Scientific Results Ledger

This ledger compiles all mathematical theorems, proof classifications, numerical simulation bounds, calibrated parameter retrodictions, and negative experimental benchmarks across the Science of Fabric Reality corpus.

┌─────────────────────────────────────────────────────────────────────────────┐
│                    MATHEMATICAL PROOF TAXONOMY MATRIX                       │
├────────────────────────────────┬──────────────────────────┬─────────────────┤
│ Result Category                │ Taxonomy Classification  │ Registered Count│
├────────────────────────────────┼──────────────────────────┼─────────────────┤
│ Authorial Novel Theorems       │ AUTHORIAL_NOVEL_RESULT   │ 0 Theorems      │
│ Authorial Specializations      │ AUTHORIAL_SPECIALIZATION │ 2 Theorems      │
│ Established Math Applications  │ ESTABLISHED_APPLICATION  │ 2 Theorems      │
│ Conditional Derivations        │ DERIVED_CONDITIONALLY    │ 2 Theorems      │
│ Open Formalization Targets     │ FORMALIZATION_TARGET     │ 5 Targets       │
│ Machine-Verified Proofs        │ MACHINE_VERIFIED         │ 0 Verified      │
│ Constitutional Axioms (Lean 4) │ Machine Type-Checked     │ 5 Axioms        │
│ Formal Definitions (Lean 4)    │ Machine Type-Checked     │ 3 Definitions   │
│ Calibrated Retrodictions       │ P2 Retrodiction (DOF=0)  │ 3 Lepton Masses │
│ Statistical Parity Null Tests  │ Negative Benchmark       │ p_null = 0.62   │
│ Active P4 Sealed Predictions   │ Pre-Registered Blind     │ 0 Registered    │
└────────────────────────────────┴──────────────────────────┴─────────────────┘

---

1. Mathematical Proof Taxonomy & Specific Theorem Registry

Every mathematical theorem in the corpus is cataloged in the [Analytic Proof Registry](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/analytic-proof-registry.json) with portable receipts in reports/current/proofs/:

1.1 Invariant Transport Under Relational Transitions (THM-INV-01)

• Taxonomy Classification: AUTHORIAL_SPECIALIZATION (PROVEN_IN_CORPUS)
• Statement: For any admissible discrete relational state $\mathcal{S}=(X,R,\partial ,\mathcal{I})$, topological boundary cycle invariants $I_k$ are preserved under the automorphism transformation group $\mathrm{Aut}(\mathcal{S})$ across relational state transitions.
• Exact Hypotheses: 5-regular relational graph carrier, admissible transport morphism preserving cycle boundary structure, unitary transition mapping on boundary cycle space.
• Proof Location: Formalization Batch 1 & Foundations of the Science of Fabric Reality, Vol. 1.
• Portable Receipt: [reports/current/proofs/THM-INV-01.json](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/proofs/THM-INV-01.json).

1.2 Constrained Variational Stationarity (THM-FFE-STAT-01)

• Taxonomy Classification: AUTHORIAL_SPECIALIZATION (PROVEN_IN_CORPUS)
• Statement: Stationary action variation $D{S}_{\text{dyn}}(u)+DC(u{)}^{\ast }\mathrm{\Lambda }=J$ subject to the exact constraint $C(u)=0$ on metric-measure spaces characterizes critical points of the constrained action functional.
• Exact Hypotheses: Banach configuration space ${\mathcal{X}}_{\mathcal{F}}$ over metric-measure Dirichlet space, ${\mathcal{C}}^2$ dynamical action functional $S_{\text{dyn}}$, ${\mathcal{C}}^1$ constraint map $C$ with surjective Fréchet derivative $DC(u)$.
• Proof Location: Fabric Field Equation.
• Portable Receipt: [reports/current/proofs/THM-FFE-STAT-01.json](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/proofs/THM-FFE-STAT-01.json).
• Epistemic Demarcation: First-order stationarity defines the constrained Karush-Kuhn-Tucker system; covariant divergence conservation is not inferred from stationarity alone.

1.3 Noetherian Divergence Conservation Under Continuous Symmetry (THM-FFE-NOETHER-01)

• Taxonomy Classification: DERIVED_CONDITIONALLY
• Statement: If the unconstrained action $S_{\text{dyn}}$ and constraint map $C$ are invariant under a 1-parameter continuous Lie group transformation, then the corresponding on-shell dynamical current satisfies covariant divergence conservation ${\mathrm{\nabla }}_{\mu }{J}^{\mu }=0$.
• Exact Hypotheses: 1-parameter continuous Lie group symmetry of $S_{\text{dyn}}$ and $C$, ${\mathcal{C}}^2$ field regularity across metric-measure manifold charts, vanishing boundary flux surface integrals at spatial infinity.
• Proof Location: Fabric Field Equation.
• Portable Receipt: [reports/current/proofs/THM-FFE-NOETHER-01.json](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/proofs/THM-FFE-NOETHER-01.json).

1.4 Finite-Dimensional KKT Multiplier Invertibility (THM-FFE-02)

• Taxonomy Classification: ESTABLISHED_THEOREM_APPLICATION (PROVEN_IN_CORPUS)
• Statement: Under strictly positive definite Hessian matrix $H=D^2{S}_{\text{dyn}}(u)>0$ and full row rank constraint Jacobian $DC(u)$, the coupled Karush-Kuhn-Tucker (KKT) block operator is uniquely invertible.
• Exact Hypotheses: Finite-dimensional Hilbert configuration space, positive definite Hessian, surjective constraint Jacobian.
• Proof Location: Fabric Field Equation.
• Portable Receipt: [reports/current/proofs/THM-FFE-02.json](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/proofs/THM-FFE-02.json).

1.5 Generic Resolvent Existence for Self-Adjoint Operators (THM-KP-GENERIC-01)

• Taxonomy Classification: ESTABLISHED_THEOREM_APPLICATION (PROVEN_BY_ESTABLISHED_THEOREM)
• Statement: For any densely-defined non-negative self-adjoint operator $L$ on Hilbert space $\mathcal{H}$, the resolvent operator $R_{\lambda }=(L+\lambda I{)}^{-1}$ exists as a bounded linear operator for all $\lambda >0$ with $\parallel R_{\lambda }\parallel \le 1/\lambda$.
• Exact Hypotheses: Densely-defined self-adjoint operator $L=L^{\ast }$, non-negative spectrum $inf\mathrm{Spec}(L)\ge 0$, positive real spectral parameter $\lambda >0$.
• Proof Location: Standard Spectral Theory (Reed-Simon, Vol. 1).
• Portable Receipt: [reports/current/proofs/THM-KP-GENERIC-01.json](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/proofs/THM-KP-GENERIC-01.json).

1.6 KP Generator Self-Adjointness Hypothesis (THM-KP-OP-01)

• Taxonomy Classification: FORMALIZATION_TARGET
• Statement: The proposed non-local graph generator ${\mathcal{L}}_{KP}=-\mathrm{\Delta }+\alpha \mathrm{\nabla }\cdot (K\ast \mathrm{\nabla })+V$ is self-adjoint on its dense maximal domain $\mathcal{D}({\mathcal{L}}_{KP})$ in $L^2(X,\mu )$ under symmetric non-local kernel $K$.
• Status: Open mathematical target; self-adjoint domain closure is not presumed unconditionally.
• Portable Receipt: [reports/current/proofs/THM-KP-OP-01.json](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/proofs/THM-KP-OP-01.json).

1.7 KP Resolvent Existence Conditioned on Domain Closure (THM-KP-01)

• Taxonomy Classification: DERIVED_CONDITIONALLY
• Statement: Conditioned on the self-adjointness closure of ${\mathcal{L}}_{KP}$ (THM-KP-OP-01), the resolvent $R_{\lambda }=({\mathcal{L}}_{KP}+\lambda I{)}^{-1}$ exists as a bounded self-adjoint Green operator on $L^2(X,\mu )$ for all $\lambda >-inf\mathrm{Spec}({\mathcal{L}}_{KP})$.
• Proof Location: KP-Field Operator.
• Portable Receipt: [reports/current/proofs/THM-KP-01.json](file:///c:/gilc.us.mesh.repos/ivanpasev.com/reports/current/proofs/THM-KP-01.json).

---

2. Interactive Theorem Proving (Lean 4 Status)

The formal proof suite in [docs/04-mathematics/lean/](file:///c:/gilc.us.mesh.repos/ivanpasev.com/docs/04-mathematics/lean/) is compiled using leanprover/lean4:v4.3.0:

• Lean 4 Source Files: 5 files compiled without syntax errors.
• Constitutional Axioms: 5 axioms formalized and type-checked.
• Formal Definitions: 3 definitions verified for syntactic well-formedness.
• Open Formalization Targets: 4 target theorems under active formalization.
• Machine-Verified Theorems: 0 (No theorem proof is claimed as machine-verified until accepted by Lean without heuristic assumptions).

---

3. Calibrated Physical Retrodictions (P2 Calibration)

A strict distinction is maintained between post-hoc parameter alignments and prospective predictions:

3.1 Charged-Lepton Mass Hierarchy

• Model Formulation: $m_k^2=m_0^2+{\mathrm{\Lambda }}^2{\lambda }_k^{D_H^{\ast }/2}$ evaluated on the 5-regular Shifted Laplacian eigenvalues ${\lambda }_k\in \{2.403,5.000,7.597\}$.
• Calibration Parameters: $m_0=0.51099895\text{MeV}$, $\mathrm{\Lambda }=105.145\text{MeV}$, $D_H^{\ast }\approx 27.6$.
• Epistemic Classification: P2 Calibrated Retrodiction ($N_{\text{data}}=3,N_{\text{param}}=3,\text{DOF}=0$).
• Epistemic Boundary: Because all 3 available data points ($m_e,m_{\mu },m_{\tau }$) are consumed to fit the 3 free parameters, this formula possesses zero residual degrees of freedom and does not constitute empirical validation.

---

4. Negative Findings & Statistical Null Benchmarks

To eliminate publication bias, null results are preserved permanently in the primary registry:

4.1 Three-Band Null Ensemble Search

• Test Formulation: Statistical audit of non-local parity violations across discrete multiscale graph configurations.
• Result: $p_{\text{null}}=0.62$ ($>0.05$ threshold).
• Conclusion: No statistically significant departure from the specified null ensemble detected. Standard Model relational parity is preserved on the unperturbed vacuum.
• Canonical Record: Recorded in reports/current/experimental-uncertainty-registry.json.

---

5. Pre-Registered Empirical Interfaces

• Hydrogen Ground-State Spectroscopy: Consensus ionization energy $13.598434599702(12)\text{eV}$ [NIST ASDv5.12].
• Helium Correlation Benchmark: Two-electron nonrelativistic ground state $E_{\text{nonrel}}\approx -2.903724377034{\text{E}}_{\text{h}}$ [Drake2006].
• LPFR Cutoff Scaling: Strong-field recollision cutoffs $\hslash {\omega }_c=I_p+3.17U_p$ [Lewenstein1994] [L'Huillier1993].
• FSR Vacuum Radiometry: Directional spectral emissivity metrology ($-{40}^{\circ }\text{C}$ to ${2300}^{\circ }\text{C}$) [PTB WG 7.352024].

---

6. Canonical Continuations