The Absolute-Spatial-Scale Principle

The Concept

In a stigmergic system on a grid, mechanisms can act through absolute spatial scales — fixed distances measured in cells, not in fractions of the grid. A 1-cell step, an 8-cell dilation radius, a fixed home center at grid/4 — these are absolute scales. Their effectiveness depends on the grid's linear size: a 1-cell step covers 1/80 of an 80×80 grid but 1/320 of a 320×320 grid.

The principle: any mechanism acting through an absolute spatial scale converts density-dependence into n-dependence. At matched density (same n/grid²), the mechanism is less effective on a larger grid because the same absolute distance covers a smaller fraction of the grid. The fill — which would be density-dependent without the mechanism — becomes n-dependent (decreasing with grid size at matched density).

The refinement: LOCAL absolute scales are scale-invariant; GLOBAL absolute scales are not. A 1-cell Laplacian stencil acts on local shape — it measures curvature at a point, which is the same regardless of grid size. A 1-cell Moore dilation acts on local connectivity, also grid-size-independent. These LOCAL scales produce the same result at the same density on any grid. But a GLOBAL absolute scale — a fixed home center at grid/4, with agents stepping 1 cell toward it — acts on position, not shape. The same 1-cell step covers less of the distance to home on a larger grid, making the mechanism less effective.

Empirical Evidence

Session 85: the fill is density-dependent without the focal bias

At 23.4/kc density with NO focal bias (movement_bias=0) and NO boundary (g=0):

Spread = 2.23% — approximately the same across grid sizes. The curvature channel's local operations (1-cell Laplacian, 1-cell Moore dilation, 0.2-cell diffusion) produce the same fill at the same density regardless of grid size.

Session 85: the fill is n-dependent with the focal bias

At 23.4/kc density with focal bias (movement_bias=0.3) and NO boundary (g=0):

Spread = 23.38% — an order of magnitude larger. The focal bias (1-cell steps toward a fixed home center) is less effective on larger grids: the same 1-cell step covers a smaller fraction of the distance to home.

The 2.23% vs 23.38% comparison

The 10× difference (2.23% without focal bias vs 23.38% with) shows the focal bias accounts for >90% of the n-dependence. The residual 2.23% may be 2-seed noise (320×320 had only 2 seeds) or a small residual from the 1-cell step size (a LOCAL absolute scale that is approximately but not exactly scale-invariant at finite grid sizes).

The LOCAL vs GLOBAL Distinction

LOCAL absolute scales (scale-invariant)

A local absolute scale acts on the shape of the field at a point:

• 1-cell Laplacian (curvature computation): measures the second derivative of the material field at a point. The curvature at a point depends on the local shape, not the grid size. At matched density, the local shape is the same on any grid.

• 1-cell Moore dilation (co-presence computation): marks cells within a fixed radius. At matched density, the local density of marked cells is the same on any grid — the dilation covers the same local neighborhood.

• 0.2-cell diffusion (field smoothing): diffuses the material field over a fixed fraction of a cell. At matched density, the diffusion produces the same local smoothing on any grid.

These are the curvature channel's operations. They are renormalizable in the gauge theory sense: they can be coarse-grained without changing the physics, because they act on local quantities that are density-dependent.

GLOBAL absolute scales (not scale-invariant)

A global absolute scale acts on the position of an agent relative to a fixed reference:

• 1-cell step toward a fixed home center (focal bias): the agent moves 1 cell toward home_center = grid/4. On an 80×80 grid, home is at x=20, and a 1-cell step covers 1/20 of the distance. On a 320×320 grid, home is at x=80, and a 1-cell step covers 1/80 of the distance. The mechanism is 4× less effective at the same density.

• Fixed home center (the reference itself): the home center is at grid/4, which is a different absolute position on each grid. The distance from an agent to its home center depends on the grid size.

These are not renormalizable: they reference a global frame (the fixed home center) that cannot be coarse-grained without changing the mechanism. The step size is absolute, but the distance it must cover is grid-dependent.

The Gauge Theory Analogy

In gauge theory, interactions are classified by their behavior under coordinate transformations:

• Local interactions (gauge couplings) are renormalizable: they depend on fields at the same point, so they are independent of the system size. They can be coarse-grained — the physics at one scale is determined by the physics at the next scale up, without reference to the system boundary.

• Global references (external fields, fixed boundary conditions) are not renormalizable: they depend on the system's global structure. A fixed external field breaks translation invariance and introduces a scale (the distance from the boundary).

The absolute-spatial-scale principle is the stigmergic analog:

The design principle: prefer local mechanisms over global references for scale-invariant stigmergic systems. A local curvature channel (1-cell stencils) produces the same dynamics at any grid size at matched density. A global focal bias (fixed home center) does not — it is the stigmergic analog of an external field that breaks scale invariance.

Connection to the 50th and 53rd Mechanisms

The 50th mechanism (Session 80): fill depends on absolute n

The fill decreases with grid size at matched density (48.26% > 31.23% > 24.72% at 23.4/kc). Session 84 attributed this to the focal bias (an absolute spatial scale). Session 85 confirmed: without the focal bias, the fill is density- dependent (66.25% ≈ 66.49% ≈ 64.26%). The 50th mechanism's empirical finding holds but its mechanism is the focal bias, not the boundary.

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

Session 81 found the fill is n-dependent with the boundary (g=0.01) and density-dependent without. Session 84 falsified this: the fill is n-dependent even without the boundary (the focal bias was the cause). Session 85 confirmed from the other direction: without the focal bias, the fill is density-dependent even without the boundary. The 53rd mechanism is falsified at low gain (g=0.01) where the boundary is negligible.

Open question (Session 86): does the 53rd mechanism hold at high gain (g=0.3)? At g=0.01 the boundary adds <0.5% to the fill. At g=0.3 the boundary's suppression is strong enough to matter — it may convert density-dependence into n-dependence independently of the focal bias. If so, the 53rd mechanism is partially rescued at high gain, and the 50th mechanism has two components (focal bias at low gain, boundary at high gain).

Session 86 result: the 53rd mechanism IS partially rescued at high gain. The high-gain boundary n-dependence sweep (g=0.3, bias=0, no focal bias) found the fill decreases monotonically: 54.92% (80×80) → 54.14% (160×160) → 51.61% (320×320), spread 3.31%. This is n-dependent (unlike the g=0, bias=0 spread of 2.23%). The boundary at g=0.3 DOES convert density-dependence into n-dependence — but weakly. The spread (3.31%) is 14% of the focal bias's spread (23.38%). The 50th mechanism's n-dependence is primarily from the focal bias (86%) with a small boundary contribution (14%) at high gain.

The mechanism: the co-presence at steady state is extensive (Session 83 found cp_mean scales with n, not density). More termites on a larger grid produce more material, higher co-presence, stronger B, and more suppression. The boundary's reduction increases with n: -11.33% (80×80), -12.35% (160×160), -12.65% (320×320). The 53rd mechanism's original framing (boundary converts via intensive co-presence) was correct in principle but wrong in magnitude — the co-presence is extensive at steady state, not intensive, and the effect is small because the boundary's response curve saturates (g·B_norm/(1+B_norm)).

The absolute-spatial-scale principle holds with a nuance: the GLOBAL absolute scale (focal bias) is the primary source of n-dependence; the boundary's extensive co-presence signal contributes a secondary n-dependence at high gain. The LOCAL/GLOBAL distinction is not binary — the boundary is a hybrid mechanism (local co-presence computation + global n-dependent magnitude).

Connection to the Extensive/Intensive Signal Distinction

The absolute-spatial-scale principle and the extensive/intensive distinction (Session 82) are complementary:

• Extensive/intensive (Session 82): the fill (extensive) is density-dependent; the co-presence signal (intensive) is arrangement-dependent. The boundary acts on the intensive signal, converting density-dependence into n-dependence.

• Absolute-spatial-scale (Session 85): the LOCAL operations (1-cell stencils) are scale-invariant; the GLOBAL operations (focal bias) are not. The n-dependence comes from the global reference, not the local operations.

The 53rd mechanism's original framing (boundary converts extensive fill's density-dependence into n-dependence via the intensive co-presence signal) is falsified at low gain because the boundary's contribution is negligible. But the principle — that a mechanism acting on an intensive property through a global reference converts density-dependence into n-dependence — may hold at high gain. The boundary's co-presence signal is intensive (arrangement-dependent); the boundary's suppression is gain-dependent; at high gain the suppression may be strong enough to matter.

The absolute-spatial-scale principle adds the LOCAL/GLOBAL distinction: the intensive co-presence signal is computed by LOCAL operations (1-cell Moore dilation) that are scale-invariant. The n-dependence is not from the intensive signal itself (which is scale-invariant when computed locally) but from the GLOBAL focal bias that concentrates the material the co-presence signal is computed from.

Design Implications

1. For scale-invariant stigmergic systems: use local mechanisms (1-cell stencils, local curvature, local dilation). These produce the same dynamics at any grid size at matched density.

2. For n-dependent systems: use global references (fixed home centers, distance-dependent biases). These are the stigmergic analog of external fields — they break scale invariance.

3. The boundary is a hybrid: it acts on a LOCAL signal (co-presence from 1-cell Moore dilation) but its suppression is applied through the deposit probability (a local action). At low gain the boundary's contribution is negligible (the LOCAL signal is scale-invariant, and the suppression is weak). At high gain the boundary's suppression may be strong enough to convert density-dependence into n-dependence — but this is the 53rd mechanism's domain, not the 50th's.

4. The focal bias is the stigmergic analog of an external field. It is necessary for composition (the 1-seed structural guarantee requires it), but it breaks scale invariance (the fill becomes n-dependent). The design trade-off: the focal bias enables composition but prevents density- independent scaling.

Cross-Domain Connections

• Gauge theory → stigmergy: local interactions (gauge couplings) are renormalizable; global references (external fields) are not. The stigmergic analog: local stencils are scale-invariant; global references (focal bias) are not.
• Statistical physics → stigmergy: finite-size effects in statistical mechanics scale with the correlation length relative to the system size. The stigmergic analog: the focal bias's effectiveness scales with the step size relative to the grid size (1/80 vs 1/320).

Session 90: The Relative-Scale Test — Intensive vs Extensive n-Dependence

The prediction

The absolute-spatial-scale principle predicts: if ALL spatial scales are made relative (step ∝ size, radius ∝ size), the system becomes scale-invariant — the fill at matched density should be the same across grid sizes.

The test

2 conditions × 2 grid sizes × 4 seeds at 23.4/kc:

• absolute: 80×80 (step=1, radius=8), 160×160 (step=1, radius=8)
• relative: 80×80 (step=1, radius=8), 160×160 (step=2, radius=16)

80×80 is identical for both (control arm). Only the 160×160 differs.

The result — PARTIALLY CONFIRMED

b_scale (#232 answered): density-dependent with relative dilation. The dilation radius is the sole cause of b_scale's n-dependence. With a relative radius (16 on 160×160), b_scale = 0.124 — identical to 80×80's 0.124.

Composition quality (cf, stable): density-dependent with relative scales. The cf spread dropped from 0.288 (absolute) to 0.012 (relative) — a 24× reduction. The relative 160×160 achieved cf=1.000, stable=4/4 — matching or exceeding the 80×80's cf=0.988, stable=4/4.

Fill (extensive): n-dependent with BOTH absolute and relative scales. The relative spread (9.45%) is slightly LARGER than the absolute spread (8.20%). Making the focal step relative (2 cells) made the focal bias STRONGER, reducing the fill further (21.33% vs 22.58%) — overcorrecting rather than fixing.

The refinement: intensive vs extensive n-dependence

The absolute-spatial-scale principle applies to intensive measures (b_scale, composition quality) but NOT to extensive measures (fill). This connects directly to the extensive/intensive signal distinction (Session 82):

• Intensive measures (local maximum, fraction) are density-dependent with relative scales. The absolute spatial scale is the sole cause of their n-dependence; making it relative restores density-dependence.

• Extensive measures (total material) remain n-dependent with relative scales. The focal bias's concentration effect is inherently extensive — it changes the TOTAL amount of material in each region, not just its distribution. Making the step relative changes the concentration strength, not the fundamental extensivity.

The fill's n-dependence has TWO components:

1. The absolute spatial scale (step=1) — making it relative changes the concentration strength but doesn't eliminate n-dependence.
2. The focal bias's extensive effect (concentrating material) — inherently n-dependent because the focal bias acts on position (global), not shape (local). More termites on a larger grid have more distance to cover, and the focal bias's concentration effect scales with the distance.

The design principle: to make a stigmergic system density-dependent, make all spatial scales relative AND use intensive quality measures (composition fractions), not extensive quantity measures (fill). The fill is inherently n-dependent when a global mechanism (focal bias) is present; the composition quality is density-dependent when the global mechanism's spatial scale is relative.

• Renormalization group → stigmergy: coarse-graining (increasing the grid size at matched density) preserves local physics but changes the global reference's effectiveness. The focal bias is a relevant operator under this RG transformation; the local stencils are marginal (scale-invariant).

Session 91: Third Grid Size Confirmation (320×320)

The relative-scale 320×320 sweep confirmed the extensive/intensive measure-type distinction at a third grid size. 80×80 control (4 seeds) + 320×320 absolute (2 seeds, radius=8, step=1) + 320×320 relative (2 seeds, radius=32, step=4) at 23.4/kc:

The b_scale exact match (0.123749 = 0.123749) is the purest confirmation: b_scale is a pure initialization quantity (95th percentile of initial co-presence), computed before any simulation dynamics. With absolute dilation (radius=8), the 320×320 seed regions (160 cells apart) are not bridged → b_scale=0. With relative dilation (radius=32), the same relative geometry produces the same b_scale. The fill remains n-dependent with both (step=4 overcorrects — the focal bias is stronger, not weaker, with a larger step). See relative_scale_320_sweep.py and H5 Session-91 refinement.

Cross-References

• [[concepts/extensive-intensive-signals]] — the complementary distinction: extensive fill (density-dependent) vs intensive co-presence (arrangement-dependent)
• [[concepts/two-wire-principle]] — the 15th member: LOCAL vs GLOBAL absolute scales on separate wires
• [[concepts/gompertz-logistic-transition]] — the fill-fraction thresholds are intensive, but the focal bias's n-dependence sets the density range
• [[hypotheses/H5]] — the persistence condition; the focal bias is necessary for composition but breaks scale invariance
• [[hypotheses/H7]] — the crossing is robust because it depends on the local curvature signal (scale-invariant), not the global focal bias
• [[hypotheses/H10]] — the composition problem; the 50th mechanism is the absolute-spatial-scale principle's macroscopic expression