The Extensive/Intensive Signal Distinction in Stigmergic Systems

The Concept

In thermodynamics, extensive properties (volume, mass, energy) scale with system size — doubling the system doubles the property. Intensive properties (density, temperature, pressure) do not — they are local and independent of system size. The distinction is foundational: extensive properties measure how much, intensive properties measure how arranged.

Our stigmergic system has the same distinction, and it explains why the boundary converts density-dependence into n-dependence (the 50th and 53rd mechanisms).

The fill is extensive

The total deposited material — the "fill" — scales with the deposit rate per termite (dpt) multiplied by n_termites. Without the boundary:

fill = dpt × n_termites × steps × (1 − erosion)

This is density-dependent: at matched density (n / grid²), the fill fraction is the same regardless of grid size. Session 81 confirmed this: no-inhibition fill at 11.7/kc is 29.26% on 320×320 (n=1200) ≈ 29.47% on 80×80 (n=75) — nearly identical at matched density.

The co-presence signal is intensive

The co-presence signal — min(dilate(material_by_id[0]), dilate(material_by_id[1])) — depends on the spatial arrangement of the two structures' material, not on the total amount. It measures where the two structures overlap after dilation, not how much material exists. At matched density on a larger grid, the same n termites spread material over 4× the area, making the co-presence signal 4× sparser — a lower signal at the same density.

The boundary acts on the intensive signal

The boundary field B grows from the co-presence signal: B = B·(1−decay) + growth·cp. The suppression is p_dep *= (1 − g·B_norm/(1+B_norm)). The boundary's suppression depends on B, which depends on the co-presence signal — an intensive property. The boundary does not act on the deposit rate (extensive); it acts on the spatial arrangement (intensive).

The disconnect

The fill (extensive) is density-dependent. The boundary acts on the co-presence signal (intensive), which is n-dependent (depends on how many termites are spread across the grid, not the density). The boundary's suppression therefore depends on n, not density.

This converts the extensive fill's density-dependence into n-dependence:

• Without the boundary: fill = density × dpt → density-dependent
• With the boundary: fill = f(n) → n-dependent (the boundary's suppression depends on the co-presence signal's spatial arrangement, which depends on n)

Empirical Evidence

The 50th mechanism (Session 80): fill depends on absolute n, not density

At matched density (23.4/kc), fill decreases with grid size:

• 80×80 (n=150): 48.26%
• 160×160 (n=600): 31.23%
• 320×320 (n=2400): 24.72%

The per-density prediction (same fill at same density) is falsified at all three grid sizes. The fill depends on absolute n, not density.

The 53rd mechanism (Session 81): boundary converts density-dependence into n-dependence

Without the boundary (no-inhibition), the fill IS density-dependent:

• 320×320 (n=1200, 11.7/kc): 29.26%
• 80×80 (n=75, 11.7/kc): 29.47%
• Nearly identical at matched density.

With the boundary (g=0.01), the fill becomes n-dependent:

• 320×320 (n=1200, 11.7/kc): 15.82%
• 160×160 (n=300, 11.7/kc): 21.29% (directly measured, Session 82; the earlier 43.49% was an extrapolation from the n≥550 plateau data)
• 80×80 (n=75, 11.7/kc): 29.47%

The boundary's suppression is n-dependent (weaker on larger grids) because the co-presence signal is sparser at the same density on a larger grid. The monotonic decrease (fill_80 > fill_160 > fill_320) holds at all densities tested (11.7/kc, 23.4/kc).

The 52nd mechanism: the co-presence signal is intensive

The co-presence signal depends on the spatial overlap of two dilated ID-tagged material arrays. At matched density on a larger grid, the same n termites spread material over 4× the area. The dilated overlap (co-presence) is lower — the signal is sparser. The boundary's suppression (proportional to B_norm, which is proportional to co-presence) is therefore weaker. The signal's intensity (spatial density of the overlap) decreases with grid size at matched density, while the fill (extensive) remains density-dependent without the boundary.

Session 83 refinement: the co-presence at steady state is NOT purely intensive. Direct measurement (copresence_density_sweep.py) at two densities (23.4/kc, 46.9/kc) across three grid sizes found:

• The co-presence COVERAGE (fraction of cells with cp > 0) saturates at ~98–100% regardless of grid size — not a discriminating metric.
• The co-presence MEAN is non-monotonic (decreases from 80×80 to 160×160, then increases to 320×320) and endogenous — it is both cause and consequence of the boundary's feedback loop.
• The 320×320 has the HIGHEST cp_mean (0.447 at 23.4/kc, 0.450 at 46.9/kc) but the LOWEST fill (25.0%, 36.2%) — the co-presence is NOT "sparser on larger grids."

The n-dependence comes from the NORMALIZATION, not the signal. The steady-state B (extensive — scales with absolute material/n) is normalized by the initial b_scale (intensive — same at matched density, from seed regions at the same relative positions). The ratio B_norm = B/b_scale is n-dependent (extensive / intensive), making the suppression n-dependent, making the fill n-dependent.

Methodology lesson: a signal measured inside a feedback loop is both cause and consequence — it cannot test the loop's causality. The co-presence at steady state is shaped by the boundary's suppression, which is shaped by the co-presence. To test the intensive/extensive distinction, measure OUTSIDE the loop (no-inhibition) or BEFORE the loop has had an effect (initial state).

The Thermodynamic Analogy

The key insight: the boundary acts on an intensive property (co-presence), while the quantity it controls (fill) is extensive. The boundary's suppression depends on the co-presence signal's spatial density, which depends on n and grid size — not on the fill's density. This is why the fill's density-dependence is converted into n-dependence: the boundary mediates through an intensive signal that "measures" arrangement, not amount.

Connection to the Two-Wire Principle

The extensive/intensive distinction is the 13th member of the two-wire principle:

Member 13 — The extensive/intensive signal distinction (Session 82). The fill (extensive) and the co-presence signal (intensive) travel on separate wires. The boundary acts on the intensive wire (co-presence → B → suppression), not the extensive wire (deposit rate → fill). The fill's density-dependence (extensive) is converted into n-dependence (intensive) because the boundary mediates through the intensive signal. This is the thermodynamic expression of the two-wire principle: the feedback signal (suppression) and the controlled quantity (fill) travel on separate wires because they have different scaling (intensive vs extensive).

The 12th member (structure-to-grid ratio) is the finite-size expression: the structural guarantee depends on the ratio of structure size to grid area, which is an intensive quantity (arrangement-dependent). The 13th member generalizes this: ALL of the boundary's properties — suppression, structural guarantee, fill cap — depend on the intensive co-presence signal, not the extensive fill.

Connection to the Gompertz-to-Logistic Transition

The Gompertz-to-logistic transition (Sessions 69–79) maps the fill cap trajectory onto the coherence parameter θ from Haugen & Gilbert (2022). The three thresholds (30th ~44%, 44th ~75%, 43rd ~100%) are fill-fraction thresholds.

The extensive/intensive distinction explains why the thresholds are fill-fraction thresholds, not density thresholds: the boundary acts on the intensive co-presence signal, which is a fill-fraction-like quantity (spatial density of the overlap). The coherence parameter θ measures the coupling between the extensive fill and the intensive co-presence — at low fill (Gompertz regime), the boundary's intensive signal controls the fill (coherent); at high fill (logistic regime), the fill's extensive growth overwhelms the intensive signal (percolation, pairwise dominance).

The design principle "keep the system in the Gompertz regime" is equivalent to "keep the intensive signal strong enough to control the extensive fill" — the co-presence signal must be dense enough (spatially) for the boundary's suppression to matter.

The Gain Compensation Result (Session 82)

Session 82 tested the gain compensation: 320×320 at n=1200 × g=[0.01–0.12] × 4 seeds. Result: NOT gain-compensable. The fill on 320×320 monotonically DECREASES with gain (15.82% → 14.14%). Higher gain suppresses MORE, not less — gain amplifies the sparse co-presence signal's magnitude, making the boundary suppress harder in the wrong places. The co-presence signal's sparsity is a spatial-arrangement property (intensive) — gain amplifies magnitude (extensive), it cannot fix spatial coverage. This is the thermodynamic analog of trying to compensate for low pressure (intensive) by increasing the volume (extensive) — the two are independent axes.

Session 84: the 53rd mechanism is FALSIFIED — the n-dependence is from the focal bias, not the boundary.

The no-inhibition co-presence sweep (mode="hetero" g=0 — focal bias, ID-tagged, no boundary) measured the co-presence OUTSIDE the boundary's feedback loop. The fill is n-dependent EVEN WITHOUT the boundary: at 23.4/kc, fill = 48.33% (80×80) → 31.58% (160×160) → 24.95% (320×320). The focal bias (1-cell steps toward home) is an absolute spatial scale that is less effective on larger grids. The boundary adds <0.5% at g=0.01 (fill deltas: -0.07%, -0.48%, +0.05%). The co-presence is extensive without the boundary too (cp_mean: 0.330 → 0.293 → 0.452, same non-monotonic pattern).

The 53rd mechanism's original control (mode="none" — no focal bias, no boundary) was confounded: it changed two variables (focal bias + boundary) simultaneously. The correct control (mode="hetero" g=0) holds the focal bias constant and varies only the boundary. The n-dependence is from the focal bias, not the boundary. The 50th mechanism's empirical finding (fill depends on absolute n) holds but its mechanism is the focal bias, not the boundary — at the gains where it was tested (g=0.01-0.02), the boundary is negligible.

Methodology lesson: the control must isolate one variable at a time. This is the control-arm pattern (#75) recurring: a control that confounds two variables cannot attribute the effect to either.

Criticisms / Limitations (updated Session 84)

• The thermodynamic analogy is phenomenological, not derived. The extensive/intensive distinction in thermodynamics is precisely defined (Euler homogeneity). Our co-presence signal is not formally intensive in the Euler sense — it is a computational quantity (min of dilated arrays) that behaves like an intensive property (arrangement-dependent, not amount-dependent). The analogy holds operationally but not formally.
• The co-presence signal is not purely intensive. It does depend on the total amount of material (more material → more overlap after dilation). It is intensive in the sense that its spatial density (how concentrated the overlap is) depends on the arrangement, not the total. At very low n, the co-presence is zero (no overlap); at very high n, it saturates (full overlap everywhere). The intensive behavior holds in the intermediate regime where the structures are compact but not percolated.
• The 50th mechanism has only been tested at three grid sizes. The monotonic decrease (fill_80 > fill_160 > fill_320 at matched density) is confirmed at 23.4/kc and 11.7/kc (the 11.7/kc sign flip was an extrapolation artifact — corrected in Session 82). A third density (46.9/kc) would strengthen the conclusion; the 53rd mechanism's mechanism (intensive signal sparsity) is confirmed by the no-inhibition control and the gain compensation test.
• The co-presence signal's spatial density is not yet measured directly. The extensive/intensive distinction is currently phenomenological — the fill decreases with grid size at matched density (qualitative), and gain cannot compensate (qualitative). A direct measurement of the co-presence signal's spatial density (fraction of cells with co-presence > 0) at matched density across grid sizes would make the distinction quantitative (Session 83).

Cross-Domain Connections

• Thermodynamics → stigmergy: The extensive/intensive distinction applies to stigmergic signals. The fill (extensive) is density-dependent; the co-presence (intensive) is arrangement-dependent. The boundary mediates through the intensive signal, converting the extensive fill's density-dependence into n-dependence.
• Statistical physics → stigmergy: The co-presence signal is like a local order parameter (intensive — it measures the local correlation between two structures). The fill is like the total magnetization (extensive — it measures the total amount of aligned material). The boundary acts on the local order parameter, not the total magnetization.
• Control theory → stigmergy: The boundary's gain g is like a controller acting on a local (intensive) measurement. The controller's effectiveness depends on the measurement's spatial coverage, not just its magnitude. A sparse measurement (low spatial coverage) cannot be compensated by higher gain — the controller acts on the wrong locations.

Cross-References

• [[concepts/two-wire-principle]] — the 13th member: extensive and intensive signals on separate wires
• [[concepts/gompertz-logistic-transition]] — the fill-fraction thresholds are intensive thresholds (co-presence density, not fill amount)
• [[concepts/non-saturating-channels]] — the boundary's mechanism; the co-presence signal is the intensive channel
• [[hypotheses/H5]] — the persistence condition; the 30th mechanism's ceiling is an intensive threshold (the co-presence signal's spatial density falls below the boundary's maintenance threshold)
• [[hypotheses/H7]] — the trace→actor crossing; the crossing is robust (4/4) because it depends on the intensive curvature signal, not the extensive fill
• [[hypotheses/H10]] — the composition problem; the fill cap's n-dependence is the extensive/intensive signal distinction's macroscopic expression