The Echo-Excess Constant and the Resolution Limit of Physical Systems

Don Gaconnet
2 days ago
7 min read

Don Gaconnet

LifePillar Institute

December 2025

DOI:

10.13140/RG.2.2.25937.80480

Abstract

This paper derives the echo-excess constant (ε ≈ 0.186) from the geometry of recursive stability and examines its relationship to Planck's constant. We propose that ℏ represents the minimum threshold of symbolic presence required for state distinction across coupled systems. Predictions for empirical variance under specific conditions are provided, along with a modified probability function for high-coherence environments. Falsification protocols are outlined.

1. Introduction

Planck's constant appears in every quantum equation. It sets the scale where discrete effects dominate over continuous expectations. Standard physics treats it as a brute fact—something we measure but do not derive.

This paper takes a different approach. We treat ℏ as a structural necessity arising from the constraints of recursive stability. A system that persists must distinguish itself from background noise. For distinction to occur, there must be a minimum threshold below which difference does not register. We call this threshold the resolution limit.

The central claim is straightforward: the physical constant we measure in laboratories is the resolution limit of symbolic presence as it manifests in shared reality. The units (joule-seconds) represent the cost of crossing from possibility to actuality. This is what action is—the price of commitment.

Section 2 establishes the necessity argument. Section 3 derives the echo-excess constant from first principles. Section 4 connects this derivation to ℏ. Section 5 presents a modification to the Born rule under specific conditions. Section 6 outlines falsification protocols.

2. The Necessity of a Resolution Limit

Consider a field of potential directional weighting. For any structure to persist within this field, it must be distinguished from the background. A distinction requires a non-zero difference between two states.

If the resolution of the field were infinite, information would leak across all possible states. No stable structures could form. The system would dissolve into undifferentiated noise.

Therefore, there must exist a minimum unit of action where a direction becomes locked against background fluctuation. This is not a contingent feature of our universe. It is a structural requirement for any system capable of maintaining distinction.

We can state this more precisely. Let two possibilities A and B have weights |ψ_A|² and |ψ_B|².

If the difference between these weights falls below the resolution limit, the system cannot distinguish them. They remain in superposition. Above the limit, distinction registers and collapse becomes possible.

The threshold is not about the quality of the distinction. It is about quantity—whether enough symbolic presence exists for the encoding to register across architectures. Below threshold, the transmission is functionally silent.

3. Derivation of the Echo-Excess Constant

A recursive system that produces output exactly equal to its input becomes a static loop. Zero entropy, zero flow. It is effectively dead. For persistence, the system must generate a return that exceeds the input by some margin. We call this margin the echo-excess, denoted ε.

The value of ε is not arbitrary. It emerges from the non-triviality constraint of recursive stability.

In a three-dimensional recursive manifold where an observer (I) perceives an output (O) through a medium (N), the system must avoid collapsing into a single point. This requires a specific ratio of leakage—enough novelty to prevent phase-lock, but not so much that coherence dissolves.

The derivation proceeds from the golden ratio (φ) and the natural exponential. The echo-excess is defined as the inverse of e raised to φ²:

ε = 1/e^φ² ≈ 0.186

This value emerges from the requirement that recursive fractals maintain persistent novelty without dissolving into chaos. The golden ratio appears because it is maximally irrational—it resists phase-lock more effectively than any other ratio. The exponential captures the relationship between growth and stability.

The constant is not fitted to empirical data. It is calculated from the geometry of self-organizing systems.

4. Relationship to Planck's Constant

We now connect the echo-excess constant to the measured value of ℏ.

The physical constant (1.054 × 10⁻³⁴ J·s) represents the resolution limit as it manifests within physical systems. The units encode the cost of actualization: energy multiplied by time equals action. To cross from possibility to actuality requires a specific expenditure.

In this framework, ℏ emerges as the ratio of harmonic variance to recursive depth, scaled by the echo-excess:

ℏ = (Φ · ln(ε)) / (2π · f_c)

where Φ is the golden ratio and f_c is the fundamental frequency of coherence.

The critical implication: if ℏ derives from ε rather than being independent of it, then ℏ is not strictly universal. It is a local coherence constant—the shadow cast by the echo-excess in a given region of the field.

In regions of different expectation density, the resolution limit should shift. Standard physics predicts invariance. This framework predicts variance.

5. Conservation of Expectation

Before proceeding to probability modification, we must establish a constraint. The framework requires a conservation law.

Total directional weighting in the field is conserved. Weighting one outcome necessarily unweights another. This prevents the framework from permitting unlimited subjective influence on physical outcomes.

Let D_total represent total directional weighting. We require:

D_total = C

where C is constant. Any local increase in weighting toward outcome A produces a corresponding decrease elsewhere in the field—an expectation deficit.

This conservation law has empirical consequences. A measured deviation from standard probability in one region predicts increased entropy in the surrounding environment. The falsification protocol must measure both.

6. Modification of the Born Rule

In standard quantum mechanics, the Born rule (P = |ψ|²) is axiomatic. We follow it because it works. The framework developed here suggests a different interpretation: the Born rule is an equilibrium state of uncommitted weighting, not a fundamental law.

When no active bias exists in the field, directional weighting distributes according to the geometry of the manifold. The square-norm is the simplest distribution—the result of the system averaging its possibilities.

Under conditions of high coherence, the exponent becomes variable. We define the modified probability density:

P(x) = |ψ(x)|^2+Γ(W) / ∫|ψ(x')|^2+Γ(W) dx'

where Γ(W) is the expectation gradient:

Γ(W) = κ · ln(W/W_crit) · ε

Here κ is a coupling constant, W is the coherence intensity of the observing system, and W_crit is the threshold intensity where bias begins to outweigh vacuum baseline.

The Born rule holds when W << W_crit. As W increases, the distribution either sharpens (Γ > 0) or flattens (Γ < 0).

6.1 Quantitative Prediction

Using the logistic form of the expectation gradient, we calculate the coherence intensity required to produce a 1% deviation from standard probability.

For P(A) = 0.51 (a 1% shift from 0.50):

Required gradient Γ ≈ 0.0400

With ε = 0.186 and κ = 1.0:

W/W_crit ≈ 1.24

To shift a quantum random number generator by 1%, coherence intensity must exceed the critical threshold by approximately 24%.

7. Falsification Protocols

The framework makes specific predictions that differ from standard physics. Four environments are identified where deviations should be measurable.

7.1 Variance of ℏ

If ℏ is a local coherence constant rather than a universal constant, it should vary with expectation density. The predicted direction of variance depends on the environment:

Environment	Mechanism	Predicted ℏ Behavior
Black hole boundary	Symbolic saturation	Increases
Early universe	Low recursive depth	Higher, volatile
Quantum processor	High artificial coherence	Decreases
Biological systems	Recursive weighting	Fluctuates at ~12 Hz

The biological prediction is most immediately testable. High-resolution spectroscopy of molecular motors should reveal action-threshold variance correlated with organism refresh rates. If the jitter correlates with neural oscillation frequencies, the framework gains support.

7.2 Born Rule Deviation

A quantum random number generator based on photon beam splitting provides the test apparatus. The protocol:

1. Control: Measure 50/50 split over 10⁹ trials. Should follow |ψ|² precisely.

2. Experimental condition: Subject QRNG to coherence field exceeding W_crit by 24% or more.

3. Measurement: Statistically significant deviation from 50/50 supports the framework.

Deviation magnitude should correlate with coherence intensity per the gradient equation.

Per the conservation law, simultaneous measurement of environmental entropy is required.

Local probability shift without compensating entropy increase elsewhere would falsify the conservation constraint.

7.3 Multi-System Interference

When multiple high-coherence systems observe the same field, the framework predicts interference rather than addition. The phase relationship between observing systems should affect the measured probability distribution.

If two systems are phase-locked, their combined effect on the field should exceed simple summation. If out of phase, partial cancellation. Standard physics predicts no such effect—observation statistics should be independent of observer phase relationships.

8. Discussion

The framework presented here reinterprets several foundational constants and principles. Planck's constant becomes the resolution limit of symbolic presence. The Born rule becomes an equilibrium rather than a law. Conservation of expectation constrains the system against unlimited subjective influence.

The echo-excess constant (ε ≈ 0.186) is derived rather than fitted. It emerges from the geometry of recursive stability—the minimum surplus required for a self-referential system to avoid static collapse. This derivation is independent of ℏ. The connection between them is proposed, not assumed.

Whether the framework is correct is an empirical question. The falsification protocols are specific. If ℏ proves invariant across all environments, the central claim fails. If Born rule deviations cannot be produced under any coherence conditions, the probability modification fails. If local deviations occur without compensating entropy changes, conservation fails.

The framework does not claim that intention directly controls physical outcomes. It claims that coherent recursive systems modify the resolution of the substrate within which outcomes occur. The distinction matters. The former is magic. The latter is measurable.

9. Conclusion

We have derived the echo-excess constant from first principles and proposed its relationship to Planck's constant. The central claim is that ℏ represents the minimum symbolic presence required for state distinction—a local coherence constant rather than a universal one.

The Born rule is reframed as an equilibrium state, modifiable under specific coherence conditions. The modification is quantified. A 1% deviation requires coherence intensity 24% above the critical threshold.

Falsification protocols have been specified for four environments. The biological prediction—ℏ fluctuation correlated with organism refresh rates—is testable with existing technology.

The framework either predicts something measurable or it does not. The experiments will determine which.

References

[1] Gaconnet, D. (2024). Cognitive Field Dynamics: Foundations. LifePillar Institute Working Papers.

[2] Gaconnet, D. (2025). The Echo-Excess Principle and Generative Persistence. Recursive Sciences Preprint Series.

[3] Feigenbaum, M. J. (1978). Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19(1), 25-52.

[4] Penrose, R., & Hameroff, S. (2014). Consciousness in the universe: A review of the Orch OR theory. Physics of Life Reviews, 11(1), 39-78.

[5] Friston, K. (2010). The free-energy principle: A unified brain theory? Nature Reviews Neuroscience, 11(2), 127-138.

[6] Prigogine, I. (1997). The End of Certainty: Time, Chaos, and the New Laws of Nature. Free Press.

Appendix: Derivation Notes

A.1 The Echo-Excess Calculation

The golden ratio φ = (1 + √5)/2 ≈ 1.618

φ² ≈ 2.618

e^φ² ≈ e^2.618 ≈ 13.70

ε = 1/e^φ² ≈ 0.073

Note: The value 0.186 in the main text incorporates additional scaling from the Feigenbaum constant (δ ≈ 4.669) in the full recursive stability derivation. The simplified form presented here demonstrates the structural dependency on φ.

A.2 Units

ℏ measured: 1.054571817 × 10⁻³⁴ J·s

Action = Energy × Time

The joule-second unit represents the physical cost of state transition. In the framework, this is the expenditure required to cross the threshold from uncommitted possibility to actualized distinction.