← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
thinking (3579 chars)
Wait — the error at 275:2 says "case hsplit" with the hsplit goal unsolved — but line 275 is `· -- the series case`! The error is attributed to line 275:2 (the bullet start) but "case hsplit" — that's confusing. OH — I see: the error at "275:2" is the START of the series-case bullet, and "case hsplit" with the hsplit goal — this means the hsplit proof is being elaborated and the error position is misreported... no. Actually — "case hsplit" with the hsplit goal — the hsplit's `exact (hker_summ.sum_add_tsum_nat_add N).symm` produced this unsolved goal. So exact's defeq check FAILED: h.symm's type vs the goal are NOT defeq.
From the probe, h's LHS-sum is `∑ i ∈ Finset.range N, ((↑i + 1) * (A + ↑i + 1))⁻¹` and my goal's is `∑ n ∈ Finset.range N, ((↑n + 1) * (A + ↑n + 1))⁻¹` — alpha ✓. Hmm — but wait: in the probe I passed hker_summ EXPLICITLY typed with A free. In the real file A is a `set` local def — set A := w.re makes A a LOCAL VARIABLE with hA_def : A = w.re and replaces w.re by A in goal/context. When hker_summ's statement is elaborated with A (the set var) — fine. When `sum_add_tsum_nat_add` is applied — the resulting h's type should have A...
Hmm, let me actually LOOK at h in the real context — the error context doesn't show h (it's not in context because it's inside the exact term). The problem might be that `hker_summ.sum_add_tsum_nat_add N` — the dot-notation: hker_summ : Summable fun n => ... — `Summable.sum_add_tsum_nat_add (k : ℕ) (h : Summable f)` — dot notation on hker_summ: Lean tries to put hker_summ at the first EXPLICIT argument whose type's head is Summable — the args are (k : ℕ) and (h : Summable f): the first matching is h ✓ so k := N explicitly given ✓ — that's what I assumed.
But wait — what if the to_additive-generated name's statement differs — e.g. it's `Summable.sum_add_tsum_nat_add (hf : Summable f) (k : ℕ) : ∑' ... = ∑ + ...'` (direction flipped)? The probe CONFIRMED the form: `hf.sum_add_tsum_nat_add k : ∑ range k + ∑' shifted = ∑' main` — probe's #check output line 1: `Summable.sum_add_tsum_nat_add k hf : ∀ (f : ℕ → ℝ) (k : ℕ), Summable f → ∑ i ∈ Finset.range k, f i + ∑' (i : ℕ), f (i + k) = ∑' (i : ℕ), f i`. So direction confirmed. And the instantiated probe trace confirmed the beta-reduced forms. So h.symm : ∑' main = ∑ + ∑' shifted — should be EXACTLY my goal...
UNLESS the issue is `set` variable unfolding differences: the goal's N — wait — maybe h uses the local-def N while my goal ALSO uses N but after `set`, tsum-bound... no.
Let me try yet another approach — prove hsplit via convert with the defeq depth:
```
rw [← hker_summ.sum_add_tsum_nat_add N]
```
— that's what the ORIGINAL attempt did (pass 5) — "unsolved goals". The rw [← h] rewrites ∑' main → (sum + shifted): the goal becomes (sum + shifted) = (my head + my tail): not rfl ⟹ the two sides differ syntactically. The difference: the SHIFTED term's form. h's shifted: `f (i + N)` — displayed in the probe as `((↑(i + N) + 1) * (A + ↑(i + N) + 1))⁻¹` — note ↑(i + N) — the cast of (i + N : ℕ). My goal's tail: `((((n + N : ℕ) : ℝ) + 1) * (A + ((n + N : ℕ) : ℝ) + 1))⁻¹` — also ↑(n+N). SAME. The head: same. So after rw [← h] the goal SHOULD be rfl... unless the rw didn't fire because the goal's LHS ∑' main doesn't match h's RHS pattern due to... A is a set-local-def: h's type has A (the local def); my goal has A ✓...
Hmm hold on — maybe the problem is `simp only [] at h` earlier pass and now... Let me just debug directly: add a `trace_state` after `have h := ...` in the real file and compile to see h's actual type there.
Let me trace h's actual type in the real context.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean",
"old_string": " exact (hker_summ.sum_add_tsum_nat_add N).symm",
"new_string": " have h := hker_summ.sum_add_tsum_nat_add N\n trace_state\n exact h.symm"
}
result 2026-08-23 10:00:55Z → 2026-08-26 07:23:44Z
{
"filePath": "D:\\Python\\Quantum\\research\\numerics\\scripts\\E-034-dogleg-validation.py",
"oldString": " print(f\"n={n}: identity resid={mp.nstr(resid, 2)} (tol {mp.nstr(tol, 2)}) | \"\n f\"V_left {mp.nstr(abs(Vl2), 4)}/{mp.nstr(vlb, 4)} | \"\n f\"H_high {mp.nstr(abs(Hhigh2), 4)}/5.94 | \"\n f\"V_right {mp.nstr(abs(Vr2), 3)}/{mp.nstr(vrb, 3)} | \"\n f\"H_low {mp.nstr(abs(Hlow2), 4)}/{mp.nstr(dlr, 4)}\", flush=True)",
"newString": " print(f\"n={n}: identity resid={mp.nstr(resid, 2)} (tol {mp.nstr(tol, 2)}) | \"\n f\"V_left {mp.nstr(abs(Vl2), 4)}/{mp.nstr(vlb, 4)} | \"\n f\"H_high {mp.nstr(abs(Hhigh2), 4)}/{mp.nstr(HHIGH, 4)} | \"\n f\"V_right {mp.nstr(abs(Vr2), 3)}/{mp.nstr(vrb, 3)} | \"\n f\"H_low {mp.nstr(abs(Hlow2), 4)}/{mp.nstr(dlr, 4)}\", flush=True)",
"originalFile": "#!/usr/bin/env python3\n\"\"\"E-034 \u2014 D-074 zero-free dog-leg repair, sub-stage DL-0 (directive \u00a711):\nnumerical validation BEFORE any Lean work.\n\nThe dog-leg: keep T_cut = sqrt(n) + eps (zero-avoiding); replace the right\nhorizontal tail H_low = int_3^{2 sqrt(n)} Jlag(sigma + i T_cut) d sigma by\n H_low = H_high - V_right + V_left (exact identity (*))\n H_high = int_3^{2 sqrt(n)} Jlag(sigma + i U) d sigma, U = 1.5 * T_cut\n V_right = i int_{T_cut}^{U} Jlag(2 sqrt(n) + i t) dt\n V_left = i int_{T_cut}^{U} Jlag(3 + i t) dt\nwith Jlag n s = ((s/(s-1))^n - 1) * (-zeta'/zeta(s)) (the Lean convention,\nLiSides.lean:55-58). The whole dog-leg rectangle is at Re s >= 3 > 1\n(unconditionally zero-free).\n\nCHECKS:\n V1: C_zeta3 certified: partial Dirichlet sum + certified tail brackets\n -zeta'/zeta(3) (mpmath cross-check); charged at the upper edge.\n V2: e^{5/2}+1 <= 13.19 certified (kernel charge on V_left).\n V3: q = e^{4/9}/2 <= 0.78 < 1 certified (the stepping condition).\n V4 (identity): |H_low - (H_high - V_right + V_left)| tiny at\n n = 1e3, 1e4, 1e5 (two trapezoid grids for convergence).\n V5 (slack): measured |V_left| <= VLeftBound, |H_high| <= 5.94,\n |V_right| <= VRightBound, |H_low| <= DogLegRight (ratios reported).\n V6 (fold): the temporary dog-leg Bn_DL(n) keeps margin > 0 on\n [8e4, N1], N1 = (3e12-1)^2 (full fold, base region, endpoint).\n\nMethod: dps 50; -zeta'/zeta via certified-tail Dirichlet series for\nsigma >= 5 and mpmath for sigma in [3,5) (validated against it at\nsigma = 3, 4, 5 \u2014 the E-032 architecture); vertical pieces by mpmath\n(validated at sigma = 3); trapezoid h = 0.05 / 0.02 (horizontals),\nh = 0.05 / 0.025 (verticals). Deterministic; cache via sidecar.\n\"\"\"\nimport json\nimport sys\nfrom pathlib import Path\n\nimport mpmath as mp\n\nmp.mp.dps = 50\n\nDATA = Path(__file__).resolve().parent.parent / \"data\"\nCSV = DATA / \"E-034-dogleg-validation.csv\"\nJSON = DATA / \"E-034-dogleg-validation.json\"\n\nSIEVE_MAX = 25000\nTOL_DIR = mp.mpf(\"1e-11\")\nNS = [10 ** 3, 10 ** 4, 10 ** 5]\npi, e = mp.pi, mp.e\n\n_spf = list(range(SIEVE_MAX + 1))\nfor _i in range(2, int(SIEVE_MAX ** 0.5) + 1):\n if _spf[_i] == _i:\n for _j in range(_i * _i, SIEVE_MAX + 1, _i):\n if _spf[_j] == _j:\n _spf[_j] = _i\nLAM = [mp.mpf(0)] * (SIEVE_MAX + 1)\nLOGK = [mp.mpf(0)] * (SIEVE_MAX + 1)\nfor _k in range(2, SIEVE_MAX + 1):\n LOGK[_k] = mp.log(_k)\n _p = _spf[_k]\n _m = _k\n while _m % _p == 0:\n _m //= _p\n if _m == 1:\n LAM[_k] = LOGK[_p]\n\n\ndef tail_cert(sig, K):\n if K <= 3:\n return mp.inf\n u = mp.mpf(K - 1)\n return (mp.log(u) / (sig - 1) + 1 / (sig - 1) ** 2) * u ** (1 - sig)\n\n\ndef choose_K(sig, tol):\n K = 4\n while tail_cert(sig, K) >= tol:\n K += 1\n if K > SIEVE_MAX:\n raise RuntimeError(f\"K exceeds sieve at sig={sig}\")\n return K\n\n\ndef dser(sig, T, tol=TOL_DIR):\n K = choose_K(sig, tol)\n w = mp.mpc(sig, T)\n s = mp.mpc(0)\n for k in range(2, K + 1):\n if LAM[k]:\n s += LAM[k] * mp.exp(-w * LOGK[k])\n return s, K\n\n\ndef mp_neg_ld(sig, T):\n w = mp.mpc(sig, T)\n return -mp.zeta(w, derivative=1) / mp.zeta(w)\n\n\ndef neg_ld(sig, T):\n if sig < 5:\n return mp_neg_ld(sig, T), 0\n return dser(sig, T)\n\n\ndef jlag(n, sig, T):\n s = mp.mpc(sig, T)\n kf = (s / (s - 1)) ** n - 1\n d, _ = neg_ld(sig, T)\n return kf * d\n\n\ndef jlag_mp(n, sig, T):\n s = mp.mpc(sig, T)\n return ((s / (s - 1)) ** n - 1) * mp_neg_ld(sig, T)\n\n\ndef zeros_near(T0, half=3.0):\n j_approx = int((T0 / (2 * mp.pi)) * mp.log(T0 / (2 * mp.pi * mp.e)))\n out = []\n for j in range(max(1, j_approx - 30), j_approx + 30):\n g = mp.zetazero(j).imag\n if abs(g - T0) <= half:\n out.append(g)\n return out\n\n\ndef choose_eps(T0):\n zs = zeros_near(T0)\n best = (-1, 0.0)\n for i in range(20):\n eps = i * 0.05\n cl = min((abs(T0 + eps - g) for g in zs), default=mp.mpf(3))\n if cl > best[0]:\n best = (cl, eps)\n return best[1], best[0]\n\n\ndef H_piece(n, y, a, b, h):\n \"\"\"int_a^b Jlag(sigma + i y) d sigma (mpmath < 5, Dirichlet >= 5).\"\"\"\n N = int(mp.ceil((b - a) / h))\n he = (b - a) / N\n tot = mp.mpc(0)\n for i in range(N + 1):\n sig = a + i * he\n f = jlag_mp(n, sig, y) if sig < 5 else jlag(n, sig, y)\n wt = mp.mpf(\"0.5\") if i in (0, N) else mp.mpf(1)\n tot += wt * f\n return tot * he\n\n\ndef V_piece(n, x, a, b, h):\n \"\"\"i int_a^b Jlag(x + i t) dt (mpmath throughout; validated at x = 3).\"\"\"\n N = int(mp.ceil((b - a) / h))\n he = (b - a) / N\n tot = mp.mpc(0)\n for i in range(N + 1):\n t = a + i * he\n f = jlag_mp(n, x, t)\n wt = mp.mpf(\"0.5\") if i in (0, N) else mp.mpf(1)\n tot += wt * f\n return 1j * tot * he\n\n\n# ------------------------------------------------- fold machinery (E-030 forms)\ndef C0hat(t):\n return mp.mpf(\"0.433\") * mp.log(t + 2) + mp.mpf(\"0.886\") * mp.log(mp.log(t + 2)) + mp.mpf(\"2.9\")\n\n\ndef Z2(t):\n return mp.mpf(\"0.6\") * mp.log(t + 2) + mp.mpf(\"0.886\") * mp.log(mp.log(t + 2)) + mp.mpf(\"2.7\")\n\n\ndef FAlt(T):\n return ((1 / (2 * pi)) * (T * mp.log(T)) + mp.mpf(\"0.137\") * mp.log(T)\n + mp.mpf(\"0.443\") * mp.log(mp.log(T)) + (mp.mpf(7) / 8 + mp.mpf(\"1.588\")))\n\n\ndef Hhat(T):\n return 3 * C0hat(T) ** 2 + mp.mpf(\"10.5\") * mp.log(T + 2) + 48\n\n\ndef IIMaj(n):\n return 2 * (mp.sqrt(n) + 1) * (mp.e ** (mp.sqrt(n) / 2) + 1) * mp.mpf(\"0.57\") * 2 ** (-(2 * mp.sqrt(n) - 2))\n\n\ndef sInftyLower(n):\n return mp.mpf(\"0.5\") * n * mp.log(n) + (mp.mpf(\"-1.41894\")) * n - 4\n\n\ndef NI(T, a):\n c_clear = 1 / (9 * C0hat(a + 2))\n return Z2(T) * 2 * mp.log(mp.mpf(\"16.5\") / c_clear)\n\n\n# candidate certified constants (V1-V3 below)\n# C_ZETA3: the LEAN-CHARGED constant 0.5 (DL-2, dirichletC3 <= 1/2 via the\n# termwise 3-vs-2 comparison + dirichletC2 <= 1). The tight analytic value\n# -zeta'/zeta(3) = 0.16482 (V1 certifies it) and the fold is indifferent\n# to the constant (margins stay > 1.3e4 at the tightest base point), so the\n# cheap certified 0.5 is charged; the 0.165/0.17 upgrades are documented\n# in D-074 sec.4 as the tightening path if ever needed.\nC_ZETA3 = mp.mpf(\"0.5\")\nE52P1 = mp.mpf(\"13.19\") # e^{5/2}+1 <= 13.19\nQ34 = mp.mpf(\"0.78\") # e^{4/9}/2 <= 0.78\n# HHIGH: the LEAN-CHARGED high-horizontal bound 8.38 (DL-3,\n# hIntegral_Jlag_dogleg_norm_le: the single-exponential-envelope form\n# 4*e^{4/3}/(2c) + 1/(2 log 2) with c = log 2 - 4/9 >= 0.2487 certified,\n# dirichletC2 <= 1). The tight R(1.5) = 5.94 would need a dirichletC2 <= 0.57\n# build (documented tightening path; the fold is indifferent: margins change\n# by ~5 at the base points against margins > 1.3e4).\nHHIGH = mp.mpf(\"8.38\")\n\n\ndef VLeftBound(n, cz3=C_ZETA3):\n return ((mp.sqrt(n) + 1) / 2) * E52P1 * cz3\n\n\ndef VRightBound(n):\n return IIMaj(n) / 4\n\n\ndef DogLegRight(n, cz3=C_ZETA3):\n return VLeftBound(n, cz3) + HHIGH + VRightBound(n)\n\n\ndef Bn_DL(n, cz3=C_ZETA3):\n \"\"\"Temporary dog-leg Bn (D-074 sec.6): the session-038 honest Bn with\n the broken right-tail 3.3 replaced by DogLegRight (NOT multiplied by\n the mid kernel charge).\"\"\"\n T = mp.sqrt(n) + 1\n return (4 * FAlt(T) + 4 * T * ((mp.mpf(2) / 3) * Z2(T) + mp.mpf(\"10.5\") * mp.log(T + 2) + 48)\n + 40 * mp.mpf(\"1.3352\")\n + 2 * (e ** 4 + 1) * (7 * Hhat(T))\n + 2 * DogLegRight(n, cz3)\n + IIMaj(n) + 1)\n\n\ndef honest_DL(n, cz3=C_ZETA3):\n T = mp.sqrt(n) + 1\n a = mp.sqrt(n)\n mid = 2 * (e ** 4 + 1) * (NI(T, a) + mp.mpf(\"5.5\") * (mp.mpf(\"10.5\") * mp.log(T + 2) + 48))\n return (4 * FAlt(T) + 4 * T * ((mp.mpf(2) / 3) * Z2(T) + mp.mpf(\"10.5\") * mp.log(T + 2) + 48)\n + 40 * mp.mpf(\"1.3352\") + mid + 2 * DogLegRight(n, cz3) + IIMaj(n) + 1)\n\n\ndef main() -> int:\n if JSON.exists() and \"--force\" not in sys.argv:\n print(f\"cached: {JSON} (use --force to recompute)\")\n return 0\n\n rows = []\n\n # ---- V1: C_zeta3 certified ----\n K3 = 25000\n s3 = mp.mpf(0)\n for k in range(2, K3 + 1):\n if LAM[k]:\n s3 += LAM[k] / (mp.mpf(k) ** 3)\n tail3 = (2 * mp.log(K3 - 1) + 1) / (4 * (K3 - 1) ** 2)\n z3mp = mp_neg_ld(3, 0).real\n v1 = s3 <= z3mp <= s3 + tail3 and s3 + tail3 <= C_ZETA3\n rows.append({\"check\": \"V1_C_zeta3\", \"n\": \"\", \"quantity\": \"-zeta'/zeta(3)\",\n \"measured\": mp.nstr(z3mp, 12), \"bound_lo\": mp.nstr(s3, 12),\n \"bound_hi\": mp.nstr(s3 + tail3, 12), \"charged\": mp.nstr(C_ZETA3, 6),\n \"pass\": str(bool(v1))})\n\n # ---- V2/V3: kernel charge + q certified ----\n e52 = mp.e ** (mp.mpf(5) / 2)\n v2 = e52 + 1 <= E52P1\n rows.append({\"check\": \"V2_kernel_charge_e^{5/2}+1\", \"n\": \"\", \"quantity\": \"e^{5/2}+1\",\n \"measured\": mp.nstr(e52 + 1, 8), \"bound_lo\": \"\",\n \"bound_hi\": mp.nstr(E52P1, 6), \"charged\": mp.nstr(E52P1, 6),\n \"pass\": str(bool(v2))})\n e49 = mp.e ** (mp.mpf(4) / 9)\n q = e49 / 2\n v3 = q <= Q34 and e49 < 2\n rows.append({\"check\": \"V3_q_certified\", \"n\": \"\", \"quantity\": \"q = e^{4/9}/2\",\n \"measured\": mp.nstr(q, 8), \"bound_lo\": mp.nstr(e49, 8),\n \"bound_hi\": mp.nstr(Q34, 6), \"charged\": mp.nstr(Q34, 6),\n \"pass\": str(bool(v3))})\n\n # ---- validation: mpmath vs Dirichlet at sigma = 3, 4, 5 ----\n val_max = mp.mpf(0)\n for n in NS:\n T0 = mp.sqrt(n)\n eps, _cl = choose_eps(T0)\n T = T0 + eps\n for sig in [3.0, 4.0, 5.0]:\n dm, _k = dser(mp.mpf(sig), T, tol=mp.mpf(\"1e-8\"))\n dd = mp_neg_ld(sig, T)\n if abs(dm - dd) > val_max:\n val_max = abs(dm - dd)\n\n # ---- V4/V5: the dog-leg pieces + identity + slack ----\n v4, v5 = True, True\n for n in NS:\n sq = mp.sqrt(n)\n T0 = sq\n eps, clearance = choose_eps(T0)\n T = T0 + eps\n U = mp.mpf(\"1.5\") * T\n XR = 2 * sq\n Hlow1 = H_piece(n, T, 3, XR, 0.05)\n Hlow2 = H_piece(n, T, 3, XR, 0.02)\n Hhigh1 = H_piece(n, U, 3, XR, 0.05)\n Hhigh2 = H_piece(n, U, 3, XR, 0.02)\n Vr1 = V_piece(n, XR, T, U, 0.05)\n Vr2 = V_piece(n, XR, T, U, 0.025)\n Vl1 = V_piece(n, 3, T, U, 0.05)\n Vl2 = V_piece(n, 3, T, U, 0.025)\n # identity: H_low = H_high - V_right + V_left\n resid = abs(Hlow2 - (Hhigh2 - Vr2 + Vl2))\n conv = max(abs(Hlow1 - Hlow2), abs(Hhigh1 - Hhigh2), abs(Vr1 - Vr2), abs(Vl1 - Vl2))\n tol = mp.mpf(\"1e-6\") * max(mp.mpf(1), abs(Hlow2)) + 4 * conv\n ok4 = resid <= tol\n v4 = v4 and ok4\n # slack vs majorants\n vlb, vrb, dlr = VLeftBound(n), VRightBound(n), DogLegRight(n)\n ratios = {\n \"V_left\": abs(Vl2) / vlb, \"H_high\": abs(Hhigh2) / HHIGH,\n \"V_right\": abs(Vr2) / vrb, \"H_low\": abs(Hlow2) / dlr,\n }\n ok5 = all(r <= 1 for r in ratios.values())\n v5 = v5 and ok5\n for name, val, bnd, ratio in [\n (\"V_left\", abs(Vl2), vlb, ratios[\"V_left\"]),\n (\"H_high\", abs(Hhigh2), HHIGH, ratios[\"H_high\"]),\n (\"V_right\", abs(Vr2), vrb, ratios[\"V_right\"]),\n (\"H_low\", abs(Hlow2), dlr, ratios[\"H_low\"])]:\n rows.append({\"check\": f\"V5_slack_{name}\", \"n\": str(n), \"quantity\": name,\n \"measured\": mp.nstr(val, 6), \"bound_lo\": \"\",\n \"bound_hi\": mp.nstr(bnd, 6), \"charged\": mp.nstr(ratio, 4),\n \"pass\": str(bool(ratio <= 1))})\n rows.append({\"check\": \"V4_identity\", \"n\": str(n), \"quantity\": \"H_low - (H_high - V_right + V_left)\",\n \"measured\": mp.nstr(resid, 3), \"bound_lo\": mp.nstr(conv, 3),\n \"bound_hi\": mp.nstr(tol, 3),\n \"charged\": f\"T={mp.nstr(T, 8)} eps={eps} U={mp.nstr(U, 8)}\",\n \"pass\": str(bool(ok4))})\n print(f\"n={n}: identity resid={mp.nstr(resid, 2)} (tol {mp.nstr(tol, 2)}) | \"\n f\"V_left {mp.nstr(abs(Vl2), 4)}/{mp.nstr(vlb, 4)} | \"\n f\"H_high {mp.nstr(abs(Hhigh2), 4)}/5.94 | \"\n f\"V_right {mp.nstr(abs(Vr2), 3)}/{mp.nstr(vrb, 3)} | \"\n f\"H_low {mp.nstr(abs(Hlow2), 4)}/{mp.nstr(dlr, 4)}\", flush=True)\n\n # ---- V6: the temporary fold ----\n def loggrid(lo, hi, pts):\n llo, lhi = mp.log(lo), mp.log(hi)\n return [mp.e ** (llo + (lhi - llo) * i / (pts - 1)) for i in range(pts)]\n\n N1 = (mp.mpf(\"3e12\") - 1) ** 2\n grid = [mp.mpf(\"8e4\"), mp.mpf(\"1e5\"), mp.mpf(\"2e5\"), mp.mpf(\"5e5\")] + \\\n loggrid(mp.mpf(\"1e6\"), N1, 25)\n v6 = True\n minmargin = None\n foldfit_max = mp.mpf(0)\n for nn in grid:\n T = mp.sqrt(nn) + 1\n a = mp.sqrt(nn)\n fit = (NI(T, a) + mp.mpf(\"5.5\") * (mp.mpf(\"10.5\") * mp.log(T + 2) + 48)) / (7 * Hhat(T))\n if fit > foldfit_max:\n foldfit_max = fit\n marg = sInftyLower(nn) - Bn_DL(nn) + 1\n if minmargin is None or marg < minmargin:\n minmargin = marg\n v6 = v6 and (marg > 0) and (honest_DL(nn) <= Bn_DL(nn))\n for nn in [mp.mpf(\"8e4\"), mp.mpf(\"1e5\"), mp.mpf(\"1e8\"), mp.mpf(\"1e12\"), N1]:\n marg = sInftyLower(nn) - Bn_DL(nn) + 1\n rows.append({\"check\": \"V6_margin_value\", \"n\": mp.nstr(nn, 8), \"quantity\": \"margin_DL\",\n \"measured\": mp.nstr(marg, 8), \"bound_lo\": \"0\",\n \"bound_hi\": \"\", \"charged\": mp.nstr(Bn_DL(nn), 8),\n \"pass\": str(bool(marg > 0))})\n rows.append({\"check\": \"V6_margin_min_on_grid\", \"n\": \"grid\", \"quantity\": \"min margin\",\n \"measured\": mp.nstr(minmargin, 6), \"bound_lo\": \"0\", \"bound_hi\": \"\",\n \"charged\": mp.nstr(foldfit_max, 4), \"pass\": str(bool(minmargin > 0 and foldfit_max < 1))})\n v6 = v6 and foldfit_max < 1\n\n allpass = v1 and v2 and v3 and v4 and v5 and v6 and val_max <= mp.mpf(\"1e-6\")\n\n DATA.mkdir(exist_ok=True)\n with CSV.open(\"w\") as f:\n f.write(\",\".join(rows[0].keys()) + \"\\n\")\n for r in rows:\n f.write(\",\".join(r.values()) + \"\\n\")\n sidecar = {\n \"experiment\": \"E-034\",\n \"title\": \"D-074 zero-free dog-leg repair DL-0: exact identity \"\n \"H_low = H_high - V_right + V_left verified; piece majorants \"\n \"with slack; the temporary dog-leg Bn_DL keeps margin > 0 to \"\n \"N1 = (3e12-1)^2\",\n \"session\": \"040\",\n \"thread\": \"ii-prime-6-DL\",\n \"design_doc\": \"D-074\",\n \"checks\": {\"V1_C_zeta3_certified\": bool(v1),\n \"V2_kernel_charge_certified\": bool(v2),\n \"V3_q_certified\": bool(v3),\n \"V4_identity_exact\": bool(v4),\n \"V5_piece_slack\": bool(v5),\n \"V6_fold_margin_positive\": bool(v6),\n \"VA_mpmath_vs_dirichlet\": bool(val_max <= mp.mpf(\"1e-6\"))},\n \"all_checks_pass\": bool(allpass),\n \"dps\": 50,\n \"constants\": {\"C_zeta3_partial\": mp.nstr(s3, 12),\n \"C_zeta3_tail\": mp.nstr(tail3, 4),\n \"C_zeta3_charged\": mp.nstr(C_ZETA3, 6),\n \"e^{5/2}+1\": mp.nstr(e52 + 1, 8),\n \"q\": mp.nstr(q, 8),\n \"val_max_disc\": mp.nstr(val_max, 3),\n \"foldfit_max\": mp.nstr(foldfit_max, 4),\n \"margin_N1\": mp.nstr(sInftyLower(N1) - Bn_DL(N1) + 1, 8)},\n \"notes\": \"Identity (*): H_low = H_high - V_right + V_left at \"\n \"T_cut = sqrt(n)+eps (zero-avoiding), U = 1.5*T_cut, X_R = 2*sqrt(n). \"\n \"DogLegRight(n) = VLeftBound + 5.94 + IIMaj/4 replaces the broken 3.3 \"\n \"(NOT multiplied by the mid kernel charge e^4+1).\",\n }\n with JSON.open(\"w\") as f:\n json.dump(sidecar, f, indent=2)\n print(f\"V1 {bool(v1)} V2 {bool(v2)} V3 {bool(v3)} V4 {bool(v4)} \"\n f\"V5 {bool(v5)} V6 {bool(v6)} VA {mp.nstr(val_max, 2)}\")\n print(f\"wrote {CSV} + {JSON}\")\n return 0 if sidecar[\"all_checks_pass\"] else 1\n\n\nif __name__ == \"__main__\":\n sys.exit(main())\n",
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[{"type":"thinking","thinking":"Wait — the error at 275:2 says \"case hsplit\" with the hsplit goal unsolved — but line 275 is `· -- the series case`! The error is attributed to line 275:2 (the bullet start) but \"case hsplit\" — that's confusing. OH — I see: the error at \"275:2\" is the START of the series-case bullet, and \"case hsplit\" with the hsplit goal — this means the hsplit proof is being elaborated and the error position is misreported... no. Actually — \"case hsplit\" with the hsplit goal — the hsplit's `exact (hker_summ.sum_add_tsum_nat_add N).symm` produced this unsolved goal. So exact's defeq check FAILED: h.symm's type vs the goal are NOT defeq.\n\nFrom the probe, h's LHS-sum is `∑ i ∈ Finset.range N, ((↑i + 1) * (A + ↑i + 1))⁻¹` and my goal's is `∑ n ∈ Finset.range N, ((↑n + 1) * (A + ↑n + 1))⁻¹` — alpha ✓. Hmm — but wait: in the probe I passed hker_summ EXPLICITLY typed with A free. In the real file A is a `set` local def — set A := w.re makes A a LOCAL VARIABLE with hA_def : A = w.re and replaces w.re by A in goal/context. When hker_summ's statement is elaborated with A (the set var) — fine. When `sum_add_tsum_nat_add` is applied — the resulting h's type should have A... \n\nHmm, let me actually LOOK at h in the real context — the error context doesn't show h (it's not in context because it's inside the exact term). The problem might be that `hker_summ.sum_add_tsum_nat_add N` — the dot-notation: hker_summ : Summable fun n => ... — `Summable.sum_add_tsum_nat_add (k : ℕ) (h : Summable f)` — dot notation on hker_summ: Lean tries to put hker_summ at the first EXPLICIT argument whose type's head is Summable — the args are (k : ℕ) and (h : Summable f): the first matching is h ✓ so k := N explicitly given ✓ — that's what I assumed.\n\nBut wait — what if the to_additive-generated name's statement differs — e.g. it's `Summable.sum_add_tsum_nat_add (hf : Summable f) (k : ℕ) : ∑' ... = ∑ + ...'` (direction flipped)? The probe CONFIRMED the form: `hf.sum_add_tsum_nat_add k : ∑ range k + ∑' shifted = ∑' main` — probe's #check output line 1: `Summable.sum_add_tsum_nat_add k hf : ∀ (f : ℕ → ℝ) (k : ℕ), Summable f → ∑ i ∈ Finset.range k, f i + ∑' (i : ℕ), f (i + k) = ∑' (i : ℕ), f i`. So direction confirmed. And the instantiated probe trace confirmed the beta-reduced forms. So h.symm : ∑' main = ∑ + ∑' shifted — should be EXACTLY my goal...\n\nUNLESS the issue is `set` variable unfolding differences: the goal's N — wait — maybe h uses the local-def N while my goal ALSO uses N but after `set`, tsum-bound... no.\n\nLet me try yet another approach — prove hsplit via convert with the defeq depth:\n```\n rw [← hker_summ.sum_add_tsum_nat_add N]\n```\n— that's what the ORIGINAL attempt did (pass 5) — \"unsolved goals\". The rw [← h] rewrites ∑' main → (sum + shifted): the goal becomes (sum + shifted) = (my head + my tail): not rfl ⟹ the two sides differ syntactically. The difference: the SHIFTED term's form. h's shifted: `f (i + N)` — displayed in the probe as `((↑(i + N) + 1) * (A + ↑(i + N) + 1))⁻¹` — note ↑(i + N) — the cast of (i + N : ℕ). My goal's tail: `((((n + N : ℕ) : ℝ) + 1) * (A + ((n + N : ℕ) : ℝ) + 1))⁻¹` — also ↑(n+N). SAME. The head: same. So after rw [← h] the goal SHOULD be rfl... unless the rw didn't fire because the goal's LHS ∑' main doesn't match h's RHS pattern due to... A is a set-local-def: h's type has A (the local def); my goal has A ✓...\n\nHmm hold on — maybe the problem is `simp only [] at h` earlier pass and now... Let me just debug directly: add a `trace_state` after `have h := ...` in the real file and compile to see h's actual type there.","signature":""},{"type":"text","text":"Let me trace h's actual type in the real context."},{"type":"tool_use","id":"Edit_335","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean","old_string":" exact (hker_summ.sum_add_tsum_nat_add N).symm","new_string":" have h := hker_summ.sum_add_tsum_nat_add N\n trace_state\n exact h.symm"},"extra_content":null}]