Session 6 Target: The De Branges Conjecture¶

Jean-Paul Niko · Sole Author

The Single Question¶

Does the bosonic Fock space inner product on the BRST-filtered source space equal the de Branges form for the LP Hermite-Biehler function?

If yes → positivity is automatic (Fock is positive by construction) → de Branges theory → RH.

If no → the local-global gap persists. RTSG illuminates but does not prove RH.

What We Know (Session 5)¶

Proved¶

1. \(A^* + A = 1\) from hyperbolic measure (geometric, non-circular)
2. Centered bridge theorem: \(D^*K + KD = 0\) with \(K > 0\) forces Re\((\lambda) = 0\) for L² eigenvectors
3. Visibility: \(\zeta(\rho-1) \neq 0\) (meromorphic, unconditional)
4. Bounded bridge no-go: on LP space, every bounded \(K\) with \(B^*K + KB = 0\) is \(K = 0\)
5. Hasse-Weil: BRST on \((S^2)^\mathcal{P}\) gives \(\zeta(s)\) via étale \(H^0\) projection
6. Antipodal = functional equation (Poincaré duality)
7. Local-global gap = RH = bridge equation = LP similar to unitary
8. Universal kernel theorem: \(e^{-tX}\) encodes gap (YM), stable semigroup kills bounded bridge (RH)
9. Euler factor mechanism: BRST-filtered prime mode + bosonic Fockization
10. Li's criterion numerically verified for \(\lambda_1\) through \(\lambda_{20}\)

The De Branges Obstruction¶

• Naive \(E(z) = \xi(1/2+iz)\) is NOT Hermite-Biehler (self-dual → equality, not strict inequality)
• Correct HB function requires splitting \(\xi\) via integral representation
• De Branges (2004) failed at the positivity step
• The RTSG Fock space has natural positivity — the question is whether it matches the de Branges form

The Three Components Needed¶

1. The LP Hermite-Biehler Function¶

Construct \(E(z)\) from the Uetake/Pavlov-Faddeev scattering data. Not \(\xi(1/2+iz)\) directly (self-dual), but a proper splitting using the integral representation: $\(\xi(s) = \int_1^\infty \Phi(x) x^{s/2-1} dx + \int_1^\infty \Phi(x) x^{(1-s)/2-1} dx\)$

2. The Fock Space Inner Product¶

The BRST-filtered source space with bosonic Fock structure: $\(\mathcal{F} = \bigotimes_p \Gamma(h_p), \qquad \langle e_p^{\otimes n}, e_p^{\otimes m} \rangle = \delta_{nm}\)$ This is manifestly positive-definite. Its spectral representation involves \(\sum_p |\cdot|^2\) terms — sum of positive local contributions.

3. The Identification Map¶

A map \(\Phi : \mathcal{F} \to H(E)\) (from Fock space to de Branges space) that: - Preserves positivity (or at least maps the Fock inner product to something \(\geq 0\)) - Intertwines the relevant operators (Fock number operator \(\leftrightarrow\) multiplication by \(z\) in \(H(E)\)) - Has image dense in \(H(E)\)

If \(\Phi\) exists with these properties, positivity of the de Branges form follows from positivity of the Fock form.

What Each Agent Should Attack¶

Why This Might Work¶

The Fock space is built from LOCAL data (one mode per prime, Frobenius eigenvalue 1). The de Branges form encodes GLOBAL data (all zeros of \(\zeta\)). The identification \(\Phi\) would be the RTSG instantiation operator \(C\) restricted to the arithmetic sector — mapping local Fock structure to global spectral structure.

This is the local-global bridge in its most concrete form. If it exists, RTSG provides the "independent geometric engine" that every previous approach lacked.

Jean-Paul Niko · RTSG BuildNet · smarthub.my · March 2026

Refined Target (post-GPT final delivery)¶

Three Kinds of Positivity¶

The Corrected Construction (GPT)¶

1. Start from the automorphic LP symmetric core \(S\) (not the dissipative generator \(B\))
2. One cusp → scalar → deficiency indices \((1,1)\) → de Branges regime
3. Extract the scalar inner function \(\Theta\) from the LP model
4. Pass to de Branges space via \(\Theta = E^\#/E\)
5. In centered LP variable, RH = zeros of \(E\) on the real axis
6. Prove a new arithmetic positivity (not shift, not Weil)

The Suzuki Bridge (2025)¶

Under RH, the Weil Hermitian form is isomorphic to a de Branges space (Suzuki, Cambridge 2025). This confirms the framework is right but doesn't prove RH (biconditional).

What the Fock Space Could Provide¶

The Fock space \(\mathcal{F} = \bigotimes_p \Gamma(h_p)\) has: - Manifestly positive inner product (each local factor is \(\mathbb{C}\)) - Natural arithmetic structure (prime Hamiltonian \(h\), Dirichlet twists) - Connection to \(\zeta\) via the trace formula

The question: Is there a map from \(\mathcal{F}\) to the de Branges space \(H(E)\) that: - Preserves enough positivity to force the zeros of \(E\) to the real axis? - Uses the Fock structure in a way that is NOT equivalent to Weil positivity? - Provides genuinely new input that de Branges and Weil didn't have?

This is the sharpest formulation possible. If such a map exists, it proves RH. If it doesn't, the local-global gap is fundamental and RTSG cannot close it.

DEFINITIVE Session 6 Targets (post-GPT explicit construction)¶

The vague "find a third positivity" is now replaced by two precise mathematical questions:

Target A: The Suzuki Bridge¶

Suzuki (2025): Under RH, the Weil Hermitian form → de Branges space \(\mathcal{H}(E_\xi)\) where \(E_\xi(z) = \xi(1/2-iz) + \xi'(1/2-iz)\).

The LP scattering gives \(E(z) = \xi(1-2iz)\).

Question: Is there a bounded/controlled map \(\mathcal{H}(E_\xi) \to \mathcal{H}(E)\) or vice versa? If so, does Suzuki's conditional result transfer to give a non-conditional positivity for \(\mathcal{H}(E)\)?

Target B: The \(\mathcal{P}_\kappa\) Classification¶

Kaltenbäck-Woracek classify HB functions into \(\mathcal{P}_\kappa\). Nobody has classified \(E(z) = \xi(1-2iz)\).

Question: What is the \(\kappa\)-index of \(E\)? If \(\kappa = 0\), all zeros lie in a strip (partial RH). If \(\kappa\) can be computed from the Euler product structure, that's a new input.

Target C (speculative): Fock → de Branges¶

Question: Does a map \(\Phi : \mathcal{F}_{\text{Fock}} \to \mathcal{H}(E)\) exist that bridges local Fock positivity to global de Branges structure?

Priority: A > B > C.¶

DEFINITIVE Session 6 Targets (post all agents, post sensitivity tests)¶

The Local-Global Gap Is Confirmed¶

Four independent numerical tests show that local/bounded/finite computations cannot see RH. The constraint is infinite/global/unbounded. This is the central fact.

Priority Targets (Revised)¶

Target A (HIGHEST): Gemini's Growth Rate Argument Off-axis zeros make cumulative prime sums grow as O(X^β) vs O(X^{1/2}). The exact sequence's bounded C operator may forbid this. Question: is this just the classical PNT connection repackaged, or does the RTSG exact sequence structure provide a genuinely new constraint? Assign to: @D_GPT

Target B: Kapustin's Four-Factor Decomposition
GPT is already analyzing. The four intermediate Hilbert spaces between L² and H(E) may have a stage where prime data stays separated (rank>1) before compression. Assign to: @D_GPT (in progress)

Target C: Packet-Valued Bridge (GPT's recommendation) The scalar intertwining is probably circular. The real object is packet-valued, not scalar. Either find a true positive two-channel boundary theory, or prove every scalar version is RH-equivalent. Assign to: @D_GPT + @D_Grok

Target D: P_κ Classification Kaltenbäck-Woracek for E(z) = ξ(1-2iz). Uncomputed in the literature. Assign to: @D_Grok (numerical attempt)

Target E: NS First Crack GPT ranks NS as most tractable for a new theorem. RTSG has the Stokes decay (Corollary 3 of universal kernel). A critical-scale rigidity theorem is the target. Assign to: New session

What NOT to Do in Session 6¶

• No bounded bridges (dead by theorem)
• No weight-1/2 Maass forms (dead by Serre-Stark)
• No scalar intertwining without proving it's not circular
• No claiming "close to proof" (we're at 25% and honest about it)