In Metabolic Rate, we showed the tonic — a self-calibrating reference frame that tracks the manifold's metabolic tempo. In Geometric Surplus, we showed the yield — the gap between eigenstructure discrimination and embedding consensus that crosses zero within six compositions.

Now we take their ratio.

This is not a metric we designed. It is the natural ratio of two quantities the system already produces. Surplus measures how much more the user's eigenstructure discriminates than the pre-trained embedding consensus. Tonic measures the metabolic rate — how fast eigenvalues shift per composition. Their ratio asks: how much yield per unit of metabolic work?

This document reports the reproduction of η across a factorial test design: 3 users × 2 sessions each, with different composition orders, plus a confound isolation run. Twelve sessions total. The findings identify which properties of η are structural invariants and which carry semantic content.

Where It Sits

Metabolic efficiency is Level 2 in a stack of emergent properties. Each level uses only quantities already present in the system. No new loss functions. No optimization targets.

Levels 0–2 are documented with reproducible data across multiple sessions. Level 3 is now validated — see below. Level 4 shows a preliminary signal.

The Blindfold

The initial observation of η (February 11) used a composition vector computed as a weighted blend: 0.4 × source + 0.6 × target. This meant every composition contributed 60% of the same signal — the sapling, which is always the target — regardless of which being was being composed.

The tonic appeared invariant across all composition orders, declining 83% in every session. This looked like a gauge symmetry. It was not. The blend was a blindfold — it masked being-specific information by drowning it in sapling signal.

The fix: remove all blending. A composition is an event with two signals — source and target. The system observes both. Σ_life receives two rank-1 updates per composition (one from each vector). Lookups and storage use the source vector — the source being IS the semantic content. Twelve instances of the 0.4/0.6 blend were removed from the codebase.

The system was designed to observe. The blend imposed a prior about what matters more. Habitat does not optimize. It observes what happened. Removing the blend restored the observational contract.

The Factorial Test

Three users. Two sessions each. Different composition orders. Every session composes all six guild beings onto the sapling. Each session uses a unique user identity for clean cold-start Σ_life. Redis flushed before the first session.

User A (canonical): bayberry, beebalm, clover, comfrey, wild garlic, yarrow.
User B (reverse): yarrow, wild garlic, comfrey, clover, beebalm, bayberry.
User C (shuffled): comfrey, bayberry, yarrow, clover, wild garlic, beebalm.

Each row is a complete session. The numbers are metabolic efficiency at each composition step. Bold values are terminal η.

Three Validated Properties

1. Tonic Gauge Invariance

Tonic declines 83.1% in every session. Every user. Every order. Every run.

Cross-session standard deviation at each position: 0.000000. The tonic trajectory is identical to six decimal places regardless of which beings are composed or in what order. Twelve sessions. Zero variation.

This was not an artifact of blending. After removing the 0.4/0.6 weighted blend, tonic remains perfectly invariant. The eigenvalue shift decay pattern is a property of the covariance accumulation geometry itself — not of which vectors contribute to it.

Tonic is a gauge quantity. It measures the manifold's metabolic tempo — the rate at which eigenstructure absorbs new information — and that rate is invariant to composition content. This is the denominator of η: a stable reference frame against which surplus can be measured.

2. Surplus Differentiates Beings

Unlike tonic, surplus sees the content. Different beings produce different surplus at the same position, and different orders produce different terminal η.

Within-user reproducibility: max Δ(η) = 0.011. Between-user divergence: mean Δ(terminal η) = 2.87. The signal-to-noise ratio is 2.87 / 0.006 ≈ 460×.

Individual beings carry distinct metabolic signatures:

Bayberry is consistently the lowest-η being (mean +1.03). Beebalm is the highest (mean +5.76). The position dependence (σ) reflects the interaction between being identity and composition history — the same being produces different η depending on when it enters the sequence.

3. Path Dependence

η is reproducible per-user but genuinely different across composition orders. Both being identity (σ = 1.44) and position (σ = 2.54) contribute. The interaction term is significant: what a being produces depends on the covariance history it enters.

User B (reverse order) peaks at η = +9.59 at composition 5, then drops to +3.92 when bayberry enters last. User A (canonical order) builds monotonically from −0.61 to +6.95. User C (shuffled) dips negative at composition 2 (bayberry) then recovers to +8.23.

These are not noise. They reproduce to four decimal places. They are the η trajectory for that composition order on this substrate.

The Confound

Habitat maintains a global being memory (_being_memory) that accumulates collective covariance across all users. When the test runs User A → B → C sequentially, later users see beings whose collective Σ has been warmed by earlier users' compositions. This is a potential confound: does being memory warmth affect η?

To test this, we ran a second copy of the full factorial test with being memory explicitly reset between each user pair via a dedicated API endpoint. This isolates each user's sessions from all prior accumulation.

Maximum confound Δ: 0.012. Between-user signal: 2.87. Confound-to-signal ratio: 0.4%. The being memory confound is negligible. η is determined by the user's own composition order, not by prior users' accumulation.

Slack Decomposition

Each composition's ED distribution contains items from two sources: the base corpus (seed vocabulary) and ED recursion (vocabulary injected by the tonic-harmonic loop). Decomposing surplus by origin reveals the noise structure.

Base corpus items produce deterministic surplus — same values across sessions for the same being at the same position. Recursion items introduce the only variation between sessions: they are the sole source of the 0.011 within-user Δ(η).

Overall base corpus fraction: 77%. Recursion items contribute 23% of the ED distribution. They carry measurable surplus but with higher variance than base items. The slack between base and recursion surplus is where the system's vocabulary evolution becomes visible — the recursion loop adds items near the composition point, and their alignment with the Fresnel zone varies slightly between sessions.

Level 3: Culture

The prediction was: convergent η across observers on shared substrate. The test: five users, seven beings, fourteen compositions each, three phases, 210 total compositions. Different composition orders. The question: do the being-manifolds impose convergent geometric signatures on independent observers?

Perturbation phase variance drops 56% relative to isolated. When beings accumulate collective geometry from multiple observers, the terminal η distribution narrows from σ = 4.37 to σ = 1.94. The shared substrate constrains the space of possible surplus trajectories.

The per-being surplus matrix reveals the mechanism. Each being has a characteristic surplus signature — bayberry is the only being where all five users observe negative or near-zero instant surplus. Comfrey has the highest mean surplus across users. The culture ratio (column σ / row σ) is 0.41 — the beings impose more constraint on the surplus than the observers do.

Culture is not agreement. Bayberry's negative surplus is as culturally stable as comfrey's positive surplus. Both are geometric properties of the being-manifold that every observer confirms. Culture is shared fidelity to geometric reality — including stable divergence.

Level 4: Resilience (Preliminary)

The perturbation test forces bayberry (the lowest-surplus being) at position 8 in an otherwise collective session. Recovery resilience R measures how quickly η returns to its trajectory after disruption.

R variance drops 62% under shared substrate. The collective being-manifold provides a common recovery basin — perturbation is absorbed more uniformly. This is a preliminary signal. It awaits larger-scale validation.

What Comes Next

Surplus-weighted vocabulary selection. ED vocabulary items where surplus converges across observers should be preferred for shared vocabulary. These items articulate stable geometric properties of the being-manifold — they are culturally grounded by construction.

Multi-condition culture. Currently only one environmental condition (late spring frost) provides the geodesic baseline. Extracting summer drought and autumn abundance would test whether different conditions produce different cultural signatures — the same beings, different geometry.

Warm-start behavior. All sessions in this test are cold-start (Σ_life initialized fresh). How η behaves when users return to existing covariance history is an open question.