RESULTS: cutoff/, gap (c), product-ball cutoff against the module cutoff

Worker cutoff/, 2026-09-23, branch teal-sea/weil-c4-s2. Nothing here is a claim about RH. The conclusion on c ∈ [2, 3) is known (Zhu arXiv:2608.24827); this folder only asks which cutoff the C4 trace term can use. Grades follow the AGENTS.md ladder. No kernel-checked statement is made (AXLE not attempted, §7). Every number below is pinned in test_cutoff.py; raw values in cutoff_cells.json.

0. Conventions and sources

S = {∞, 2}, A_S = ℝ × Q_2, Γ = Γ_S = {±2^n}, X_S = A_S/Γ, C_S = (ℝ^* × Q_2^)/Γ, module |x|S = |x∞| · |x_2|_2, K_S = its kernel (≅ Z_2^). The C4 trace terms only see the K_S-invariant sector H := L²(X_S)^{K_S}. As in CCM arXiv:2310.18423 s4:

• w_S : H → L²(ℝ_+^*, d^u) is unitary, w_S(ξ)(u) = u^{1/2} ξ(1 × u) (fundamental domain x_2 ∈ Z_2^, x_∞ = u > 0, where |x|_S = u);
• w_∞(f)(u) = u^{1/2} f(u) on L²(ℝ)^{ev};
• F_μ(w)(s) = ∫ w(u) u^{−is} d^*u, and U_S = F_μ ∘ w_S, U_∞ = F_μ ∘ w_∞;
• the scaling action ϑ(g) of the archimedean ℝ_+^* becomes multiplication by ĝ(s) := F_μ(g)(s) under both U_∞ and U_S.

Windows are centred as in Connes-Consani: g ∈ C_c^∞ with support in [c^{−1/2}, c^{1/2}], so g ∗ g^* lives on [c^{−1}, c] and the only prime power inside is 2. The C4 class also imposes ĝ(±i/2) = ĝ(0) = 0, which removes the pole terms; on it Q = Q_∞ − W_2 with W_2 = √2 log 2 · (the shift form H of §5).

Sources read 2026-09-23 (PDFs in the session scratchpad, not committed): Connes arXiv:math/9811068 s VII (eq. (5), (12), (13), Thm 4, and its proof eq. (27)-(29)); Connes arXiv:2602.04022 s7.1-7.4, eq. (19), (22), footnote 11; Connes-Consani arXiv:2006.13771 Thm 1, Thm 4.7 (eq. (83)-(84)); Connes-Consani-Moscovici arXiv:2310.18423 s4 (Prop 4.1-4.7, Def 4.4-4.5, Thm 4.6). The last one was found during this task and is the key input: it constructs the module Sonin space and relates it to product data.

1. The two cutoffs, exactly (task 1)

1.1 Module cutoff (Connes)

Connes 1999 s VII (12): P_λ^S is multiplication by 1{|x|_S ≤ λ} on L²(X_S); P̂_λ^S = F_S P_λ^S F_S^* with F_S = F_∞ ⊗ F_2 (self-dual, 1_{Z_2} fixed), unitary on L²(X_S) by his Lemma 1(b). It is Γ-invariant because |γ|_S = 1 for γ ∈ Γ. Via w_S it is the plain interval cutoff 1_{(0, λ]}(u) on L²(ℝ_+^*); all the arithmetic sits in F_S. The module Sonin space is S_λ(X_S) = {ξ ∈ H : ξ = 0 and F_S ξ = 0 on |x|_S < λ} (CCM Def 4.5), with orthogonal projection Σ^mod. This is the cutoff of (22): Connes 2026 footnote 11 says the projections there are "defined using the module".

Trace term: T^mod(g) = Tr(ϑ(g) Σ^mod ϑ(g)^*) on H, λ = 1.

Where the module is used in the proof of the trace formula: Connes 1999 eq. (28)-(29). After Parseval the cutoff region {|x| ≤ Λ, |ξ| ≤ Λ} is rewritten with u = xξ as |u|/Λ ≤ |x| ≤ Λ, which depends on x only through |x|. That step is what a product ball does not allow.

1.2 Product-ball cutoff (theory §7.3 item 2)

On L²(A_S) = L²(ℝ) ⊗ L²(Q_2): P = P_λ^∞ ⊗ M_B and P̂ = F P F^* = P̂_λ^∞ ⊗ M̂_B, with M_B multiplication by 1_B for a ball B at 2 and M̂_B = F_2 M_B F_2^*. At ∞ open and closed balls differ by a null set; at 2 they do not, so both are recorded. Exact facts at 2 (radial functions, in the finite model of BallModel, which is the space of radial functions on the finite group 2^{−K}Z_2 / 2^K Z_2 and so exact; K = 3 and 5 pinned):

(ε_n is the indicator of |x|2 = 2^n. The open-ball row is CCM Prop 4.5, rechecked here exactly.) So theory §7.3 item 2 ("1{Z_p} is its own Fourier transform, so time and frequency limiting to Z_p are the same projection") is false as an operator statement. What is true: 1_{Z_2} is the common fixed vector, and the two projections commute with product |1_{Z_2}⟩⟨1_{Z_2}|. Grade: exact.

Not Γ-invariant (exact). x = (3/4, 1) is in the product ball B_1 = {|x_∞| ≤ 1, |x_2|_2 ≤ 1}; 2x = (3/2, 2) is not; |2x|_S = |x|_S = 3/4. So P is a projection on L²(A_S) that does not commute with Γ and defines no operator on L²(X_S). Two exact facts tie it to the module ball: the Γ-saturation of B_1 is the module ball, and the Γ-sum of its indicator is n(x) = ⌊log_2(1/|x|_S)⌋ + 1 on |x|_S ≤ 1 (pinned on five points), a function of the module that is not an indicator, so it is not a cutoff projection on X_S.

1.3 The trace term each reading of "product ball" defines

ϑ(g) acts on L²(A_S) only through a section of C_S → ℝ^* × Q_2^*; the level-0 section u ↦ (u, 1) gives ϑ(g) = ϑ_∞(g) ⊗ 1 on radial functions.

Grade: ordinary argument, unreviewed. The +∞ rows use only that a convolution operator on ℝ_+^* with a nonzero kernel is not Hilbert-Schmidt.

2. What transfers: the Sonin subspace (prove, for reading X)

CCM Prop 4.6-4.7 and Thm 4.6: θ_S(f) := class of σ_2 ⊗ f in L²(X_S) is a bounded bijection with bounded inverse L²(ℝ)^{ev} → H, and it maps the archimedean Sonin space S_λ(ℝ) onto the module Sonin space S_λ(X_S). The class map E_S applied to the tensor product of the local Sonin spaces (open ball at 2, reading A) is θ_S by definition. So after the Γ-quotient the product-ball Sonin space and the module Sonin space are the same closed subspace of H, and the orthogonal projection onto it gives the same trace: T^(X) = T^mod exactly.

Grade: ordinary argument (the identification of reading A's image with θ_S), unreviewed, on CCM Thm 4.6 used as published. What was rechecked here, at S = {∞, 2} with f = exp(−πx²), dps 30, by two routes (route 1: the Γ-sum from the definition, Mellin transform by quadrature; route 2: the local Euler factor times the closed-form archimedean transform ½ π^{−z/2} Γ(z/2), z = 1/2 − is):

(relative deviations; the quadrature of the oscillatory integral limits both columns, worst at the largest s.) Grade: hardened for these identities at these points; Thm 4.6 itself is not rechecked.

3. What does not transfer: the projection (the exact gap)

In the Mellin picture let Ŝ := U_∞(S_1(ℝ)^{ev}) and a(s) := 1 − 2^{−1/2−is}. From CCM (57), (47) and Thm 4.6:

• U_S(module Sonin space) = a · Ŝ;
• U_S(η_S(S_1)) = ā^{−1} · Ŝ, since L_2(1/2 − is) = 1/conj(a(s));
• every trace term above is T_W(g) = Tr(M_ĝ Π_W M_ĝ^) = ‖Π_W M_ĝ^‖²_HS for a closed subspace W, and T_∞ = T_Ŝ.

The product side is an oblique projection onto the module space. By CCM Prop 4.7(iii), ⟨θ_S f, η_S h⟩ = ⟨f, h⟩, so θ_S^{−1} = η_S^. Then Π^obl := θ_S S_∞ η_S^ is an idempotent with range θ_S(S_1) = S_1(X_S), the module Sonin space, and since θ_S commutes with ϑ(g), Tr(ϑ(g) Π^obl ϑ(g)^) = Tr(ϑ_∞(g) S_∞ ϑ_∞(g)^) = T_∞(g). So reading (A) and reading (X) are traces against two projections onto the same subspace: oblique (A) and orthogonal (X).

The same multiplier is the atom on window functions (exact). In the additive variable θ_S g = g − 2^{−1/2} g(· − log 2) (CCM Prop 4.6 proof), so for g on the window ‖θ_S g‖² = (3/2)‖g‖² − √2 ⟨g, H g⟩ with H the shift form of §5. Hence on the window space

Gram(θ_S) = 3/2 − √2 H = 3/2 − W_2 / log 2 (2 < c < 4).

The non-isometry of θ_S is what separates the orthogonal from the oblique projection, and compressed to window functions it is exactly the 2-adic atom. The trace terms see a different compression of the same multiplier, G = Π_Ŝ M_m Π_Ŝ on the Sonin space, which is not computed here. Checked by direct quadrature of Gram(θ_S) (N = 2, the three cells, dps 30) against 3/2 − √2 H (at most 5.9e−31) and against the prime block of weil_trunc/galerkin.py (at most 1.7e−31): hardened, three routes. two_adic/ (commit f1e912d) derived and pinned this identity first; this folder rechecks it by direct quadrature as a third route.

When they coincide. In Mellin form Π^obl = M_a Π_Ŝ M_a^{−1}; it is self-adjoint iff Π_Ŝ commutes with M_m, m = |a|² = 3/2 − √2 cos(s log 2), i.e. iff S_1(ℝ) is invariant under D_2 + D_{1/2} (the unitary dilations by 2 and 1/2). On |x| < 1, D_2 f vanishes but D_{1/2} f(x) = √2 f(2x) vanishes only if f = 0 on |x| < 2; the Fourier side is the same with f̂. So invariance forces S_1 = S_2. Grade: ordinary argument, unreviewed. Whether the Sonin chain is strict between λ = 1 and 2 was not checked here, and operator equality is not necessary for equal traces on window functions: a trace term only sees the diagonal of Π_W, i.e. a distribution τ_W with T_W(g) = ⟨τ_W, g ∗ g^*⟩, and two cutoffs give the same T on window c iff their τ agree on (c^{−1}, c).

Which term fails to transfer, conditionally. Connes-Consani Thm 4.7 gives τ_Ŝ = W_∞ + ε_∞ with ε_∞ (their ε, eq. (84)) a bounded function. The semilocal analogue, obtained by running their argument on (22) with the module projections, would read τ_mod = W_∞ − W_2 + ε_S. It needs the exact finite-cutoff form of (22) at S = {∞, 2} and the spectral decomposition of the module pair (P^S, P̂^S), neither of which exists yet (CCM say the semilocal prolate operator is still a candidate). If it holds,

T^mod(g) − T_∞(g) = −W_2(g ∗ g^) + ⟨ε_S − ε_∞, g ∗ g^⟩,

and on c ∈ (2, 3) the atom at u = 2 cannot be cancelled by a bounded function, so T^mod ≠ T_∞ there: the 2-adic atom is the term that fails to transfer. Grade: derivation, conditional on the unproved semilocal analogue; not checked numerically.

4. The bound (task 2)

Lemma. Let W ⊂ L²(ℝ, ds) be closed, b measurable with 0 < β₋ ≤ |b|² ≤ β₊. Then for every g,

(β₋/β₊) T_W(g) ≤ T_{bW}(g) ≤ (β₊/β₋) T_W(g).

Proof. A := M_b Π_W has closed range bW and A^A = Π_W M_{|b|²} Π_W =: G with β₋ ≤ G ≤ β₊ on W, so Π_{bW} = A G^{−1} A^ lies between β₊^{−1}AA^* and β₋^{−1}AA^*. Conjugating by M_ĝ, which commutes with M_b, and taking traces of positive operators: T_{bW}(g) ≤ β₋^{−1}‖Π_W M_{b̄ ĝ̄}‖²_HS = β₋^{−1} Tr(Π_W M_{|b|²|ĝ|²} Π_W) ≤ (β₊/β₋) T_W(g). The lower bound is the same with the roles swapped. ∎

With b = a (module) and b = ā^{−1} (η), β₊/β₋ = max m / min m for both:

κ = (3/2 + √2)/(3/2 − √2) = (3 + 2√2)² = 17 + 12√2 ≈ 33.97, κ^{−1} T_∞ ≤ T^mod ≤ κ T_∞, κ^{−1} T_∞ ≤ T^η ≤ κ T_∞, so T^mod/T^η ∈ [κ^{−2}, κ²].

Grade: ordinary argument, unreviewed; κ and the range of m exact. Finite-dimensional sanity check (n = 40, random 12-dimensional W, random |b|² ∈ [min m, max m] and ĝ, seed 7, 400 trials): every ratio T_{bW}/T_W lies in [0.8099, 1.1624], well inside [1/κ, κ] (measured). The bound is far from tight on random data; it is what survives the worst case of m.

What the bound does not give: control of the inertia of R_S. The difference T^mod − T_∞ is not of finite rank, so the negative index of Q − T^mod and of Q − T_∞ are not related by it.

5. Consequence for C4: the product-side term is refuted

C4 (theory §7.2) asks for Q(g) ≥ T_S(g) − R_S(g) on the class with R_S of bounded rank. Take T_S = T_∞, reading (A), which is what a T_S built on L²(A_S) with the level-0 section computes. On the class,

R := Q − T_∞ = (W_∞ − T_∞) − W_2 = −⟨ε_∞, g ∗ g^*⟩ − √2 log 2 · ⟨g, H g⟩.

1. The first term is Hilbert-Schmidt on L²(window): Connes-Consani eq. (84), with ξ_n and ζ_n unit vectors (their Prop 4.5) and the prolate values 0 < λ(n) < 1 decaying to 0, gives |ε_∞| ≤ Σ_n λ(n)(1 − λ(n)²)^{−1/2} < ∞, a bounded kernel on a compact square. (Read off their formula, used as published.)
2. H is the shift form v ↦ Re ∫ f(y) conj f(y − log 2) dy on L²[0, ℓ]. For log 2 < ℓ < log 4 the collars I₁ = [0, ℓ − log 2] and I₂ = [log 2, ℓ] are disjoint, the shift is unitary from L²(I₁) onto L²(I₂), and H acts as ½[[0, S^*], [S, 0]] there and as 0 on the middle interval. Its spectrum is {−1/2, 0, 1/2}, each of infinite multiplicity (exact argument).
3. By Weyl's theorem the essential spectrum of R contains −(√2/2) log 2 = −0.4901. So R has infinite negative index, also on the finite-codimension C4 class. No finite-rank R_S can make the inequality hold.

Candidate refuted (ALIGNMENT s5, disposition 1): C4 with the product-side trace term, on every window c ∈ (2, 3). Grade: ordinary argument, unreviewed; its inputs are Connes-Consani Thm 4.7 (used as published) and the assumption that Q and T_∞ are in the same normalization (the one C4 itself presupposes). The prime atom enters Q and is absorbed by no trace term: this is the arithmetic failure of reading (A).

Galerkin measurement (dps 40, shared basis). H on the mission cells; two routes to the matrix (the closed form in shift_form_matrix, and the prime block of weil_trunc/galerkin.py) agree to at most 5.0e−40 entrywise at N = 8 and 16 (N = 32 not run through the second route). The negative directions of −W_2 are the eigenvalues of H above 1/4:

The counts grow with N at every c and track the collar fraction within 1 (measured; counts unchanged at dps 25 for N = 16). Prediction for checker/, not a measurement: if two_adic/'s T_S reduces to kernel/'s T_inf_matrix on the ζ data, the negative inertia of R_S = Q − T_S will grow with N at every cell at about this rate, up to a bounded number coming from the Hilbert-Schmidt part.

6. Matrix comparison on the cells (task 3): not computed

T^mod − T_∞ on the shared basis needs an orthonormal basis of a truncation of Ŝ, which is kernel/'s gap (a) and was not delivered in this box (kernel/ held only its brief). The cost once it exists: with J Sonin vectors φ_j, one J × J Gram matrix G_{jk} = ⟨φ_j, m φ_k⟩ (m acts as 3/2 − (D_2 + D_{1/2})/√2, so each φ_j is needed on the dilated range [c^{−1/2}/2, 2c^{1/2}]), one inversion, and then T^mod_{nm} = Σ_{jk} ⟨U_n-image, a φ_j⟩ (G^{−1})_{jk} ⟨a φ_k, U_m-image⟩ against T_∞'s Σ_j ⟨·, φ_j⟩⟨φ_j, ·⟩. That is the same order of work as T_∞ itself plus 2J² extra inner products per cell, so no CI proposal is needed; it is a follow-up for whoever holds S_∞.

7. Lean / AXLE (task 4): not attempted

The only clean lemma is §4's, and its content is operator theory (ranges of multiplication operators, traces of positive operators) whose Mathlib statement would take longer than the box. The scalar core (3/2 − √2 ≤ 3/2 − √2 cos t ≤ 3/2 + √2 and κ = 17 + 12√2) is pinned exactly by sympy in the tests instead.

8. Grading summary and status

ALIGNMENT s5 status.

• C4 with the product-side trace term (reading A): candidate refuted, every window c ∈ (2, 3), for the stated reason.
• Readings A'', A''', X'' (the theory formula as written): no finite trace term exists, so there is nothing to test.
• C4 with the module trace term T^mod (reading X, the one in (22)): attempt unresolved. Its remainder is −⟨ε_S, ·⟩ if the semilocal analogue holds, and whether that has bounded negative rank is exactly the semilocal prolate problem.
• Nothing here closes C4 as a family, and nothing is claimed about RH.

Original versus known. The module Sonin space, θ_S, η_S and Thm 4.6 are CCM's. Original to this folder: the table of readings (§1.3), the identification of the product side as an oblique projection (§3), the comparison bound (§4) and the refutation of the product-side reading (§5). The window identity Gram(θ_S) = 3/2 − W_2/log 2 is two_adic/'s (f1e912d), rechecked here. Novelty was not searched.

9. Reproduction

PYTHONPATH=$PWD .venv/bin/python hunts/weil_propagation/c4_s2/cutoff/cutoff.py # about 100 s, writes cutoff_cells.json PYTHONPATH=$PWD .venv/bin/python -m pytest -q -n 2 hunts/weil_propagation/c4_s2/cutoff tests/test_hunt_probe_discipline.py tests/test_docs_numbering.py