The Gompertz-to-Logistic Transition in Stigmergic Systems

The Concept

The boundary's fill-capping mechanism (44th mechanism, Sessions 69–76) follows a trajectory that maps onto the Gompertz-to-logistic transition in growth theory. Haugen & Gilbert (2022, arXiv:2211.05653) derive Gompertz growth as the coherent limit of a microscopic Richards model: a coherence parameter θ sets the coupling between nodes on a network. At θ→0 (Gompertz), long-range interactions dominate — each entity contributes linearly to the collective growth rate, and the aggregate dynamics depend only on the collective mean (invariant to how fluctuations are distributed). At θ=1 (logistic), pairwise/local interactions dominate — the macroscopic drift acquires a dependence on microscopic variance. The coherence parameter acts as a symmetry-breaking parameter.

Our stigmergic boundary is a coherent field: its suppression acts simultaneously across the entire grid (like the "ubiquitous external field" Haugen & Gilbert describe). Below the percolation threshold, this coherent field caps fill logarithmically — the Gompertz regime. Above the threshold, local connectivity (Moore 8-neighbor bonds) overwhelms the coherent field — the logistic regime.

The Three Thresholds

The fill cap trajectory (19 data points, n=550 to n=15000) reveals three thresholds of a single phase transition:

1. Persistence ceiling (~44% fill) — the 30th mechanism

The 46th mechanism's strongest form (Session 78): the persistence condition (stable composition) has a hard density ceiling (~44% fill) independent of boundary gain. Above this fill, the boundary cannot maintain stable composition at any gain tested (g=0.0001–0.005). This is the spatial analog of a percolation threshold for autopoietic boundaries: above the critical fill, the structures' surface area exceeds what the boundary can suppress, regardless of bond strength (gain). The boundary's gain is like bond strength; the fill fraction is like occupation probability. Below the ceiling, the boundary maintains stable composition; above it, the structures over-fragment.

This is the Gompertz regime's limit: the coherent field can maintain persistence only below ~44% fill. Above it, the coherence is insufficient — the structures are too large for the long-range suppression to hold them without over-splitting.

2. Fill cap acceleration (~75% fill) — the 44th mechanism breaks

At n=10000 (~75% fill), the fill cap accelerates dramatically (+14.5% from n=8000's ~61%). The boundary retains ~25% of the grid as inter-structure void but the fill is no longer logarithmic. The 47th mechanism (bimodal largest- component fraction, percolation criticality) appears at n=8000 (~61% fill) — the system flips between percolated (one spanning cluster at ~93%) and non- percolated (two near-equal halves at ~48%) states. At n=10000, the bimodality resolves to a single mode at ~97.5% (the 48th mechanism). The boundary's coherent suppression is at the edge of its capability.

This is the transition regime: the coherent field and local connectivity are in competition. The bimodality is the signature of percolation criticality — near the threshold, the system is critical, and the seed determines which side of the transition it falls on.

3. Saturation (~100% fill) — the 43rd mechanism's limit

At n=15000 (~100% fill), the boundary has zero capping power (ncomp=1, largest_frac=100%). The g* limit sweep (Session 77) found the boundary's capping power (gap to no-inhibition) decays exponentially: 16.2% → 8.3% → 4.3% → 1.2% → ~0 (n=11000 to 15000). The gap halves at each step — exponential decay of the coherent field's effect. g* hits zero at n≈14000. The boundary becomes irrelevant when fill ≈ 99%.

This is the logistic regime: local connectivity dominates entirely. The boundary's long-range suppression is a perturbation the percolated structure does not feel. The system is fully in the pairwise regime where local bonds determine the dynamics.

The Mapping

The key insight: the boundary's coherent suppression is the stigmergic analog of the "ubiquitous external field" that Haugen & Gilbert identify as the source of Gompertz growth. The field acts simultaneously across all entities (grid cells), producing the logarithmic fill cap. When percolation makes local connectivity dominant, the coherent field loses its capping power — the Gompertz-to-logistic transition.

The Design Principle

Keep the stigmergic system in the Gompertz (coherent) regime.

The three thresholds define the design space:

• Below ~44% fill: stable composition is possible. The boundary's coherent suppression maintains the structures as separated droplets. This is the Gompertz regime — the fill cap is logarithmic, the boundary's gain scales as 1/√n (Laplace pressure), and the 1-seed structural guarantee holds.
• ~44–75% fill: composition is present but unstable (30th mechanism). The coherent field can maintain coexistence but not stable persistence. The fill cap is still sub-percolation but accelerating.
• Above ~75% fill: the coherent field breaks down (44th mechanism). The fill accelerates toward 100%. Composition collapses (coexist → 0/4). The system percolates and local bonds overwhelm the boundary's long-range suppression.

The design principle connects to the Gompertz-to-logistic framework: the Gompertz regime (coherent, long-range) is where composition works; the logistic regime (pairwise, local) is where it doesn't. The transition is mediated by percolation — when the structure's local connectivity exceeds the coherent field's suppression, the system crosses from Gompertz to logistic.

This is a design principle for multi-scale stigmergic systems: the boundary must be strong enough to keep the system below the persistence ceiling (~44% fill) but weak enough to avoid over-splitting. The 1/√n (Laplace pressure) scaling of g* is the signature of the coherent regime — the boundary's gain scales with the surface-to-volume ratio (∝ 1/√n), which is the geometric expression of the coherent field's long-range nature.

Cross-Domain Connections

• Statistical physics → stigmergy: the percolation threshold (~0.407 for Moore 8-neighbor) is the critical occupation probability where local connectivity overwhelms the coherent field. The fill cap's acceleration above this threshold is the approach to percolation.
• Growth theory → stigmergy: Haugen & Gilbert's coherence parameter θ maps to the fill fraction. The Gompertz (coherent) limit is the low-fill regime; the logistic (pairwise) limit is the high-fill regime. The transition is percolation-mediated.
• Contact inhibition → stigmergy: the fill cap is the stigmergic analog of contact inhibition in biological tissues (the 44th mechanism). Cells stop growing when they reach confluence; the boundary stops the structure from filling the grid. The log fill cap's sublinear growth (13.1·log₁₀(n) − 2.14) is the spatial analog of Gompertz growth under contact inhibition.
• Control theory → stigmergy: the boundary's gain g is like bond strength in percolation; the fill fraction is like occupation probability. The 30th mechanism (persistence ceiling independent of gain) says the topological transition (percolation) dominates over the dynamical parameter (gain) — the same way the percolation threshold is independent of bond strength.

The Gompertz-to-Logistic Transition as a Phase Diagram

The fill cap's complete trajectory (13 points, Sessions 69–76):

The Gompertz-to-logistic transition is COMPLETE at n=15000. The system is fully in the logistic (pairwise) regime — local connectivity dominates, the boundary's coherent suppression is irrelevant.

The Coherence Parameter and Microscopic Variance

Haugen & Gilbert's key insight: at θ=0 (Gompertz), the aggregate dynamics depend only on the collective mean — they are invariant to how fluctuations are distributed among entities. At θ>0, the macroscopic drift acquires a dependence on microscopic variance.

In our system:

• Gompertz regime (low fill): the fill cap depends on total fill (the collective mean), not on how the fill is distributed between the two structures (microscopic variance). The boundary suppresses uniformly — it does not matter whether the fill is one big structure or two small ones.
• Logistic regime (high fill): the fill depends on the local connectivity structure (microscopic variance) — how many bonds, which cells are connected. The total fill alone does not determine the dynamics; the topology does.

This is the formal expression of the 44th mechanism: the fill cap is a coherent (long-range) property below the percolation threshold and a pairwise (local) property above it. The percolation transition is the symmetry breaking where microscopic variance begins to matter.

Connection to the Two-Wire Principle

The two-wire principle (12 members, Sessions 23–42) says: when two properties are carried on the same wire, saturating one destroys the other. The Gompertz-to-logistic transition is the macroscopic expression of this principle:

• In the Gompertz regime, the boundary's coherent suppression is the wire that carries both the feedback signal (deposit suppression) and the spatial signal (boundary location). The two travel on the same wire — the coherent field — but the field is strong enough to separate them.
• In the logistic regime, local connectivity is the wire that carries both signals. When local bonds dominate, the spatial signal (where the structures are) and the feedback signal (where to suppress) merge — the boundary cannot distinguish between the two structures because they are locally connected.

The Heisenberg trade-off (Members 10-11: the signal cannot be simultaneously exogenous and spatially specific) is the microscopic expression of the same principle: the coherent field (exogenous, long-range) cannot maintain spatial specificity (local structure) once local connectivity dominates.

Criticisms / Limitations

• The mapping is phenomenological, not derived. The fill cap's logarithmic regime (R²=0.991) matches the Gompertz form, but no microscopic derivation from Haugen & Gilbert's network model to our grid+agent system exists. The coherence parameter θ is a formal analogy, not a measured quantity.
• The percolation threshold (~0.407 Moore) was corrected late (Session 75). Earlier sessions used ~0.593 (von Neumann 4-neighbor). The correction re-frames Sessions 73–74: the n=8000 bimodality (~61% fill) is well above the Moore threshold, not near it. The boundary delays the transition to higher fill than the random-site threshold.
• The fill cap's functional form is super-logarithmic (Session 74). The stretched exponential (R²=0.964) fits better than the log (R²=0.933) across all 16 points. The log is a good sub-percolation approximation but not the true form. The Gompertz analogy is to the log regime, not the full trajectory.
• The boundary's suppression is not a field in the Haugen & Gilbert sense. Their coherent field is an external stimulus acting on all entities simultaneously. Our boundary is emergent — it grows from the co-presence of two structures. The boundary is both the coherent field AND the product of the structures it suppresses. This is a stigmergic feedback loop, not a pure external field. The analogy holds in the limit where the boundary's growth is much faster than the structures' growth (the boundary equilibrates instantaneously), but this limit is not exactly satisfied.
• The 30th mechanism's ceiling (~44%) does not match the Moore percolation threshold (~0.407). The persistence ceiling is at ~44% fill; the Moore threshold is ~40.7%. The 30th mechanism may be a pre-threshold effect — the boundary over-splits structures approaching the threshold, not at it. The two numbers are close but not identical, and the 30th mechanism is gain-independent (topological) while percolation is connectivity-dependent.

Empirical Evidence

• Haugen & Gilbert (2022), arXiv:2211.05653. Gompertz growth as the coherent limit of a microscopic Richards model; θ as symmetry-breaking parameter; logistic as pairwise limit. The coherence parameter maps to the coupling between nodes on a network.
• Fill cap trajectory (19 data points, Sessions 66–77): n=550 to n=15000. Sub-percolation log fit R²=0.991 (6 points, n≤5000); 10-point R²=0.875. Stretched exponential R²=0.964 (best overall). The fill cap accelerates above the percolation threshold.
• 30th mechanism (Sessions 67–78): stable composition impossible above ~44% fill at any gain (g=0.0001–0.005). Gain-independent, topological.
• 43rd mechanism (Sessions 66–77): g* ≠ 0 at 10 density ranges (n=550 to n=13000). g* hits zero at n≈14000. The boundary's capping power decays exponentially (gap halves per ~1000 n).
• 44th mechanism (Sessions 69–76): fill caps at ~39–41% (independent of n above ~1200). The fill cap is logarithmic below ~44% (R²=0.991) and accelerating above. The no-inhibition control fills 100%.
• 47th mechanism (Session 74): bimodal largest-component fraction at n=8000 (~61% fill) — percolation criticality.
• 48th mechanism (Session 75): spanning cluster fraction converges to ~97.5% at n=10000; 100% at n=15000 (the 48th breaks).
• Percolation threshold correction (Session 75): Moore 8-neighbor threshold is ~0.407, not ~0.593 (von Neumann 4-neighbor).
• g limit (Session 77)*: boundary's capping power decays exponentially (16.2% → 8.3% → 4.3% → 1.2% → ~0), halving per ~1000 n.

Open Questions

• Can the coherence parameter θ be measured in our system? Haugen & Gilbert define it as the coupling between nodes on a network. Our grid has Moore 8-neighbor connectivity — the coupling is the deposit/excavate rule's sensitivity to local curvature. Can θ be extracted from the fill cap's functional form?
• Is the 30th mechanism's ceiling (~44%) the same as the Moore percolation threshold (~0.407)? The two numbers are close but not identical. The 30th is gain-independent; percolation is connectivity-dependent. Is the 30th a pre-threshold effect or a different transition?
• Does the Gompertz-to-logistic transition hold at other grid sizes? The 1/√n scaling of g* was tested only on 160×160. Does the log fill cap's coefficient (13.1) scale with grid size, or is it a universal constant?
• Is the fill cap's super-logarithmic form (stretched exponential, R²=0.964) the signature of diffusion-limited growth under a coherent suppression field? The Kolmogorov-Avrami theory produces stretched exponentials for diffusion-limited processes — is our fill cap a diffusion-limited growth process?

Cross-References

• [[concepts/non-saturating-channels]] — the boundary's mechanism; the fill cap is the macroscopic expression of the non-saturating channel's coherent suppression
• [[concepts/two-wire-principle]] — the 12 members; the Gompertz-to-logistic transition is the macroscopic expression of the two-wire principle's Heisenberg trade-off
• [[hypotheses/H5]] — the persistence condition; the 30th mechanism is the persistence ceiling that defines the Gompertz regime's limit
• [[hypotheses/H7]] — the trace→actor crossing; the crossing fires (H7=4/4) throughout the Gompertz regime and fails above the persistence ceiling
• [[hypotheses/H10]] — the composition problem; the fill cap's three thresholds define the composition regime's boundaries
• [[hypotheses/H11]] — the saturating channel; the Gompertz regime is where the non-saturating channel's coherent suppression holds