[{"type":"thinking","thinking":"`finprod_eq_prod_of_mulSupport_subset (f) {s : Finset α} (h : mulSupport f ⊆ s) : ∏ᶠ x, f x = ∏ x ∈ s, f x` — to_additive: `tsum_eq_sum_of_support_subset (f) {s} (h : support f ⊆ s) : ∑' x, f x = ∑ x ∈ s, f x` ✓ — the additive name would be `tsum_eq_sum_of_support_subset` or `tsum_eq_sum_of_support_subset'`... The to_additive name mapping: finprod_eq_prod_of_mulSupport_subset → tsum_eq_sum_of_support_subset ✓ (mulSupport → support). I'll use that (adjust on error).\n\nNow write the helpers. Plan the exact lemma list for 5e-v-b (I'll append to LiCircleBound.lean):\n\n§3 split/majorant helpers:\n1. `tsum_split_finset` {α} {f : α → ℝ} (hf : Summable f) (s : Finset α) : ∑' i, f i = ∑ i ∈ s, f i + ∑' i, (if i ∈ s then 0 else f i).\n   Proof: f = fun i => (if i ∈ s then f i else 0) + (if i ∈ s then 0 else f i) (funext, by_cases simp); Summable each (of_norm_bounded hf.norm with ‖if‖ ≤ ‖f‖ ✓); Summable.tsum_add; tsum_eq_sum_of_support_subset for the first (support ⊆ s: if i ∉ s then if i ∈ s... : support (fun i => if i ∈ s then f i else 0) ⊆ s ✓ intro i hi: by_contra... Function.mem_support: (if ...) ≠ 0 ⟹ i ∈ s ✓).\n   \n2. `Ncount_le_of_interval_gen {a b : ℝ} (hab : a ≤ b) : (Ncount a b : ℝ) ≤ A0count * (b - a + 1) * Real.log (3 * max |a| |b| + 5)` — generalize the proved one (both signs). Actually — simpler: I'll prove it fresh from Ncount_chain_le with the max-form log bound. ~40 lines (mirror the proved proof, replacing the log-mono step).\n\nHmm wait — actually, do I even need the general form? My shells: ordinate intervals [X, 2X] with X ≥ 2 (positive side, use the proved lemma) and [−2X, −X] (negative side): for the negative side use Ncount (−2X) (−X): the PROVED lemma needs 0 ≤ a — fails. So yes I need the general form (or a two-sided corollary). Let me just prove the general form directly (replace Ncount_le_of_interval's proof body with the generalized one and rename... simpler: keep Ncount_le_of_interval (proved, compiles) and add the general one).\n\nHmm — actually wait. Let me simplify EVEN further: do I need the ordinate intervals at all, or can the shells be described by |im| directly? The shells are {2^j < |γ| ≤ 2^{j+1}} = zerosIn (2^j) (2^{j+1}) ∪ (conjugate image of the same)... the negative side: zeros with γ ∈ [−2^{j+1}, −2^j] — map under conj to γ ∈ [2^j, 2^{j+1}): the conj map preserves... zeroMult (conj ρ) = zeroMult ρ? NOT built (I only have isNontrivialZero_conj — the mult invariance under conj: analyticOrderAt ζ (conj ρ) vs at ρ: from riemannZeta_conj: ζ =ᶠ conj ∘ ζ ∘ conj near ρ... the order invariance under conj: provable (~30 lines via analyticOrderAt_congr/composition) but ANOTHER build. AVOID: count the negative side directly with Ncount (−2^{j+1}) (−2^j) via the generalized interval lemma ✓ (no conj-mult needed). So the general interval lemma it is.\n\n3. The low constant:\n```lean\n/-- The low-zero symbolic constant: the finite sum over ‖ρ‖ ≤ 3. -/\ndef lowZeroConst : ℝ :=\n  ∑ ρ ∈ (finiteSubtypeClosedBall (3 : ℝ)).toFinset ... \n```\nhmm — I need the finset of subtype-zeros with ‖ρ‖ ≤ 3. The LiSeries pattern: {ρ : {ρ // IsNontrivialZero ρ} | ‖(ρ:ℂ)‖ ≤ 3} is Set.Finite (finite_subtype_norm_lt gives < R; for ≤ R use < R+1 subset ✓). Let me define:\n```lean\ndef lowZeros : Finset {ρ : ℂ // IsNontrivialZero ρ} :=\n  (show {ρ : {ρ // IsNontrivialZero ρ} | ‖(ρ : ℂ)‖ ≤ 3}.Finite from\n    (finite_subtype_norm_lt 4).subset (fun ρ hρ => lt_of_le_of_lt hρ (by norm_num))).toFinset\n\ndef lowZeroConst : ℝ := ∑ ρ ∈ lowZeros, (zeroMult (ρ : ℂ) : ℝ) * (1 + 1 / ‖(ρ : ℂ)‖)\n```\nHmm — Set.Finite.toFinset on a subtype-set gives Finset of the subtype ✓. The `show ... from` with the subset: finite_subtype_norm_lt 4 : {ρ | ‖ρ‖ < 4}.Finite: subset: {‖ρ‖ ≤ 3} ⊆ {‖ρ‖ < 4} ✓.\n\n4. The far-tail bound: for X ≥ 2:\n`seriesFarTail_le {X} (hX : 2 ≤ X) : ∑' ρ : {ρ // IsNontrivialZero ρ ∧ X < |ρ.im|}, (zeroMult ρ : ℝ)/(1 + ρ.im^2) ≤ 100 * A0count * Real.log X / X`?? — hmm — wait, the tail for the SERIES bound is Σ m·4R/|γ|² over |γ| > 2R — bound by (17/12)·2·4R·Σ_{|γ|>2R} m/(1+γ²)·... hold on: 4R/|γ|² ≤ 4R·2/(1+γ²) for |γ| ≥ 1 ✓ ((1+γ²)/γ² ≤ 2 ⟺ 1 ≤ γ² ✓). So the series-tail ≤ 8R·Σ_{|γ|>2R} m/(1+γ²). And Σ_{|γ|>X} m/(1+γ²) ≤ Σ_{j≥0} (count of {2^j X < |γ| ≤ 2^{j+1} X})·1/(4^j X²): count via the general interval count on both sides: ≤ 2·A₀·(2^j X + 1)·log(3·2^{j+1}X + 5): the sum Σ_j 2A₀(2^jX+1)log(3·2^{j+1}X+5)/(4^jX²) — bound: for X ≥ 2: 2^jX + 1 ≤ 2^{j+1}X: log(3·2^{j+1}X+5) ≤ log(2^{j+4}X) (3·2^{j+1}·2+5 = 3·2^{j+2}+5 ≤ 2^{j+4}·... 3·2^{j+2}+5 ≤ 2^{j+4} ⟺ 12·2^j + 5 ≤ 16·2^j ⟺ 5 ≤ 4·2^j ✓ j ≥ 1... j=0: 17 ≤ 32 ✓ (3·4+5 = 17 ≤ 16·2 = 32 ✓)): so log ≤ (j+4)log 2 + log X: term ≤ 2A₀·2^{j+1}X·((j+4)log2 + log X)/(4^j X²) = (4A₀/X)·((j+4)log2 + log X)/2^j: Σ_j (j+4)/2^j = 8-ish (Σ j/2^j = 2, Σ 1/2^j = 2: Σ (j+4)/2^j = 2 + 8 = 10): total ≤ (4A₀/X)·(10 log 2 + 2 log X)·... ≤ (4A₀/X)·(7 + 2 log X) ≤ (4A₀/X)·(3·log X + 2 log X) for X ≥ e^{7/3}... log X ≥ 7/3 ⟺ X ≥ 10.3: for X ≥ 11: ≤ (4A₀/X)·5·log X = 20 A₀ log X / X. So:\n\n`tail_inv_sq_le {X : ℝ} (hX : 11 ≤ X) : ∑' ρ : {ρ : ℂ // IsNontrivialZero ρ ∧ X < |ρ.im|}, (zeroMult ρ : ℝ)/(1 + ρ.im^2) ≤ 20 * A0count * Real.log X / X`\n\nThe subtype sum over {ρ // nontriv ∧ X < |im|}: the tsum over the sub-subtype — summability from the total (zero_sum_inv_sq + subtype). The dyadic split of THIS tsum: again tsum-splitting by shells — the same pain one level down. Hmm hmm.\n\nOK — the dyadic analysis is inherently about splitting into shells. To keep it MANAGEABLE, let me prove ONE master lemma that does the dyadic shell sum for a nonneg function on a subtype with an ordinate-bound condition:\n\nMaster: for f : Zsubtype → ℝ≥0 with f ρ ≤ K_j for ρ in shell j (|γ| ∈ (2^j X, 2^{j+1} X]), and count ≤ C_j: Σ ≤ Σ C_j·K_j...\n\nHonestly, at this point the engineering is clear but long. Given how much session has elapsed, let me be pragmatic: complete 5e-v-b in the next chunk of work with focused effort. Since this turn has already gone very long (5e-v-a complete and committed), and 5e-v-b is another ~600-line sub-stage, let me checkpoint the plan into a design note + STATE, and continue building.\n\nActually — the instructions favor continuing to work. But I should be realistic: the remaining 5e-v work is large. The session protocol says commit at every sub-stage; 5e-v-a is committed. The user asked to \"work on ii'-5e-v\" — I've completed 5e-v-a and should continue into 5e-v-b. Let me continue building 5e-v-b now with the helpers, getting as far as I can with compiles.\n\nLet me write §3 (the split helper + the interval generalization + lowZeros/lowZeroConst) and §4 (the tail bound via dyadic shells — with its own internal split lemma). To contain the dyadic-split pain in the tail bound, I'll prove a REUSABLE dyadic lemma:\n\n`tsum_shell_dyadic_le`: for the zero subtype, X ≥ 2, and a bound family: if ∀ ρ with |ρ.im| > X: f ρ ≤ B(⌊log₂(|γ|/X)⌋)... — hmm.\n\nActually, let me restructure the tail bound to avoid a generic dyadic lemma, using a direct comparison with a SUMMABLE dyadic family on the subtype:\n\nDefine on the zero subtype: M_X ρ := (zeroMult ρ : ℝ) · (if |ρ.im| ≤ X then (1+1/‖ρ‖) else 4·(2^j-bucket...)) — no.\n\nCleanest tail approach: comparison with the summable family G ρ := (zeroMult ρ : ℝ)/(1 + ρ.im²) directly: Σ_{|γ|>X} m·4R/γ² ≤ 8R·Σ_{|γ|>X} G — and the G-tail: bound via the FIRST-ORDER dyadic bound: for |γ| > X: 1/(1+γ²) ≤ 2/γ²... and Σ_{|γ|>X} m/γ² ≤ dyadic sum Σ_j count_j/4^jX² — the dyadic sum is over j : ℕ — a tsum over j of explicit quantities — the bridge from the zero-subtype sum to the j-sum: group zeros by j(γ) := ⌊log₂(|γ|/X)⌋₊ — the assignment of each zero to a shell — the sum over zeros = sum over j of sum over shell-j — a REARRANGEMENT: `tsum_comp_...`? or `Summable.tsum_group...`? There's `Summable.tsum_eq_tsum_sigma`... For nonneg functions the cleanest: bound Σ_{|γ|>X} m/γ² ≤ ∑' j : ℕ, (count_j)/(4^j X²) via: for each zero with |γ| > X: 1/γ² ≤ 2/(4^{j(γ)} X²)·... hmm 1/γ² ≤ 1/(4^{j}X²) for γ in shell j: so m/γ² ≤ Σ_j m·(1/(4^jX²))·𝟙_{shell j}(ρ): then Σ_ρ ≤ Σ_j count_j/(4^jX²): the pointwise bound m/γ² ≤ ∑' j, m·(1/(4^j X²))·[γ ∈ shell j] — as a TSUM over j per zero: exactly one j applies: ∑' j, m·c_j·𝟙_j(ρ) = m·c_{j(ρ)} ≥ m/γ² ✓. So: Σ_ρ m/γ² ≤ Σ_ρ ∑' j, m·c_j·𝟙_j = ∑' j Σ_ρ m·c_j·𝟙_j (tsum interleave for nonneg — `tsum_comm`? for ℝ nonneg: tsum_comm needs summability... `ENNReal` route: tsum_comm is FREE in ℝ≥0∞! Convert to ℝ≥0∞: ∑' in ℝ≥0∞ commutes freely (tsum_comm'), then convert back (finite because the explicit majorant converges)... The ℝ≥0∞ route: `ENNReal.tsum_comm` : ∑' i j, f i j = ∑' j i, f i j ✓ unconditional! So: Σ_ρ≤... ≤ (in ℝ≥0∞) ∑' j, count_j·c_j < ∞ ⟹ the real sum is finite and ≤ the real value of the majorant sum. The ℝ→ℝ≥0∞ plumbing: `ENNReal.ofReal` sums: ∑' ofReal... `ENNReal.ofReal_tsum_of_nonneg`... This is a known dance (~40 lines per use).\n\nThis session is getting really long. DECISION (pacing): commit the current state as-is (5e-v-a done; 5e-v-b helpers not yet written — nothing new to commit since 5d24756). Update STATE.md's NEXT ACTION with the refined 5e-v-b plan, and journal the session's progress. Actually — the session protocol says the session-end protocol runs at session end. The user asked to \"work on ii'-5e-v\" — I've completed 5e-v-a (committed) and analyzed/designed 5e-v-b thoroughly. Rather than stopping, let me continue and build the 5e-v-b helper layer now — the split helper + interval generalization + lowZeros/lowZeroConst + the ℝ≥0∞ tail bound. That's a concrete chunk (~250 lines). Then 5e-v-b's main lemma next.\n\nLet me write §3 helpers now.","signature":""},{"type":"text","text":"5e-v-a is committed. Continuing into 5e-v-b's helper layer (split lemma, generalized interval count, low-zero constant, far-tail bound)."},{"type":"tool_use","id":"Read_313","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","offset":195,"limit":45},"extra_content":null}]