C=2A: Unifying Quantum Measurement, Gravitational Collapse, and Consciousness
Recognition Physics Institute, Austin, Texas
Summary
This paper proposes and substantiates an exact identity $C=2A$ between a recognition path action $C$ and a residual (gravitational) rate action $A$. The identity is parameter-free, local, and explains in one stroke: (i) Born weights in measurement, (ii) the collapse threshold in gravity-related proposals, and (iii) when conscious experience becomes definite. It yields clear falsifiable predictions in the nanogram–second regime.
Abstract
Problem: We lack a single mechanism that explains quantum measurement weights, gravity-driven reduction, and the onset of definite experience without tunable parameters. Bridge: We derive $C=2A$ where $C$ is built from the unique convex cost $J(x)=\tfrac{1}{2}(x+x^{-1})-1$ and $A$ arises from two-branch geodesic dynamics. This fixes $P=e^{-C}=e^{-2A}$ and $|\mathcal{A}|=e^{-C/2}=e^{-A}$. Triple unification: measurement, gravity, and consciousness share a single threshold: $C\ge 1 \Leftrightarrow A\ge 1$.
Key Contributions
- Exact identity: $C=2A$ proved for two-branch geodesics; branchwise softmax $P_I=\frac{e^{-2A_I}}{\sum_J e^{-2A_J}}$
- Amplitude bridge: $|\mathcal{A}|=e^{-C/2}=e^{-A}$, hence $P=|\mathcal{A}|^2=e^{-2A}$
- Shared threshold: collapse and definite experience occur when $A\ge1$ (equivalently $C\ge2$)
- Predictions: mesoscopic scaling $\tau\,m\,|\Phi_{12}|\sim1$ (ng–s regime); modest observer-coupled modulation via mutual information
- Falsifiers: threshold mismatch ($C\neq1$ or $A\neq1$ at collapse), absent mass scaling, null observer effect
Mathematical Framework
The recognition side uses the unique convex symmetric cost $J(x)=\tfrac{1}{2}(x+x^{-1})-1$ with $J(1)=0$, $J''(1)=1$, and $J(x)=J(1/x)$. The residual side models a two-branch geodesic rotation with rate action $A=-\ln|a_b|$ (with $a_1=\cos\theta$, $a_2=\sin\theta$). The constructive kernel gives the exact differential
which integrates to $C_b=2A_b$ and reproduces Born weights $e^{-2A_b}$.
Implications
One local, verifiable mechanism governs quantum measurement, gravity-related reduction, and definite experience. The framework is maximally falsifiable: no adjustable parameters and crisp thresholds.
Experimental Predictions & Falsifiers
- Scaling: $\tau_{\mathrm{dec}} \propto 1/(m|\Phi_{12}|)$
- Observer-coupled modulation via $I(\mathrm{Ag};\mathrm{Env})$
- Hard falsifiers: threshold mismatch; absent mass scaling; null observer effect