• 1 Overview ▶
    • 1.1 What this project is
    • 1.2 What it is trying to find out
    • 1.3 Shadow holography
    • 1.4 Goals, in order of ambition
    • 1.5 How to read this blueprint
    • 1.6 Coverage caveat
  • 2 Haar Measure Foundation ▶
    • 2.1 Self-Duality on Compact Groups
  • 3 Functional Equation ▶
    • 3.1 Completed Riemann Zeta Function
  • 4 Shadow Symmetry = Time Reversal ▶
    • 4.1 The Three-Step Proof
  • 5 Zero Pairing and the Positivity Reduction (RH is not proved here) ▶
    • 5.1 Two-Zeros Argument
    • 5.2 Temperedness
  • 6 Link 6 as a Named Hypothesis
  • 7 Celestial Holography ▶
    • 7.1 Shadow Discontinuity = Loop Integrand
    • 7.2 Tree-to-Loop Topology (Shadow-Pair Sewing)
    • 7.3 Loops from Cuts (canonical replacement series)
    • 7.4 Dispersion Reconstruction: the Mechanism Behind sewing_identity
  • 8 Adèlic \(L^2\) Regularization
  • 9 Spectral Weil Formula
  • 10 Cesàro Regularization and the Weil Positivity Criterion
  • 11 L2 Constraint ▶
    • 11.1 \(L^2(K^1)\) Unitarity and the Critical Line: what is proved and what is open
  • 12 The Half-Flip: CPT, Time Reversal, and Complete Positivity ▶
    • 12.1 Wigner time reversal squares to \(-1\)
    • 12.2 Antiunitary conjugation is transposition
    • 12.3 Transposition is positive but not completely positive
    • 12.4 Entanglement bounds (same module family)
  • 13 Standard Model — Three Generations ▶
    • 13.1 Division Algebra Tower
  • 14 Standard Model — Dark Matter Abundance
  • 15 Arithmetic and Quantum-Information Miscellanea (from On the Nature of Nature v5.2.1.3)
  • 16 Quantum Gravity — Born Rule from Haar
  • 17 Quantum Gravity — Celestial Spectral Weight and All-Loop Bound
  • 18 General Relativity — Rigidity
  • 19 Thread HT — Prime–Archimedean Heat Trace (Elementary Layer) ▶
    • 19.1 Complete Monotonicity and Laplace Bookkeeping
    • 19.2 Subordination
    • 19.3 Bridge to Thread L, and the Archimedean Laplace Transform
    • 19.4 Honest Boundary
  • 20 The Digamma Function ▶
    • 20.1 The closing third of the First Moment Theorem
  • 21 Local-Field Shadow Kernels ▶
    • 21.1 The \(d=2\) Cut vs. the Spherical Weyl Coefficient
    • 21.2 The Global Eisenstein Coefficient
  • 22 The Golden Ratio as the Minimal Hyperbolic Sector of \(PSL_2(\mathbb {Z})\)
  • 23 Local Shadow Kernels and the Finite-Prime Weil Kernel
  • 24 Thread Weil-Semiboundedness
  • 25 Formalized Content Not Yet Written Up Individually ▶
    • 25.1 Riemann Hypothesis — Supporting Tower
    • 25.2 Riemann Hypothesis — Codex Workbench Formalizations (2026-09-28)
    • 25.3 Thread Weil-Parity
    • 25.4 Riemann Hypothesis — Gaussian Rigidity and the Hardy Kernel (2026-09-28)
    • 25.5 Riemann Hypothesis — Schur Gap Transfer and BPY Coercivity (2026-09-28)
    • 25.6 Riemann Hypothesis — The Conformal Casimir and the TFD GCD Kernel (2026-09-28)
    • 25.7 Riemann Hypothesis — Why One Half: the Prime-Torus Edge and the Casimir Trichotomy
    • 25.8 Thread Weil-Parity — Metric Positivity of the Cross-Resolvent
    • 25.9 Riemann Hypothesis — Mathlib’s Statement in the Casimir Variable
    • 25.10 Riemann Hypothesis — Prime Cayley Two-Channel Identities
    • 25.11 Riemann Hypothesis — Centered Principal-Series Casimir for Self-Dual L-Functions
    • 25.12 Riemann Hypothesis — Wall/Krylov Coefficients and the Local Prime Poisson Identity
    • 25.13 Riemann Hypothesis — Prime TFD Weights and the Weighted Lax–Phillips Target
    • 25.14 Riemann Hypothesis — Prime Cayley Channel: Golden Point and Cayley Inversion
    • 25.15 Riemann Hypothesis — Prime TFD as an SU(1,1) Coherent State: Euler Factor, Blaschke Channel, Mass
    • 25.16 Riemann Hypothesis — The \(k=\tfrac 12\) Character and Blaschke Defect Kernels
    • 25.17 Riemann Hypothesis — SU(1,1) Lightcone Excess and Prime Tail Thresholds
    • 25.18 Riemann Hypothesis — Antipodal Phases, Trivial-Zero Ladder, Heat Trace
    • 25.19 Riemann Hypothesis — Finite Zeta-Graph Metric and Its Dual Norm
    • 25.20 Riemann Hypothesis — Prime Powers as Schur-Composed Massive Edges
    • 25.21 Riemann Hypothesis — Prime Defect as a TFD-Dressed Delay Line
    • 25.22 Riemann Hypothesis — Local Möbius Whitening Operator and Its Symbol
    • 25.23 Riemann Hypothesis — Hilbert–Pólya Lemma, Growth Bounds and Unitary Defect Identities
    • 25.24 Riemann Hypothesis — Prime Channel as a Householder Reflection and the Julia Colligation
    • 25.25 Riemann Hypothesis — Rank-One Pole Thresholds, Sobolev-Trace DtN, Midpoint Uniqueness, Theta Whitening
    • 25.26 Riemann Hypothesis — Zero Quartet and the Two Spectral Half-Flips
    • 25.27 Riemann Hypothesis — Tomita Modular Flow on Ratio States
    • 25.28 Riemann Hypothesis — Stieltjes–Mellin Dispersion Kernel
    • 25.29 Riemann Hypothesis — Finite Bost–Connes Divisibility Projections: Two No-Go Results
    • 25.30 Riemann Hypothesis — Bost–Connes Midpoint Algebra: Torsion Average, Two-Point Function, Jump Energy
    • 25.31 Riemann Hypothesis — No-Go: Prime Inner Channels Have a Non-Globalizable Zero Divisor
    • 25.32 Riemann Hypothesis — Critical Zeta Gibbs Ensemble: The Escaping Energy is Exponential
    • 25.33 Riemann Hypothesis — Self-Dual Divisor Geometry: Subgroup Gram Matrix and Mobius Parity
    • 25.34 Riemann Hypothesis — Dual Mobius Obstruction in the Finite Zeta-Graph Metric
    • 25.35 Riemann Hypothesis — Half-Line Resolvent Commutator: Rank-One Kernel
    • 25.36 Riemann Hypothesis — Local Odd-Transfer Contraction and the Cross-Mode Krein Obstruction
    • 25.37 Riemann Hypothesis — Two-Logistic Density as a Ward Operator on the Bose Trace
    • 25.38 Riemann Hypothesis — Prime-Power Current as Excess Group Delay
    • 25.39 Riemann Hypothesis — Primitive Renormalization and Finite Half-Derivative Energy
    • 25.40 Riemann Hypothesis — SU(1,1) Transfer Discriminant as the Finite-Place Mass Coordinate
    • 25.41 Riemann Hypothesis — Regularized Prime Scattering and the Schatten Strip Hierarchy
    • 25.42 Riemann Hypothesis — TFD Boundary Cayley Transform, Parity Determinants, Schur Complement
    • 25.43 Riemann Hypothesis — Centered Divisor Coordinate and Hodge Reflection
    • 25.44 Riemann Hypothesis — Finite Fourier Duality Exchanges Subgroup States
    • 25.45 Riemann Hypothesis — Pullback Metric from an Ambient Self-Adjoint Intertwiner
    • 25.46 Riemann Hypothesis — No Diagonal Reweighting of the Prime Hilbert Space
    • 25.47 Riemann Hypothesis — Prime TFD Phase: Caratheodory Function, Local Variance, Fisher Distance
    • 25.48 Riemann Hypothesis — Prime-Power Valuation Chains: Gram Matrix and Euler Green Function
    • 25.49 Riemann Hypothesis — Haar Valuation Law and the Mean Mass Series
    • 25.50 Riemann Hypothesis — The totient Volterra kernel: exact zero mean and divisor regrouping
    • 25.51 Riemann Hypothesis — The logistic density, the \(\tanh (x/2)\) substitution and Möbius boosts
    • 25.52 Number Theory — Arithmetic Layer
    • 25.53 Celestial Holography — Further Results
    • 25.54 Quantum Gravity — Further Results
    • 25.55 Standard Model — Further Results
    • 25.56 General Relativity — Further Results
    • 25.57 Cosmology
    • 25.58 String Theory
    • 25.59 Thread S — Signature and Inertia
    • 25.60 Yang–Mills
    • 25.61 Core Theorems and Grassmannian Geometry
  • 26 Axiom Inventory ▶
    • 26.1 Physics Inputs (now hypotheses, not axioms)
    • 26.2 Mathlib-Gap Axioms
    • 26.3 Retired Axioms
    • 26.4 The True-Stub Convention (not axioms, not sorries)
  • Dependency graph
  • 1 Overview
    • 1.1 What this project is
    • 1.2 What it is trying to find out
    • 1.3 Shadow holography
    • 1.4 Goals, in order of ambition
    • 1.5 How to read this blueprint
    • 1.6 Coverage caveat
  • 2 Haar Measure Foundation
    • 2.1 Self-Duality on Compact Groups
  • 3 Functional Equation
    • 3.1 Completed Riemann Zeta Function
  • 4 Shadow Symmetry = Time Reversal
    • 4.1 The Three-Step Proof
  • 5 Zero Pairing and the Positivity Reduction (RH is not proved here)
    • 5.1 Two-Zeros Argument
    • 5.2 Temperedness
  • 6 Link 6 as a Named Hypothesis
  • 7 Celestial Holography
    • 7.1 Shadow Discontinuity = Loop Integrand
    • 7.2 Tree-to-Loop Topology (Shadow-Pair Sewing)
    • 7.3 Loops from Cuts (canonical replacement series)
    • 7.4 Dispersion Reconstruction: the Mechanism Behind sewing_identity
  • 8 Adèlic \(L^2\) Regularization
  • 9 Spectral Weil Formula
  • 10 Cesàro Regularization and the Weil Positivity Criterion
  • 11 L2 Constraint
    • 11.1 \(L^2(K^1)\) Unitarity and the Critical Line: what is proved and what is open
  • 12 The Half-Flip: CPT, Time Reversal, and Complete Positivity
    • 12.1 Wigner time reversal squares to \(-1\)
    • 12.2 Antiunitary conjugation is transposition
    • 12.3 Transposition is positive but not completely positive
    • 12.4 Entanglement bounds (same module family)
  • 13 Standard Model — Three Generations
    • 13.1 Division Algebra Tower
  • 14 Standard Model — Dark Matter Abundance
  • 15 Arithmetic and Quantum-Information Miscellanea (from On the Nature of Nature v5.2.1.3)
  • 16 Quantum Gravity — Born Rule from Haar
  • 17 Quantum Gravity — Celestial Spectral Weight and All-Loop Bound
  • 18 General Relativity — Rigidity
  • 19 Thread HT — Prime–Archimedean Heat Trace (Elementary Layer)
    • 19.1 Complete Monotonicity and Laplace Bookkeeping
    • 19.2 Subordination
    • 19.3 Bridge to Thread L, and the Archimedean Laplace Transform
    • 19.4 Honest Boundary
  • 20 The Digamma Function
    • 20.1 The closing third of the First Moment Theorem
  • 21 Local-Field Shadow Kernels
    • 21.1 The \(d=2\) Cut vs. the Spherical Weyl Coefficient
    • 21.2 The Global Eisenstein Coefficient
  • 22 The Golden Ratio as the Minimal Hyperbolic Sector of \(PSL_2(\mathbb {Z})\)
  • 23 Local Shadow Kernels and the Finite-Prime Weil Kernel
  • 24 Thread Weil-Semiboundedness
  • 25 Formalized Content Not Yet Written Up Individually
    • 25.1 Riemann Hypothesis — Supporting Tower
    • 25.2 Riemann Hypothesis — Codex Workbench Formalizations (2026-09-28)
    • 25.3 Thread Weil-Parity
    • 25.4 Riemann Hypothesis — Gaussian Rigidity and the Hardy Kernel (2026-09-28)
    • 25.5 Riemann Hypothesis — Schur Gap Transfer and BPY Coercivity (2026-09-28)
    • 25.6 Riemann Hypothesis — The Conformal Casimir and the TFD GCD Kernel (2026-09-28)
    • 25.7 Riemann Hypothesis — Why One Half: the Prime-Torus Edge and the Casimir Trichotomy
    • 25.8 Thread Weil-Parity — Metric Positivity of the Cross-Resolvent
    • 25.9 Riemann Hypothesis — Mathlib’s Statement in the Casimir Variable
    • 25.10 Riemann Hypothesis — Prime Cayley Two-Channel Identities
    • 25.11 Riemann Hypothesis — Centered Principal-Series Casimir for Self-Dual L-Functions
    • 25.12 Riemann Hypothesis — Wall/Krylov Coefficients and the Local Prime Poisson Identity
    • 25.13 Riemann Hypothesis — Prime TFD Weights and the Weighted Lax–Phillips Target
    • 25.14 Riemann Hypothesis — Prime Cayley Channel: Golden Point and Cayley Inversion
    • 25.15 Riemann Hypothesis — Prime TFD as an SU(1,1) Coherent State: Euler Factor, Blaschke Channel, Mass
    • 25.16 Riemann Hypothesis — The \(k=\tfrac 12\) Character and Blaschke Defect Kernels
    • 25.17 Riemann Hypothesis — SU(1,1) Lightcone Excess and Prime Tail Thresholds
    • 25.18 Riemann Hypothesis — Antipodal Phases, Trivial-Zero Ladder, Heat Trace
    • 25.19 Riemann Hypothesis — Finite Zeta-Graph Metric and Its Dual Norm
    • 25.20 Riemann Hypothesis — Prime Powers as Schur-Composed Massive Edges
    • 25.21 Riemann Hypothesis — Prime Defect as a TFD-Dressed Delay Line
    • 25.22 Riemann Hypothesis — Local Möbius Whitening Operator and Its Symbol
    • 25.23 Riemann Hypothesis — Hilbert–Pólya Lemma, Growth Bounds and Unitary Defect Identities
    • 25.24 Riemann Hypothesis — Prime Channel as a Householder Reflection and the Julia Colligation
    • 25.25 Riemann Hypothesis — Rank-One Pole Thresholds, Sobolev-Trace DtN, Midpoint Uniqueness, Theta Whitening
    • 25.26 Riemann Hypothesis — Zero Quartet and the Two Spectral Half-Flips
    • 25.27 Riemann Hypothesis — Tomita Modular Flow on Ratio States
    • 25.28 Riemann Hypothesis — Stieltjes–Mellin Dispersion Kernel
    • 25.29 Riemann Hypothesis — Finite Bost–Connes Divisibility Projections: Two No-Go Results
    • 25.30 Riemann Hypothesis — Bost–Connes Midpoint Algebra: Torsion Average, Two-Point Function, Jump Energy
    • 25.31 Riemann Hypothesis — No-Go: Prime Inner Channels Have a Non-Globalizable Zero Divisor
    • 25.32 Riemann Hypothesis — Critical Zeta Gibbs Ensemble: The Escaping Energy is Exponential
    • 25.33 Riemann Hypothesis — Self-Dual Divisor Geometry: Subgroup Gram Matrix and Mobius Parity
    • 25.34 Riemann Hypothesis — Dual Mobius Obstruction in the Finite Zeta-Graph Metric
    • 25.35 Riemann Hypothesis — Half-Line Resolvent Commutator: Rank-One Kernel
    • 25.36 Riemann Hypothesis — Local Odd-Transfer Contraction and the Cross-Mode Krein Obstruction
    • 25.37 Riemann Hypothesis — Two-Logistic Density as a Ward Operator on the Bose Trace
    • 25.38 Riemann Hypothesis — Prime-Power Current as Excess Group Delay
    • 25.39 Riemann Hypothesis — Primitive Renormalization and Finite Half-Derivative Energy
    • 25.40 Riemann Hypothesis — SU(1,1) Transfer Discriminant as the Finite-Place Mass Coordinate
    • 25.41 Riemann Hypothesis — Regularized Prime Scattering and the Schatten Strip Hierarchy
    • 25.42 Riemann Hypothesis — TFD Boundary Cayley Transform, Parity Determinants, Schur Complement
    • 25.43 Riemann Hypothesis — Centered Divisor Coordinate and Hodge Reflection
    • 25.44 Riemann Hypothesis — Finite Fourier Duality Exchanges Subgroup States
    • 25.45 Riemann Hypothesis — Pullback Metric from an Ambient Self-Adjoint Intertwiner
    • 25.46 Riemann Hypothesis — No Diagonal Reweighting of the Prime Hilbert Space
    • 25.47 Riemann Hypothesis — Prime TFD Phase: Caratheodory Function, Local Variance, Fisher Distance
    • 25.48 Riemann Hypothesis — Prime-Power Valuation Chains: Gram Matrix and Euler Green Function
    • 25.49 Riemann Hypothesis — Haar Valuation Law and the Mean Mass Series
    • 25.50 Riemann Hypothesis — The totient Volterra kernel: exact zero mean and divisor regrouping
    • 25.51 Riemann Hypothesis — The logistic density, the \(\tanh (x/2)\) substitution and Möbius boosts
    • 25.52 Number Theory — Arithmetic Layer
    • 25.53 Celestial Holography — Further Results
    • 25.54 Quantum Gravity — Further Results
    • 25.55 Standard Model — Further Results
    • 25.56 General Relativity — Further Results
    • 25.57 Cosmology
    • 25.58 String Theory
    • 25.59 Thread S — Signature and Inertia
    • 25.60 Yang–Mills
    • 25.61 Core Theorems and Grassmannian Geometry
  • 26 Axiom Inventory
    • 26.1 Physics Inputs (now hypotheses, not axioms)
    • 26.2 Mathlib-Gap Axioms
    • 26.3 Retired Axioms
    • 26.4 The True-Stub Convention (not axioms, not sorries)