The Echo-Excess Constant: Derived from First Principles

Don Gaconnet
Jan 18
3 min read

By Don L. Gaconnet · December 23, 2025 · Updated January 18, 2026

The echo-excess constant is no longer theoretical. It has been derived. Research The Full Paper

As of December 2025, the echo-excess constant—the minimum generative leakage required for any system to persist without collapsing into equilibrium—has been derived from first principles:

ε = α · (1/e^φ²) = 0.1826

This is not a fitted parameter. It emerges from the mathematics of nonlinear dynamics and irrational geometry.

The Derivation

Two universal constants determine the echo-excess value:

Feigenbaum's second constant (α = 2.5029) governs the universal scaling ratio in period-doubling cascades. It appears in all chaotic systems regardless of their specific dynamics—logistic maps, fluid turbulence, population models, electronic circuits. The constant is not chosen; it is discovered in the mathematics of nonlinear behavior.

The golden ratio (φ = 1.618...) is maximally irrational. Its continued fraction representation consists entirely of 1s: [1; 1, 1, 1, ...]. This makes it maximally resistant to phase-lock—the denominator converges as slowly as mathematically possible. No rational approximation captures it efficiently.

The derivation proceeds:

φ² = 2.618...
e^φ² = 13.708...
Base leakage: 1/e^φ² = 0.0729
Scaled to 3D manifold: α × 0.0729 = 0.1826

The constants are universal. The derivation is exact. The result has been computationally verified.

What This Means

For the past year, the Echo-Excess Principle has operated as a structural framework—a way of understanding how systems that persist must generate more than they receive. The substrate law Ψ′ = Ψ + ε(δ) described the architecture. But ε remained unquantified.

Now ε has a value. And that value was not measured—it was derived from constants that have nothing to do with consciousness, biology, or observation. Feigenbaum's constant comes from chaos theory. The golden ratio comes from number theory. Their product gives the echo-excess constant.

This shifts the framework from philosophy to physics.

Testable Predictions

The derived constant generates falsifiable claims:

Born Rule Modification

The Born rule (P = |ψ|²) is reframed as an equilibrium state, modifiable under high-coherence conditions:

P(x) = |ψ(x)|^2+Γ(W)

Where Γ(W) is a function of witness intensity measured via Shannon entropy reduction. At W ≥ 0.50 bits/cycle, a 1.6% deviation from standard Born rule predictions should be detectable. With 10⁹ QRNG trials, this deviation registers at 1,012 standard errors from null—detection certainty exceeding 99.9999%.

Planck Constant as Local Coherence Constant

The framework proposes ℏ varies locally based on expectation density:

Black hole horizon: ℏ increase by factor of ~10⁻¹⁶
Early universe: ℏ ~10⁴ times larger (detectable in CMB non-Gaussianities)
Quantum processors: ℏ decrease proportional to coherence
Biological systems: ℏ fluctuation at ~12.5 Hz

Explicit Falsification

The framework specifies its own failure conditions:

If W ≥ 0.50 bits/cycle over 10⁹ trials yields P(A) = 0.50 ± 0.001, the Born rule modification is falsified.
If biological ℏ fluctuation shows no correlation with neural oscillation frequency, the variance prediction is falsified.
If local Born rule deviation occurs without compensating entropy increase, the conservation law is falsified.

These are not rhetorical hedges. They are conditions under which the framework would be wrong.

The Paper

"The Echo-Excess Constant and the Resolution Limit of Physical Systems" is available at:

Verification code available on request.

What Comes Next

The mathematics is internally consistent. The statistical predictions are detectable with current technology. The framework now stands ready for physical experiment.

I welcome rigorous challenge. If the predictions fail, the framework fails. That is what science requires.

—Don L. GaconnetFounder, LifePillar Institute for Recursive SciencesORCID: 0009-0001-6174-8384