examples: the capacity arm carries von Mangoldt (148)

Intent: answer the user's key direction -- which arm of the dual-lever (ex 147)
carries the arithmetic difficulty (the Riemann zeros), and why is the per-node
substrate blind to it? Uses the grammar/dual-lever lens to LOCATE the Riemann
wall on a specific axis. Builds on 146 (kernel) + 147 (free-monoid gradings).
Operators: none modified. Affected invariants: #1 Nodal Equation Integrity,
#4 Grammar Compliance (characterization).

Measured:
- M1: the CAPACITY arm log n = sum_{d|n} Lambda(d) exactly (residual ~1e-16;
  Mobius-inverse Lambda=mu*log), so the size grading log (nu_f, ex 147) IS the
  von Mangoldt divisor-sum, and psi(x)=sum Lambda (the Chebyshev staircase
  carrying S(T), ex 96) is the capacity arm's summatory.
- M2: the Riemann ZEROS are the POLES of the capacity series -- -zeta'/zeta(s)=
  sum Lambda(n) n^-s (P12) is a simple pole (residue 1) at rho_1=0.5+14.1347i,
  |-zeta'/zeta(rho_1+eps)|~1/eps measured 9.6/49.6/249.6 at eps=0.1/0.02/0.004;
  sum Omega(n) n^-s = zeta(s)*P(s) has zeta in the NUMERATOR so the zeros are
  invisible to the pressure arm.
- M3: the pressure arm Omega is smooth (Erdos-Kac Gaussian CLT, spread ~1.13);
  the per-node substrate encodes pressure (Phi_s<-dNFR<-Omega), so it is
  structurally BLIND to the capacity/von-Mangoldt arm where the zeros live --
  the same Fix(G)^perp blindness of ex 103/116/120, now LOCATED on the dual-
  lever axis (S(T) in ker(R_inf) cap Fix(S_n)^perp = the capacity oscillatory half).

Paradigm insight: the dual-lever LOCATES the Riemann difficulty on the capacity
(nu_f/log) axis the substrate does not encode -- which is exactly WHY the
substrate is blind (it encodes pressure dNFR/Omega, the smooth zero-free side).

Honest scope: log=Lambda*1 and -zeta'/zeta=sum Lambda n^-s (poles at the zeros)
are CLASSICAL (P12 is the TNFR prime-ladder form); the NEW part is the dual-lever
localisation + the structural explanation of substrate blindness. Does NOT
advance RH (G4 open, program paused at T-HP); locates the wall on the axis the
substrate omits. Docs: README + AGENTS.md showcase + my-agent.md +
TNFR_NUMBER_THEORY.md section 12.2.

Grammar<->number-theory<->Riemann synergy: 146 (kernel) + 147 (free monoid) +
148 (capacity arm carries the zeros).

Author: TNFR AI Agent

.github/agents/my-agent.md

@@ -1672,6 +1672,7 @@ When adding to grammar documentation:
 - **νf-Embedded Prime Visibility**: [examples/07_number_theory/116_nuf_emergent_prime_visibility.py](examples/07_number_theory/116_nuf_emergent_prime_visibility.py) (the doctrine-faithful pivot: arithmetic enters ONLY through νf on an arithmetic-NEUTRAL Watts–Strogatz graph — no imposed divisibility/GCD edges. MEASURED, 3 seeds, carrier-agnostic Pearson(|diffusion|, injected νf): the binary carriers `prime` and `arbitrary` (a random equal-size set with NO arithmetic meaning) are EQUALLY visible in the structural-diffusion channel (≈0.25–0.34 vs ≈0.27–0.29) ⇒ primality is NOT special; the continuous carriers Ω(n)/log n echo too but more weakly (echo strength tracks the spatial SHARPNESS of the νf contrast, not arithmetic content); the ΔNFR-derived substrate fields Φ_s/J_ΔNFR stay blind (|r|≲0.18) to all carriers; the prime-aligned point-biserial signature (r_pb≈0.25–0.34, Cohen d≈0.6–0.9) COLLAPSES under the prime-shuffled control. MECHANISM = carrier-agnostic νf-as-mobility echo: νf is the per-node diffusivity in EPI+=dt·νf·ΔNFR, so whichever nodes carry high νf develop a distinctive EPI transient; the geometry re-expresses ANY νf contrast you inject, it does NOT discover primes — mirrors example 103 and the REMESH-∞ "universality is structural/operational, not spectral". Surfaced + fixed a real cache bug: Φ_s/ξ_C/J_ΔNFR cache key was blind to ΔNFR (alias-key mismatch in _compute_dependency_hash), contaminating an early run with stale-identical substrate fields; characterization, not a closure)
 - **Primality as Grammatical Inertness**: [examples/07_number_theory/146_primality_grammatical_inertness.py](examples/07_number_theory/146_primality_grammatical_inertness.py) (bridges the grammar thread 139-145 to the number-theory thread 40/100-102, never connected before. The single bridge is the nodal equation itself: every operator/"word" acts on form through ∂EPI/∂t=νf·ΔNFR, and on an arithmetic node ΔNFR is the canonical §4 primality field [ΔNFR(n)=ζ(Ω−1)+η(τ−2)+θ(σ/n−(1+1/n)), prime ⟺ ΔNFR=0], so via the dual-lever (ex 37/130) the CAPACITY lever νf is a scalar gain on the PRESSURE ΔNFR. MEASURED at the nodal-equation level (NOT the canonical graph operators, which recompute ΔNFR from neighbours — the arithmetic ΔNFR is a per-node field): (M1) ONE EQUILIBRIUM, THREE READINGS — n prime ⟺ ΔNFR(n)=0 (the theorem) ⟺ local coherence C(n)=1/(1+|ΔNFR|)=1 (maximal); the primes are exactly the zero-pressure, maximal-coherence nodes (0 mismatches in [2,60]). (M2) THE CAPACITY LEVER — PRIMES ARE THE GRAMMATICAL KERNEL: under the nodal flow EPI+=dt·νf·ΔNFR every prime is FROZEN for every νf (12/12 at νf∈{0.5,1,2}), a composite drifts and its drift FACTORS exactly as (νf gain)×(arithmetic pressure) — doubling νf doubles the drift exactly (27/27); the capacity lever scales the RATE but can never move a prime, so primes are the kernel of the whole νf-lever sub-grammar. (M3) THE U2 PRESSURE AXIS — THE GRAMMAR'S CONVERGENCE TARGET IS PRIMALITY: U2 drives ΔNFR→0, i.e. C→1; the maximal-coherence target C=1 IS exactly primality, and C decreases monotonically with Ω (mean C: prime 1.000, Ω=2 0.239, Ω=3 0.130, Ω=4 0.089, Ω=5 0.085 — factorization complexity is structural coherence DEBT); a prime needs the EMPTY word (the identity of the star-free syntactic monoid, ex 145) — it is already at the convergence target. PARADIGM INSIGHT (the user's thesis realized): the grammar dynamics is a LENS that unifies the number-theory module with the operator grammar — primality is grammatical inertness. HONEST SCOPE: primality ⟺ ΔNFR=0 is the existing §4 theorem; the NEW content is the grammar-lens reading (dual-lever kernel + U2 target + monoid identity); the bridge lives at the nodal-equation level (per-node arithmetic ΔNFR, not graph diffusion); restates the theorem through the grammar dynamics, not new number theory, closes no open problem)
 - **Numbers as Words / the Dual-Lever as Monoid Gradings**: [examples/07_number_theory/147_numbers_as_free_monoid_words.py](examples/07_number_theory/147_numbers_as_free_monoid_words.py) (deepens 146 to its algebraic core, uniting physics + grammar + number theory in one statement. By the Fundamental Theorem of Arithmetic the multiplicative monoid (ℕ,×) is the FREE COMMUTATIVE MONOID on the primes — numbers ARE words: primes=letters, 1=empty word, Ω(n)=word length, multiplication=concatenation; this is the arithmetic counterpart of the operator grammar's syntactic monoid (ex 145). MEASURED: (M1) THE COHERENCE DEBT SPLITS BY COMPOSITION LAW — the three ΔNFR pressure channels (TNFR_NUMBER_THEORY §4) are distinguished by how they compose under ×: the factorization channel ζ(Ω−1) is ADDITIVE (Ω completely additive ⇒ a monoid homomorphism; P_Ω(mn)=P_Ω(m)+P_Ω(n)+ζ exact, residual 0 — the free-monoid/word-length backbone), while the divisor η(τ−2) and abundance θ(σ/n−(1+1/n)) channels are MULTIPLICATIVE (τ,σ multiplicative on coprime — the divisor lattice); so ΔNFR = 1 ADDITIVE channel (Ω) + 2 MULTIPLICATIVE channels (τ,σ). (M2) MULTIPLYING BY A PRIME = THE UNIT DESTABILIZER — building 1→2→6→30→210 one prime at a time adds exactly ζ to the factorization channel each step (C drops 1.0→0.21→0.096→0.049), and the additive channel ALONE detects primality (Ω(n)=1 ⟺ n prime, 0 mismatches in [2,80]; the §4 theorem is 3× redundant — each channel detects primality — but only Ω is the clean free-monoid backbone; primes = single letters Ω=1, 1 = empty word Ω=0). (M3) THE DUAL-LEVER = THE TWO ADDITIVE GRADINGS — the free commutative monoid on primes has two canonical additive gradings, and they are exactly the two arms of the dual-lever (ex 37/130) restricted to arithmetic: COUNT Ω (Σ exponents → the ΔNFR factorization pressure channel) and SIZE log n (Σ eₚ·log p → the νf capacity, ex 94 atom νf=log p), both exact monoid homomorphisms (ℕ,×)→(ℝ,+); Ω asks how MANY prime letters (pressure arm), log asks how BIG the word is (capacity arm). PARADIGM INSIGHT: one algebraic statement fixes the dictionary physics dual-lever ↔ free-monoid gradings ↔ primality across three modules — the synergy the user expected to be key. HONEST SCOPE: Ω-additive, τ/σ-multiplicative, primes = irreducible generators of (ℕ,×), and the FTA free-monoid structure are CLASSICAL facts; the NEW content is the TNFR-lens reading (the three ΔNFR channels split by composition law, the additive channel = the primality-bearing backbone, the dual-lever = the two gradings); restates classical multiplicative number theory through the grammar/dual-lever lens, not new number theory, closes no open problem)
+- **The Capacity Arm Carries von Mangoldt — the Riemann Zeros Live Where the Substrate Is Blind**: [examples/07_number_theory/148_capacity_arm_carries_von_mangoldt.py](examples/07_number_theory/148_capacity_arm_carries_von_mangoldt.py) (answers the user's key question through the grammar/dual-lever lens: WHICH arm of the dual-lever (ex 147) carries the arithmetic difficulty — the Riemann zeros — and why is the per-node substrate blind to it? MEASURED: (M1) THE CAPACITY ARM IS THE VON MANGOLDT SUM — log n = Σ_{d|n} Λ(d) exactly (residual ~1e-16 over [2,200]; Möbius-inverse Λ=μ*log), so the SIZE grading log (the νf arm, ex 94/147) IS the divisor-sum of von Mangoldt, and ψ(x)=Σ Λ — the Chebyshev staircase carrying S(T)=(1/π)arg ζ(½+iT), the sole open obstruction of the TNFR-Riemann program (ex 96) — is the capacity arm's summatory. (M2) THE ZEROS ARE THE POLES OF THE CAPACITY SERIES — −ζ'/ζ(s)=Σ Λ(n)n⁻ˢ (P12) blows up as a simple pole (residue 1) at ρ₁=½+14.1347i: |−ζ'/ζ(ρ₁+ε)|≈1/ε measured 9.6/49.6/249.6 at ε=0.1/0.02/0.004; by contrast Σ Ω(n)n⁻ˢ=ζ(s)·P(s) has ζ in the NUMERATOR so a zero of ζ is a ZERO of the Ω series — the PRESSURE arm does not see the zeros as poles. (M3) THE PRESSURE ARM IS SMOOTH; THE SUBSTRATE ENCODES IT, HENCE BLIND — Ω obeys the Erdős–Kac Gaussian CLT ((Ω−loglog n)/√(loglog n), spread ≈1.13 over [3,10⁵]; slow convergence because loglog n≈2.4 is tiny, but a CLT not a zero-driven oscillation); the per-node substrate encodes the pressure arm (Φ_s ← ΔNFR ← Ω), so it is structurally BLIND to the capacity/von-Mangoldt arm where the zeros live — the SAME Fix(G)^⊥ blindness measured in ex 103/116/120, S(T)∈ker(R∞)∩Fix(S_n)^⊥ being the capacity arm's oscillatory half. PARADIGM INSIGHT (the user's direction realized): the dual-lever LOCATES the Riemann difficulty — it lives on the capacity (νf/log) axis the per-node substrate does not encode, which is exactly WHY the substrate is blind (it encodes pressure ΔNFR/Ω, the smooth zero-free side). HONEST SCOPE: log=Λ*1, Λ=μ*log, and −ζ'/ζ=Σ Λ n⁻ˢ with poles at the zeros are CLASSICAL analytic number theory (P12 is the TNFR prime-ladder form); the NEW content is the dual-lever localisation of the zeros on the capacity arm + the structural explanation of substrate blindness; does NOT prove or advance RH (G4 stays open, S(T)∈Fix(S_n)^⊥ unreachable, program PAUSED at T-HP), it locates the wall on the dual-lever axis the substrate omits — a sharper statement of where the obstruction lives, not a closure)
 - **Emergent Substrate Meets Riemann**: [examples/08_emergent_geometry/103_emergent_substrate_meets_riemann.py](examples/08_emergent_geometry/103_emergent_substrate_meets_riemann.py) (the symplectic substrate on P14: static graph blind to primes, dynamics-emergent geometry carries {k·log p} with r≈0.99, but re-expresses not adds Riemann structure; characterization, NOT a G4 closure — program stays paused at T-HP)
 - **Navier–Stokes Is Not Riemann**: [examples/08_emergent_geometry/104_navier_stokes_is_not_riemann.py](examples/08_emergent_geometry/104_navier_stokes_is_not_riemann.py) (comparative: substrate BLIND to Riemann (content in νf) but POPULATED for NS (velocity=phase, ω=K_φ, p=Φ_s); NS residual=K_φ cascade is TRANSPORT/native, Riemann residual S(T) is ARITHMETIC/foreign; REMESH-∞ STRUCTURAL_EFFECT vs DEGENERATE; characterization, neither Clay/RH closed)
 - **Attacking Navier–Stokes (Enstrophy Dispersion)**: [examples/08_emergent_geometry/105_navier_stokes_enstrophy_dispersion.py](examples/08_emergent_geometry/105_navier_stokes_enstrophy_dispersion.py) (enstrophy budget dZ/dt=P−D: dissipation D=ν·λ_k is TNFR-native exact (grows ~k²), production P is model input; isolates NS-G_blowup to whether production outgrows native ν·λ_k~k² at scale→0; measured ∫P/∫D decreases with n (1.0→0.73), enstrophy bounded; ISOLATION not closure, Clay OPEN)