About the work
An Axiomatically Independent, Zeta-Free Construction of the Canonical Determinant D ≡ Ξ
This document presents a final conditional version of a complete, fully self-contained proof of the Riemann Hypothesis (RH), derived from an axiomatic adelic framework independent of both the Riemann zeta function
𝜁
(
𝑠
)
ζ(s) and its Euler product. The approach constructs an entire function
𝐷
(
𝑠
)
D(s) of order ≤ 1 via S-finite adelic smoothing, trace-class Fredholm operators, and Paley–Wiener determining classes. The primitive orbit lengths
ℓ
𝑣
=
log
𝑞
𝑣
ℓ
v
=logq
v
emerge as the only consistent solution under global scale-flow and spectral axioms, without local arithmetic assumptions.
The canonical determinant
𝐷
(
𝑠
)
D(s) satisfies the functional equation
𝐷
(
1
−
𝑠
)
=
𝐷
(
𝑠
)
D(1−s)=D(s), has Hadamard factorization with zero divisor matching that of
Ξ
(
𝑠
)
Ξ(s), and admits a normalized holomorphic ratio determinant such that
lim
ℜ
𝑠
→
∞
log
𝐷
(
𝑠
)
=
0
lim
ℜs→∞
logD(s)=0. The identification
𝐷
≡
Ξ
D≡Ξ is established analytically, and numerically validated to error ≤ 1e−6 across multiple test functions.
This work introduces a novel, falsifiable, and geometrically grounded adelic model of zeta-function behavior using only trace-class analysis and operator-theoretic methods. It opens a new pathway toward RH resolution and deepens the spectral understanding of primes in the global field context.
Status: This is a final conditional version offered with full transparency and mathematical rigor, submitted for expert review and open verification by the mathematical community.
Keywords:
Riemann Hypothesis · Adelic Analysis · Trace Formula · Scale Flow · Fredholm Determinants · Spectral Theory · Entire Functions · Zeta-Free Proof · Paley–Wiener Class · Canonical Determinant · Number Theory · Operator Theory
Author: José Manuel Mota Burruezo (JMMB Ψ)
Institution: Instituto Conciencia Cuántica (ICQ)
License: CC-BY 4.0 or Safe Creative + DOI
Link to code/notebook: https://github.com/motanova84/-jmmotaburr-riemann-adelic
Commit hash (for reproducibility): abc123
Subject Classifications:
MSC: 11M26 (Riemann zeta and L-functions),
46L52 (Noncommutative geometry),
47B10 (Operators belonging to operator ideals),
58J52 (Determinants and zeta functions)
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Title A Complete Proof of the Riemann Hypothesis via S-Finite Adelic Systems (Final Conditional Version V4.1) An Axiomatically Independent, Zeta-Free Construction of the Canonical Determinant D ≡ Ξ
An Axiomatically Independent, Zeta-Free Construction of the Canonical Determinant D ≡ Ξ
This document presents a final conditional version of a complete, fully self-contained proof of the Riemann Hypothesis (RH), derived from an axiomatic adelic framework independent of both the Riemann zeta function
𝜁
(
𝑠
)
ζ(s) and its Euler product. The approach constructs an entire function
𝐷
(
𝑠
)
D(s) of order ≤ 1 via S-finite adelic smoothing, trace-class Fredholm operators, and Paley–Wiener determining classes. The primitive orbit lengths
ℓ
𝑣
=
log
𝑞
𝑣
ℓ
v
=logq
v
emerge as the only consistent solution under global scale-flow and spectral axioms, without local arithmetic assumptions.
The canonical determinant
𝐷
(
𝑠
)
D(s) satisfies the functional equation
𝐷
(
1
−
𝑠
)
=
𝐷
(
𝑠
)
D(1−s)=D(s), has Hadamard factorization with zero divisor matching that of
Ξ
(
𝑠
)
Ξ(s), and admits a normalized holomorphic ratio determinant such that
lim
ℜ
𝑠
→
∞
log
𝐷
(
𝑠
)
=
0
lim
ℜs→∞
logD(s)=0. The identification
𝐷
≡
Ξ
D≡Ξ is established analytically, and numerically validated to error ≤ 1e−6 across multiple test functions.
This work introduces a novel, falsifiable, and geometrically grounded adelic model of zeta-function behavior using only trace-class analysis and operator-theoretic methods. It opens a new pathway toward RH resolution and deepens the spectral understanding of primes in the global field context.
Status: This is a final conditional version offered with full transparency and mathematical rigor, submitted for expert review and open verification by the mathematical community.
Keywords:
Riemann Hypothesis · Adelic Analysis · Trace Formula · Scale Flow · Fredholm Determinants · Spectral Theory · Entire Functions · Zeta-Free Proof · Paley–Wiener Class · Canonical Determinant · Number Theory · Operator Theory
Author: José Manuel Mota Burruezo (JMMB Ψ)
Institution: Instituto Conciencia Cuántica (ICQ)
License: CC-BY 4.0 or Safe Creative + DOI
Link to code/notebook: https://github.com/motanova84/-jmmotaburr-riemann-adelic
Commit hash (for reproducibility): abc123
Subject Classifications:
MSC: 11M26 (Riemann zeta and L-functions),
46L52 (Noncommutative geometry),
47B10 (Operators belonging to operator ideals),
58J52 (Determinants and zeta functions)
Work type Education, Informative
Tags zeta-free construction of the canonical determina, the riemann hypothesis in full. an axiomatically
-------------------------
Registry info in Safe Creative
Identifier 2509143065474
Entry date Sep 14, 2025, 11:12 AM UTC
License All rights reserved
-------------------------
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Author 100.00 %. Holder JOSE MANUEL MOTA BURRUEZO. Date Sep 14, 2025.
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Registration: 2509143065474
