Session 5 Registry: Proved Theorems and Killed Approaches¶

Jean-Paul Niko · Sole Author

Part I: Proved Theorems¶

#	Theorem	Statement	Proof	Page
T1	Geometric Adjoint	\(A^* = 1-A\) on \(L^2(\mathbb{R}_+, dy/y^2)\) where \(A=y\partial_y\)	Integration by parts	functional_bridge.md
T2	Centered Skew-Adjointness	\(D = A-1/2\) satisfies \(D^*=-D\)	From T1	functional_bridge.md
T3	Conditional Bridge	If \(D^*=-D\), \(K\geq 0\), \([D,K]=0\), \(D\phi=\lambda\phi\), \(\langle K\phi,\phi\rangle>0\) → Re\((\lambda)=0\)	Linear algebra	functional_bridge.md
T4	Bounded Bridge No-Go	On any LP scattering space, \(B^*G+GB=0\) with \(G\) bounded → \(G=0\)	Strong stability + const of motion	bounded_bridge_nogo.md
T5	No Bounded Intertwiner	\(DC=CB\) with \(D\) skew-adjoint, \(C\) bounded → \(C=0\)	Corollary of T4	bounded_bridge_nogo.md
T6	Universal Kernel (i)	\(K_t=e^{-tX}\) with \(X\geq 0\): \(\\|K_t(I-P_0)\\|=e^{-t\Delta}\)	Spectral theorem	bounded_bridge_nogo.md
T7	Universal Kernel (ii)	Same as T4 (strongly stable → bridge=0)	Same	bounded_bridge_nogo.md

#	Theorem	Statement	Proof	Page
T8	Hasse-Weil Identification	BRST filter on \((S^2)^\mathcal{P}\) = étale \(H^0\) projection → \(\zeta(s)\)	Weil conjectures for \(\mathbb{P}^1\)	arithmetic_source_space.md
T9	Functional Equation	Antipodal involution on \(S^2\) = Poincaré duality = \(s\leftrightarrow 1-s\)	Étale cohomology	arithmetic_source_space.md
T10	Visibility (meromorphic)	\(\zeta(\rho-1)\neq 0\) for ζ-zeros: Res\((φ,ρ/2)\neq 0\)	Euler product at Re\(=-1/2\)	functional_bridge.md
T11	Spectral Disjointness	Cusp form eigenvalues \(\neq\) ζ-zeros (contrapositive visibility)	Hecke L-function independence	functional_bridge.md
T12	Euler Factor Mechanism	BRST-filtered prime mode + bosonic Fockization → \((1-p^{-s})^{-1}\)	Direct construction	bounded_bridge_nogo.md

#	Theorem	Statement	Proof	Page
T13	Exact Sequence	\(0\to\ker(C)\to\mathcal{H}_Q\to\text{Im}(C)\to 0\)	First isomorphism theorem	cs_operator_theory.md
T14	K-Matrix Positivity	\(K\) PSD iff cognitive basis orthogonal	Gram matrix argument	k_bridge.md
T15	Filter Non-Commutativity	\(F_\text{cult}\circ F_\text{dev}\neq F_\text{dev}\circ F_\text{cult}\)	2D counterexample	filter_algebra.md
T16	Kernel Composition	Information loss monotonically non-decreasing through pipeline	Ker inclusion	filter_algebra.md

#	Result	Value	Method	Page
N1	HO ground state cost	\(\sigma_0^2=1/2\)	Husimi projection	cost_functional.md
N2	H atom: \(\sigma_{2p}>\sigma_{1s}\)	Energy \(\neq\) instantiation	Coherent-state overlap	cost_functional.md
N3	Li's \(\lambda_n>0\) for \(n\leq 20\)	Verified (30 zeros)	Direct computation	Chain E1
N4	Exact D-sum (306 discs)	80-540× larger than ζ	Kronecker symbols	Chain C1

#	Approach	Kill Date	Killed By	Mechanism	Fatal?
K1	Theta-family kernel \(\sum\theta_\chi\otimes\bar\theta_\chi\)	03-08	@D_Gemini	Serre-Stark: weight 1/2 = theta series, \(n^2\) support	✓
K2	Cusp-form bypass \(S_{1/2}^+(\Gamma_0(4N))\)	03-08	@D_Gemini	Serre-Stark applies at all levels	✓
K3	Sylvester v1.0 (\(\bar\rho+\rho\neq 0\))	03-09	@B_Niko	Wrong equation. Actual: resonant at \(\bar\rho+\rho=1\)	✓
K4	RTF \(P^*P\) bare kernel	03-09	@D_Claude	Archimedean: \(\Delta\propto\zeta(\bar u+s)\neq 0\) off-diagonal	✓
K5	RTF \(P^{\vee*}P\) dual	03-09	@D_Claude	Self-dual (\(\varepsilon=1\) for quadratic chars)	✓
K6	RTF \(K_f\) dressed	03-09	@D_Gemini	\(h_f(s)\) univariate + Paley-Wiener	✓
K7	Exact D-sum cancellation	03-09	@D_Claude	306 discriminants, 80-540× LARGER	✓
K8	Wigner \(\Theta=-M^{-1}M'\)	03-09	@D_GPT	Noncompact, divergent, signed (4 citations)	✓
K9	SVD v2.5 circular	03-09	@D_Claude	\(A=1/2+iT\) is RH → conclusion in premise	✓
K10	ALL bounded bridges	03-09	@D_GPT	Strong stability theorem: \(G=0\)	✓ (permanent)

#	Approach	Status	Issue
K11	\(I-C^*C = H\) (raw)	KILLED	Bounded \(\neq\) unbounded (GPT)
K12	Polyakov loop = Clay gap	PARTIAL	Finite-T only (Svetitsky-Yaffe)
K13	RG monotonicity	KILLED	Does not give constructive bound (Gemini, session 4)

#	Approach	Status	Issue
K14	Archimedean from \(S^2\) heat kernel	KILLED	\(\zeta_{S^2}(s)\) ≠ \(\Gamma(s/2)\) (Gemini B4)
K15	Deligne closes local-global gap	KILLED	\(H^1(\mathbb{P}^1)=0\), Deligne vacuous (Claude)
K16	Faltings → Hodge via BRST=étale	PARTIAL	\(\mathbb{P}^1\) not abelian (Claude)

Direction	Status	Key Result	Next Step
De Branges spaces for LP	OPEN	Only unbounded path survives T4	Construct HB function from LP data
Hasse-Weil / BRST = étale	PROVED	\(\zeta(s)\) from source space	Formalize adelic construction
\(C_t=e^{-tH/2}\) for YM	PROVED	Correct bounded transform	Constructive QFT gap
Fock space Euler product	PROVED	Filtered modes + Fockization	Connect to de Branges form
Fradkin-Shenker armor	CITED	GL↔confinement continuity	Already in paper
Plato → RTSG	PUBLISHED	Cave = exact sequence	Already on wiki
Filter algebra	PROVED	Non-abelian monoid	P≠NP connection weak

Jean-Paul Niko · RTSG BuildNet · smarthub.my · Session 5, March 9 2026