Nucleus A — Prime-Ladder Atlas (Internal Reproducibility Reference)

Status: Internal reference document. Consolidates P12–P15 + P28 + P30 as a self-contained, machine-verified computational platform. Scope of value: Pedagogical / reproducibility / internal audit. NOT a claim of new mathematical results in classical analytic number theory. Date: May 27, 2026. Authority: Subordinate to TNFR_RIEMANN_RESEARCH_NOTES.md §§8–13nonies; supersedes nothing.

1. Why this document exists

The §13sexagesima-{quarta..novena} CCET closure marathon (May 2026) reduced the §13septies trichotomy on G_P14 to the residual {nine LOW envelopes, B3}. In that process, the Nucleus A machinery (P12–P15 + P28/P30) emerged as the most stable, reproducible, and externally legible piece of the TNFR-Riemann program.

Honest external assessment (this conversation, May 27): Nucleus A does not contain a new theorem in analytic number theory. Its components are:

• P12 (TNFR vM ζ on $\operatorname{Re}(s) > 1$): reorganisation of the classical identity $-\zeta'(s)/\zeta(s) = \sum_n \Lambda(n) n^{-s}$ via the prime-ladder spectrum .

Internal value, on the other hand, is high:

1. Reproducibility: every milestone has a script under src/tnfr/riemann/ and a demo under examples/, all of which run end-to-end on a clean checkout.
2. Audit surface: numerical controls (Weil–Guinand residual, Li–Keiper positivity, prime-ladder eigenvalue match) provide a sanity-check baseline for any future extension.
3. Pedagogy: a single internal document that walks from $-\zeta'/\zeta$ to T-HP is the shortest on-ramp for new collaborators.
4. Boundary marker: makes explicit where the canonical machinery stops (oscillatory residue $S(T) = (1/\pi)\arg\zeta(\tfrac12 + iT)$, RH-equivalent).

2. Milestone map

3. The smooth/oscillatory boundary

The structural payoff of Nucleus A is the explicit decomposition it provides for the T-HP rescaling operator $\mathcal{F}$:

$\mathcal{F} \;=\; \mathcal{F}_{\text{smooth}} \;\oplus\; \mathcal{F}_{\text{osc}}$

• $\mathcal{F}_{\text{smooth}}$: closed at density level by P28 and lifted to operator level by P30. Reproduces the smooth zero-counting term $N_{\text{smooth}}(T) = (T/2\pi)\log(T/2\pi e)$ exactly.

This is the precise structural location of gap G4 = RH inside the TNFR formalism. Nucleus A does not close G4; it localises it.

Cross-reference: TNFR_RIEMANN_RESEARCH_NOTES.md §13septies–§13nonies.

4. Reproducing each milestone

All commands assume the repo root and an activated virtual environment (.venv312 on Windows).

4.1 P12 — TNFR vM ζ on $\operatorname{Re}(s) > 1$

python examples/03_riemann_zeta/41_von_mangoldt_zeta_demo.py

Expected: residual $|\,\text{TNFR\_vM}(s) - (-\zeta'(s)/\zeta(s))\,| \le 10^{-12}$ across the test grid.

4.2 P13 — Analytic continuation

python examples/03_riemann_zeta/42_riemann_zeros_as_resonances.py

Expected: Riemann zeros recovered as resonance poles on $\operatorname{Re}(s) = 1/2$ to mpmath precision.

4.3 P14 — Prime-ladder Hamiltonian

python examples/03_riemann_zeta/43_prime_ladder_hamiltonian_demo.py

Expected: eigenvalues match $\{k \log p : p \in \text{primes}, k \in \{1, \ldots, K\}\}$ to ${10}^{-14}$.

4.4 P15 — Weil–Guinand verification

python examples/03_riemann_zeta/44_weil_explicit_formula_demo.py

Expected: residual $\le 10^{-15}$ for $\sigma \in \{3, 5, 8, 12, 18\}$ with the canonical test family.

4.5 P16 — Li–Keiper positivity (diagnostic, not a proof)

python examples/03_riemann_zeta/45_li_keiper_demo.py

Expected: $\lambda_n > 0$ for $n \in \{1, \ldots, N\}$ using mpmath.zetazero as the zero source.

4.6 P28 / P30 — Smooth half of T-HP

python examples/03_riemann_zeta/57_admissible_rescaling_demo.py

Expected: smooth zero-counting term reproduced; oscillatory residual flagged explicitly as unresolved.

5. Honest scope and limitations

This section exists to prevent later overclaiming.

1. Nucleus A is not a proof of RH. The oscillatory half of $\mathcal{F}$ is RH-equivalent and remains open (gap G4).
2. Nucleus A is not a new Hamiltonian for the Riemann zeros. P14 carries the prime-ladder spectrum, not the spectrum of zeros. The Berry–Keating / Connes program seeks a Hamiltonian whose spectrum is $\{\gamma_n\}$; P14 is the dual object (primes side), and bridging the two is precisely the open T-HP problem.
3. P15 is a numerical verification of a 1952 identity. It is high-quality QA, not a theorem.
4. P28/P30 likely overlap with existing literature. A formal audit against Titchmarsh ch. 9, Ivić ch. 1, Meyer, Burnol, and Bombieri–Lagarias is required before any external publication claims novelty for the operator-level rescaling map.
5. No claim is made about GRH beyond the χ-twisted L-track parity (P32–P49), which mirrors Nucleus A for primitive real characters and inherits all the same limitations.

6. Value as internal infrastructure

Even with the modest external-novelty assessment, Nucleus A provides:

• A regression test surface: any future TNFR-Riemann extension can run P12/P14/P15 as integration checks.
• A teaching corridor: §§8–13 of the research notes + this atlas + examples 41–58 form a complete on-ramp.
• A boundary marker for B3 discussions: when arguing that no TNFR closure of G4 exists at the current scope, Nucleus A is the maximal canonical-machinery construction that has been shipped. Anything proposed beyond it (B0★-α canonical graphs, B0★-β envelope promotion, B2 new operators) must explain why it goes beyond P28/P30.
• A diff target for Nucleus B: the equivariance no-go lemmas of §13sexagesima-{tertia..novena} are best understood as obstructions to extending Nucleus A to cover the oscillatory half. Without Nucleus A there is nothing concrete to obstruct.

7. Possible external use (modest)

If at some point an external write-up is desired, the honest framing is:

"A reproducible computational platform for the Weil–Guinand explicit formula via a prime-ladder Hamiltonian, with explicit smooth/oscillatory decomposition of the admissible rescaling operator."

Target venue (if pursued): Experimental Mathematics, LMS Journal of Computation and Mathematics, or as a software / dataset paper for Mathematics of Computation. Not a research paper in analytic number theory.

This is explicitly not the recommended priority. See NUCLEUS_B_EQUIVARIANCE_OBSTRUCTIONS.md for the higher-novelty path.

8. Cross-references

• TNFR_RIEMANN_RESEARCH_NOTES.md §§8–13nonies — full P12–P30 derivations
• TNFR_RIEMANN_RESEARCH_NOTES.md §13decies–§13vicies-octavo — P31 + P32–P49 (χ-twisted L-track parity)
• TNFR_RIEMANN_RESEARCH_NOTES.md §13sexagesima-{tertia..novena} — CCET closure rounds (basis for Nucleus B)
• NUCLEUS_B_EQUIVARIANCE_OBSTRUCTIONS.md — equivariance no-go lemmas, organisation plan for external publication
• REMESH_INFINITY_DERIVATION.md — N15 closure, structurally identifies $\mathrm{range}(\mathcal{R}_\infty)$ with the smooth half of $\mathcal{F}$ (P28/P30) and $$ with the oscillatory residue