Urban Planning & Quantum: The (G) Law 📊Analyzing TfL London mobility, I found a 65% convergence toward the constant (G) (0.618). Flows self-organize via gnomonic proportions; deviations at 88 "BETA-nodes" predict chaos before it hits.Same pattern in IBM Quantum chips: 39.7% of "noise" follows (G). Instead of heavy filters, we "clean" data by isolating structural truth from thermal noise.From cities to atoms, (G) is the universal coordinate for stability. 🌍⚛️

Testing the G constant (0.618...) via HoTT (Homotopy Type Theory) and 10k Monte Carlo iterations.Key Findings:Necessity (BOX): 33% absolute stability under entropy.Scientific Validity: G is the unique topological invariant that minimizes noise interference.Native R validation confirms: logical identity remains resilient within stochastic chaos.#PsiU #HoTT #RStats #LogicEngine #DataScience
#UrbanPlanning #SmartCities #Mobility #Transport #London #QuantumComputingHub #QuantumComputing #DataScience #Physics #Complexity
https://doi.org/10.5281/zenodo.20322505

Structural Refactoring Objectives

https://doi.org/10.5281/zenodo.20138853

https://github.com/lombardisedr-dev/PsiU-Protocol-HoTT/blob/main/README.md

The recent optimization phase was dedicated to three fundamental pillars:Homotopic Formalization: A rigorous, clean, and clear theorization at the formal level, ensuring the \(X_{n}\) hierarchy is logically transparent.Protocol Optimization: The creation of an enhanced version of the PSIU Protocol, including its full integration with YML-based verification (CI/CD) for automated auditing.Comprehensive Reporting: Detailed documentation of results and architectural modifications, providing a traceable path for peer-review and future scalability.

This report documents the definitive formal verification of the PSIU Protocol using Agda. By implementing a strict semisimplicial universe without Axiom K, we resolve the higher-dimensional coherence challenges (X3 tetrahedral level), ensuring total stability and well typedness of the system.

The development of this protocol involved a critical refactoring phase. Previous iterations contained imprecisions in the mapping of higher-order coherences between X2 and X3. These were resolved within a 24-hour sprint by rewriting the core transition formulas, leading to the current verified state.

The PSIU Protocol successfully achieves ”Crystallized” stability. The proof is automated via CI/CD and is HoTT-compliant, marking a definitive step in verified semisimplicial structures.

#HoTT #TypeTheory #UnivalentFoundations
#Agda #LeanProver

Certified PSIU Protocol - SemiSimplicial Types (SST)
PSIU-Protocol-Final-Verification

https://github.com/lombardisedr-dev/PsiU-Protocol-HoTT

https://doi.org/10.5281/zenodo.19703879

Overview
PSIU-Protocol-Final-Verification

This project provides a formal verification of the PSIU Protocol within the framework of SemiSimplicial Types (SST). It is implemented in Agda and specifically designed to solve the coherence problem in higher dimensions without relying on Axiom K.

Formal Verification Details
The implementation is certified with the following rigorous Agda options:

{-# OPTIONS --without-K --safe #-}
No Postulates: All proofs are purely constructive, utilizing the J-rule via pattern matching on refl.
Higher Dimensions: Coherence is verified from
X0 (points) up to X4 (pentatopes).

Core Structure
The protocol is defined by the following layers:

X₀ - X₁: Points and Path-based edges.
X₂: Triangular face coherence (Path composition).
X₃: Tetrahedral volume coherence.
X₄: Pentatope hyper-volume coherence.
Omega Stability: Ensuring the structure remains "crystallized" through induction.

Usage
To verify the formal proof locally, ensure you have Agda 2.6.4+ installed and run:

agda --no-libraries -i . --without-K --safe Certified_PSIU_Protocol.agda

#Agda, #HoTT, #FormalVerification, #TypeTheory

This week the #HoTTEST seminar presents:

Freek Geerligs

Synthetic Stone duality

The talk is at 11:30am EDT (15:30 UTC) on Thursday, April 16. The talk will be 60 minutes long, followed by up to 30 minutes for questions. See https://hottest-seminar.github.io/ for the Zoom link and a list of all upcoming talks.

All are welcome!

Abstract:

In this talk, we will give an overview of Synthetic Stone duality (https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2024.3). We will then discuss some related work in progress.

Synthetic Stone duality is an extension of homotopy type theory with four axioms. These axioms are strong enough to decide Bishop's omniscience principles. We introduce a (synthetic) topology on any type, such that all functions are continuous. We are interested in Stone spaces and compact Hausdorff spaces, where the topology behaves as one would expect. In particular, we can define the (topological) interval and show that all functions are continuous in the epsilon-delta sense.

Currently, we are working on a paper with a method for calculating cohomology with countably presented coefficients for compact Hausdorff spaces. We are also interested in a correspondence between homotopical concepts defined using traditional topology (using paths from the topological interval) and homotopy type theory (using identity types).

This talk will contain joint work with Reid Barton, Felix Cherubini, Thierry Coquand, and Hugo Moeneclaey.

#HoTT @carloangiuli @emilyriehl @de_Jong_Tom

This week the #HoTTEST seminar presents:

Szumi Xie

The groupoid-syntax of type theory is a set

The talk is at 11:30am EDT (15:30 UTC) on Thursday, April 2. The talk will be 60 minutes long, followed by up to 30 minutes for questions. See https://hottest-seminar.github.io/ for the Zoom link and a list of all upcoming talks. (This should be back to the "usual" time for Europeans who are now on summer time.)

All are welcome!

Abstract:

Categories with families (CwFs) have been used to define the semantics of type theory in type theory. In the setting of homotopy type theory, one of the limitations of the traditional notion of CwFs is the requirement to set-truncate types, which excludes models based on univalent categories, such as the standard set model. To address this limitation, I will introduce the notion of groupoid categories with families (GCwFs), which truncates types at the groupoid level and incorporates coherence equations.

I will demonstrate that the initial GCwF for a type theory with some type formers is set-truncated, using a technique called α-normalization. This allows us to utilize the conventional intrinsic syntax of type theory while enabling interpretations in semantically richer and more natural models.

I will also present a generalization of GCwFs and discuss its relation to comprehension categories.

This talk is based on joint work with Thorsten Altenkirch and Ambrus Kaposi (https://doi.org/10.4230/LIPIcs.CSL.2026.40).

#HoTT @carloangiuli @emilyriehl @de_Jong_Tom

This week the #HoTTEST seminar presents:

Astra Kolomatskaia

Displayed Type Theory, intervals, and analytic higher categorical structures

The talk is at 11:30am EDT (15:30 UTC) on Thursday, March 19. The talk will be 60 minutes long, followed by up to 30 minutes for questions. See https://hottest-seminar.github.io/ for the Zoom link and a list of all upcoming talks. (Note that we recently started daylight time in North America, so the local time may have changed for you.)

All are welcome!

Abstract:

I have historically encountered a number of difficulties in communicating my work to others. The process of preparing this talk has thus involved engaging with throughlines in the type theory literature and has helped me identify places in which building bridges was necessary.

My joint work with Mike Shulman introduced Displayed Type Theory [dTT], which syntactically admits a construction of semi-simplicial types in a way that then semantically admits interpretation into arbitrary Grothendieck (∞,1)-topoi. This result is not novel as stated: First, it is not a syntactic construction in Book HoTT. Second, syntactic constructions of SSTs were a foremost consideration in the development of 2LTT, and Elif Üsküplü's analysis shows that the inner layer of 2LTT, when enriched with an axiom of cofibrant exo-nats, is general with respect to Grothendieck (∞,1)-topos semantics. [...]

[Full abstract too long for even two toots, so follow the link to the seminar page to see it all.]

#HoTT @carloangiuli @emilyriehl @de_Jong_Tom