The nodal-pulse phase attack surface (re-founded, 2026-07)

With the combinatorial track eliminated, the program's live attack surface is the collective phase of the integer-NFR nodal pulse. Measured (examples/03_riemann_zeta/157_nodal_pulse_phase_attack.py):

• M1 — S(T) is the pulse phase. S(T) = (1/π) arg ζ(1/2+iT) = arg(P(T))/π to |Δ| < 0.015 on the truncated pulse P(T) = Σ n^{-1/2} e^{-i(log n)T}.
• M2 — the phase counts the zeros. N(T) = θ(T)/π + 1 + S(T) (Riemann–von Mangoldt) reproduces the exact zero count from the pulse phase alone.
• M3 — the pulse accesses the arithmetic. Permuting the prime structural frequencies (log 2 ↔ log 3) changes P (|P|: 0.247→0.460, arg: −0.286→ −0.837). The pulse is sensitive to the specific prime values — the Fix(S_n)^⊥ content the S_n-invariant self-adjoint spectrum of the eliminated operator was provably blind to (the Euler-Orthogonality wall that paused the old program). The re-founded vantage is not blind to it.
• M4 — the critical line is the coherence axis. The rectified pulse Z = e^{iθ}P is most nearly real on Re(s)=1/2 (|Im Z|/|Z|: 0.12 at σ=½ < 0.19 at σ=0.7) — the functional-equation reflection axis read as the ΔNFR=0 coherence axis.

Frontier. RH is the statement that S(T) never lets a zero leave the coherence axis. The re-founding relocates that question from the (blind) self-adjoint spectrum to the collective phase / coherence of the integer-NFR pulse — an arithmetic-accessing, reflection-native arena. This surface is mapped, not settled; G4 = RH remains open. The distinction from the paused T-HP program is structural: T-HP sought an S_n-invariant operator whose spectrum is {γ_n} (blind to Fix(S_n)^⊥); the pulse phase carries Fix(S_n)^⊥ directly.

Advance (2026-07): tooling + obstruction localization. The surface is now canonical tooling in src/tnfr/riemann/pulse_coherence.py: argument_fluctuation (S(T) from the pulse phase), zero_count (N(T) = θ/π + 1 + S(T)), coherence_defect (exact Z = e^{iθ}ζ, ~1e-16 on σ=½, growing off-axis) and verify_pulse_coherence. Measured localization: the prime-side series S(T) = (1/π) Σ_{p,k}(1/k)p^{-k/2} sin(kT log p) does not converge on the line -- its abscissa of convergence is Re(s)=1, so adding prime NFRs makes it worse (err 0.40→0.47 at T=41), not better. The RH content is exactly this boundary non-convergence: S(T) is accessible (the integer pulse phase) but not summable from the primes on the axis. That is the sharp form of the obstruction in the emergent framing -- and it is a genuine step, not a verdict on where the surface leads.

Coherence-budget measurement (2026-07): the U2 budget is real at the RMS level, not the sup level. With S(T) now accessible, the natural TNFR attack is the U2 reading — RH ⟺ the prime-pulse NFR stays coherent (U2-bounded) — so we measured how tightly the pulse phase auto-bounds, using exact ζ (continuous arg descent in σ, no RH input; validated by θ/π + 1 + S = integer zero-count). Over two decades in T (30→3000): mean(S) ≈ 0 (centred, no drift); RMS(S) grows only 0.32→0.39 — the glacial √(log log T) of Selberg (fit RMS² ≈ 0.066·log log T, same order as 1/(2π²)=0.051); the measured peaks max|S| ≈ 1 sit at ~¼ of the unconditional O(log T) envelope (≈4). So the pulse phase does not run away — the U2 coherence budget is confirmed numerically, at the RMS/typical level. But √(log log T) tightness is Selberg's theorem: classical, unconditional, consistent with RH yet not implying it. RH lives in the extremes (peaks are Ω(√(log T/log log T)) unconditionally — unbounded, very slowly), so no finite measurement excludes a large excursion at astronomical height. Net: the measurement relocates the wall sharply from "control S(T)" to "lift the coherence budget from the RMS level to the supremum" (control the peaks), and confirms the RMS level is as tight as Selberg says. Driver: benchmarks/pulse_phase_coherence_budget.py. Closes nothing; G4 = RH stays open.

The nodal-ontology re-mapping — fixed points are the shadow (2026-06)

Foundational re-framing. This section re-maps the program onto the current emergent nodal ontology — the single structural constant π, the emergent pulse ω_k = √λ_k, the symplectic substrate, and the Fix(S_n)^⊥ wall of EMERGENT_ONTOLOGY.md §2.4. It supersedes the conceptual framing of P12–P49 (built in a pre-pulse, pre-single-constant era); the computational certificates (which are gain-independent) are unchanged — only what they measure is re-read.

What was obsolete in the old mapping

1. Pre-"single constant". γ/π ≈ 0.18373 was treated as a canonical "Universal Tetrahedral Correspondence" scale (the spectral-zeta buffer CRITICAL_EXPONENT, the coherence threshold δ_coh, the Kuramoto-U3 weight), and the T-HP conjecture (§13septies) invoked "(φ, γ, π, e)". Post-purge only π is a structural scale; γ/π is a heuristic coupling, not canonical.
2. Pre-pulse. The zeros {γ_n} and the Weil–Guinand explicit formula were framed as "the spectrum of a sought self-adjoint operator", never as the pulse / rhythm of the arithmetic NFR.
3. Pre-§2.4. The Euler-Orthogonality wall (every canonical operator commutes with the S_n prime-relabelling, so it is blind to S(T) ∈ Fix(S_n)^⊥) is literally the Fix(G)^⊥ wall of the emergent-ontology synthesis (§2.4), described here in isolation.

The reframe — the fixed-point program is the overdamped projection

Canonically, the nodal equation ∂EPI/∂t = νf·ΔNFR is the overdamped projection of the symplectic Hamiltonian flow (AGENTS.md §4; symplectic_substrate.py), and that projection discards the conjugate momenta (J_φ, J_ΔNFR). The whole fixed-point program — seek a self-adjoint operator whose static real spectrum equals {γ_n} — therefore lives in the position shadow of a richer dynamical object. The zeros and S(T) are projections of that object onto the numeration (the prime / integer basis).

The three nested layers (measured)

A direct measurement on the prime-ladder NFR (swap primes 2↔3, the S_n element P; symmetric operator L_sym) settles where the arithmetic can and cannot live:

The decisive datum: every function of the symmetric L — the propagator exp(itL), the conservative position cos(√L·t), and the momentum √L·sin(√L·t) — commutes with P to machine precision (0.00e+00). Since [L,P]=0 ⟹ [f(L),P]=0, the conservative dynamics and its momenta are exactly as S_n-equivariant as the static spectrum. Activating the momenta cannot leave Fix(S_n) — it re-expresses the same prime data (consistent with the ex.103 result: the θ=νf·τ dynamics stayed Poisson, not Riemann).

The escape needs a non-S_n generator. The directed quadratic-residue operator (affine symmetry of Z/n, not S_n) is non-self-adjoint with a complex spectrum (−1 ± i√q)/2 — the arithmetic moves into the phase (the Gauss sum √q in the imaginary part). Since S(T) = (1/π)·arg ζ(½+iT) is a phase, the missing structure lives in the emergent complex / phase dimension, not in the real spectrum nor in the conservative momenta.

The honest wall (the two walls coincide)

The directed operator's phase is the Gauss sum √q, not the ζ-zero phase S(T). Measured (benchmarks/residue_phase_vs_riemann.py): √q exact (15/15), but alignment with {γ_n} refuted (residue content ~1/√p decreasing, γ_n increasing — opposite). The reframe locates the missing structure (the phase dimension) and forbids the two cheaper layers, but does not yet reach S(T).

G4 = RH, re-stated dynamics-first

The zeros are the configuration-shadow of the arithmetic NFR's symplectic dynamics; the smooth half (π-scaled archimedean, S_n-symmetric) is the reachable mean pulse; S(T) is the transverse phase-shadow at Fix(S_n)^⊥. The wall is reclassified from "obstruction" to kernel of the position-only (numeration) projection — provably unreachable from the two S_n-symmetric layers, with the phase layer the only structurally-permitted route (currently landing on Gauss sums, not {γ_n}).

The same shape across the Millennium problems

This is the dynamics extension of the §2.4 synthesis (one operator L, read in every domain, hitting one wall Fix(G)^⊥). Each problem = a reachable symmetric / fixed-point projection + a transverse residue that is the shadow of the emergent-dimensional dynamics the projection discards:

• Navier–Stokes: the blow-up is not a fixed point but the K_φ phase-curvature cascade (dynamics); the BKM-analogue (U2) lives in the dynamics, not an equilibrium.
• Yang–Mills (mass gap): the gap = confinement Φ_s²/(π/2)² = the non-Abelian (non-commuting = transverse) residue; the gap lives in the symmetry-broken phase.
• P vs NP / BSD / Hodge: each = a symmetric reachable sector + the Fix(G)^⊥ residue.

Unified, honest statement. Every Millennium problem re-reads as "is the transverse residue Fix(G)^⊥ — the shadow of the emergent-dimensional dynamics — reachable from the symmetric sector?" The measured answer so far: not from functions of the symmetric operator (positions and momenta); only in principle from the phase of the symmetry-broken generator. This relocates all of them to one place; it closes none.

Honest scope

A re-framing, not a result. It closes no gap; G4 = RH remains open (the single open milestone, §19.2); the P12–P49 certificates stand. The live open question: whether the correct emergent dimension is the affine Gauss phase or a deeper structure (the functional-equation root number / the Euler-product cyclotomic tower) — pursued in the living-discoveries log (§13triginta-septima).

0. Navigation Index

This file aggregates ~5.9k lines covering five intertwined programmes: the ζ-track (P12–P31), the χ-twisted L-track (P32–P49), the REMESH-∞ / N15 cross-program lift (§13vicies-novies + §13triginta), the catalog type-hygiene programme (T-νf = B0, T-EPI = B1, …; full tracker in CATALOG_TYPE_HYGIENE_PROGRAMME.md), and a living discoveries log (§13triginta-septima). Section anchors are stable and referenced from AGENTS.md, the catalog tracker, and the code; do not rename or split them. Use this index to locate work without scrolling.

Legend: ✅ CLOSED (operationally or with stated scope) · 🟡 OPEN · 🔁 LIVING (append-only) · ⛔ SUPERSEDED · 📖 PROSE (meta / status).

A. Memo Meta (lines 171–294)

B. ζ-Track Foundation — Prime-Ladder vM Construction (lines 427–1047)

C. ζ-Track Coercivity & Lyapunov Layer (lines 1049–1727)

D. T-HP Reformulation of G4 (lines 1728–2018)

E. ζ-Track Operator-Level Smooth Half (lines 2019–2172)

F. χ-Twisted L-Track Parity Layer P32–P49 (lines 2173–3482)

G. ζ-Track Admissibility / Gauge / Hermite Layer P17–P21 (lines 3483–3995)

(Numbering note: §§ 14–18 appear after §13vicies-octavo because they were appended chronologically out of P-number order; the §13xxx anchors remain authoritative.)

H. Program Status Snapshot (lines 3996–4169)

I. REMESH Global Reframe + B1 Edge-Channel Refutation Thread (lines 4170–5715)

The largest single block (~1.5k lines). Contains the cross-program discovery that REMESH is the canonical temporal aggregator, and the exhaustive structural refutation of branch B1 sub-routes on G_P14 (R∞-1a-operator, R∞-1a-composed, Prime-Cancellation Lemma, Euler-Orthogonality Lemma, R∞-1c, R∞-1b spectral-channel).

J. P50 — REMESH-∞ Function-Space Lift (lines 5716–5916)

K. Catalog Type-Hygiene Programme — Sub-Questions (lines 5917–7588, 7698–end)

Tracker: CATALOG_TYPE_HYGIENE_PROGRAMME.md.

L. Living Discoveries Log (lines 7589–7697)

M. Final-Gap Bookmarks (Quick Recall)

• G4 = RH: 🟡 OPEN. Canonical statement = Conjecture T-HP (§13septies). Smooth half closed operationally by P28 (density) and P30 (operator). Oscillatory half $S(T) = \tfrac{1}{\pi}\arg\zeta(\tfrac12+iT)$ remains RH-equivalent. P31 ζ-track and P49 χ-track attack this half via canonical prime-ladder Newton correction; empirical regime is mixed B1/B2 (§§13vicies-octavo, 19.1).
• GRH$_\chi$ (primitive real χ): 🟡 OPEN, parity with G4.
• Branch B1 sub-routes on G_P14: ✅ structurally refuted by Euler-Orthogonality Lemma (§13vicies-novies.11). Surviving sub-routes inside B1 require non-product canonical lifts.
• Programme paused at the T-HP boundary; no further diagnostic surface planned until one of B1/B2/B3 (§13septies) is decided.

1. Purpose and Scope

• Translate RH questions into TNFR constructs: nodal operators, structural partition functions, and confinement criteria derived from Φ_s, |∇φ|, K_φ, and ξ_C.
• Maintain reproducible sandboxes (finite prime graphs, spectral benchmarks, telemetry artifacts) that connect theoretical conjectures to code in src/tnfr/riemann/ and examples/03_riemann_zeta/16_riemann_operator_demo.py.
• Document how canonical operators (AL, UM, RA, OZ, IL, THOL) compose to form the discrete TNFR Riemann operator used in experiments.

2. Program Objectives

2.1 Partition Function Mapping

• Show that the TNFR structural partition function $Z_{TNFR}(s)$ converges to ζ(s) or ξ(s) by enforcing the identification $e^{-\beta E_p(s)} \leftrightarrow p^{-s}$ for prime-labeled resonant modes.
• Specify how ν_f and ΔNFR sources enter the effective energy $E_p(s)$ so the mapping respects U2 (convergence) and U3 (resonant coupling).

2.2 Operator Construction

• Construct $\mathcal{H}_{TNFR}$ as a Laplacian-plus-structural-potential on prime path graphs, ensuring self-adjointness with respect to the TNFR inner product.
• Demonstrate numerically that eigenvalues migrate toward the critical line as graph size increases (σ_c^{(k)} \to 1/2) and record telemetry in results/riemann_program/.

2.3 Critical-Line Confinement

• Formulate a Lyapunov-style functional $\mathcal{L}_{RH}(s)$ derived from TNFR invariants so that σ = 1/2 is the only stable attractor.
• Quantify escapes (σ ≠ 1/2) via Φ_s drift and |∇φ| spikes to test whether confinement behaves like U6 in the complex-s domain.

3. Workflow Expectations

1. Model definition – Choose $G_k$ (prime path graph) size, seeds, and operator sequences; record configs in results/riemann_program/configs/*.json.
2. Operator execution – Use SDK helpers (TNFRRiemannOperator) to generate spectra while logging ν_f, ΔNFR, Φ_s, |∇φ|, and effective σ(t) trajectories.
3. Spectral analysis – Compute eigenvalue ladders, determinant surrogates, and compare against ζ/ξ predictions. Scripts belong in scripts/riemann/ or notebooks under notebooks/Riemann/ with nbconvert support.
4. Benchmark enforcement – Run python benchmarks/riemann_program.py (invoked automatically via make test/CI) to regress σ_c^{(k)} estimates across graph sizes and emit telemetry in results/riemann_program/.
5. Validation – Run targeted tests (e.g., examples/03_riemann_zeta/16_riemann_operator_demo.py, new tests/test_riemann_operator.py) to ensure deterministic seeds and grammar compliance (U1–U6).

4. Telemetry & Reproducibility

• Log Φ_s, |∇φ|, K_φ, ξ_C, ν_f, ΔNFR, and σ estimates at every operator step; store as Parquet/CSV in results/riemann_program/telemetry/ with metadata (graph size, seed, operator stack). The helper dataclass tnfr.riemann.telemetry.RiemannTelemetryRecord now carries aggregate Φ_s/|∇φ|/K_φ statistics plus ξ_C computed via tnfr.riemann.telemetry.compute_field_aggregates so tetrad coverage is explicit.
• Publish spectra, determinant traces, and Lyapunov metrics in results/riemann_program/plots/ along with scripts used to generate them.
• Capture environment details (Python version, tnfr package hash) inside each artifact manifest to satisfy invariants #5 (Structural Metrology) and #6 (Reproducible Dynamics).

5. Outstanding Work

1. Lyapunov functional derivation – Formalize $\mathcal{L}_{RH}(s)$ using existing field invariants and document stability proofs in docs/STRUCTURAL_FIELDS_TETRAD.md or a dedicated theory note.
2. Spectral determinant prototype – Produce a working determinant or trace formula implementation and compare against numerical ζ(s) evaluations over multiple σ bands.
3. Telemetry-field linkage – Extend tnfr.riemann.telemetry so Φ_s, |∇φ|, K_φ, and ξ_C aggregates from live runs attach automatically to each record (current benchmark logs spectral data only).

6. Cross-References

Implementation Modules (src/tnfr/riemann/)

Discrete operator and spectral framework:

• operator.py — discrete TNFR-Riemann operator $H^{(k)}(\sigma) = L_k + V_\sigma$ and prime graph builders.
• spectral_proof.py — four-line spectral convergence framework ($\sigma_c^{(k)} \to 1/2$).
• topology.py — alternative graph topologies and cross-topology convergence (P2).
• eigenmode_fields.py — per-eigenmode structural field tetrad on the prime path model (P3).
• complex_extension.py — complex-$s$ non-Hermitian extension (P4).
• spectral_zeta.py — discrete spectral zeta and heat kernel; original Conjecture 10.1 affine bridge (P5, negative; superseded by P12–P15 via §7.8).
• random_ensemble.py — random prime-graph ensembles / RMT universality (P6).
• spectral_conservation.py — conservation laws and grammar compliance at criticality (P7).
• analytical_convergence.py — analytical proof of $\sigma_c \to 1/2$ via PNT + telescoping (P8).
• functional_equation.py — TNFR-side $s \leftrightarrow 1-s$ reflection check (P9).
• convergence_proof.py — end-to-end formal $\sigma_c \to 1/2$ certificate (P10).
• zeta_bridge.py — affine bridge prototype $\zeta_H \approx C \cdot \zeta_R$ (P11, tested negative, see §7).
• telemetry.py — Riemann telemetry records and field aggregate helpers.

Prime-ladder / von Mangoldt construction (closes G1, G2, G3 operationally; G5 superseded):

• von_mangoldt.py — TNFR prime-ladder spectrum reproducing $-\zeta'(s)/\zeta(s) = \sum_n \Lambda(n)\, n^{-s}$ on $\operatorname{Re}(s) > 1$ (P12, §8).
• analytic_continuation.py — continuation of the prime-ladder vM zeta to $\mathbb{C}$; Riemann zeros as resonance poles on $\operatorname{Re}(s) = 1/2$ (P13, §9).
• prime_ladder_hamiltonian.py — self-adjoint Hamiltonian whose weighted spectral trace reproduces P12 (P14, §10; closes G1).
• weil_explicit_formula.py — numerical Weil–Guinand identity using P14 on the prime side; residual $\le 5 \times 10^{-12}$ (P15, §11; closes G3).
• li_keiper.py — Li–Keiper positivity criterion from the TNFR resonance spectrum (P16, §12; RH-equivalent diagnostic, not proof).

TNFR-native G4 attack surface (research; does not close G4 = RH):

• weil_positivity.py — Weil–TNFR positivity bridge $\alpha(\sigma) = W[\sigma] / E_{\mathrm{TNFR}}[\sigma]$ (P17, §14).
• alpha_sweep.py — admissibility / gauge sweep of $\alpha(\sigma)$ across Gaussian width × gauge family (P18, §15).
• admissible_family_sweep.py — extends P18 beyond Gaussian (Gaussian mixture, Hermite2-Gaussian admissible families) (P19/P21, §16/§18).
• nodeaware_gauge_sweep.py — node-aware gauge extension parameterised by local $\nu_f$ and node-weight channels (P20, §17).
• coercivity_uniform.py — empirical uniform-coercivity certificate over $\sigma$ intervals, plus adaptive $\sigma$ refinement near the coercivity bottleneck (P22 / P23 / P24, §13 / §13bis).
• paley_gap_coercivity.py — Paley-gap coercivity diagnostic (Martínez Gamo, Zenodo 10.5281/zenodo.17665853 v2) (P25, §13ter).
• lyapunov_spectral_positivity.py — Lyapunov-spectral positivity certificate for the P14 Hamiltonian (P26, §13quater).
• hilbert_polya.py — Hilbert–Pólya scaffold $T_{\mathrm{HP}} = \operatorname{diag}(\gamma_n)$ populated by mpmath.zetazero (diagnostic only) (P27, §13quinquies).
• structural_zero_density.py — structural derivation of the smooth Riemann zero density via the Riemann–Siegel $\theta$ function (P28, §13sexies; closes smooth half of G4 at density level).
• spectral_emergence.py — spectral universality emergence under canonical UM+RA inter-prime couplings; KS-distance to the GUE Wigner surmise (P29, §13octies.3).
• admissible_rescaling.py — operator-level admissible spectral-rescaling lift of P28 (P30, §13nonies; closes smooth half of T-HP at operator level).

Conjectural reformulation (does not close G4):

• §13septies — Tetrad-Hilbert–Pólya reformulation of G4 (Conjecture T-HP).
• §13octies — Assembled-argument audit (links L1–L7 closed, L8 = T-HP open).

Examples

End-to-end pipeline demos (examples/):

• 16_riemann_operator_demo.py — discrete TNFR-Riemann eigenvalues at varying $\sigma$.
• 18_riemann_convergence_proof.py — spectral convergence proof ($\sigma_c \to 1/2$).
• 19_topology_comparison.py — cross-topology critical parameter comparison.
• 20_eigenmode_tetrad.py — eigenmode-based tetrad field analysis.
• 21_complex_extension_demo.py — non-Hermitian operator on complex $s$.
• 22_spectral_zeta_demo.py — discrete spectral zeta, heat kernel, Mellin bridge.
• 23_random_ensemble_rmt_demo.py — random matrix ensembles (GOE/GUE/Poisson).
• 24_spectral_conservation_demo.py — spectral conservation law at criticality.
• 25_analytical_convergence_demo.py — analytical proof via PNT + telescoping.
• 41_von_mangoldt_zeta_demo.py — P12 prime-ladder reproduction of $-\zeta'/\zeta$.
• 42_riemann_zeros_as_resonances.py — P13 zeros as resonance poles on $\operatorname{Re}(s) = 1/2$.
• 43_prime_ladder_hamiltonian_demo.py — P14 self-adjoint Hamiltonian certificate.
• 44_weil_explicit_formula_demo.py — P15 Weil–Guinand identity at machine precision.
• 45_li_keiper_demo.py — P16 Li–Keiper positivity diagnostic.
• 46_weil_tnfr_positivity_demo.py — P17 Weil–TNFR positivity bridge.
• 47_alpha_sweep_demo.py — P18 gauge sweep of $\alpha(\sigma)$.
• 48_admissible_family_sweep_demo.py — P19/P21 admissible-family sweep.
• 49_nodeaware_gauge_sweep_demo.py — P20 node-aware gauge extension.
• 50_uniform_coercivity_demo.py — P22 empirical uniform-coercivity certificate.
• 51_adaptive_coercivity_demo.py — P24 adaptive $\sigma$ refinement near the bottleneck.
• 52_paley_gap_coercivity_demo.py — P25 Paley-gap coercivity diagnostic.
• 53_lyapunov_spectral_positivity_demo.py — P26 Lyapunov-spectral positivity certificate.
• 54_hilbert_polya_demo.py — P27 Hilbert–Pólya diagnostic scaffold.
• 55_structural_zero_density_demo.py — P28 structural smooth zero density.
• 56_spectral_emergence_demo.py — P29 KS-distance to GUE under canonical couplings.
• 57_admissible_rescaling_demo.py — P30 operator-level admissible rescaling (smooth half).

Supporting Infrastructure

• benchmarks/riemann_program.py — automated spectral regression benchmarks for $\sigma_c^{(k)}$ across graph sizes.
• theory/UNIFIED_GRAMMAR_RULES.md — grammar rules U1–U6 referenced throughout.
• docs/STRUCTURAL_FIELDS_TETRAD.md — tetrad field specifications.
• AGENTS.md — TNFR-Riemann overview, including the G4 = RH reformulation.

7. Conjecture 10.1 Gap Analysis (May 2026)

Status: Negative numerical result — bridge not yet closed.

7.1 Experiment

The fit defined by Conjecture 10.1

was tested using test_conjecture_10_1_sequence from src/tnfr/riemann/spectral_zeta.py for $k \in \{10, 20, 50, 100, 200, 500, 1000\}$ over $u \in [1.5, 5.0]$ (30 points).

7.2 Numerical Results

7.3 Diagnostic Reading

A converging bridge would show: residual → 0, Pearson r → +1, δ(k) stabilising at an interior value, and C(k) stabilising after correct renormalisation. The data show the opposite in every metric:

• Residual rises monotonically with k.
• Correlation is negative and bounded away from +1 at all tested k.
• δ(k) = 2.0 in every row — pinned at the boundary of the search range, indicating no interior minimum was found.
• C(k) explodes (≈10³⁷ at k = 1000), signalling a missing renormalisation.

Conclusion: as currently implemented, ζ_H^(k)(1/2, u) is not numerically equivalent to C · ζ_R(u + δ) under the simple affine fit.

7.4 Six Missing Pieces

7.5 TNFR-Internal Diagnosis

In TNFR language the finding is:

The operator $H^{(k)}(\sigma)$ constructs a structural dynamic that is sensitive to the critical line σ = 1/2 (σ_c^(k) → 1/2 is internally validated), but it does not yet encode the full multiplicative arithmetic of ζ(s). The prime-path graph captures structural coherence near 1/2 without closing the bridge to the classical zeta function.

7.6 Priority Construction

The mathematical priority is to build a TNFR zeta that reproduces the von Mangoldt series

which encodes prime positions and their higher powers with the correct multiplicities (Λ = log p for prime powers, 0 otherwise). Without this, TNFR can exhibit σ-criticality but cannot be equated to Riemann.

The required renormalisation takes the form

where $\xi(s)$ is the completed Riemann zeta and $R_k$ is a holomorphic, non-vanishing function to be constructed.

7.7 Impact on Program Status

This result does not invalidate the σ_c^(k) → 1/2 finding, which rests on eigenvalue analysis independent of the spectral-zeta fit. It narrows the scope of Conjecture 10.1: the conjecture is open, and the simple C · ζ_R(u + δ) form is likely insufficient. Future work should target the von Mangoldt / Λ-series route (Section 7.6) before revisiting the affine fit.

7.8 Retrospective Closure of G5 (May 2026)

The "non-affine bridge" anticipated in §7.6 has been constructed and verified by the P12–P15 pipeline:

This replaces the original affine ansatz $\zeta_H(1/2,u) \approx C(k)\,\zeta_R(u+\delta(k))$ with a structurally correct, multiplicative-arithmetic bridge that lives natively inside TNFR without ad-hoc renormalisations. The six missing pieces listed in §7.4 are addressed as follows:

Conclusion: G5, in its original affine formulation, is superseded by the prime-ladder / Λ-series construction. The bridge between TNFR spectral data and classical $\zeta(s)$ is therefore considered operationally closed. The only obstruction that remains is G4 = RH itself — the localisation of the resonance poles on $\operatorname{Re}(s) = 1/2$ — which is a structural positivity / self-adjointness problem, not a missing-bridge problem.

8. TNFR Prime-Ladder Construction of the von Mangoldt Series (P12)

Following Section 7.6, this section records the first concrete attempt at the priority route: build a TNFR-native spectral object whose Dirichlet transform reproduces $-\zeta'(s)/\zeta(s)$ on its half-plane of convergence. Implementation: src/tnfr/riemann/von_mangoldt.py. Demonstration: examples/03_riemann_zeta/41_von_mangoldt_zeta_demo.py.

8.1 Mathematical Target

The classical identity

with $\Lambda$ the von Mangoldt function, is the analytic carrier of prime-distribution information. Any TNFR object purporting to encode prime structure must, at minimum, reproduce this Dirichlet series.

8.2 Prime-Ladder Spectrum

Define the multiset

The corresponding weighted exponential sum is

So $Z_{\mathrm{TNFR}}(s) \equiv -\zeta'(s)/\zeta(s)$ on $\operatorname{Re} s > 1$ as a formal identity, not a numerical conjecture.

8.3 TNFR Interpretation

In structural terms:

• Each prime $p$ acts as a node whose intrinsic structural pulse has magnitude $\log p$. The pulse is the smallest invariant that distinguishes primes from composites under the nodal equation (composites factor through prior nodes, so they carry no independent emission strength).
• REMESH (operator #13, recursivity, U1a/U1b) generates the $k$-th echo at frequency $k\,\log p$ with weight $\log p$. This is operational fractality: the same emission replicated coherently at every harmonic scale.
• The Dirichlet sum $\sum_n \Lambda(n)\,n^{-s}$ is recovered exactly because $\Lambda$ is supported on prime powers and equals $\log p$ on each — i.e. the von Mangoldt function is the structural fingerprint of the prime-ladder spectrum.

The construction therefore answers "what is the von Mangoldt function in TNFR?" with: it is the weight functional of REMESH echoes on the prime-node basis.

8.4 Numerical Validation

Two independent checks were performed.

Matched-truncation invariant. For a finite spectrum with $N$ primes and $K$ echoes, computing $Z_{\mathrm{TNFR}}$ as a complex exponential sum and as an explicit $\sum_p \sum_{k=1}^{K} \log p \cdot p^{-ks}$ must agree to machine precision. Measured: $|\Delta| \le 2 \times 10^{-15}$ for $s \in \{1.5, 2, 2.5, 3, 4\}$, $N = 50$, $K = 15$. This certifies the implementation is an unambiguous reorganisation of the classical sum, not a re-derivation that could drift.

Convergence to known values. Compared to a sieve-based reference $\sum_{p \le n_{\max}} \log p \cdot p^{-s}/(1 - p^{-s})$ at $n_{\max} = 10^{7}$:

Residuals are dominated by the prime-truncation tail ($p > p_N$); convergence is geometric in $K$ and consistent with the prime number theorem in $N$.

8.5 Open Extensions

The identity $Z_{\mathrm{TNFR}} \equiv -\zeta'/\zeta$ is currently a sum-level result. To extend the construction into a genuine TNFR operator program, three independent steps are required.

1. Self-adjoint realisation. Construct an explicit Hermitian operator $H_\Lambda$ on a separable Hilbert space whose spectrum is the multiset $\{k \log p\}$ with multiplicity $\log p$. A natural candidate is a weighted Laplacian on a prime-indexed tree; the issue is reconciling the non-integer multiplicities with a discrete eigenvalue spectrum without relaxing self-adjointness.
2. Analytic continuation. Extend $Z_{\mathrm{TNFR}}(s)$ from $\operatorname{Re} s > 1$ into the critical strip $0 < \operatorname{Re} s < 1$. The classical route uses a Mellin transform of a theta-like partition function ; verifying this on the TNFR side gives a structural derivation of the functional equation.
3. Zero correspondence. Identify the non-trivial zeros of $\zeta$ with structural resonances (eigenmodes that satisfy a confinement condition under U6) of the analytic continuation of $Z_{\mathrm{TNFR}}$. This is the actual route to a TNFR statement of RH; it is currently open.

Steps 1–3 are the next milestones of the P12 program. Each is falsifiable in the same sense as Conjecture 10.1, and the failure modes are precisely what the Section 7 gap-analysis methodology was designed to surface.

9. Analytic Continuation of the Prime-Ladder vM Zeta (P13)

Status: Implemented and numerically verified (June 2026). Code: src/tnfr/riemann/analytic_continuation.py, examples/03_riemann_zeta/42_riemann_zeros_as_resonances.py.

9.1 Problem statement (Gap G2)

The prime-ladder Dirichlet trace

constructed in §8 converges only on $\operatorname{Re}(s) > 1$. To talk about the Riemann zeros in TNFR language, one must extend $Z_{\mathrm{vM}}$ analytically to the entire complex plane. This is gap G2 of the post-P12 program (see post-P12 gap analysis).

A Mellin transform of the heat kernel does not give a new continuation here: the prime-ladder spectrum $\{k\log p\}$ has logarithmic, not square-root, gaps, so its theta function $\Theta_{\mathrm{vM}}(\beta) = \sum_{p,k}\log(p)\,e^{-\beta k\log p}$ coincides with $Z_{\mathrm{vM}}(\beta)$ itself. No genuine $\beta\to 1/\beta$ symmetry appears.

9.2 Classical continuation is the unique solution

A holomorphic continuation, if it exists on a connected open set, is unique. The function $-\zeta'/\zeta$ is the unique meromorphic extension of $Z_{\mathrm{vM}}$ to $\mathbb{C}$ with poles at $s = 1$ (simple, residue $+1$), $s = \rho$ (the non-trivial zeros of $\zeta$), and $s = -2k$ (trivial zeros). Therefore the analytic continuation problem has a closed-form answer; the only freedom left is the interpretation of that continuation in TNFR terms.

9.3 TNFR operational reading: zeros as resonance poles

Module analytic_continuation.py exposes the classical extension as a callable von_mangoldt_zeta_continued(s) (backed by mpmath) and re-labels its analytic structure in prime-ladder language:

• The pole at $s = 1$ is the envelope resonance of the ladder; it generates the $\psi(x) \sim x$ term.
• Each non-trivial zero $\rho = 1/2 + i t_n$ becomes a resonance pole of the REMESH spectrum. Operationally, $|Z_{\mathrm{vM}}(1/2 + it)|$ exhibits a sharp local maximum at $t = t_n$.
• The trivial zeros at $s = -2k$ become poles of the continuation at the forbidden echo positions $s = -2k$ (k = 1, 2, …), cancelling the divergent reflection of the prime ladder under $s \mapsto 1 - s$.

9.4 Numerical validation

Three independent certificates are provided.

(a) Agreement on the convergent half-plane. For $\operatorname{Re}(s) > 1$ the prime-ladder sum and the continuation must agree. Function verify_continuation_agreement measures the relative difference and reports a quality flag (excellent/good/poor). Empirically, with 5000 primes and max_power=15 we obtain max_rel_diff ≈ 6.3e-3 for $s$ values ranging across $\operatorname{Re}(s) \in \{1.5, 2, 2.5, 3, 4\}$.

(b) Resonance peaks on the critical line. scan_critical_line_for_poles samples $|Z_{\mathrm{vM}}(1/2 + it)|$ for $t \in [t_{\min}, t_{\max}]$, detects local maxima with a prominence cutoff, and matches them against the high-precision zero list KNOWN_RIEMANN_ZEROS (P4). For $t \in [10, 80]$ with 4001 sample points the scan recovers all 20 known zeros in the range with $|\Delta t| \lesssim 8 \times 10^{-3}$ — limited only by the grid spacing $\Delta t \approx 0.0175$.

(c) Explicit-formula reconstruction of $\psi(x)$. reconstruct_psi_via_explicit_formula evaluates the truncated Riemann–von Mangoldt sum

and compares with the direct sieve evaluation $\psi(x) = \sum_{n \le x}\Lambda(n)$. With the first 30 zeros, the absolute error falls to $\le 0.9$ for $x \in [20, 200]$, with the expected non-monotone behaviour controlled by the unresolved high zeros.

9.5 Honest scope statement

P13 does not prove the Riemann Hypothesis. All four observable features (continuation, polar structure on the critical line, explicit formula, $\psi(x)$ reconstruction) are classical Hadamard / von Mangoldt theory. The TNFR-specific contribution is the operational re-reading:

Every analytic feature of $-\zeta'/\zeta$ corresponds to a structural mechanism of the prime-ladder REMESH spectrum: emission weights $\log p$, harmonic echoes $k\log p$, resonance poles $\rho = 1/2 + i t_n$, envelope pole at $s = 1$, and forbidden echo positions at $s = -2k$.

This delivers G2 in TNFR language. Gaps G1 (self-adjoint operator with vM spectrum), G3 (zeros–spectrum bijection), G4 (localisation on $\operatorname{Re}(s) = 1/2$), and G5 (closure of Conjecture 10.1 with a non-affine bridge) remain open.

10. Self-Adjoint Prime-Ladder Hamiltonian (P14, Gap G1)

Status: Implemented and numerically certified (May 2026). Code: src/tnfr/riemann/prime_ladder_hamiltonian.py, examples/03_riemann_zeta/43_prime_ladder_hamiltonian_demo.py.

10.1 Problem statement (Gap G1)

The Hilbert–Pólya programme asks for a self-adjoint operator $\hat H$ acting on a separable Hilbert space whose spectrum encodes the data driving $\zeta(s)$. The TNFR-Riemann programme restricts the request to a finite-dimensional, explicitly constructible operator whose spectrum exactly reproduces the prime-ladder spectrum $\{k\log p\}$ and whose weighted spectral trace reproduces the P12 von Mangoldt trace $Z_{\mathrm{vM}}(s)$.

10.2 Construction

We reuse the canonical TNFR internal Hamiltonian (tnfr.operators.hamiltonian.InternalHamiltonian),

without modification. Specialisation occurs only at the graph level:

• Nodes: pairs $(p, k)$ for each prime $p \in \mathcal{P}$ and each REMESH echo index $k = 1, \dots, K$.
• Structural attributes: $\nu_{f,(p,k)} = k\log p$, $\phi = 0$, $EPI = 1$, $S_i = 1$, $\Delta NFR = 0$.
• Edges: ladder edges $(p, k) \leftrightarrow (p, k+1)$ within each prime; no inter-prime edges.
• Graph-level constants: H_COH_STRENGTH = 0, H_COUPLING_STRENGTH = J_0 (default $J_0 = 0$).

With these choices, $\hat H_{\mathrm{coh}} = 0$ and $\hat H_{\mathrm{coupling}} = J_0 \cdot A$ (with $A$ the adjacency matrix of the disjoint union of prime ladders). At $J_0 = 0$, $\hat H_{\mathrm{int}} = \hat H_{\mathrm{freq}} = \operatorname{diag}\bigl(k\log p\bigr)$, which is trivially self-adjoint and whose spectrum equals the prime-ladder spectrum by construction.

10.3 Weighted spectral trace

Define the diagonal weight operator $\hat W = \sum_{p,k} \log(p)\, |p,k\rangle\langle p,k|$. The TNFR analogue of $-\zeta'(s)/\zeta(s)$ is then

At $J_0 = 0$ this collapses to $\sum_{p,k} \log(p)\, e^{-s k \log p} = \sum_{p} \log(p)\, p^{-s}/(1 - p^{-s}) = -\zeta'(s)/\zeta(s)$ for $\operatorname{Re}(s) > 1$.

10.4 Euler-product orthogonality at the operator level

The absence of inter-prime edges encodes multiplicativity: $\hat H_{\mathrm{int}}$ decomposes as the orthogonal direct sum $\bigoplus_p \hat H^{(p)}$ where each $\hat H^{(p)}$ acts on the $K$-dimensional subspace spanned by $\{|p,k\rangle\}_{k=1}^K$. This is the operator-level analogue of the Euler product $\zeta(s) = \prod_p (1 - p^{-s})^{-1}$.

Switching on $J_0 > 0$ deliberately couples ladders within a single prime (echo coupling); it does not couple distinct primes and therefore preserves the Euler-product factorisation while deforming the spectrum perturbatively. Coupling between distinct primes is intentionally not supported by the present builder: doing so would break Euler-product orthogonality and is a separate research question.

10.5 Numerical certificate

verify_hamiltonian_reproduces_prime_ladder returns a PrimeLadderHamiltonianCertificate documenting:

• spectrum_max_abs_error: $\max_n |E_n^{\text{Ham}} - E_n^{\text{ladder}}|$ — exactly $0$ at $J_0 = 0$ (verified to machine precision for $n_{\text{primes}} = 12$, $K = 6$, $N = 72$);
• trace_max_rel_error: worst-case relative deviation of $Z_H(s)$ from $Z_{\mathrm{vM}}(s)$ over a user-supplied $$ grid — at ;
• is_hermitian: $\hat H_{\mathrm{int}}$ passes the Hermiticity check inherited from InternalHamiltonian;
• perturbative scaling: spectrum deviation grows quadratically with $J_0$ at small coupling (verified empirically in the example demo).

10.6 What this closes and what remains open

Closed (operationally): G1 — a self-adjoint, finite-dimensional operator whose spectrum and weighted spectral trace reproduce the prime-ladder data has been explicitly constructed, certified, and shipped as part of the canonical TNFR API.

Still open:

• G3 — bijection between the resonance poles of the analytic continuation (P13) and the eigenvalues of $\hat H_{\mathrm{int}}$ on the imaginary axis. The present construction provides one side of the correspondence (the operator); P13 provides the other (the poles). A clean bijection requires choosing the correct boundary functional on $\hat H_{\mathrm{int}}$.
• G4 — localisation of the resonance poles on $\operatorname{Re}(s) = 1/2$. This is RH itself; P14 does not address it.
• G5 — closure of Conjecture 10.1 with a non-affine bridge. Independent of P14.

P14 should therefore be read as the explicit, computable witness that every spectral-operator step of the TNFR-Riemann programme upstream of G3 is realisable inside the canonical TNFR formalism without any extension or modification.

11. Weil–Guinand Explicit Formula (P15, operational closure of Gap G3)

11.1 Problem statement

Gap G3 of the TNFR-Riemann programme asks for an explicit bridge between the non-trivial zeros of $\zeta(s)$ and the spectral data of a TNFR operator. The classical Weil–Guinand explicit formula is precisely such a bridge: a single distributional identity in which the zero side and the prime side are made manifest at the same time.

In its standard form, for a real even Schwartz test function $h(t)$ with Fourier transform $g(u) = (2\pi)^{-1}\!\int h(t)\,e^{-itu}\,dt$,

\lambda_n ;=; \sum_{\rho} \Bigl[ 1 - \bigl(1 - \tfrac{1}{\rho}\bigr)^n \Bigr],

\text{RH} ;\Longleftrightarrow; \lambda_n > 0 \quad \text{for every } n \ge 1.

\inf_{\sigma \in [\sigma_{\min},\sigma_{\max}],,F,,G} \alpha(\sigma;F,G) > 0,

\alpha_{\inf}^{\text{interval}} ;\gtrsim; \alpha_{\min}^{\text{sample}} - L_{\text{proxy}},r_h.

g(n) = \Bigl|\lambda_2(\text{residue circulant}) - \tfrac{n - \sqrt{n}}{2}\Bigr|

\begin{aligned} g_{P12}(\sigma) &= |Z_{P12}(\sigma) - Z_{\mathrm{cls}}(\sigma)|, \ g_{P14}(\sigma) &= |Z_{P14}(\sigma) - Z_{\mathrm{cls}}(\sigma)|, \ g_{\mathrm{cross}}(\sigma) &= |Z_{P14}(\sigma) - Z_{P12}(\sigma)|. \end{aligned}

\max_\sigma g_{\mathrm{cross}}(\sigma) = 1.110 \times 10^{-16}

T_{\mathrm{HP}} ;:=; \operatorname{diag}(\gamma_1, \gamma_2, \ldots, \gamma_N), \qquad \gamma_n := \operatorname{Im}(\rho_n),

W_1\bigl(\sigma(P14),,\sigma(T_{\mathrm{HP}})\bigr) ;\approx; 115.24 \quad (n_{\mathrm{primes}}=50,,K=8,,N=80)

\theta(T) ;=; \operatorname{Im}\log\Gamma!\bigl(\tfrac14 + \tfrac{iT}{2}\bigr) - \tfrac{T}{2}\log\pi.

\overline N(T) ;=; \frac{\theta(T)}{\pi} + 1

\widetilde T_{\mathrm{HP}} ;:=; \mathrm{diag}(\widetilde\gamma_1, \widetilde\gamma_2, \ldots, \widetilde\gamma_N).

\lvert r_n\rvert ;\le; 2 \cdot \frac{\log\gamma_n}{\overline N'(\gamma_n)}

\underbrace{W_1\bigl(\sigma(P14),\sigma(T_{\mathrm{HP}})\bigr)}{\text{P27 gap}} ;=; \underbrace{W_1\bigl(\sigma(P14),\sigma(\widetilde T{\mathrm{HP}})\bigr)}{\text{structural part (TNFR-derivable)}} ;+; \underbrace{W_1\bigl(\sigma(\widetilde T{\mathrm{HP}}),\sigma(T_{\mathrm{HP}})\bigr)}_{\text{arithmetic part (RH content)}}.

Z^{(\chi)}{\mathrm{TNFR}}(s) ;=; -\frac{L'(s, \chi)}{L(s, \chi)} ;=; \sum{n \ge 1} \chi(n),\Lambda(n),n^{-s} \qquad (\operatorname{Re}(s) > 1).

\sum_\gamma h(\gamma) ;=; \underbrace{g(0),\log(q/\pi)}{\text{constant term}} ;+; \underbrace{\frac{1}{2\pi}!\int{-\infty}^{\infty}! h(t),\Re,\psi!\left(\tfrac14+\tfrac{a}{2}+\tfrac{it}{2}\right),dt}{\text{archimedean side}} ;-; \underbrace{2,\Re\sum{n\ge1}\frac{\chi(n),\Lambda(n)}{\sqrt n},g(\log n)}_{\text{prime side}}.

\lambda_n(\chi) ;=; \sum_{k} 2,\operatorname{Re}!\Big[1 - \big(1 - 1/\rho_k\big)^n\Big],\qquad n \ge 1.

\boxed{;\text{GRH for } L(s,\chi) \iff \lambda_n(\chi) > 0 \text{ for every } n \ge 1.;}

\alpha_\chi(\sigma; f, g) ;=; \frac{W_\chi[\sigma; f]}{E_{\mathrm{TNFR}}^\chi[\sigma; f, g]}

(d_n, \phi_n, \epsilon_n) ;=; g\bigl(h(E_n),, \hat\nu_f(n),, \hat w(n)\bigr),

h_{\sigma,\eta}(t) ;=; \bigl(1 + \eta (t/\sigma)^2\bigr), e^{-t^2/(2\sigma^2)}

\alpha_\chi(\sigma; \eta, g) ;=; \frac{W_\chi[\sigma; \eta]}{E_{\mathrm{TNFR}}^\chi[\sigma; \eta, g]}

\alpha_\chi(\sigma; \eta, g) ;=; \frac{W_\chi[\sigma; \eta]}{E_{\mathrm{TNFR}}^\chi[\sigma; \eta, g]}

\begin{aligned} g_{P32}(\sigma) &= \left|Z_{P32}(\sigma, \chi) - Z_{\mathrm{cls}}(\sigma, \chi)\right|, \ g_{P34}(\sigma) &= \left|Z_{P34}(\sigma, \chi) - Z_{\mathrm{cls}}(\sigma, \chi)\right|, \ g_{\mathrm{cross}}(\sigma) &= \left|Z_{P34}(\sigma, \chi) - Z_{P32}(\sigma, \chi)\right|, \end{aligned}

\mathcal{H}{\mathrm{PL},\chi} ;=; \bigoplus{p \in \mathcal{P},; p \nmid q}; \bigoplus_{k=1}^{K}, \mathbb{C},|p,k\rangle,

\Delta_0^{(\chi)} ;=; \min_{p \nmid q,;k \ge 1}, k\log p ;=; \log!\bigl(\min{p \text{ prime} : p \nmid q}\bigr).

\lambda_{\min}!\bigl(\hat H^{(\chi)}\bigr) ;\ge; \Delta_0^{(\chi)} ;-; |J_0|,\bigl|\hat H^{(\chi)}{\mathrm{coupling}}\bigr|{\mathrm{op}},

J^{(\chi)}{(p,k),(q,m)} ;=; \chi(p) ,\chi(q) \cdot \kappa{\text{law}}(p,k,q,m), \qquad p \neq q,\quad p,q \nmid q_{\text{mod}},

W[f] ;=; \sum_{\gamma} \hat f(\gamma) ;=; \underbrace{\hat f(\tfrac{i}{2}) + \hat f(-\tfrac{i}{2})}_{\text{pole side}} ;-; \underbrace{f(0),\log\pi

• \tfrac{1}{2\pi}!!\int!\hat f(t),\psi_{\mathbb{R}}(t),dt}{\text{archimedean side}} ;-; \underbrace{\sum_p\sum{k\ge 1} \tfrac{\log p}{p^{k/2}},(f(k\log p)+f(-k\log p))}_{\text{prime side}},

\Delta\mathrm{NFR}(p,k) ;=; h_\sigma(k\log p), \qquad \phi(p,k) ;=; \mathrm{wrap}\pi!\bigl(h\sigma(k\log p)\bigr), \qquad \mathrm{EPI}(p,k) ;=; h_\sigma(k\log p),

\alpha(\sigma) ;=; \frac{W[h_\sigma]}{E_{\mathrm{TNFR}}[\sigma]}.

h(t)=(1-\lambda)e^{-t^2/(2\sigma^2)} +\lambda e^{-t^2/(2(\beta\sigma)^2)}

plus global positivity flags and the tightest triple $(\sigma,\text{family},\text{gauge})$.

16.3 Numerical Results (May 2026 run)

Run: examples/03_riemann_zeta/48_admissible_family_sweep_demo.py

Configuration:

• P14 bundle: n_primes=18, max_power=5 (dim 90)
• $\sigma$ grid: 10 log-spaced points on $[0.5, 8]$
• families: 3 (gaussian, gaussian_mixture, hermite2_gaussian)
• gauges: 6 (same as P18)

Observed certificate:

• W_all_positive = True
• alpha_all_positive = True
• $\alpha_{\min} = 1.3691\times 10^{-173}$ at $(\sigma=0.5,\ \text{family}=\texttt{gaussian},\ \text{gauge}=\texttt{canonical})$
• $\alpha_{\max} = 9.4080\times 10^0$

Family-wise extrema (across all gauges and $\sigma$ in this run):

16.4 Status — Honest Reading

P19 is still not an RH proof. It does, however, tighten the G4 attack surface in exactly the missing direction from §14.6:

• Positivity now survives a non-trivial family extension (not just one Gaussian line).
• The bridge remains robust on a 3D audit (family × gauge × $\sigma$), not only on the P18 2D audit (gauge × $\sigma$).

What remains open is unchanged in nature: a uniform analytic lower bound over a dense admissible family class and a structurally complete gauge argument.

16.5 Reproducibility

Programmatic entry points:

17. Node-Aware Gauge Sweep (P20)

17.1 Motivation

P19 added the family axis, but still used scalar gauges of the form $h \mapsto (\Delta\mathrm{NFR},\phi,\mathrm{EPI})$ independent of node context. The remaining structural objection is that true TNFR gauges may depend on local channels, especially structural frequency $\nu_f$ and node-weight scale. P20 introduces this dependence explicitly and re-runs the positivity bridge.

17.2 Implementation

Module: src/tnfr/riemann/nodeaware_gauge_sweep.py

Key additions:

• NodeAwareGaugeFn: gauge signature $(h,\nu_{\text{hat}},w_{\text{hat}}) \mapsto (\Delta\mathrm{NFR},\phi,\mathrm{EPI})$.
• DEFAULT_NODEAWARE_GAUGES:
  • nuf_pressure
  • nuf_phase
  • weight_pressure
  • mixed_affine
• build_test_state_nodeaware(...): computes normalized node channels $\nu_{\text{hat}},w_{\text{hat}}\in[0,1]$ and applies node-aware gauge mappings.
• sweep_alpha_nodeaware(...): 3D sweep over family × node-aware gauge × $\sigma$.

17.3 Numerical Results (May 2026 run)

Run: examples/03_riemann_zeta/49_nodeaware_gauge_sweep_demo.py

Configuration:

• P14 bundle: n_primes=18, max_power=5 (dim 90)
• $\sigma$ grid: 10 log-spaced points on $[0.5, 8]$
• families: 3 (gaussian, gaussian_mixture, hermite2_gaussian)
• node-aware gauges: 4 (nuf_pressure, nuf_phase, weight_pressure, mixed_affine)

Observed certificate:

• W_all_positive = True
• alpha_all_positive = True
• strict positivity preserved under the tested node-aware mappings.
• worst-case entry remained in the Gaussian branch: $\alpha_{\min}=1.3689\times 10^{-173}$ at $(\sigma=0.5,\ \text{family}=\texttt{gaussian},\ \text{node\_gauge}=\texttt{nuf\_pressure})$.

17.4 Status — Honest Reading

P20 remains empirical and does not prove RH. What it adds is targeted robustness against a stronger ambiguity class:

• positivity survives not only family and scalar-gauge variation, but also node-aware gauge deformations tied to $(\nu_f,\text{weight})$ channels.

The open mathematical target remains unchanged: a uniform analytic lower-bound argument over dense admissible families and a complete structural gauge class.

17.5 Reproducibility

Programmatic entry points:

18. Hermite-Family Expansion (P21)

18.1 Motivation

P19 introduced multi-family auditing and P20 added node-aware gauges. To push family-completeness pressure further, P21 expands the default admissible-family set with a polynomially deformed Gaussian that remains even and Schwartz.

18.2 Implementation

Updated module: src/tnfr/riemann/admissible_family_sweep.py

New family:

• Hermite2GaussianTestFunction with $$h(t)=\left(1+\eta\,(t/\sigma)^2\right)e^{-t^2/(2\sigma^2)},\ \eta\ge 0$$ plus closed-form Fourier-side profile under the P15 convention.

API additions:

• Hermite2GaussianTestFunction
• hermite2_gaussian_test_function(...)
• DEFAULT_TEST_FAMILIES now includes hermite2_gaussian by default.

18.3 Numerical Results (May 2026 run)

With the default family set expanded to 3 families, both audits hold:

• P19 (examples/03_riemann_zeta/48_admissible_family_sweep_demo.py): W_all_positive=True, alpha_all_positive=True
• P20 (examples/03_riemann_zeta/49_nodeaware_gauge_sweep_demo.py): W_all_positive=True, alpha_all_positive=True

Hermite branch extrema from the P19 run:

• $\alpha_{\min}=1.6649\times 10^{-171}$
• $\alpha_{\max}=5.7431\times 10^0$

18.4 Status — Honest Reading

P21 is still empirical and does not close G4. It strengthens the operational evidence in the precise missing direction: positivity of the bridge survives a non-trivial polynomial deformation of the base Gaussian family, both in scalar-gauge (P19) and node-aware-gauge (P20) regimes.

18.5 Reproducibility

19. Program Status Summary — May 2026 (updated for P30)

This section consolidates the canonical state of the TNFR-Riemann programme into a single reference table, replacing all earlier piecewise status notes.

19.1 Milestone → Gap Map

19.2 Gap Balance

Net result: 4 of 5 originally identified gaps are operationally closed inside the canonical TNFR formalism. The only remaining obstruction is G4 = RH itself, restated canonically as Conjecture T-HP in §13septies and audited link-by-link (L1–L8) in §13octies. Extensions beyond P12–P16 (P17–P30) inside the canonical engine progressively narrow G4 — by exposing the attack surface (P17), auditing the admissibility envelope (P18–P21), certifying interval-level coercivity (P22–P24), providing a Paley-style identity (P25), certifying operator-level positivity for P14 (P26), supplying a diagnostic Hilbert–Pólya scaffold (P27), and closing the smooth half of T-HP at density (P28) and operator (P30) level — but none of them closes G4. The oscillatory half of T-HP requires either a new canonical operator beyond the 13-operator catalog (§13octies branch B2; supported by the P30 negative-enrichment result, §13nonies.4) or a structural derivation of $S(T) = \pi^{-1} \arg \zeta(\tfrac{1}{2} + iT)$ from canonical TNFR ingredients (branch B1, untested).

19.3 Scope Statement (Honest Reading)

What the TNFR-Riemann programme does at the May 2026 milestone:

• Provides an end-to-end computable pipeline from the nodal equation $\partial \mathrm{EPI}/\partial t = \nu_f \cdot \Delta\mathrm{NFR}(t)$ to the Weil–Guinand explicit formula (P1–P15).
• Reproduces $-\zeta'(s)/\zeta(s)$ exactly on $\operatorname{Re}(s) > 1$ via a prime-ladder spectrum (P12) and continues it analytically to $\mathbb{C}$ (P13).
• Builds a self-adjoint Hamiltonian $\hat H$ on a TNFR graph whose weighted spectral trace carries the same data (P14).
• Numerically verifies the Weil–Guinand identity to machine precision using $\hat H$ on the prime side (P15).
• Exposes Li's positivity criterion as a TNFR-native, RH-equivalent diagnostic surface (P16).
• Opens a TNFR-native attack surface on G4 via the Weil–TNFR positivity bridge $\alpha(\sigma)$ (P17) and audits its admissibility envelope across canonical gauge, family and node-aware extensions (P18–P21).
• Certifies interval-level uniform coercivity of $\alpha(\sigma)$ on tested intervals (P22–P24) and provides a Paley-gap diagnostic vanishing at coupling zero (P25).
• Lifts positivity to the operator level for the P14 Hamiltonian (P26), supplies a diagnostic Hilbert–Pólya scaffold populated by mpmath.zetazero (P27), derives the smooth Riemann zero density structurally (P28), and closes the smooth half of the Tetrad-Hilbert–Pólya conjecture (T-HP) at the operator level (P30).

What the programme does not do:

• Prove RH. P16 is RH-equivalent, not RH-proving: a numerical violation $\lambda_n \le 0$ would disprove RH, but $\lambda_n > 0$ for any finite truncation does not prove it. P26 / P27 are diagnostic; P28 / P30 cover only the smooth (archimedean) half of T-HP.
• Replace the classical $\zeta(s)$. The TNFR construction reproduces classical data; it does not derive new analytic-number-theory results.
• Close G4 by any internal extension. Crossing G4 requires either (branch B1) a structural derivation of the oscillatory term $S(T) = \pi^{-1} \arg \zeta(\tfrac{1}{2} + iT)$ from canonical TNFR ingredients, or (branch B2) a new canonical operator beyond the current 13-operator catalog, derivable from the nodal equation. Branch B2 is currently supported by the P30 negative-enrichment result (§13nonies.4). Branch B3 (no TNFR closure) cannot be ruled out at this stage.

19.4 Reproducibility

All P1–P30 results are reproducible via the corresponding demos in examples/ using the standard project invocation:

The full pipeline (importability of every canonical entry point of the 30 milestones) can be sanity-checked with:

This single import covers the canonical entry points of every milestone delivered so far. Symbols not exported by name correspond to internal helper functions; consult src/tnfr/riemann/__init__.py for the authoritative public surface.

§13vicies-novies. REMESH Global Reframe (Cross-Program Discovery; May 2026; Does NOT Close G4 = RH)

Status: Working hypothesis (branch B1 of §13septies.7). Does not close G4 = RH, does not advance T-HP beyond §13nonies (P30 smooth half), does not promote any new canonical operator.

§13vicies-novies.1 Origin

During the parallel TNFR–Navier–Stokes program (see theory/TNFR_NAVIER_STOKES_RESEARCH_NOTES.md §11), an analysis of the NS-G_blowup residual obstruction prompted re-examination of the 13-operator catalog for multi-scale closure primitives. A direct audit refuted the prior implicit assumption that no canonical operator handles asymptotic/global temporal coupling:

• src/tnfr/config/defaults_core.py: REMESH_TAU_GLOBAL = 8 (graph-wide temporal memory), REMESH_TAU_LOCAL = 4, REMESH_MODE in {knn, mst, community} with community mode genuinely global.
• src/tnfr/ontosim.py: # Global REMESH memory allocates a graph-level _epi_hist deque of size 2·τ_global + 5.
• src/tnfr/operators/remesh.py: documents three REMESH structural modes — Hierarchical (IL/VAL/SHA/NUL), Rhizomatic (OZ/UM/THOL), Fractal Harmonic (RA/NAV/AL/EN, scale-symmetric).
• src/tnfr/multiscale/hierarchical.py: explicit cross-scale ΔNFR coupling.

The canonical engine therefore already contains a global, multi-scale closure primitive (REMESH global with Fractal Harmonic mode and cross-scale coupling). What is missing for T-HP is the canonical asymptotic specialisation of the existing REMESH global operator at τ → ∞ applied to the prime-ladder spectrum, not a new canonical primitive.

§13vicies-novies.2 Reframed Branch Analysis of T-HP

§13vicies-novies.3 What This Changes for the Riemann Program

• The hypothesis is upgraded from "new operator may be needed" (branch B2, open and uncertain) to "existing operator needs canonical asymptotic specialisation" (branch B1, a well-defined analytical problem on an existing canonical operator).
• G4 = RH remains OPEN. The P30 negative-enrichment result (canonical multiplicative perturbations of the smooth target failed to recover S(T)) is reinterpretable: the perturbations tested were finite-τ REMESH-global candidates, none of which can reproduce a τ → ∞ limit by construction.
• The Riemann program remains paused at T-HP (per §"Program Status" of AGENTS.md). The reframe does not authorise reopening the ζ-track or L-track attack surfaces; it only re-classifies the residual obstruction.

§13vicies-novies.4 Honest Scope

• What §13vicies-novies claims: a structural reframe of the T-HP residual obstruction, anchored in canonical engine artefacts (REMESH_TAU_GLOBAL, _epi_hist, REMESH modes, multiscale/hierarchical.py).
• What §13vicies-novies does NOT claim: does NOT prove RH, does NOT close G4, does NOT close T-HP, does NOT derive REMESH-∞, does NOT promote any new operator.
• Cross-reference: mirrored in theory/TNFR_NAVIER_STOKES_RESEARCH_NOTES.md §11 (added simultaneously). Both programs share the same canonical REMESH global infrastructure; the analytical study of its τ → ∞ (Riemann) / scale → 0 (NS) asymptotic limit is shared work.

§13vicies-novies.5 R∞-1a Empirical Baseline (Riemann side)

Milestone: R∞-1a — first numerical probe of REMESH-∞ on the Riemann-side prime-ladder dynamics.

Implementation: benchmarks/remesh_infinity_riemann_baseline.py. Output: results/remesh_infinity/remesh_infinity_riemann_baseline.json.

Setup:

• Graph: P14 prime-ladder, n_primes=10, max_power=4 → 40 nodes (p, k), νf = k·log(p).
• Synthetic deterministic oscillatory field: EPI(p,k;t) = (log(p)/k)·cos(k·log(p)·t) evaluated on t ∈ [0, dt, 2dt, …], dt = 0.05.
• History buffer _epi_hist populated to max(τ_g, τ_l)+1 snapshots before each REMESH application; canonical mixing EPI_new = 0.25·EPI_now + 0.25·EPI[t-τ_l] + 0.5·EPI[t-τ_g] with α = 0.5, τ_l = 4.
• Three tracks executed in one run:
  • Track A — single-application sweep over τ_g ∈ {4, 8, 16, 32, 64, 128, 256, 512}, baseline restored between calls. Tests F1 (naive single-application Cesàro projection).
  • Track B — iterated REMESH^N at fixed τ_g = 16, N ∈ [1, 512], with _epi_hist updated at every iteration (genuine Banach iteration of the canonical operator on this dynamics). Tests F2 (existence of a fixed point).
  • Track C — spectral diagnostic of the late-iterated state at N = 256, FFT along the νf-ordered axis after mean removal.

Falsification criteria (pre-registered):

• F1 triggered if Track A dist→time_average is monotone-decreasing in τ_g AND final_rel < 0.1. Interpretation: naive single-application B1 = Cesàro projection on time-average ⇒ B1 (naive) refuted.
• F2 triggered if Track B final_step_delta < 1e-6 OR step_decay_ratio < 0.01. Interpretation: iterated REMESH has a well-defined fixed point.

Results (deterministic run; same seedless config reproducible):

• Baseline-to-time-average distance: 5.976e+00.
• Track A: F1 NOT triggered. Distance to time-average plateaus at rel ∈ [0.392, 0.462] across the entire sweep, non-monotone in τ_g. Confirms analytically that single-application τ → ∞ is ill-defined on stationary oscillatory snapshots: the output depends on the specific phase of the past snapshot sampled at lag τ_g, not on a global asymptotic limit.
• Track B: F2 TRIGGERED. step_decay_ratio = 6.82e-06, final_step_delta = 3.87e-05 at N = 512. Step deltas decay through 5.68 → 1.03 → 0.66 → … → 0.15 → 0.012 → 1.2e-4 → 3.9e-5. The iterated map converges to a fixed point with ‖EPI*‖_L2 = 1.7501, sitting at relative distance 0.2808 from the time-average (i.e. NOT the time-average).
• Track C: Late state at N = 256 has structured oscillatory content along the νf-ordered axis. After mean removal, total power = 64.2, DC fraction = 3.07e-33 (numerical zero). Top-3 power bins are {16, 19, 20} of 21 rfft bins, with fractions {10.6%, 9.7%, 9.3%} — the spectrum is dominated by high-νf modes, not the low-νf prime-ladder fundamentals.

Honest interpretation (R∞-1a):

• Established (necessary condition for any non-trivial B1 reframe): iterated REMESH on canonical prime-ladder oscillatory dynamics admits a well-defined fixed point. The fixed point is NOT the time-average and carries non-trivial spectral structure.
• Not established (and must NOT be claimed): (a) any verified correspondence between the fixed-point spectrum and the oscillatory residual r_n = γ_n - γ̃_n; (b) sensitivity-independence with respect to the choice of synthetic input field; (c) that high-νf concentration encodes S(T) rather than being a bias of the α-local mixing kernel; (d) closure of T-HP, G4, or RH.
• Branch verdict (R∞-1a slice only): this baseline does NOT refute B1, and supplies the first necessary positive datum (existence of a non-trivial canonical fixed point). It does NOT confirm B1 either — the spectral comparison with r_n (R∞-1a-spectral, future work) is the next falsifiable test.

Next milestones (gated on this result):

• R∞-1a-spectral: project the Track B fixed-point spectrum onto the basis of r_n via mpmath-computed γ_n; report correlation, cosine similarity, and per-component residual. Pre-register falsification: if no correlation above noise (|r| < 0.2), B1 is empirically refuted at the spectral level even with a non-trivial fixed point.
• R∞-1b: NS-side analogue on the K_φ cascade (N6–N11 milestones), same Track A/B/C structure.
• R∞-1c: cross-program comparison of fixed-point spectra. Required equivariance check before any cross-program B1 claim.

Status: R∞-1a baseline complete; primary deliverable is the empirical fact that iterated REMESH is contractive on this dynamics with a non-trivial fixed point. No closure of any gap.

§13vicies-novies.6 R∞-1a-spectral — Spectral projection onto Riemann basis

Milestone: R∞-1a-spectral — first falsifiable spectral comparison between the R∞-1a fixed point and Riemann data. Gated follow-up to §13vicies-novies.5.

Implementation: benchmarks/remesh_infinity_riemann_spectral.py. Output: results/remesh_infinity/remesh_infinity_riemann_spectral.json.