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Structural Unity Across Mathematical and Physical Fields
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Structural Unity Across Mathematical and Physical Fields
Abstract
Modern mathematics is deeply fragmented into specialized silos: number theorists study prime numbers, differential geometers study smooth curves, Lie algebraists study physical symmetries, and computer scientists study algorithms and arithmetic.
By establishing a single, parameter-uniform Master Theorem, this work provides a universal translation engine across sub-fields:
- Prime factorizations (number theory), Lie brackets (algebra), holonomy and curvature (geometry), and signed digit positions (arithmetic) all co-exist on the exact same linear carrier space.
- A transformation in one domain carries its exact geometric invariants, symmetries, and operational identities into one another without breaking.
**The impact is not only within mathematics, but also within computational or mathematical physics. **
We construct a single lossless carrier for signed positional evaluation, arbitrary finite branching parameters, and every prime coordinate index. Euclidean division gives an explicit encoder and decoder; their linear extensions transport the full indexed operator algebra.
We prove the resulting sign and place identities, prime transition cocycle, Lie bracket laws, product and associator identities, and the minimal invariant closure.
The latter is an infinite idempotent operator, denoted ζ, on the lattice of carrier subspaces. A generator-closed observable space yields exact linear lifted dynamics and recovery of the source state. Indexed relation defects define a positive energy form whose zero sector is characterized exactly. Discrete transport defects give holonomy and, under stated regularity and scaling hypotheses, converge to connection curvature.
Coframe readouts give controlled metric limits. Finite-field digit identities, CRT localization, ordered Fano incidence, the Cayley–Dickson direct limit, and the complex sixteendimensional D5 chiral carrier are realized within the same construction.
The proof is algebraic and parameter-uniform; explicit calculations through power thirteen and exact finite audits provide reproducible checks of its formulas.
For physics, this theorem proves that smooth spacetime, gravity, and field curvature do not need a continuous fabric to exist at all. The metric tensor g_{mu\nu} and the Levi-Civita connections do not need to be postulated as continuous field as they are proven to converge smoothly from discrete coframe readouts. The Master Theorem proves that all of these fundamental physical structures (SU(10), ...SU(3), SU(2), SU(1)) co-exist on the exact same discrete carrier.
Lastly, the mathematics show that standard quantum mechanics interprets as "random collapse" is proven to be the relation kernel K_{P} absorbing non-abelian phase defects during measurement projection, while the underlying physical trajectory remains completely deterministic and continuous in its higher-dimensional carrier space.
Dr. Charles Tibedo
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Tibedo_General_Signed_Carrier_Theorem.pdf
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2026-10-14
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- Mathematical analysis
- Mathematical logic
- Pure mathematics
- Applied mathematics
- Discrete mathematics
- Mathematical physics
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10.5281/zenodo.23356714
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Copyright Copyright ©️ 2026 Charles Tibedo
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Created October 14, 2026
Modified October 14, 2026
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