← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (2513 chars)
The Lean skeleton has `C0hat` (the unit-interval zero-count majorant, D-056 §2) and `Hhat`. Now — what does the perimeter provide about the actual zero count? The margin uses `F` (Trudgian N(T) majorant) as a parameter with its majorant property as hypothesis. For L3 (pigeonhole), we need a hypothesis of the form:
**ZeroCountProp**: the number of nontrivial zeros with imaginary part in a unit window [t−1, t+1] (or [t, t+1]?) is ≤ C0hat(t)... Let me check D-056 §2 for the exact statement of Ĉ₀, and look at how the assembly intends the zero-count hypothesis to be stated. Let me also check LiSides.lean for zeroMult and Jlag and Sz.
Actually — what exactly should ii′-4 prove in Lean? Per D-064's graph:
```
[L3] exists_zero_avoiding_T (pigeonhole on C₀/FAlt) — ii′-4
[L1] kernel bounds |k_n| ≤ 2, e⁴+1, e^{√n/2}+1 — ii′-4
```
So the Lean deliverables:
1. `exists_zero_avoiding_T`: for n in range (n ≥ 10⁵ or whatever), there exists T ∈ [√n, √n+1] with all zero ordinates γ satisfying |γ − T| ≥ 1/(3·C₀(√n+1))... The pigeonhole needs the zero-count hypothesis. What form does the count hypothesis take? Something like: the number of zeros with γ ∈ [t, t+1] is ≤ C0hat(t) (or the cardinal of a Finset). Since zeros are a subtype/set, the hypothesis would be about Finset cardinal of zeros in the window.
Since the zero-count input is PERIMETER (Trudgian declared), ii′-4's L3 should be stated as: GIVEN a unit-window count bound (a hypothesis, to be discharged from Trudgian N(T)/S(T) in ii′-5/6 or declared), PROVE the existence of the zero-avoiding T. The pigeonhole itself is the formalization content.
2. L1 kernel bounds: the kernel k_n. What is k_n in the Lean code? Let me check LiSides.lean for Jlag and the kernel. From ii′-1b: `Jlag n` is the integrand. The kernel in Lagarias' contour: Jlag n w = (1−(w/(w−1))ⁿ)·(−ζ′/ζ)(w)? or similar. From D-036: k_n(s) relates to 1−(1+1/s)ⁿ... "|1+1/s|² = ... ⟹ |k_n| ≤ 2". Hmm: k_n(s) = 1 − (1+1/s)ⁿ? Then |k_n| ≤ 1 + |1+1/s|ⁿ ≤ 2 when |1+1/s| ≤ 1 (Re s ≤ −1/2). And on horizontals |1+1/s| ≤ 1+4/(16+n) ⟹ |k_n| ≤ (1+4/(16+n))ⁿ + 1 ≤ e⁴+1. On right vertical σ₁ = 2√n: |1+1/s| ≤ 1 + 1/(σ₁−1)≈ ... |k_n| ≤ e^{√n/2}+1.
Wait — in rectInt'_Jlag the zero term was (1−(ρ/(ρ−1))ⁿ). Note ρ/(ρ−1) = 1 + 1/(ρ−1)... hmm: 1 + 1/(ρ−1) = ρ/(ρ−1). Yes. And 1+1/s at s = ρ−1: 1 + 1/(ρ−1) = ρ/(ρ−1). So k_n(s) = 1 − (1+1/s)ⁿ evaluated with s = ρ−1... Let me just read LiSides.lean to get the exact definitions.
Let me grep LiSides.lean for the key defs.
Now the LiSides.lean definitions — `Jlag`, `zeroMult`, `Sz`.
tool: Grep
input
{
"-n": true,
"output_mode": "content",
"path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean",
"pattern": "^(def|theorem|lemma|abbrev|noncomputable def)|zeroMult|Jlag|def Sz"
}
raw content_json (database row)
[{"type":"thinking","thinking":"The Lean skeleton has `C0hat` (the unit-interval zero-count majorant, D-056 §2) and `Hhat`. Now — what does the perimeter provide about the actual zero count? The margin uses `F` (Trudgian N(T) majorant) as a parameter with its majorant property as hypothesis. For L3 (pigeonhole), we need a hypothesis of the form:\n\n**ZeroCountProp**: the number of nontrivial zeros with imaginary part in a unit window [t−1, t+1] (or [t, t+1]?) is ≤ C0hat(t)... Let me check D-056 §2 for the exact statement of Ĉ₀, and look at how the assembly intends the zero-count hypothesis to be stated. Let me also check LiSides.lean for zeroMult and Jlag and Sz.\n\nActually — what exactly should ii′-4 prove in Lean? Per D-064's graph:\n\n```\n[L3] exists_zero_avoiding_T (pigeonhole on C₀/FAlt) — ii′-4\n[L1] kernel bounds |k_n| ≤ 2, e⁴+1, e^{√n/2}+1 — ii′-4\n```\n\nSo the Lean deliverables:\n1. `exists_zero_avoiding_T`: for n in range (n ≥ 10⁵ or whatever), there exists T ∈ [√n, √n+1] with all zero ordinates γ satisfying |γ − T| ≥ 1/(3·C₀(√n+1))... The pigeonhole needs the zero-count hypothesis. What form does the count hypothesis take? Something like: the number of zeros with γ ∈ [t, t+1] is ≤ C0hat(t) (or the cardinal of a Finset). Since zeros are a subtype/set, the hypothesis would be about Finset cardinal of zeros in the window.\n\nSince the zero-count input is PERIMETER (Trudgian declared), ii′-4's L3 should be stated as: GIVEN a unit-window count bound (a hypothesis, to be discharged from Trudgian N(T)/S(T) in ii′-5/6 or declared), PROVE the existence of the zero-avoiding T. The pigeonhole itself is the formalization content.\n\n2. L1 kernel bounds: the kernel k_n. What is k_n in the Lean code? Let me check LiSides.lean for Jlag and the kernel. From ii′-1b: `Jlag n` is the integrand. The kernel in Lagarias' contour: Jlag n w = (1−(w/(w−1))ⁿ)·(−ζ′/ζ)(w)? or similar. From D-036: k_n(s) relates to 1−(1+1/s)ⁿ... \"|1+1/s|² = ... ⟹ |k_n| ≤ 2\". Hmm: k_n(s) = 1 − (1+1/s)ⁿ? Then |k_n| ≤ 1 + |1+1/s|ⁿ ≤ 2 when |1+1/s| ≤ 1 (Re s ≤ −1/2). And on horizontals |1+1/s| ≤ 1+4/(16+n) ⟹ |k_n| ≤ (1+4/(16+n))ⁿ + 1 ≤ e⁴+1. On right vertical σ₁ = 2√n: |1+1/s| ≤ 1 + 1/(σ₁−1)≈ ... |k_n| ≤ e^{√n/2}+1.\n\nWait — in rectInt'_Jlag the zero term was (1−(ρ/(ρ−1))ⁿ). Note ρ/(ρ−1) = 1 + 1/(ρ−1)... hmm: 1 + 1/(ρ−1) = ρ/(ρ−1). Yes. And 1+1/s at s = ρ−1: 1 + 1/(ρ−1) = ρ/(ρ−1). So k_n(s) = 1 − (1+1/s)ⁿ evaluated with s = ρ−1... Let me just read LiSides.lean to get the exact definitions.\n\nLet me grep LiSides.lean for the key defs.","signature":""},{"type":"text","text":"Now the LiSides.lean definitions — `Jlag`, `zeroMult`, `Sz`."},{"type":"tool_use","id":"Grep_44","name":"Grep","input":{"-n":true,"output_mode":"content","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","pattern":"^(def|theorem|lemma|abbrev|noncomputable def)|zeroMult|Jlag|def Sz"},"extra_content":null}]