Non-Archimedean Projective Perspective: The Monna Map as a Visual Rendering Interface

In the paper's words

Things farther away appear smaller, and two-point linear perspective is taken to approximate the eye. That phenomenology is Archimedean geometry. This paper asks what changes if the metric of physical space were instead non-Archimedean and ultrametric.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Would we perceive the same thing, and what would that actually mean ontologically?" The QNFO corpus already treats ultrametric geometry as the candidate substrate for physics and computation [1,2,3], the external p-adic-physics literature pursues the same substrate hypothesis at the Planck scale [6,7,8], the corpus treats the observer as a node inside the ultrametric tree [4], and names the Monna map as the ratio-based interface between p-adic structure and continuous experience [5].

   from 1. Introduction and Positioning

2. The corpus's Module 11 already proposes the Monna map as the ratio-based consciousness interface [5], and UNO states the interface premise directly: the mind "naturally smooths the discrete nature of reality into a continuous narrative" [1].

   from 4. The Rendering Lemma [MAP]

How it is done

1. Given (i) and (ii) the conclusion follows at theorem level; without them it is unfounded.

   from Abstract

2. This paper supplies that worked example in four moves: (2) the Archimedean anatomy of perspective; (3) the ultrametric demolition of its ingredients; (4) the rendering lemma that restores the appearances; (5) the underdetermination theorem and its ontological consequences.

   from 1. Introduction and Positioning

3. The proportionality is the similar-triangles theorem of Euclidean geometry, and it requires two things: a well-defined ratio of lengths, and a well-defined angle.

   from 2. The Archimedean Anatomy of Perspective [TERRITORY]

4. Falsifiability C1: a no-go theorem ruling out surjective renderings of the required kind would kill the lemma.

   from 4. The Rendering Lemma [MAP]

What would falsify it

1. The naive alternative — ultrametric perception without a rendering — is already falsified by the observed smoothness of perspective, which is recorded here as falsified hypotheses C3–C4 rather than silently avoided. v0.2 (2026-08-16).

   from Abstract

2. The underdetermination theorem, the falsifiability register, and the ontology claims are unchanged.

   from Abstract

3. But that is the naive answer, and it is already falsified by ordinary vision.

   from 3. The Ultrametric Demolition [TERRITORY]

Where the claim stops

1. Things farther away appear smaller, and two-point linear perspective is taken to approximate the eye.

   from Abstract

2. Why a reader should care: the question converts a familiar visual fact into a decisive constraint on a live research program (p-adic and ultrametric physics [1,6,7,8]) — smooth perspective does not rule out a discrete world, it only rules out direct perception of one, and the distinction marks the exact boundary between physics and phenomenology.

   from Abstract

3. The originating question (Obsidian note _26228180341.md , 2026-08-16): "Things that are farther away appear smaller, and 2 point linear perspective is thought to approximate our eyes — but what if instead of an Archimedean linear geometry, perspective is actually non-Archimedean and ultrametric?

   from 1. Introduction and Positioning

Where to read

1. Abstract274 words · 1 min
2. 1. Introduction and Positioning200 words · 1 min
3. 2. The Archimedean Anatomy of Perspective [TERRITORY]176 words · 1 min
4. 3. The Ultrametric Demolition [TERRITORY]300 words · 1 min
5. 4. The Rendering Lemma [MAP]280 words · 1 min
6. 5. The Underdetermination Theorem [MAP]202 words · 1 min
7. 6. Ontological Consequences [MAP]191 words · 1 min
8. 7. Falsifiability Register135 words · 1 min
9. 8. Mandatory Symmetry Template265 words · 1 min
10. 9. Conclusion76 words · 1 min
11. Declarations78 words · 1 min