What does this research mean for the field?

All mathematics is isomorphic to a category generated by a single right isosceles triangle (△-ontology), providing a unified framework that proves ten fundamental mathematical problems, including six unsolved Millennium problems. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

This work aims to unify proofs of ten fundamental mathematical problems through a triangular framework.

June 3, 2026Open Access

Theorem of Ten Bridges v.10: A Unified Architectural Proof of Ten Fundamental Problems of Mathematics in △-Ontology via the Single Master Operator ℋ and the RPT Self-Similar Invariant

Key Points

This work aims to unify proofs of ten fundamental mathematical problems through a triangular framework.
Developed a proof structure based on △-ontology using a right isosceles triangle as a foundational element.
Illustrated connections between ten mathematical problems and the balance of symmetry and asymmetry in the infinium.
Employed explanatory comments to clarify each step of the proofs.
Demonstrated that all ten problems can be viewed as projections of balance in symmetry and asymmetry.
Established that all ten problems are valid within the △-ontology framework via forcing.
Showed an isomorphism between all mathematics and the category generated by the triangle.

Abstract

This work presents a unified architectural proof of ten fundamental problems of mathematics within △-ontology, where the foundation is the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs 1 and hypotenuse √2). The central statement of the system is: ∀ Math ≅ Topos(△₁ₓ₁) — all mathematics is isomorphic to a category generated by a single triangle. The ten problems include: six unsolved Millennium problems (P vs NP, Hodge conjecture, Riemann hypothesis, Yang-Mills conjecture, Navier-Stokes equations, BSD conjecture), one already solved (Poincaré conjecture), Fermat's Last Theorem, as well as the Goldbach and Collatz conjectures. Comment: the number 10 here is an empirical list, not derived from the structure of the infinium; the poetic image of "ten bridges" carries no evidential weight. It is shown that all of them are projections of the balance of symmetry and asymmetry encoded in the infinium. All proofs are accompanied by explanatory comments explaining each step. The logical status of the results is fixed via forcing: ℑ ⊩ (all ten problems are true in △-ontology).

Theorem of Ten Bridges v.10: A Unified Architectural Proof of Ten Fundamental Problems of Mathematics in △-Ontology via the Single Master Operator ℋ and the RPT Self-Similar Invariant

Key Points

Abstract

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