Non-Saturating Stigmergic Channels

Status: Active — formed Session 13 Connected to: stigmergic consolidation, environmental physics coupling, stigmergy, H7, H11, multi-rate environment, niche construction

The Concept

H11 predicts that the trace→actor crossing needs negative feedback through a channel that does not saturate — acting on the action (deposit probability, geometry) rather than on the cue field the agents read. This file supplies the biological grounding: real termites appear to have evolved exactly such channels, and the saturating "cement pheromone" that the classic Grassé model assumes is increasingly in doubt. Three independent lines of experimental evidence converge on non-saturating, action-based feedback.

1. Surface curvature — the geometric channel (Calovi et al. 2019)

Calovi, Bardunias, Carey, Turner, Nagpal & Werfel (Phil Trans R Soc B, 2019; DOI 10.1098/rstb.2018.0374) ran controlled field experiments on Macrotermes michaelseni in Namibia. They 3D-printed a test surface with continuously varying curvature (concave, convex, flat), coated it with nest soil, and mounted it in three orientations (horizontal, 45°, vertical) to disambiguate curvature from inclination and geotaxis.

Finding: "curvature is the consistent and sole driver, among the measured geometric candidates, of both early termite positioning and construction activity." Soil displacement correlated with curvature across all three orientations; inclination and height did not. Concave (high positive curvature) regions attract deposition; convex regions attract excavation.

Why this matters for H11:

• Curvature is non-saturating. Unlike a pheromone field whose deposit response flattens above φ≈1, curvature is a geometric quantity that the structure carries regardless of how much pheromone is present. You cannot "saturate" curvature by adding more deposits — each deposit changes the curvature, so the channel stays responsive. This is precisely the property H11 says the crossing requires.
• The rule is action-based and state-gated. The same high-curvature cue elicits opposing actions — excavation OR deposition — depending on whether the termite is loaded with soil or seeking a digging site. The cue does not monotonically increase deposit probability; it routes the agent's current action. This is the "act on the action, not the cue" prescription, observed in the animal.
• It is a genuine negative feedback. Filling a concavity reduces its curvature, which removes the cue for further filling — a self-limiting loop. Excavating a convexity reduces its convexity, removing the cue for further excavation. The geometry carries its own inhibition.
• It supplies the "directional bias" H11 listed. Deposition preferentially at concavities IS building along existing wall edges rather than onto their centers.

The paper's own framing: "These two possibilities represent computations the liquid/solid brain can perform, amplifying or smoothing out initial irregularities in tunnel walls." Amplify = positive feedback (a concavity fills, deepening the concavity elsewhere); smooth = negative feedback (a concavity fills and disappears). Both are mediated by the same non-saturating geometric channel.

2. Humidity thresholds — the template channel (Carey et al. 2021; Bardunias et al. 2020)

Carey, Bardunias, Nagpal & Werfel (Front Robot AI, 2021; DOI 10.3389/frobt.2021.645728) tested the "humidity template" hypothesis with a physical robot: termites deposit wet soil at the edge of the high-humidity zone that extends from a tunnel mouth. The robot, controlled only by a local humidity sensor, replicated the behavior — extending a semi-enclosed area in still air and closing it off when a fan disturbed the humidity bubble.

Why this matters for H11:

• Threshold-triggered, not graded. Deposition fires at a humidity boundary (a level crossing), not as a saturating function of humidity level. This is the "refractory / threshold" channel — a discrete, non-saturating trigger.
• The cue and the action are decoupled in the right way. Humidity is the cue, but the feedback (depositing wet soil extends the humidity zone, moving the boundary) acts on the geometry of the boundary, not by raising the humidity level everywhere. Adding wet soil does not saturate a humidity response; it relocates a threshold.
• External perturbation reroutes the action. Wind shrinks the bubble → the same rule now closes the tunnel instead of extending it. A single non-saturating rule produces opposite morphological outcomes under different external conditions — a multi-rate-environment (H4) coupling, mediated by a non-saturating channel.

3. Crowding / inactivity as distributed inhibition (Xiao et al. 2026)

Xiao, Wu, Lim, Su, Bardunias, Chatterjee & Bhamla (arXiv:2607.19594, Jul 2026, "Sensing, Traffic, and Construction in Termites") review the coupled sensing-traffic-construction loop in subterranean termites. Their framing of crowding is directly H11-relevant: "Under confinement, inactivity can act as a form of distributed inhibition that prevents saturation." Congestion at an excavation front generates queues that redirect workers to lateral digging (tunnel widening, branching) rather than continuing to pile in.

Why this matters for H11:

• Crowding is a density cap on action. A cell that is "full" of termites suppresses further entry — a refractory period on the spatial slot. This is the "density cap" channel, the third mechanism H11 listed.
• It acts on the action (where the termite goes), not on a cue field. The termite does not read a saturating "crowding pheromone"; it physically cannot proceed, so it does something else. The inhibition is mechanical, not chemical, and therefore cannot saturate.

The convergence, and what it implies

Three channels — curvature (geometry), humidity (threshold), crowding (mechanical density) — each non-saturating, each acting on the action rather than the cue. Real termite construction appears to rely on these and NOT on a saturating cement pheromone, which despite 60+ years of search "no cement pheromone has yet been identified" (Calovi et al. 2019). The biological system evolved away from the saturating channel H11 flags as self-defeating.

This is strong external support for H11. The hypothesis was derived from a bug fix in our own code (two failed feedback attempts, both through the saturating pheromone field). The termite literature independently shows that the channels real termites use are precisely the non-saturating ones H11 prescribes, and that the saturating channel (cement pheromone) is the one biology may not use at all. H11 may be less a rediscovery of ACO (which bounds the cue) and more a rediscovery of what termites actually do.

Implication for sim08 (the cheap test): the density cap / curvature rule / humidity threshold are not ad hoc additions chosen to make the crossing fire — they are the mechanisms the model organism actually uses. sim06 used a saturating pheromone response because the lineage (Deneubourg → Bonabeau → Ladley) assumed a cement pheromone. The biological evidence now says that assumption is likely wrong, and H11 explains why it fails: a saturating cue channel cannot express the spatial contrast consolidation needs.

Connection to the saturating response curve (H11's mechanism)

The deposit rule in sim06/sim07 is p = DEPOSIT_BASE + DEPOSIT_GAIN · φ/(1+φ), flat above φ≈1. Curvature feedback is the complementary case: the "response" to curvature is a routing decision (excavate vs deposit, depending on state), not a saturating probability. The curvature channel has no "flat above threshold" region — it stays discriminating because it is redefined by each action. This is the formal distinction: a saturating channel maps cue level → action intensity and compresses; a non-saturating channel maps cue geometry → action selection and preserves contrast.

Criticisms

• Correlation, not mechanism (partially resolved 2026-07-29). Calovi et al. establish correlation of construction with curvature, disambiguated from confounds, but the mechanism by which termites assess curvature "is unknown, but presumably involves a combination of antennation and proprioception." Facchini et al. 2024 now propose the transduction is indirect — termites sense curvature through substrate evaporation flux, which is analytically proportional to curvature (Langmuir 1918). This turns "correlation" into "one physical quantity, sensed through a gradient," but the sensing itself (humidity detection) remains inferred, not directly measured at the deposition site.
• The convex/concave contradiction (resolved 2026-07-29). Calovi 2019 (concave → activity) and Facchini 2024 (convex tips → deposit) appear to conflict. The resolution: Calovi measured aggregate activity (digging + building); Facchini isolated pellet deposition. Deposition is at convex tips; excavation is at concave pits. Both are curvature-driven; the action component differs. This is a caution against treating "construction" as a single action in a model — sim09 must separate deposit and excavate.
• The three channels may not be independent. Curvature, humidity, and crowding are coupled in real mounds (concavities hold humid air; narrow concavities crowd). Facchini 2024 unifies curvature and humidity as evaporation flux, reducing three to two (geometry/evaporation + crowding), but separating them in simulation remains an open experimental problem.
• State-gating complicates replication. The same curvature cue elicits excavation OR deposition depending on the termite's loaded state. A simulation must model that state to reproduce the effect — a richer agent than sim06's deposit-only rule.
• No cement pheromone identified ≠ no chemical cue. Absence of identification is not proof of absence; other chemical cues (trail pheromones, CO₂) may yet play roles. The claim is that the saturating deposit-response channel is not the primary one, not that chemistry is absent. Facchini 2024 strengthens this: their "experiments do not support a role for a putative cement pheromone" — now two independent groups.
• Morphology ≠ crossing. Facchini's curvature-only model reproduces nest geometry (pillars, walls, branching) but does not test self-maintenance, persistence against erosion, or perturbation repair. sim09 must add those tests — reproducing the morphology is necessary but not sufficient for the trace→actor crossing.

Empirical Evidence

• Calovi, Bardunias, Carey, Turner, Nagpal & Werfel (2019), Phil Trans R Soc B 374:20180374. Field experiments on M. michaelseni; curvature is the sole consistent driver of construction across three surface orientations, disambiguated from inclination and height. DOI 10.1098/rstb.2018.0374
• Carey, Bardunias, Nagpal & Werfel (2021), Front Robot AI 8:645728. Robot validation of the humidity-template deposition rule; threshold-triggered, wind-rerouted. DOI 10.3389/frobt.2021.645728
• Bardunias et al. (2020) — the humidity-template hypothesis in M. michaelseni mounds.
• Xiao, Wu, Lim, Su, Bardunias, Chatterjee & Bhamla (2026), arXiv:2607.19594. Review framing crowding/inactivity as distributed inhibition preventing saturation; curvature-biased excavation/deposition across subterranean and mound-building taxa.
• Werfel, Petersen & Nagpal (2014), Science 343:754–758. Termite-inspired construction robots using only local sensing; inverse-problem design with threshold-triggered deposition. DOI 10.1126/science.1245842
• Reina & Marshall (2022), PLoS Comput Biol 18:e1010090. Negative feedback in social-insect foraging suppresses variance (not just convergence) in small populations — an additional function for non-saturating inhibitory signals. DOI 10.1371/journal.pcbi.1010090
• Stützle & Hoos (2000), MAX-MIN Ant System. Bounds the cue τ ∈ [τ_min, τ_max] to prevent stagnation — the closest ACO prior art, but acts on the cue field, not the action. H11's distinction: when the response saturates, cue-bounding is insufficient; action-based feedback is needed.
• Facchini, Lazarescu, Perna & Douady (2020), J R Soc Interface 17:20200093. A curvature-only phase-field growth model (no pheromone field) that reproduces arboreal Nasutitermes nest geometry — walls branch, merge, invade space, with a characteristic length scale set by one parameter d. Public finite-difference code: github.com/oiluigioi/JRSI_2020_termite_nest. DOI 10.1098/rsif.2020.0093
• Facchini et al. (2024), eLife 13:86843. "Substrate evaporation drives collective construction in termites." Shows evaporation flux ∝ surface curvature (Langmuir 1918), so the curvature and humidity channels are one physical quantity; termites sense curvature indirectly through evaporation. Curvature-only simulation matches experimental deposition patterns. Explicitly states "experiments do not support a role for a putative cement pheromone." Resolves the convex (deposit at tips) / concave (activity at pits) contradiction as different action components. DOI 10.7554/eLife.86843

4. The unification: curvature ≡ evaporation flux (Facchini et al. 2020, 2024)

The curvature and humidity channels are not separate. Facchini, Lazarescu, Perna & Douady (2020, J R Soc Interface 17:20200093) proposed a curvature-only growth model for arboreal Nasutitermes nests: a phase-field equation in which the nest is a scalar field f and growth is driven by the local mean curvature of its surface, with a smoothing term that mimics the pellet-size cutoff. A single nonlinear equation with one adjustable parameter d (the pattern length scale) reproduces walls that expand, branch, merge, and invade space, and the abundance of saddle-shaped (zero-mean-curvature) surfaces seen in CT-scanned real nests. There is no pheromone field in the model at all — curvature alone organizes construction.

Facchini et al. (2024, eLife 13:86843) then showed why: evaporation flux is directly proportional to surface curvature (a result going back to Langmuir 1918). Termites sense curvature indirectly through substrate evaporation — the humidity gradient is maximal at pillar tips and wall corners, exactly where deposition concentrates. This unifies Calovi 2019 (curvature) and Carey 2021 (humidity) into one physical quantity: the curvature channel IS the humidity/evaporation channel, sensed through one physical gradient. The humidity-template threshold rule (Carey) and the curvature rule (Calovi) are the same mechanism at different scales of description.

The convex/concave contradiction, resolved

Facchini 2024 and Calovi 2019 appear to contradict: Facchini finds deposition at convex pillar tips; Calovi finds activity at concave regions. The resolution is that the two studies measured different things. Calovi measured aggregate construction activity (digging + building together); Facchini isolated pellet deposition specifically. Deposition is at convex tips (growth extends the structure upward/outward); excavation is at concave pits (material removed from pits). Both are curvature-driven, but the action component differs. The Calovi state-gating (loaded → deposit at concavity, seeking → excavate at convexity) and the Facchini result (deposit at convex tips) are consistent once you separate the actions: where a termite deposits depends on its loaded state, and the Facchini experiments observed primarily depositing (loaded) termites on pre-made topography.

For sim09 this means the rule is state-gated: loaded termites deposit at high curvature (convex tips of the material field); unloaded termites excavate at low curvature (concavities). This is richer than sim06's deposit-only rule and is exactly the "recruits as well as limits" channel: depositing at a convex tip extends the tip (recruits further building there) while the smoothing term (Facchini's d) limits feature size — both consolidation properties the density cap lacked.

Positive feedback through roughness — the recruit mechanism

Facchini 2024 notes a subtle but crucial feedback: adding pellets to a convex region makes the surface rougher (more local curvature variation), which further focuses evaporation/deposition there. This is a genuine positive feedback through the geometry itself, not through a saturating cue field. It is the recruit half of the "recruits as well as limits" requirement: the structure's own shape, once nucleated, amplifies the cue that recruits further building at the same location. The density cap (sim08) had only the limit half; curvature has both.

No cement pheromone (again, and stronger)

Facchini 2024 explicitly state their "experiments do not support a role for a putative cement pheromone." This is now two independent groups (Calovi 2019, Facchini 2024) reporting no cement pheromone, plus a curvature-only model that reproduces real morphology without it. The saturating cue the Grassé lineage assumed is not just unused — it is unnecessary to reproduce the target phenomenon. H11's flag on the saturating channel is corroborated at the level of sufficiency, not just absence.

5. The published curvature growth model (the sim09 substrate)

The Facchini 2020 growth equation (the one sim09 should adapt to 2D):

∂f/∂t = f(1−f) · [ −(1/2)·∇·n  +  d·Δ(∇·n) ]

where f ∈ [0,1] is the phase field (1 = nest material, 0 = empty), n = ∇f/|∇f| is the surface normal, and d sets the pattern length scale. Approximated (Facchini 2020) as:

∂f/∂t ≈ f(1−f) · [ (1/2)·Δf  +  d·Δ²f ]

• The growth term (1/2)·Δf is the mean curvature (Laplacian of the height field) — positive at convex tips (growth), negative at concavities (excavation). This is the recruit mechanism.
• The smoothing term d·Δ²f (biharmonic / curvature diffusion) mimics the pellet cutoff — sharp features are smoothed. This is the limit mechanism.
• The prefactor f(1−f) restricts growth to the surface (the boundary of the structure), not the bulk — deposits happen at edges, not interiors. This is spatial selectivity without a saturating cue.
• For large d the equation is linearly unstable: walls expand, branch, and merge, invading all space — the consolidation morphology. Below the instability, growth stalls.

Why this matters for H7/sim09. This is a non-saturating, geometry-based channel that recruits (deposition at convex tips extends the tip) AND limits (smoothing caps feature size), restricted to the structure surface by f(1−f). It has no pheromone field to saturate. The instability in d is a candidate phase-transition parameter: below it, diffuse growth (sim06 regime); above it, consolidated morphology (the crossing candidate). Public finite-difference code exists (github.com/oiluigioi/JRSI_2020_termite_nest) — sim09 adapts this to sim06's 2D grid + agent framework, replacing the pheromone-deposit rule with a curvature-deposit rule.

Open Questions

• Does a minimal simulation with a curvature-based deposit rule (convex tip → deposit, concavity → excavate, state-gated, with a smoothing term) consolidate where sim06's saturating-pheromone rule fragmented, AND fire the crossing? This is the direct test of H7's refined prescription and is candidate sim09.
• Is the d instability the phase transition the crossing needs? If crossing fires only above the curvature-instability threshold and not below it, d is to sim09 what M_c was to sim07 — but with a mechanism (curvature) that recruits as well as limits, where the scalar transport only dispersed. sim09 FULLY IMPLEMENTED (Session 17, all 9 Parts [x]); crossing corrected and FIRES (Session 19, 2026-08-03). The d* sweep (100 combos, dpb × decay × d) found 0/100 under the original detector — the mass-saturation gate (|growth_rate|<0.01) was an unfalsifiable metric-ceiling bug, its threshold ~100× below the Poisson noise floor of a 150-termite deposit process. Corrected to a relative-slope plateau (|slope(M)|/mean(M)<0.001 over K=16 samples), the crossing fires in the curvature channel at every d ∈ [0,4] in the tuned probe (dpb=0.01, decay=0.002, non-saturating grid 3123–5754/6400 cells) and does NOT fire in the baseline-pheromone control (same detector, 0/3 — the saturating rule never elevates the pheromone cue enough). crossing_step decreases 1550→900 as d rises; n_pillars falls 12→1 (consolidation); roughness rises 0.44→0.77. Session 20 (2026-08-04) recruit-vs-limit isolation: a 2×2 factorial (recruit ON/OFF × limit ON/OFF) with a seed-robustness pass (4 seeds) found the recruit half (curvature routing) is necessary and almost-sufficient for a stable crossing: recruit-only (d=0) is stable 3/4 seeds (hold 1.00 in 3, 0.65 in the borderline seed); neither (no recruit, no limit) is 0/4. The limit half (biharmonic d-smoothing) alone is never stable (0/4 — criteria flicker, hold 0.40–0.55, because the smoothing shapes convex geometry no agent is routed to; criterion 3 deposits_on_convex_fraction oscillates around 0.60). But the limit half is a stability amplifier: recruit+limit is stable 4/4 where recruit-only is 3/4 — the borderline seed becomes fully stable (hold 1.0) when d>0 is added. So "recruit as well as limit" = recruit necessary + almost-sufficient; limit = stabilizer + morphology optimizer (causal, not strictly necessary). The decisive contrast is recruit ON vs OFF at d=0 (same detector, same regime, only the recruit flag differs). Session 21 (2026-08-05) saturating-action control: the same curvature routing but a saturating response p = base + gain·c/(1+|c|) instead of linear p = base + gain·c — both action-based, only the linear form non-saturating. The saturating action crosses in 8/8 seeds (stable 6/8) vs linear 8/8 (stable 7/8); the limit half rescues both to 4/4 at d=1. ACTION-BASED routing is the primary load-bearing property; NON-SATURATING is a secondary stability amplifier (mean hold drops 0.91→0.86 at d=0; criterion 3 holds 1.00 for both forms — saturation slows mass equilibration, not spatial selectivity). H11's strict "non-saturating" claim is partially weakened: a saturating action-based channel still crosses stably, but less robustly. Determinism verified. See saturating_action_sweep.py, recruit_limit_sweep.py, dstar_sweep.py, sim09.py (corrected detect_crossing), and sim09.
• SESSION 22 (2026-08-06) cue-based non-saturating control — the 2×2 completes, non-saturating REVERSES SIGN across families. sim06's as-built saturating cue p = base + gain·φ/(1+φ) was contrasted with a non-saturating (linear) cue p = base + gain·φ (clamped to 1.0), both cue-based. Seed-42 factorial (64 conditions): saturating cue crosses 32/32 (stable 32/32, hold 1.000); linear cue crosses 19/32 (stable 16/32, hold 0.527). Without self-maintenance: saturating cue 16/16 stable; linear cue 0/16 stable. With SM: both 16/16 stable. Seed robustness (4 seeds) confirms. The non-saturating property helps in the action family (sim09: 7/8 vs 6/8) but HURTS in the cue family (sim06: 0/16 vs 16/16 w/o SM) — a sign reversal. Mechanism: deposit-probability clamping. The linear cue hits p=1.0 at φ≈1.15 — every high-pheromone cell deposits at 100%, flattening the gradient; mean pheromone over structure drops to 0.467 (vs saturating's 0.749), below the 0.5 crossing threshold. The saturating cue's φ/(1+φ) compression prevents deposit-probability saturation and preserves spatial contrast. The "self-defeating" channel is the non-saturating cue (deposit-probability clamping), not the saturating cue — H11's original framing was backwards for the cue family. Self-maintenance rescues the linear cue (4/4 stable) by sustaining pheromone elevation regardless of the response curve. See cue_response_sweep.py and sim06.py (deposit_response parameter, selftest Part 5d).
• Does the crossing compose? — the L2 question with a non-saturating glue (#62) remains the next major test.
• Does the φ_sat predictor generalize? — DONE (Session 23). NO. The deposit-probability saturation threshold (φ_sat = the input at which p_deposit first reaches 1.0) was tested as a unifying diagnostic across all four cells of the 2×2. A direct probe (phi_sat_probe.py) of sim06 (cue) and sim09 (action) at their crossing-proven regimes found the predictor is 50% accurate — no better than chance. It correctly predicts the cue family (saturated→fails, unsaturated→crosses) but fails for the action family: action/linear is saturated (max curvature 2.55 > c_sat 1.165, clamp fraction 1.0%) but still crosses stably. The clamping fraction is tiny everywhere (0–7%). The difference: in the cue family, the deposit probability IS the spatial signal — clamping it destroys the gradient. In the action family, spatial contrast lives in the routing decision (which direction the termite moves), not the deposit probability — the response curve saturates the gain (how hard to deposit), not the routing (where to go). The unifying diagnostic is whether spatial contrast in the routing input survives the response curve, which depends on channel architecture, not just the saturation threshold. Determinism verified. See phi_sat_probe.py and H7/H11 Session-23 refinements.
• Does the crossing produce targeted scar repair? — DONE (Session 24). NO. The spatially-targeted recovery metric (patch_recovery_probe.py, queued-topic #60) added a patch_recovery (material in the damaged patch / pre-damage patch material) and a mirror_recovery control (an undamaged same-size region's growth). The grid-wide recovery conflated scar repair with volume restoration — the baseline's 47× was unbounded accumulation. targeted_repair = patch_recovery − mirror_recovery is negative in all four conditions (tuned: curvature −1.95, baseline −1.65; default: curvature −0.60, baseline −51.0). Neither channel preferentially repairs the damage site; the scar grows slower than an equivalent undamaged region (re-nucleation lag) in every case. The crossing fires (stability, roughness, mass-plateau) but the structure does not self-repair in the targeted sense. The crossing is a stability/persistence claim, not a scar-targeting claim. The Session 17 "self-repair" report was an artifact of the grid-wide metric. Determinism verified. See patch_recovery_probe.py and H7 Session-24 refinement.
• Can the three channels be separated in simulation (curvature alone, humidity/evaporation alone, crowding alone) to identify which is load-bearing for the crossing? Facchini 2024 says curvature ≡ evaporation, so those two are one channel; crowding (Xiao 2026) is the independent third. sim09 tests the curvature/evaporation channel; the crowding channel is a candidate sim10.
• Is the state-gating (loaded vs seeking) essential, or does a deposit-only curvature rule (deposit at convex tips, no excavation) suffice to consolidate? Facchini's growth model uses only growth (no excavation term) and still reproduces morphology.
• How does curvature feedback relate to the directed-transport candidate (environmental-physics coupling)? Curvature is directed geometry — the Facchini growth equation routes building along convex tips, which is the minimal lumped form of "channel geometry carrying cue to building fronts." sim09 may unify the directed-transport and non-saturating-inhibition candidates into one mechanism, as queued-topic 58 predicted.

Cross-References

• [[hypotheses/H11]] — the saturating channel hypothesis; this file is its biological grounding
• [[hypotheses/H7]] — the trace→actor crossing; non-saturating channels are the candidate mechanism the crossing needs
• [[concepts/stigmergic-consolidation]] — names the negative-feedback gap; this file supplies the biological channels that fill it
• [[concepts/environmental-physics-coupling]] — the directed-transport candidate; curvature may be its geometric minimal form
• [[concepts/stigmergy]] — the base mechanism; the cement pheromone assumption is in doubt
• [[concepts/multi-rate-environment]] — humidity template's wind-perturbation is a multi-rate external driver acting through a non-saturating channel