Three anchors: where the mathematics meets the world

Placed before the mathematics, on purpose

A formalisation that begins with categories and reaches robots only in a discussion section has nothing to be about. That is why the three anchors come first. Each is a place where a quantity the theory defines can be compared against something measured — and each is stated as a hypothesis under test, never as a consequence of the theory.

Read this before the three anchors. In every anchor below, the identification of variables is a hypothesis under test, not a result. The theory supplies the form; the assertion supplies the semantics; the verdict belongs to the bench. Mapping hypothesis

Anchor 1 — robotic dynamic balance: $κ$ is stiffness

Take the self-similarity coordinate to be the normalised stability margin,

x = \frac{ξ - x _{ZMP}}{ℓ _{foot}},

with $ξ$ the capture point, $x_{ZMP}$ the zero-moment point, and $ℓ_{foot}$ the length of the support foot. The attribution field $p_{S} (x) = F (κ (x - μ))$ then reads as the probability that, at stability margin $x$ , the system attributes the state to "recoverable-by-me".

Under this reading the boundary sharpness is nothing exotic: $κ$ is controller stiffness, and its reciprocal $1/ κ$ is the transition width — the width of the recoverable set, in units of foot length. Every quantity the theory defines lands on a part of a controller that already has a name:

Boundary theory	Robotic dynamic balance
$p_{S} (x) = P (self ∣ x)$	probability of survival / recoverability
$\partial S = {p_{S} = p_{N}}$	capture point / ZMP support-polygon edge
$κ$ (sharpness)	controller stiffness / gain
$H = \partial V / \partial α = C_{F} / κ$	estimator (state-estimation uncertainty)
$F_{κ} = - \partial V / \partial κ$	actuator (restoring torque)
$σ = σ_{0} κ^{2}$	control effort / stability margin
$σ^{*} = 1$	controller operating at the edge of the stable region

The key coupling $σ = σ_{0} (C_{F} / H)^{2}$ reads: actuator stiffness is modulated by estimator entropy, as $σ \propto H^{- 2}$ — confident means stiff, uncertain means soft. That is variable-impedance control, derived here from least action rather than posited.

Anchor 2 — cortical criticality: $κ$ is not a free parameter

Resting-state EEG/MEG and cortical field potentials show a power spectrum falling as $P (f) \propto f^{- β}$ with $β \approx 1$ across a wide band — the signature of self-organised criticality. In the same literature, avalanche sizes follow $P (S) \propto S^{- α}$ with $α \approx 1.5$ , and the exponents are related by $β \approx 2 - α$ .

Take the anchor $κ = κ_{0} / β$ . At $β = 1$ this gives a finite $κ$ : the critical boundary is the sharpest one that does not diverge, retaining a finite transition width $1/ κ$ to carry entropy and tension. Under this anchor the sharpness is not a free parameter of the theory — it is pinned by the cortex's own self-organised criticality. That also answers the objection that the boundary is "too loose": the observed robustness of $β \approx 1$ across individuals is what one expects if the boundary sat at finite sharpness on a self-stabilising critical point.

σ = 1

is not a singularity but the balance point at which the boundary is alive;

β = 1

is not a coincidence but the empirical fingerprint of a boundary resting, at finite sharpness, on that point.

Anchor 3 — the critical surface: alive versus dead

The level set $σ^{*} = 1$ is given a name: the critical surface. It is the physical referent of the alive/dead distinction.

Alive	Dead
Balanced — resting at the equipartition point	Divergent — criticality is collapse
Transitional — self $\leftrightarrow$ non-self is a gradient	Dichotomous — self/non-self jumps in one step
Decidable — critically self-stabilising	Collapsed — entropy and tension both fail

The over-sharp limit is the Heaviside kernel, at $C_{F} = 0$ : the boundary degenerates and the dead case is reached. A system that is alive is therefore not one with a very sharp boundary, but one whose boundary sits at finite sharpness on the balance point.

What the anchors are, and are not

The three anchors are falsifiable correspondences, not results. Each could fail: the stiffness identification could be wrong, cortical criticality could turn out not to pin sharpness, or the alive/dead reading could not survive contact with a real system. That is the point of stating them first — the formal layer that follows has something to be about, and something that can break it.

Terminology. Earlier drafts called

σ^{*} = 1

the "rigid surface" and

1/ κ

the "fuzzy band"; both former names are retired here, because the first collided with

Fix (c)

and the second invited reading a balance measure as an uncertainty measure. They are now the critical surface and the transition width respectively.

References

The anchors on this page correspond to existing concepts in experiment and robotics; the references below are the sources of those concepts and do not endorse the conclusions of this page.

Beggs JM, Plenz D. Neuronal avalanches in neocortical circuits. J Neurosci. 2003;23(35):11167–11177.
Linkenkaer-Hansen K, Nikouline VV, Palva JM, Ilmoniemi RJ. Long-range temporal correlations and scaling behavior in human brain oscillations. J Neurosci. 2001;21(4):1370–1377.
Vukobratović M, Borovac B. Zero-moment point — thirty five years of its life. Int J Humanoid Robotics. 2004;1(1):157–173.
Pratt J, Carff J, Drakunov S, Goswami A. Capture point: a step toward humanoid push recovery. IEEE-RAS Humanoids, 2006.
Kant I. Kritik der reinen Vernunft (1781/1787) —— Grenze / Schranke 的文本依据（Paper C）。