ℓ_e|Q_e|^2−μ subject to Kirchhoff's law. Equivalently, it minimises dissipation ∑_e ℓ_eQ_e²/D_e under a metal budget ∑_e ℓ_eD_e^2/μ−1 = K, with K matched. The proof is an elementary KKT (Karush–Kuhn–Tucker) identity: eliminating D gives p_u − p_v = ℓ_e·sign(Q_e)·|Q_e|^1−μ, which is the stationarity condition, and the cost is strictly convex. The continuum analogue is Facca, Cardin & Putti's Proposition 2; on graphs, for 2/3 < μ < 1, the same functional is a proved Lyapunov function for a related ODE with the same fixed points (Lonardi et al. 2021, whose exponent is β = 3 − 2/μ; Bonifaci et al. 2022, Theorem 4, which covers 0 < β < 2); for 0 < μ ≤ 2/3 that related ODE lies outside both proofs. A mathematics sanity-check script certifies the statement exactly, in rational arithmetic, on instances built by inverse design so that the answer is rational: a ring, a Wheatstone bridge with a chord, a 3×3 grid and 300 random connected graphs. Each passing certificate is a proof for that instance only; the sweep is not evidence for the general statement, which rests on the proof. For arbitrary rational data the equilibrium is generically algebraic-irrational (on the ring with μ = 1/2, lengths (2, 3) and unit flux, D* = (3/√13, 2/√13)). Two later maths sanity checks, each resting on our own (unpublished, not citable) derivation, treat such data one instance at a time. The first, adversarially reviewed, encloses the equilibrium in a box proven to contain exactly one solution of its optimality conditions (the Krawczyk–Moore interval method). The second, not yet reviewed, describes it exactly for rational μ on small graphs, by minimal polynomials with isolating intervals (real-algebraic certificates). Neither covers arbitrary data as a family, and neither is formally verified. An earlier numerical check on a 6-edge Wheatstone bridge with a chord matched the Tero equilibrium to a numerical convex optimum to ≤ 5×10⁻⁸ (optimiser tolerance; the value varies with library version).

This is the shape of the loops-versus-trees optimal-transport-network literature, which studies adaptation dynamics as energy minimisation more generally (Hu & Cai, Phys. Rev. Lett. 111:138701, 2013, on a related but not identical adaptation model; we did not confirm that paper states this exact Tero correspondence). On a power mesh whose equilibrium carries flow on every strap, the Tero equilibrium could be computed directly by a convex solver, and the dynamics would have to earn its place by speed, locality or parallelism, not by the answer it reaches — and whether the dynamics reaches that answer at all on general graphs is the open convergence question above.

An interactive illustration of these dynamics runs in the browser. Athlone to Nottingham applies Tero-type conductance dynamics to a stylised network of OpenStreetMap motorways, trunk roads and car ferries, with one unit of demand between the two places. It illustrates the mathematics; it is not an experiment. Its costs are stylised, it uses no traffic data, and it is not a navigation tool.

Alignment against current capability

Leibniz confines language models to proposing statements and proofs. Only mechanical checkers decide: the Lean 4 kernel, Z3, or an exact procedure. A result it publishes to its ledger is a promulgated law, tagged with the tier of evidence behind it (kernel-decided, exact-procedure or cross-kernel). To get there it must pass mechanical gates. The novelty gate includes a non-triviality test that quarantines any statement one of decide, simp, omega, trivial, aesop, ring or nlinarith closes on its own. The faithfulness gate checks that the formal statement says what the claim says.

None of the propositions below is a realistic law candidate, for different reasons. The exhaustive finite checks (rows 3f, 5a below) would be closed by decide and quarantined as trivial. The multi-step lemmas — L1's induction, L2, L3's general proof — are not closed by any single tactic, so they would reach the kernel; but they are real-valued claims with no mechanical faithfulness probe of this shape, so that gate would defer them, and its only fallback, a judged route, is budget-capped. None is a published result to be re-decided, and none is new enough to pass as originated. The tier they could reach is one the record already uses heavily: a kernel-checked audit record, operator- or agent-authored, axiom-clean, promulgating nothing.

Two things follow. First, the supporting lemmas are textbook-grade, so in Leibniz they would land as audit records, never as laws, for the reasons above — not because a two-line tactic closes them (it doesn't), but because there is no route to a law for claims of this kind. Second, Leibniz's own record states that “autonomous novelty is not solved”. Its own probes and a sweep over producer methods conclude that the region that is “soundly checkable and finitely encodable” is the textbook region, and that the binding constraint is the producer — what the proposing side can encode, not the checker, the prover or compute. The mathematics that would make this thesis interesting — convergence of Tero's dynamics for μ ≠ 1 on general graphs, convergence of the stochastic agent dynamics, and a classification of pattern regimes — is the kind of frontier content the record says the producer cannot yet emit, and nothing like it appears there. The realistic route is the one the record names for human-supplied targets: a person or an agent writes the statement and the certificate, and Leibniz checks it soundly. Its automated intake only steers what the conjecturer tries; it never decides and does not build certificates.

The stochastic item is open, within the reach of our searches. We found no proof of convergence, ergodicity or a mean-field/hydrodynamic limit for Jones- or MCPM-type agents: heading-based agents turning by forward-sensor argmax (or MCPM's power-law choice) on a field they deposit, which diffuses and decays, with or without one-agent-per-cell exclusion. Nearby results include one theorem on a stripped-down lattice variant: with co-location allowed and synchronous moves, every run almost surely ‘locks’ into corridors; that model has no diffusion, resets pheromone to a maximum and decays it linearly (Vreeswijk 2017, Thm 4.3, whose author reports that with co-location forbidden, similar results are “much harder to come by”); rigorous mean-field or hydrodynamic limits for smooth chemotactic particles that produce a diffusing, degrading field, with no heading sensors and no exclusion (Stevens 2000; Wieczorek 2022; Natalini & Paul 2023); a PDE limit for a different slime-mould SDE model (fixed food potential, pairwise kernel, no trail field), derived under an assumed propagation of chaos (Göttlich, Knapp & Weber 2020); formal or heuristic macroscopic trail models (Helbing et al. 1997; Boissard, Degond & Motsch 2013; Mokhtari et al. 2022); rigorous limits for simpler graph-walker and ant models, none with all three Jones features (listed in the research record); and otherwise empirical reports.

This should not be understated in the other direction, either. Leibniz's pipeline imports all of Mathlib (pinned to Lean and Mathlib v4.34.0-rc2), and its record already includes a kernel-verified ∀k induction over a rational-matrix recurrence, general real-matrix positive-semidefinite theorems, and kernel-checked real inequalities via nlinarith/linarith — all at audit tier and agent- or operator-authored. On one of them, a positive-semidefinite lemma, an automated prover had returned no proof after about 644 seconds before agent authorship discharged it. The capacity bound, the open-loop contraction and the arbitration invariants above are the same shape of statement as those, and sit within demonstrated reach as audit records; what is missing is authorship, not a capability the record lacks. Everything in this section is sourced to the Leibniz repository, read at commit 67737a4; its reading room is codexcalculemus.com.

Local lane: mvcc

The other half of the alignment is what a local Apple-silicon machine, running mvcc (a fork of doximity/mvcc, compiling CUDA C++ to Metal), already gives and does not. It gives native 32-bit atomicMin/atomicMax and float atomicAdd, unified memory, and a documented hazard that turns out narrower than it first looks: only a lock-holding spin loop can hang under lockstep warps. A single-attempt atomicCAS, a lock-free CAS-retry loop, an atomicMin on a packed (priority, id) key, or a pure gather that scans each cell's neighbours are all safe; 64-bit integer atomics work but serialise through hashed spinlocks, which is why keys and deposits should stay 32-bit. What it does not give: grid-wide synchronisation (each phase — sense, claim, move/deposit, diffuse, decay — is its own kernel launch), a random-number library (a counter-based hash keyed on seed, agent id and step stands in), texture memory, or a radix sort. No GPU platform, not only mvcc, gives order-independent float summation of unequal deposits, which is why deposits should be 32-bit fixed point. fp32 denormals flush to zero, timing is host-side only, and floating-point results can differ from NVIDIA hardware, so a CUDA or GLSL reference cannot serve as a bit-exact oracle. A whole-word search of the repository for Physarum, Jones or Tero, and for net-routing code, finds none (the only “router” hits are the local language-model example's mixture-of-experts token router): no Physarum kernel exists in mvcc today, and building one is new CUDA C++ against this documented, narrower surface — a capability gap to close, not something already sitting there to run. The same machines run local language models (mvcc's own example decodes a 35-billion-parameter mixture-of-experts model at about 80 tokens/s on an M5 Pro); a local proposer adds throughput to the empirical lane, but it does not move the constraint Leibniz's record names, which is the producer.

Experiments the local lane would run

Pre-registered, and none of them have been run. Each fixes, before any run: the instance set and sizes, the number of seeds (at least 10), the machines (named Apple GPU models), the decision rule (the thresholds stated below, no post-hoc metric changes), and a hash of this criteria text committed with a timestamp. Each names a baseline, a pass criterion, a kill criterion, and which proposition or mechanism it tests.

1. Determinism. Control: the collision-free variant with float atomicAdd and heterogeneous per-label weights; it must diverge between runs in at least one of 50 runs, or the test is reported as inconclusive on that hardware. Test: the exclusion + gather-arbitration + fixed-point path, with 8 lattice headings and integer-only sensing (continuous headings use sin/cos, which differ between fast and precise math libraries), run 50 times across three block sizes and every Apple GPU generation in the fleet, for 10⁴ steps on a 1024×1024 grid. Pass: every trail-map hash is identical across runs and identical to a single-threaded CPU reference running the same integer rules. Kill: any mismatch. This checks that the implementation realises L3 and L4; the propositions themselves are already proved.
2. Deposit-on-success vs. deposit-always. Arms, all on the same agent router and instances: (a) withhold-on-block (Jones's rule); (b) deposit on every attempt; (c) deposit on every attempt plus an Independence-style history cost (h ← 0.9h + 0.5 on overuse). Context baseline: a PathFinder/Independence negotiated-congestion maze router on the same grids. Instances: 20 synthetic multi-net grids at three densities, 10 seeds each. Measures: wall-clock to zero overflow, and total wirelength at that point. Pass: arm (a) reaches zero overflow on at least as many instances as (b) and (c), in no more than their median wall-clock, at ≤ 1.02× their wirelength. Kill: (a) is worse than (b) or (c) on either measure on a majority of instances. L1 is asserted at run time (max T ≤ c(s) + tolerance) as a check of the implementation, not a test of the theorem.
3. Label-per-net router versus real routers. Instances: published interposer/RDL benchmarks used by the cited RDL routers where obtainable, otherwise generated instances matching those papers' reported statistics, from 10² to 10⁴ nets. Baselines: (i) the best published RDL/interposer router that can be run, or whose per-instance results can be matched; (ii) an ILP-optimal solution on instances small enough to solve; (iii) a multi-type ant-colony router with the same attract-own/repel-others sign structure, as the like-for-like bio-inspired baseline. Route extraction from the field is fixed in advance (a per-net shortest path on −log T of the net's own channel, followed by tree extraction). Pass: 100% routability with zero shorts and zero spacing violations from an independent checker, and total wirelength ≤ 1.05× baseline (i), on at least 80% of instances. Kill: any short in any final extracted route; lower routability than baseline (i) on more than 20% of instances; or per-net state that does not fit a fleet machine at 10³ nets. Wall-clock is reported, not a criterion. A first gate, not a test of the thesis at real size: passing it would justify the contest-scale comparison the thesis's “what would confirm it” requires.
4. Tero-sized strap mesh versus real PDN sizing. Topology: a fixed strap mesh with bump sources and a load-current map. Baselines: uniform sizing and Gupta & Kahng's iterative area-preserving scaling, both at equal total metal area; and the convex optimum of dissipation under L5′'s budget ∑_e ℓ_eD_e^2/μ−1 = K (which is not metal area unless μ = 1), with K matched to the Tero run; a μ<1 equilibrium with flow on every segment equals that optimum (L5′). Segments below a fixed minimum width are removed (the rule is fixed in advance). Measures: worst static IR drop; worst segment current density against the electromigration limit; worst IR drop after each single-segment removal. Pass: Tero reaches within 1% of the convex optimum's objective, with worst IR drop no worse than Gupta–Kahng's at equal metal, in less wall-clock than a standard convex solver on meshes of at least 10⁵ segments. Kill: it is slower than the convex solver at that size; any segment exceeds the electromigration limit; or its worst single-fault IR drop is worse than uniform's. Tests mechanism 6, L5's μ<1 regime, and whether the dynamics reaches L5′'s optimum in practice (convergence is open).
5. Soft-to-hard schedule (a tuning study; the mechanism is known practice). Schedule: the per-cell agent cap falls from N to 1 over S steps, or exclusion is enforced with a probability rising from 0 to 1. Baselines: hard exclusion from step one, and a placement-style penalty ramp. Measures at equal steps: network cost relative to the minimum spanning tree over the food sources (Tero 2010's TL_MST), connectivity of all sources, and steps until the state is exclusion-feasible. Pass: ≥5% lower TL_MST at equal connectivity on ≥80% of seeds. Kill: no difference beyond seed-to-seed variation. Tests mechanism 3.
6. Chemorepulsion spacing versus lattice and repulsion baselines. Baselines: a regular lattice legalised around blockages, and density-repulsion spreading after Athikulwongse et al., scored on its own thermal objective as well. Measures: violations of the w×w window-density rule, minimum spacing, and wall-clock on the same machine. Pass: fewer window-density violations than the legalised lattice at equal minimum spacing, on blockage-rich layouts. Kill: the legalised lattice matches or beats it. Tests mechanism 7.

Where it will not transfer

Signal nets want trees. A Physarum network keeps loops at μ<1 and prunes by initial conditions at μ>1. For a signal net a loop is a defect: extra capacitance, stubs, reflections. A router built this way needs an explicit per-net tree-extraction step, and the baselines above score that step, not the field.

Timing and signal integrity. Die-to-die buses on an interposer are routed to length-matching, differential-pair, impedance and shielding constraints. Nothing in the trail dynamics targets equal length, just as nothing in it targets zero skew (below).

Design rules and layers. A trail map is a 2D grid per layer. Rule-based dummy-metal fill is already a density-window constraint problem (area density within any w×w window between fixed bounds, checked by overlapping windows) — a self-organised spot pattern would have to match that constraint, not merely look regular. Width, spacing, via-enclosure and stacking rules, and the cost of changing layers, all sit outside the model and would have to be imposed by the extraction step or by per-layer cost fields.

Clock networks. Clock networks are judged on skew, the maximum difference in clock arrival time between any two sinks, and academic clock-tree-synthesis frameworks are organised around skew constraints. Every Tero regime in the mathematics above biases toward redundancy or toward the shortest path, and neither is skew balance. The exception worth naming is the clock mesh, which high-performance designs use for better robustness at a power cost; a μ<1 sizing rule is closer to that problem than to a tree, and this page does not examine it.

Memory for labels. One trail channel per net costs nets × cells of state. At 32-bit values, 2,000 nets on a 10⁶-cell layer need about 8 GB, and 20,000 nets need about 80 GB — beyond a 48 GB unified-memory machine. Past a few thousand nets, channels would have to be windowed to each net's bounding box or shared between nets that never overlap.

Single-net Steiner-tree construction is already well served. FLUTE is proved optimal up to degree 6 and verified optimal up to degree 9 on five million random nets per degree, stays highly accurate up to degree 100, has an O(n log n) bound at fixed accuracy (the released implementation is O(n²)), and is the Steiner-tree subroutine inside routers such as NTHU-Route. There is no case for reaching for a Physarum agent system to build single-net trees where FLUTE already gives a fast, near-optimal answer.

Finally, a caution the wider metaheuristics literature earns: component-level analyses have found several widely-published “novel” bio-inspired algorithms are recombinations of known methods (Camacho Villalón, Stützle & Dorigo 2020; Molina et al. 2020), and a 2023 benchmark audit found a confirmed centre-bias defect in 47 of 90 surveyed nature-inspired optimisers, including the one literally named “Slime Mould Algorithm” (Kůdela 2023). Nothing on this page is that kind of metaheuristic — Jones's model and Tero's flux law are mechanistic, spatially embedded dynamics, not black-box search — but the distinction is worth stating rather than assuming it is obvious.

Limits and what we did not check

How this was built

Four ledgers — models and mathematics, EDA practice, prior art, and capability alignment — were built by research agents from papers, specifications, contest pages and the operator's repositories, mostly primary sources and marked “secondary” where only an aggregator could be read. Four separate adversarial reviews then re-checked the ledgers entry by entry. Their verdicts: models and mathematics, 63 ledger entries (52 confirmed, 9 corrected, 1 unverifiable, 1 refuted) plus 3 entries the reviewer added (all confirmed); EDA practice, 80 entries (50 confirmed, 18 corrected, 1 refuted, 11 not re-checked); prior art, 50 entries (18 confirmed, 10 corrected, 3 refuted, 19 not re-checked) plus 21 citations the original pass had missed; capability alignment, 70 entries (62 confirmed, 8 corrected). In all, 233 of 263 ledger entries were re-opened, 30 were not, and reviewers added 24. The mathematics review checked L1–L5 line by line; the capability-alignment review re-derived the tier assignments from the two repositories directly; the 31 checks in the mathematics review's script were re-run once before drafting. The binding decisions then adopted the reviewers' corrections, including the reversal of the working guess about which Tero exponent regime keeps redundancy, and reframing “is any mechanism new” into “is this particular combination, at this scale, evaluated”. A writer drafted this page from the surviving, corrected ledger entries, and further adversarial passes reviewed the draft itself — for factual fidelity against the ledgers, for the mathematics and the alignment matrix, and for how the page renders — before this revision.

A second round, in September 2026, re-opened the record from papers the operator supplied (the Monte Carlo Physarum Machine paper, a Physarum networking-protocol paper, and later the full text of Taneichi & Rinoie's 2024 paper and of their 2025 J. Aircraft paper) and from the sources this page had listed as unread. It ran in four phases, each with its own adversarial review: the operator's two papers (63 ledger entries; 55 confirmed, 8 corrected, none refuted, 3 missed sources added); the previously unread sources, including Nath's thesis, Gowtham et al.'s tables and Jones 2010 (58 ledger entries; 38 confirmed, 20 corrected, none refuted, 1 missed source added); a wider prior-art pass over contests, Tero convergence, labelled agents and Jones/MCPM descendants (126 ledger entries, reviewed in two halves: 37 confirmed and 15 corrected, and 57 confirmed, 13 corrected and 4 not re-checked; none refuted, 4 missed sources added); and mathematics certificates for L5′ (all of its checks pass; the 31 earlier checks still pass; its review found no refutations). The decisions were then amended in a binding document, itself reviewed (0 high-severity, 8 medium and 14 low findings, all applied), and this page was revised only where that amendment lists a page change. A third, search-gap round pulled forward citations for the convergence, contest and mechanism-2 claims and read MoMaS and Erignoux (five ledgers, 63 entries; its review confirmed 40, corrected 8 and left 15 not re-checked, with none refuted, and raised 8 medium and 17 low findings). A further binding amendment, VALIDATED_R3, was itself reviewed, and the page was again revised only where that amendment lists a page change.

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