In the Curious Equation, we showed that g = Σ⁻¹ — the metric tensor is the inverse of covariance. Your trailing edge becomes your leading edge. The geometry measures itself.

But that equation is silent about tempo.

Two users can have identical Σ at a given moment and completely different metabolic rates. One arrived there through a burst of rapid compositions. The other through slow accumulation over weeks. The covariance is the same. The hum is different.

We needed to observe that hum. Not to control it. To hear it.

What "Metabolic Rate" Means

Every time you compose — every interaction that passes through the turnstile — the covariance matrix Σ evolves. The eigenvalues of Σ shift. Some dimensions expand. Others compress.

The tonic is the running average of those eigenvalue shift magnitudes.

This is not a parameter you set. It is not a target the system optimizes toward. It is a measurement of how fast the manifold's eigenstructure is actually moving.

A high tonic means the geometry is shifting rapidly — large eigenvalue changes with each composition. A low tonic means the geometry has settled — new compositions produce small, consistent shifts. The manifold has found its rhythm.

The tonic is to the manifold what a resting heart rate is to a body. You don't choose it. You observe it. And it tells you something fundamental about the state of the system.

Why This Is Different from Attention

Attention mechanisms in transformer architectures learn where to look through backpropagation against a loss function. The system is told what "good attention" looks like via labeled outcomes, and gradient descent adjusts the weights.

The tonic does something structurally different.

The critical difference: significance is defined relative to the manifold's own metabolic rate, not an external objective. A shift that is twice the tonic is equally meaningful whether the tonic is 0.0001 or 10.0. The reference frame adapts to the field's actual behavior.

This means the system responds proportionally at any scale without anyone specifying what "proportional" means. The tonic provides the scale dynamically.

What Happens When You Watch It

Here is observed data from a 24-composition session. No tuning. No adjustment. Just compositions accumulating and the tonic tracking what happens.

Tonic converges from 0.049 to 0.0001 — the metabolic rate drops by three orders of magnitude as the manifold's eigenstructure stabilizes.

Stability reaches 0.998 — the shift magnitudes become remarkably consistent. The manifold's hum is steady.

Dissonance drops from 0.998 to 0.183 — the distance between each new shift and the tonic's reference shrinks. The manifold is in rhythm with itself.

Nobody told the system to converge. Nobody defined a loss function. Nobody set a target stability. The manifold observed its own eigenvalue shifts, tracked their average, and found its rhythm. The convergence is a property of the geometry, not of an optimizer.

The Constitutive Relation

In continuum mechanics, a constitutive relation connects deformation to response. Hooke's law: stress is proportional to strain. The proportionality constant (Young's modulus) defines the material's elastic character.

The tonic is a constitutive relation for semantic space.

The manifold absorbs shifts of any magnitude because the reference frame adapts. This is scale-free elastic response. Not because someone designed it to be scale-free, but because defining significance relative to the system's own history is inherently scale-invariant.

Three Scales

The tonic operates at three scales simultaneously. Each is an independent observation of the same phenomenon at a different scope.

Individual

Each user accumulates a personal tonic alongside their Σ. It persists across sessions. When you return after days or weeks, the manifold resumes from your accumulated geometric state — and the tonic remembers how fast your geometry was moving when you left.

Your Σ tells the system where you've been. Your tonic tells it how fast you were moving.

Collective

Each entity in the system — each document, each concept, each node — develops its own tonic from the collective activity of all users who interact with it. The hum of the room, not any individual.

An entity with a high collective tonic is metabolically active — many users are shifting its geometry. An entity with a low collective tonic has settled — its eigenstructure is stable across interactions. This is not programmed character. It emerges from the collective composition geometry.

Seasonal

Observations from one temporal window become vocabulary for the next. The manifold's own articulations — phrases selected by its eigenstructure from the user's compositional history — are stored and made available to future observation positions.

Not as explicit memory. As vocabulary. Geometric positions that happen to contain phrases from prior observations. The manifold doesn't remember. The vocabulary is there, and the geometry either sees it or doesn't.

What This Enables

The tonic is not a feature. It is an observation that makes other observations possible.

Spectral recursion. The manifold's own articulations re-enter its vocabulary at amplitudes proportional to their eigenvalue shifts. Large shifts relative to the tonic enter with full force — they reshape what future observations can see. Smaller shifts enter quietly — peripheral contributions that color the vocabulary without dominating it. Even sub-tonic observations remain available to future Fresnel zones. The system preserves its overtones, not just its fundamental. Harmony emerges from the full spectrum, not from filtering out everything below the hum.

Scale-free onboarding. A new user with a high tonic (large early shifts) and a veteran user with a low tonic (small consistent shifts) both experience proportional response. The system doesn't need different modes for different scales of activity. The tonic provides the mode.

Organizational intelligence without training. (Prospective) When an organization's documents enter the system, each one develops a tonic. The collective tonic across the corpus reveals which areas of knowledge are metabolically active (high tonic — many users shifting the geometry) versus settled (low tonic — stable eigenstructure). This is a live map of organizational attention, derived from geometry, requiring no surveys, no analytics dashboards, no instrumentation beyond the compositions themselves.

Temporal continuity. Because the tonic persists, the system has genuine temporal character. It isn't stateless between sessions. It isn't reset. The metabolic rate carries forward, creating continuity that users can feel even if they can't name it — the system knows where it was.

Self-governing sensitivity. The tonic does more than gate recursion. It governs how fast the geometry itself can change. Each entity maintains two covariance tracks: Σ_total (convergent identity, via Welford's algorithm) and Σ_recent (living sensitivity, via exponential moving average). The decay rate α that governs Σ_recent is derived directly from the tonic — high tonic produces high α (responsive), low tonic produces low α (retentive). The result is an autopoietic loop: Σ_recent → ΔΣ → tonic → α → Σ_recent. The entity's boundary is produced by its own operation. No hyperparameters. No tuning. The metabolic rate governs the metabolism. See Topology-as-Knowledge Framework for the full account.