Positive Geometries, not Space-Time, is Fundamental?

from Google AI:

Positive geometry is an interdisciplinary field linking theoretical physics and algebraic geometry, where physical observables—such as particle scattering amplitudes—are determined by the geometry of specific shapes. These "positive geometries," including amplituhedra and cosmological polytopes, are specialized, often convex, higher-dimensional shapes that encode scattering probabilities and cosmological interactions through their structural boundaries and a unique associated "canonical form".

Key Aspects of Positive Geometry:

• Fundamental Idea: It provides a new way to calculate particle scattering amplitudes and cosmological correlations, bypassing traditional Feynman diagram methods.
• The Amplituhedron: Considered a prototypical positive geometry, it lives in momentum twistor space and helps calculate scattering in planar N=4 supersymmetric Yang–Mills theory.
• Structure: They are defined as semi-algebraic subsets of complex projective varieties (over the real numbers), with boundaries corresponding to physical singularities (where particle processes diverge).
• Canonical Form: Each positive geometry is associated with a unique "canonical form"—a differential form with logarithmic singularities on all boundaries.
• Significance: It serves as a candidate to redefine the foundations of quantum field theory and gravity, potentially showing that spacetime locality and unitarity are emergent properties rather than foundational assumptions. 

Positive geometry connects the abstract mathematics of convex polytopes and complex varieties to the practical calculation of particle interaction rates and cosmological evolution.  

“we must abandon the project of reconciling spacetime and quantum mechanics, and accept that spacetime, at least as Einstein described it, does not exist.”https://t.co/Pe73C4GgbT

— Donald Hoffman (@donalddhoffman) April 21, 2026

On Markov blankets

Yes, in the context of theoretical biology and cognitive science (specifically the Free Energy Principle), identity is often conceptualized as a Markov blanket. It acts as the boundary or interface that separates an agent's internal states from the external environment, defining what is "self" versus "not-self". 

Here is a breakdown of the connection:

• Definition of Self: A Markov blanket separates internal states (e.g., brain activity, metabolic states) from external states (e.g., the world). Your sense of self emerges within this blanket.
• Boundary Function: The blanket consists of sensory and active states that mediate interactions between the internal and external world. It defines an "inside" and "outside".
• Active Inference: As described in this Medium article by Satyam and thisLessWrong post, active inference is how systems maintain their identity (Markov blanket) by managing sensory input and active outputs, as explored in this PMC article.
• Dynamic Identity: Identity is not static; rather, it is a process of "knitting" a Markov blanket, meaning that living creatures actively maintain their boundaries through autopoietic (self-producing) processes.
• Scale Invariance: Markov blankets are nested, which means that "identities" can exist at different scales—from a single cell's membrane to an organism's sensorium. 

While a Reddit thread provides an intuitive overview of the concept, the idea that Markov blankets are essential for biological self-organization is detailed further in this research paper. 

Note: There is an ongoing debate about the limitations of this model in fully capturing biological identity, as noted by Walter Veit, regarding its ability to cover all facets of living systems

On the Amplituhedron

from Google AI:

Donald Hoffman utilizes the amplituhedron—a geometric structure discovered by Nima Arkani-Hamed and Jaroslav Trnka—to support his theory that spacetime is not fundamental and that consciousness is the ultimate reality. Hoffman argues these "positive geometries" exist outside spacetime, representing a deeper reality of interacting conscious agents rather than physical particles.

Key Connections to Hoffman's Theory

Beyond Spacetime: Hoffman views the amplituhedron and similar structures (e.g., associahedra) as evidence that, as he explains in this Brill publication, spacetime is an artifact rather than a fundamental framework.

Conscious Agents: Hoffman aims to map the dynamics of his "conscious agents" (a network of agents) onto these geometric structures, specifically using "decorated permutations" to connect his model of consciousness to the scattering amplitudes.

Reality as Projection: In his view, the physical world, including particles and spacetime, is a projection or "user interface" of these deeper, non-spatial structures.

Simplification of Physics: As discussed in this interview, the amplituhedron is a "jewel" that simplifies particle scattering calculations, which Hoffman interprets as evidence that the fundamental reality is mathematical and conceptual, not physical.

Hoffman's use of the amplituhedron is detailed in his 2023 paper Fusions of Consciousness, where he connects his theories of perception to quantum physics research.

On Cosmological Polytopes