← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (1575 chars)
DL-6 committed (a19172f). The D-074 analytic core is complete:
- DL-1: exact deformation identity (Lean-proved, no perimeter)
- DL-2: V_left bound
- DL-3: H_high bound (8.38)
- DL-4: V_right bound
- DL-5: total right-horizontal bound (both sides)
- DL-6: direct dirichlet bound + IIMaj/4 quarter charge (no dirichletC2 ≤ 0.57)
Now ii′-6-a2: the Bn dog-leg redefinition. This is a substantial Lean sub-stage. Let me set up the todos and start. Given the session has been very long, let me first write journal-040 as a checkpoint (crash-safe), then continue with ii′-6-a2.
Actually — per the crash-safe protocol, I should checkpoint now: journal-040 covering DL-0 through DL-6, then continue with ii′-6-a2. The journal is a session-end artifact, but the session has been extremely long with many natural breaks — the protocol says save artifacts immediately. The commits are all in place (each sub-stage committed). Let me write journal-040 now as a checkpoint and continue with ii′-6-a2 in the next block.
Actually, the cleanest: since DL-0 through DL-6 are all committed and the state is fully persisted (STATE.md, D-074, NGUYEN.md, db, all commits), the crash-safe requirement is met. The journal-040 can wait until the session actually ends or the next natural checkpoint (after ii′-6-a2 or the cherry-pick). But it's prudent to write it now given the huge amount of work. Let me write journal-040 (DL-0 through DL-6), commit it, then start ii′-6-a2 with the cherry-pick.
Let me also register session 040 in the db (it wasn't registered yet — I registered 039 earlier).
DL-6 committed (`a19172f`). Checkpointing with journal-040 (DL-0 through DL-6 complete), then starting ii′-6-a2.
tool: TodoWrite ok
input
{
"todos": [
{
"content": "Checkpoint: journal-040 (DL-0\u2013DL-6) + db session registration + commit",
"status": "in_progress",
"activeForm": "Writing journal-040 checkpoint"
},
{
"content": "Cherry-pick bracket < 4/3 + IIMaj_deriv2_lt_three onto master (from wip branch, k-shift excluded)",
"status": "pending",
"activeForm": "Cherry-picking the bracket strengthening"
},
{
"content": "ii\u2032-6-a2-i: Bn/marginTilde/marginExpanded/mK dog-leg redefinition + eq_expanded/eq_sub + contDiffOn/continuousOn",
"status": "pending",
"activeForm": "Redefining Bn at the dog-leg constants"
},
{
"content": "ii\u2032-6-a2-ii: marginTilde_convex re-proof (id-block lower bound + 1.5\u00b7IIMaj bracket) + base-point re-certification + endpoint margins",
"status": "pending",
"activeForm": "Re-proving margin convexity + base points"
},
{
"content": "Full build + axiom audit + commit ii\u2032-6-a2; then ii\u2032-6-c per the return map",
"status": "pending",
"activeForm": "Closing ii\u2032-6-a2"
}
]
}
result 2026-08-26 10:51:12Z → 2026-08-26 10:51:12Z
{
"oldTodos": [
"content='DL-6-i: dirichletCX split + tail + capstone norm_logDeriv_zeta_le_iimZeta (\u2264 0.57\u00b72^{\u2212(2\u221an\u22122)}, n \u2265 5)' status='in_progress' active_form='Building the direct zeta bound at 2\u221an'",
"content='DL-6-ii: vIntegral_Jlag_right_norm_le_quarter (V_right \u2264 (U\u2212T)\u00b7(e+1)\u00b70.57\u00b72^{\u2212(2\u221an\u22122)} = IIMaj/4-charge)' status='pending' active_form='Re-charging V_right at IIMaj/4'",
"content='DL-5 upgrade: hVr2 \u2192 IIMaj/4 via DL-6-ii; dogLegUpper \u2192 VLeft + 8.38 + IIMaj/4; E-034 re-validation at IIMaj/4' status='pending' active_form='Upgrading DL-5 to the quarter charge'",
"content='Cherry-pick bracket < 4/3 + IIMaj_deriv2_lt_three onto master (from wip branch, k-shift excluded)' status='pending' active_form='Cherry-picking the bracket strengthening'",
"content='ii\u2032-6-a2: Bn dog-leg redefinition + margin lockstep + convexity re-proof + base points + audit + commit' status='pending' active_form='Executing the Bn redefinition'"
],
"newTodos": [
"content='Checkpoint: journal-040 (DL-0\u2013DL-6) + db session registration + commit' status='in_progress' active_form='Writing journal-040 checkpoint'",
"content='Cherry-pick bracket < 4/3 + IIMaj_deriv2_lt_three onto master (from wip branch, k-shift excluded)' status='pending' active_form='Cherry-picking the bracket strengthening'",
"content='ii\u2032-6-a2-i: Bn/marginTilde/marginExpanded/mK dog-leg redefinition + eq_expanded/eq_sub + contDiffOn/continuousOn' status='pending' active_form='Redefining Bn at the dog-leg constants'",
"content='ii\u2032-6-a2-ii: marginTilde_convex re-proof (id-block lower bound + 1.5\u00b7IIMaj bracket) + base-point re-certification + endpoint margins' status='pending' active_form='Re-proving margin convexity + base points'",
"content='Full build + axiom audit + commit ii\u2032-6-a2; then ii\u2032-6-c per the return map' status='pending' active_form='Closing ii\u2032-6-a2'"
],
"cleared": false
}
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-040.md",
"content": "# Session 040 (2026-08-25) \u2014 D-074 dog-leg repair: DL-0 through DL-6 COMPLETE\n\n## Mission\nExecute PROJECT_DIRECTIVE_D-074.txt (the zero-free dog-leg repair of the\nF-D broken (6.12) stepping): keep T_cut = \u221an (\u27f9 the ORIGINAL N\u2081 =\n(T\u2080\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074), deform only the right horizontal tail [3, 2\u221an] up to\nU = (3/2)\u221an through the unconditionally zero-free region Re \u2265 3.\n\nSub-stages DL-0 \u2192 DL-6, each committed per the crash-safe protocol.\n\n## DL-0 \u2014 design + audit + E-034 + fold (commit 8f9ccb2)\n- \u00a72 coordinate audit (13-row table): the Lean contour MATCHES the\n assumed geometry \u2014 Jlag n s = (s/(s\u22121))\u207f\u22121\u00b7(\u2212\u03b6\u2032/\u03b6(s)) confirmed,\n CCW orientations pinned, T_cut/X_R/bridge/3.3-entry located.\n- \u00a73 exact derivation: identity H_low = H_high \u2212 V_right + V_left;\n analyticity unconditional at Re \u2265 3; \u00a710 dependency statement\n (nothing above T_cut).\n- \u00a74 constants: C_zeta3 = 0.165 certified (the hand-estimate 0.150 was\n caught WRONG by the smoke test); e^{5/2}+1 \u2264 13.19; q = 0.7798;\n HHigh = 5.94; VRight = IIMaj/4.\n- **E-034 ALL CHECKS PASS** (identity exact to 2\u20135e-5; piece slack\n 4\u201313\u00d7; fold margins 14 419/90 317/2.458e26 \u2014 the ORIGINAL endpoint\n RESTORED). \u00a77 A\u2013L report; **user gate APPROVED \u2192 DL-1**.\n\n## DL-1 \u2014 the exact deformation identity in Lean (commit 4c36802)\nLiDogLeg.lean (new): Jlag_differentiableAt_dogleg +\nholomorphicOn_Jlag_dogleg (Re s \u2265 3 zero-free via Mathlib\nriemannZeta_ne_zero_of_one_lt_re, NO perimeter, no pigeonhole at U);\nrectangleIntegral(_')_Jlag_dogleg_eq_zero (Cauchy via\nHolomorphicOn.vanishesOnRectangle); the capstone\n**upper_right_horizontal_eq_dogleg**: H(3\u2192X_R, T) = H(3\u2192X_R, U) \u2212\nV(X_R, T\u2192U) + V(3, T\u2192U); kFun_conj/Jlag_conj. Build 8773, audit clean.\n\n## DL-2 \u2014 the V_left bound (commit b959a31)\nkFun_three_add_it_norm_le_exp_five_halves_add_one (kernel \u2264 e^{5/2}+1,\nthe 1+5/(4+t\u00b2) computation); dirichletC3 \u2264 1/2 (termwise 3-vs-2\ncomparison + dirichletC2 \u2264 1 \u2014 **C_zeta3 charged 0.5**, fold-indifferent;\nthe tight 0.165/0.17 upgrade path documented in D-074 \u00a74);\nnorm_logDeriv_zeta_le_half; capstone **vIntegral_Jlag_three_norm_le**:\n\u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U\u2212T)(e^{5/2}+1)/2. E-034 re-validated at\nC = 0.5: ALL CHECKS PASS (margins 13 165/88 916/2.458e26).\n\n## DL-3 \u2014 the H_high stepping (commit 6eaf531)\nnorm_term/norm_logDeriv_zeta_le_two_pow (the (6.10) halving in LSeries\nform: \u2016\u03b6\u2032/\u03b6(\u03c3+it)\u2016 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7dirichletC2);\none_add_div_pow_norm_le_exp_four_ninths (the kernel envelope\ne^{(4/9)\u03c3} \u2014 SINGLE-EXPONENTIAL form replacing the unit-step geometric\nseries, the D-073 R(1.5) alternative); Jlag_dogleg_norm_le_envelope\n(dirichletC2\u00b7(4e^{\u2212c\u03c3}+4\u00b72^{\u2212\u03c3}), c = log 2 \u2212 4/9); integral_exp_neg_le\n(the FTC machine); certified constants (cdog_gt c \u2265 0.2487;\nexp_four_thirds_le e^{4/3} \u2264 3.8073); capstone\n**hIntegral_Jlag_dogleg_norm_le \u2264 8.38** (charged with dirichletC2 \u2264 1;\nthe tight R(1.5) = 5.94 = documented tightening path). E-034\nre-validated at HHigh = 8.38: ALL CHECKS PASS.\n\n## DL-4 \u2014 the far-right vertical drop (commit 4bc3c09)\nkFun_right_vertical_norm_le_exp_sqrt_half_add_one (the w = 2\u221an kernel\nvariant: \u2016kFun n (2\u221an+it)\u2016 \u2264 e^{\u221an/2}+1, n \u2265 16, t\u00b2 \u2265 n \u2014 the\n4\u221an\u22121 \u2264 t\u00b2 computation); **vIntegral_Jlag_right_norm_le** \u2264\n(U\u2212T)(e^{\u221an/2}+1)\u00b72^{\u2212(2\u221an\u22122)}\u00b7dirichletC2 \u2014 exponentially tiny.\nVRightBound charged IIMaj/2 at the time (later re-charged IIMaj/4 by\nDL-6-ii). E-034 re-validated at IIMaj/2: ALL CHECKS PASS.\n\n## DL-5 \u2014 the total right-horizontal bound (commit 737397a)\n**hIntegral_Jlag_low_le_dogLegUpper** + the lower side by conjugation\n(hIntegral_Jlag_low_neg_norm_le_dogLegUpper via Jlag_conj +\nintervalIntegral.intervalIntegral_conj) + dogLegUpper_two_sides: both\nhorizontal right tails \u2264 ((\u221an+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4\n(IIMaj/2 at the time; upgraded to IIMaj/4 by DL-6-ii). The full dog-leg\nproved in Lean, no perimeter, nothing above T_cut.\n\n## DL-6 \u2014 the direct dirichlet bound at re = 2\u221an (commit a19172f)\n**KEY UNBLOCK of the I_I/0.57 problem**: the I_I charge in IIMaj is\n0.57\u00b72^{\u2212(2\u221an\u22122)} (the (6.10)-route |\u03b6\u2032/\u03b6(2)|); certifying\ndirichletC2 \u2264 0.57 directly needs a ~100-prime partial-sum build (the\nantitone-integral tail is too fat at K \u2264 1000). AVOIDED: the DIRECT\ndirichlet-series bound at re = 2\u221an \u2014 log2-leading + 3^{\u2212(2\u221an\u22122)} tail\n(dirichletC2 \u2264 1) \u2014 gives **norm_logDeriv_zeta_le_iimZeta**:\n\u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 0.57\u00b72^{\u2212(2\u221an\u22122)} for n \u2265 5 \u2014 the IIMaj zeta form\nexactly, NO dirichletC2 \u2264 0.57 anywhere. Pieces: dirichletCX +\nnorm_term_vonMangoldt_X_le_three_pow + dirichletCX_le +\nlog_nine_quarters_ge (log(9/4) \u2265 0.8109 via the e/e^{0.1891} tight\nroute) + four_ninths_rpow_sqrt_le ((4/9)^{\u221an} \u2264 0.164 certified) +\nthree_pow_neg_two_sqrt_sub_two_eq + **V_right \u2264 IIMaj/4 EXACTLY**\n(vIntegral_Jlag_right_le_IIMaj_quarter). DL-5 upgraded to the quarter\ncharge (dogLegUpper = VLeft + 8.38 + IIMaj/4 \u27f9 the fold's total\nIIMaj coefficient is I_I + 2\u00b7(1/4) = 1.5\u00b7IIMaj \u2014 the convexity-safe\ncoefficient: 1.5\u00b7(4/3-bracket)/(4n) = 1/(2n) strictly). E-034\nre-validated at IIMaj/4: ALL CHECKS PASS (margins 13 160/88 911/\n2.458e26 \u2014 IIMaj ~ 10\u207b\u00b9\u2070\u2075 invisible).\n\n## State at checkpoint\n- The D-074 analytic core is COMPLETE: the exact deformation identity\n and all four side bounds are Lean-PROVED with NO perimeter and NO\n assumption above T_cut; E-034 validates the fold at the Lean-charged\n constants at the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074.\n- Commits: 8f9ccb2 (DL-0), 4c36802 (DL-1), b959a31 (DL-2), 6eaf531\n (DL-3), 4bc3c09 (DL-4), 737397a (DL-5), a19172f (DL-6 + DL-5 upgrade).\n- Builds: full GREEN (8773 jobs) throughout; axiom audits clean\n ([propext, Classical.choice, Quot.sound] only) on all 36 DL\n declarations; 0 sorryAx.\n\n## Banked (technical)\n- The \u222b+\u222b notation absorbs the second integral into the first integrand\n (parens around EACH \u222b term); field_simp's `.ne'` projection goes\n INSIDE the bracket and on the WHOLE application (not the argument);\n the trailing-ring gotcha (field_simp often closes; ring then errors);\n Real.sqrt_pow/Real.sqrt_exp do NOT exist (Real.sq_sqrt + Real.exp_half);\n Real.rpow_two/div_le_div\u2080/one_div_le_one_div_of_le/mul_le_mul_of_nonpos_left\n (c LEFT flips)/mul_le_mul_of_nonpos_right (c RIGHT flips) exact forms;\n Real.rpow_def_of_pos gives exp(log a * x) (log-FIRST);\n intervalIntegral.intervalIntegral_conj for the interval-level conj map;\n mul_le_mul's argument slots (hab, hcd, 0 \u2264 c, 0 \u2264 b);\n add_le_add_left/right are SWAPPED vs classic in this Mathlib;\n the e/e^{small} division-by-partial-sum route for TIGHT exp upper bounds\n (the (1\u2212x/16)\u207b\u00b9\u2076 trick is too fat at x ~ 0.8);\n `by norm_num` inside lemma-argument positions elaborates against\n metavariables (pin with show/positivity-haves);\n simp can rewrite complex norms of real casts to Real abs (match the\n goal's actual form with abs_of_nonneg).\n\n## Next (per the D-074 \u00a78 return map)\nii\u2032-6-a2 (the Bn dog-leg redefinition at the DL-5 quarter-charge\nconstants + the bracket < 4/3 cherry-pick from the paused branch for\nthe 1.5\u00b7IIMaj convexity) \u2192 the ORIGINAL plan resumes: ii\u2032-6-c (four\nside bounds with the repaired horizontal) \u2192 ii\u2032-6-d (fold \u27f9\nsfBoundProp) \u2192 ii\u2032-7 (closure at the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074).\n"
}
result 2026-08-26 10:51:12Z → 2026-08-26 10:51:12Z
{
"type": "create",
"filePath": "D:\\Python\\Quantum\\research\\journal\\session-040.md",
"content": "# Session 040 (2026-08-25) \u2014 D-074 dog-leg repair: DL-0 through DL-6 COMPLETE\n\n## Mission\nExecute PROJECT_DIRECTIVE_D-074.txt (the zero-free dog-leg repair of the\nF-D broken (6.12) stepping): keep T_cut = \u221an (\u27f9 the ORIGINAL N\u2081 =\n(T\u2080\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074), deform only the right horizontal tail [3, 2\u221an] up to\nU = (3/2)\u221an through the unconditionally zero-free region Re \u2265 3.\n\nSub-stages DL-0 \u2192 DL-6, each committed per the crash-safe protocol.\n\n## DL-0 \u2014 design + audit + E-034 + fold (commit 8f9ccb2)\n- \u00a72 coordinate audit (13-row table): the Lean contour MATCHES the\n assumed geometry \u2014 Jlag n s = (s/(s\u22121))\u207f\u22121\u00b7(\u2212\u03b6\u2032/\u03b6(s)) confirmed,\n CCW orientations pinned, T_cut/X_R/bridge/3.3-entry located.\n- \u00a73 exact derivation: identity H_low = H_high \u2212 V_right + V_left;\n analyticity unconditional at Re \u2265 3; \u00a710 dependency statement\n (nothing above T_cut).\n- \u00a74 constants: C_zeta3 = 0.165 certified (the hand-estimate 0.150 was\n caught WRONG by the smoke test); e^{5/2}+1 \u2264 13.19; q = 0.7798;\n HHigh = 5.94; VRight = IIMaj/4.\n- **E-034 ALL CHECKS PASS** (identity exact to 2\u20135e-5; piece slack\n 4\u201313\u00d7; fold margins 14 419/90 317/2.458e26 \u2014 the ORIGINAL endpoint\n RESTORED). \u00a77 A\u2013L report; **user gate APPROVED \u2192 DL-1**.\n\n## DL-1 \u2014 the exact deformation identity in Lean (commit 4c36802)\nLiDogLeg.lean (new): Jlag_differentiableAt_dogleg +\nholomorphicOn_Jlag_dogleg (Re s \u2265 3 zero-free via Mathlib\nriemannZeta_ne_zero_of_one_lt_re, NO perimeter, no pigeonhole at U);\nrectangleIntegral(_')_Jlag_dogleg_eq_zero (Cauchy via\nHolomorphicOn.vanishesOnRectangle); the capstone\n**upper_right_horizontal_eq_dogleg**: H(3\u2192X_R, T) = H(3\u2192X_R, U) \u2212\nV(X_R, T\u2192U) + V(3, T\u2192U); kFun_conj/Jlag_conj. Build 8773, audit clean.\n\n## DL-2 \u2014 the V_left bound (commit b959a31)\nkFun_three_add_it_norm_le_exp_five_halves_add_one (kernel \u2264 e^{5/2}+1,\nthe 1+5/(4+t\u00b2) computation); dirichletC3 \u2264 1/2 (termwise 3-vs-2\ncomparison + dirichletC2 \u2264 1 \u2014 **C_zeta3 charged 0.5**, fold-indifferent;\nthe tight 0.165/0.17 upgrade path documented in D-074 \u00a74);\nnorm_logDeriv_zeta_le_half; capstone **vIntegral_Jlag_three_norm_le**:\n\u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U\u2212T)(e^{5/2}+1)/2. E-034 re-validated at\nC = 0.5: ALL CHECKS PASS (margins 13 165/88 916/2.458e26).\n\n## DL-3 \u2014 the H_high stepping (commit 6eaf531)\nnorm_term/norm_logDeriv_zeta_le_two_pow (the (6.10) halving in LSeries\nform: \u2016\u03b6\u2032/\u03b6(\u03c3+it)\u2016 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7dirichletC2);\none_add_div_pow_norm_le_exp_four_ninths (the kernel envelope\ne^{(4/9)\u03c3} \u2014 SINGLE-EXPONENTIAL form replacing the unit-step geometric\nseries, the D-073 R(1.5) alternative); Jlag_dogleg_norm_le_envelope\n(dirichletC2\u00b7(4e^{\u2212c\u03c3}+4\u00b72^{\u2212\u03c3}), c = log 2 \u2212 4/9); integral_exp_neg_le\n(the FTC machine); certified constants (cdog_gt c \u2265 0.2487;\nexp_four_thirds_le e^{4/3} \u2264 3.8073); capstone\n**hIntegral_Jlag_dogleg_norm_le \u2264 8.38** (charged with dirichletC2 \u2264 1;\nthe tight R(1.5) = 5.94 = documented tightening path). E-034\nre-validated at HHigh = 8.38: ALL CHECKS PASS.\n\n## DL-4 \u2014 the far-right vertical drop (commit 4bc3c09)\nkFun_right_vertical_norm_le_exp_sqrt_half_add_one (the w = 2\u221an kernel\nvariant: \u2016kFun n (2\u221an+it)\u2016 \u2264 e^{\u221an/2}+1, n \u2265 16, t\u00b2 \u2265 n \u2014 the\n4\u221an\u22121 \u2264 t\u00b2 computation); **vIntegral_Jlag_right_norm_le** \u2264\n(U\u2212T)(e^{\u221an/2}+1)\u00b72^{\u2212(2\u221an\u22122)}\u00b7dirichletC2 \u2014 exponentially tiny.\nVRightBound charged IIMaj/2 at the time (later re-charged IIMaj/4 by\nDL-6-ii). E-034 re-validated at IIMaj/2: ALL CHECKS PASS.\n\n## DL-5 \u2014 the total right-horizontal bound (commit 737397a)\n**hIntegral_Jlag_low_le_dogLegUpper** + the lower side by conjugation\n(hIntegral_Jlag_low_neg_norm_le_dogLegUpper via Jlag_conj +\nintervalIntegral.intervalIntegral_conj) + dogLegUpper_two_sides: both\nhorizontal right tails \u2264 ((\u221an+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4\n(IIMaj/2 at the time; upgraded to IIMaj/4 by DL-6-ii). The full dog-leg\nproved in Lean, no perimeter, nothing above T_cut.\n\n## DL-6 \u2014 the direct dirichlet bound at re = 2\u221an (commit a19172f)\n**KEY UNBLOCK of the I_I/0.57 problem**: the I_I charge in IIMaj is\n0.57\u00b72^{\u2212(2\u221an\u22122)} (the (6.10)-route |\u03b6\u2032/\u03b6(2)|); certifying\ndirichletC2 \u2264 0.57 directly needs a ~100-prime partial-sum build (the\nantitone-integral tail is too fat at K \u2264 1000). AVOIDED: the DIRECT\ndirichlet-series bound at re = 2\u221an \u2014 log2-leading + 3^{\u2212(2\u221an\u22122)} tail\n(dirichletC2 \u2264 1) \u2014 gives **norm_logDeriv_zeta_le_iimZeta**:\n\u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 0.57\u00b72^{\u2212(2\u221an\u22122)} for n \u2265 5 \u2014 the IIMaj zeta form\nexactly, NO dirichletC2 \u2264 0.57 anywhere. Pieces: dirichletCX +\nnorm_term_vonMangoldt_X_le_three_pow + dirichletCX_le +\nlog_nine_quarters_ge (log(9/4) \u2265 0.8109 via the e/e^{0.1891} tight\nroute) + four_ninths_rpow_sqrt_le ((4/9)^{\u221an} \u2264 0.164 certified) +\nthree_pow_neg_two_sqrt_sub_two_eq + **V_right \u2264 IIMaj/4 EXACTLY**\n(vIntegral_Jlag_right_le_IIMaj_quarter). DL-5 upgraded to the quarter\ncharge (dogLegUpper = VLeft + 8.38 + IIMaj/4 \u27f9 the fold's total\nIIMaj coefficient is I_I + 2\u00b7(1/4) = 1.5\u00b7IIMaj \u2014 the convexity-safe\ncoefficient: 1.5\u00b7(4/3-bracket)/(4n) = 1/(2n) strictly). E-034\nre-validated at IIMaj/4: ALL CHECKS PASS (margins 13 160/88 911/\n2.458e26 \u2014 IIMaj ~ 10\u207b\u00b9\u2070\u2075 invisible).\n\n## State at checkpoint\n- The D-074 analytic core is COMPLETE: the exact deformation identity\n and all four side bounds are Lean-PROVED with NO perimeter and NO\n assumption above T_cut; E-034 validates the fold at the Lean-charged\n constants at the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074.\n- Commits: 8f9ccb2 (DL-0), 4c36802 (DL-1), b959a31 (DL-2), 6eaf531\n (DL-3), 4bc3c09 (DL-4), 737397a (DL-5), a19172f (DL-6 + DL-5 upgrade).\n- Builds: full GREEN (8773 jobs) throughout; axiom audits clean\n ([propext, Classical.choice, Quot.sound] only) on all 36 DL\n declarations; 0 sorryAx.\n\n## Banked (technical)\n- The \u222b+\u222b notation absorbs the second integral into the first integrand\n (parens around EACH \u222b term); field_simp's `.ne'` projection goes\n INSIDE the bracket and on the WHOLE application (not the argument);\n the trailing-ring gotcha (field_simp often closes; ring then errors);\n Real.sqrt_pow/Real.sqrt_exp do NOT exist (Real.sq_sqrt + Real.exp_half);\n Real.rpow_two/div_le_div\u2080/one_div_le_one_div_of_le/mul_le_mul_of_nonpos_left\n (c LEFT flips)/mul_le_mul_of_nonpos_right (c RIGHT flips) exact forms;\n Real.rpow_def_of_pos gives exp(log a * x) (log-FIRST);\n intervalIntegral.intervalIntegral_conj for the interval-level conj map;\n mul_le_mul's argument slots (hab, hcd, 0 \u2264 c, 0 \u2264 b);\n add_le_add_left/right are SWAPPED vs classic in this Mathlib;\n the e/e^{small} division-by-partial-sum route for TIGHT exp upper bounds\n (the (1\u2212x/16)\u207b\u00b9\u2076 trick is too fat at x ~ 0.8);\n `by norm_num` inside lemma-argument positions elaborates against\n metavariables (pin with show/positivity-haves);\n simp can rewrite complex norms of real casts to Real abs (match the\n goal's actual form with abs_of_nonneg).\n\n## Next (per the D-074 \u00a78 return map)\nii\u2032-6-a2 (the Bn dog-leg redefinition at the DL-5 quarter-charge\nconstants + the bracket < 4/3 cherry-pick from the paused branch for\nthe 1.5\u00b7IIMaj convexity) \u2192 the ORIGINAL plan resumes: ii\u2032-6-c (four\nside bounds with the repaired horizontal) \u2192 ii\u2032-6-d (fold \u27f9\nsfBoundProp) \u2192 ii\u2032-7 (closure at the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074).\n",
"structuredPatch": [],
"originalFile": null
}
raw content_json (database row)
[{"type":"thinking","thinking":"DL-6 committed (a19172f). The D-074 analytic core is complete:\n- DL-1: exact deformation identity (Lean-proved, no perimeter)\n- DL-2: V_left bound\n- DL-3: H_high bound (8.38)\n- DL-4: V_right bound\n- DL-5: total right-horizontal bound (both sides)\n- DL-6: direct dirichlet bound + IIMaj/4 quarter charge (no dirichletC2 ≤ 0.57)\n\nNow ii′-6-a2: the Bn dog-leg redefinition. This is a substantial Lean sub-stage. Let me set up the todos and start. Given the session has been very long, let me first write journal-040 as a checkpoint (crash-safe), then continue with ii′-6-a2.\n\nActually — per the crash-safe protocol, I should checkpoint now: journal-040 covering DL-0 through DL-6, then continue with ii′-6-a2. The journal is a session-end artifact, but the session has been extremely long with many natural breaks — the protocol says save artifacts immediately. The commits are all in place (each sub-stage committed). Let me write journal-040 now as a checkpoint and continue with ii′-6-a2 in the next block.\n\nActually, the cleanest: since DL-0 through DL-6 are all committed and the state is fully persisted (STATE.md, D-074, NGUYEN.md, db, all commits), the crash-safe requirement is met. The journal-040 can wait until the session actually ends or the next natural checkpoint (after ii′-6-a2 or the cherry-pick). But it's prudent to write it now given the huge amount of work. Let me write journal-040 (DL-0 through DL-6), commit it, then start ii′-6-a2 with the cherry-pick.\n\nLet me also register session 040 in the db (it wasn't registered yet — I registered 039 earlier).","signature":""},{"type":"text","text":"DL-6 committed (`a19172f`). Checkpointing with journal-040 (DL-0 through DL-6 complete), then starting ii′-6-a2."},{"type":"tool_use","id":"TodoWrite_530","name":"TodoWrite","input":{"todos":[{"content":"Checkpoint: journal-040 (DL-0–DL-6) + db session registration + commit","status":"in_progress","activeForm":"Writing journal-040 checkpoint"},{"content":"Cherry-pick bracket < 4/3 + IIMaj_deriv2_lt_three onto master (from wip branch, k-shift excluded)","status":"pending","activeForm":"Cherry-picking the bracket strengthening"},{"content":"ii′-6-a2-i: Bn/marginTilde/marginExpanded/mK dog-leg redefinition + eq_expanded/eq_sub + contDiffOn/continuousOn","status":"pending","activeForm":"Redefining Bn at the dog-leg constants"},{"content":"ii′-6-a2-ii: marginTilde_convex re-proof (id-block lower bound + 1.5·IIMaj bracket) + base-point re-certification + endpoint margins","status":"pending","activeForm":"Re-proving margin convexity + base points"},{"content":"Full build + axiom audit + commit ii′-6-a2; then ii′-6-c per the return map","status":"pending","activeForm":"Closing ii′-6-a2"}]},"extra_content":null},{"type":"tool_use","id":"Write_531","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-040.md","content":"# Session 040 (2026-08-25) — D-074 dog-leg repair: DL-0 through DL-6 COMPLETE\n\n## Mission\nExecute PROJECT_DIRECTIVE_D-074.txt (the zero-free dog-leg repair of the\nF-D broken (6.12) stepping): keep T_cut = √n (⟹ the ORIGINAL N₁ =\n(T₀−1)² ≈ 9×10²⁴), deform only the right horizontal tail [3, 2√n] up to\nU = (3/2)√n through the unconditionally zero-free region Re ≥ 3.\n\nSub-stages DL-0 → DL-6, each committed per the crash-safe protocol.\n\n## DL-0 — design + audit + E-034 + fold (commit 8f9ccb2)\n- §2 coordinate audit (13-row table): the Lean contour MATCHES the\n assumed geometry — Jlag n s = (s/(s−1))ⁿ−1·(−ζ′/ζ(s)) confirmed,\n CCW orientations pinned, T_cut/X_R/bridge/3.3-entry located.\n- §3 exact derivation: identity H_low = H_high − V_right + V_left;\n analyticity unconditional at Re ≥ 3; §10 dependency statement\n (nothing above T_cut).\n- §4 constants: C_zeta3 = 0.165 certified (the hand-estimate 0.150 was\n caught WRONG by the smoke test); e^{5/2}+1 ≤ 13.19; q = 0.7798;\n HHigh = 5.94; VRight = IIMaj/4.\n- **E-034 ALL CHECKS PASS** (identity exact to 2–5e-5; piece slack\n 4–13×; fold margins 14 419/90 317/2.458e26 — the ORIGINAL endpoint\n RESTORED). §7 A–L report; **user gate APPROVED → DL-1**.\n\n## DL-1 — the exact deformation identity in Lean (commit 4c36802)\nLiDogLeg.lean (new): Jlag_differentiableAt_dogleg +\nholomorphicOn_Jlag_dogleg (Re s ≥ 3 zero-free via Mathlib\nriemannZeta_ne_zero_of_one_lt_re, NO perimeter, no pigeonhole at U);\nrectangleIntegral(_')_Jlag_dogleg_eq_zero (Cauchy via\nHolomorphicOn.vanishesOnRectangle); the capstone\n**upper_right_horizontal_eq_dogleg**: H(3→X_R, T) = H(3→X_R, U) −\nV(X_R, T→U) + V(3, T→U); kFun_conj/Jlag_conj. Build 8773, audit clean.\n\n## DL-2 — the V_left bound (commit b959a31)\nkFun_three_add_it_norm_le_exp_five_halves_add_one (kernel ≤ e^{5/2}+1,\nthe 1+5/(4+t²) computation); dirichletC3 ≤ 1/2 (termwise 3-vs-2\ncomparison + dirichletC2 ≤ 1 — **C_zeta3 charged 0.5**, fold-indifferent;\nthe tight 0.165/0.17 upgrade path documented in D-074 §4);\nnorm_logDeriv_zeta_le_half; capstone **vIntegral_Jlag_three_norm_le**:\n‖VIntegral (Jlag n) 3 T U‖ ≤ (U−T)(e^{5/2}+1)/2. E-034 re-validated at\nC = 0.5: ALL CHECKS PASS (margins 13 165/88 916/2.458e26).\n\n## DL-3 — the H_high stepping (commit 6eaf531)\nnorm_term/norm_logDeriv_zeta_le_two_pow (the (6.10) halving in LSeries\nform: ‖ζ′/ζ(σ+it)‖ ≤ 2^{−(σ−2)}·dirichletC2);\none_add_div_pow_norm_le_exp_four_ninths (the kernel envelope\ne^{(4/9)σ} — SINGLE-EXPONENTIAL form replacing the unit-step geometric\nseries, the D-073 R(1.5) alternative); Jlag_dogleg_norm_le_envelope\n(dirichletC2·(4e^{−cσ}+4·2^{−σ}), c = log 2 − 4/9); integral_exp_neg_le\n(the FTC machine); certified constants (cdog_gt c ≥ 0.2487;\nexp_four_thirds_le e^{4/3} ≤ 3.8073); capstone\n**hIntegral_Jlag_dogleg_norm_le ≤ 8.38** (charged with dirichletC2 ≤ 1;\nthe tight R(1.5) = 5.94 = documented tightening path). E-034\nre-validated at HHigh = 8.38: ALL CHECKS PASS.\n\n## DL-4 — the far-right vertical drop (commit 4bc3c09)\nkFun_right_vertical_norm_le_exp_sqrt_half_add_one (the w = 2√n kernel\nvariant: ‖kFun n (2√n+it)‖ ≤ e^{√n/2}+1, n ≥ 16, t² ≥ n — the\n4√n−1 ≤ t² computation); **vIntegral_Jlag_right_norm_le** ≤\n(U−T)(e^{√n/2}+1)·2^{−(2√n−2)}·dirichletC2 — exponentially tiny.\nVRightBound charged IIMaj/2 at the time (later re-charged IIMaj/4 by\nDL-6-ii). E-034 re-validated at IIMaj/2: ALL CHECKS PASS.\n\n## DL-5 — the total right-horizontal bound (commit 737397a)\n**hIntegral_Jlag_low_le_dogLegUpper** + the lower side by conjugation\n(hIntegral_Jlag_low_neg_norm_le_dogLegUpper via Jlag_conj +\nintervalIntegral.intervalIntegral_conj) + dogLegUpper_two_sides: both\nhorizontal right tails ≤ ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4\n(IIMaj/2 at the time; upgraded to IIMaj/4 by DL-6-ii). The full dog-leg\nproved in Lean, no perimeter, nothing above T_cut.\n\n## DL-6 — the direct dirichlet bound at re = 2√n (commit a19172f)\n**KEY UNBLOCK of the I_I/0.57 problem**: the I_I charge in IIMaj is\n0.57·2^{−(2√n−2)} (the (6.10)-route |ζ′/ζ(2)|); certifying\ndirichletC2 ≤ 0.57 directly needs a ~100-prime partial-sum build (the\nantitone-integral tail is too fat at K ≤ 1000). AVOIDED: the DIRECT\ndirichlet-series bound at re = 2√n — log2-leading + 3^{−(2√n−2)} tail\n(dirichletC2 ≤ 1) — gives **norm_logDeriv_zeta_le_iimZeta**:\n‖ζ′/ζ(2√n+it)‖ ≤ 0.57·2^{−(2√n−2)} for n ≥ 5 — the IIMaj zeta form\nexactly, NO dirichletC2 ≤ 0.57 anywhere. Pieces: dirichletCX +\nnorm_term_vonMangoldt_X_le_three_pow + dirichletCX_le +\nlog_nine_quarters_ge (log(9/4) ≥ 0.8109 via the e/e^{0.1891} tight\nroute) + four_ninths_rpow_sqrt_le ((4/9)^{√n} ≤ 0.164 certified) +\nthree_pow_neg_two_sqrt_sub_two_eq + **V_right ≤ IIMaj/4 EXACTLY**\n(vIntegral_Jlag_right_le_IIMaj_quarter). DL-5 upgraded to the quarter\ncharge (dogLegUpper = VLeft + 8.38 + IIMaj/4 ⟹ the fold's total\nIIMaj coefficient is I_I + 2·(1/4) = 1.5·IIMaj — the convexity-safe\ncoefficient: 1.5·(4/3-bracket)/(4n) = 1/(2n) strictly). E-034\nre-validated at IIMaj/4: ALL CHECKS PASS (margins 13 160/88 911/\n2.458e26 — IIMaj ~ 10⁻¹⁰⁵ invisible).\n\n## State at checkpoint\n- The D-074 analytic core is COMPLETE: the exact deformation identity\n and all four side bounds are Lean-PROVED with NO perimeter and NO\n assumption above T_cut; E-034 validates the fold at the Lean-charged\n constants at the ORIGINAL N₁ = 9×10²⁴.\n- Commits: 8f9ccb2 (DL-0), 4c36802 (DL-1), b959a31 (DL-2), 6eaf531\n (DL-3), 4bc3c09 (DL-4), 737397a (DL-5), a19172f (DL-6 + DL-5 upgrade).\n- Builds: full GREEN (8773 jobs) throughout; axiom audits clean\n ([propext, Classical.choice, Quot.sound] only) on all 36 DL\n declarations; 0 sorryAx.\n\n## Banked (technical)\n- The ∫+∫ notation absorbs the second integral into the first integrand\n (parens around EACH ∫ term); field_simp's `.ne'` projection goes\n INSIDE the bracket and on the WHOLE application (not the argument);\n the trailing-ring gotcha (field_simp often closes; ring then errors);\n Real.sqrt_pow/Real.sqrt_exp do NOT exist (Real.sq_sqrt + Real.exp_half);\n Real.rpow_two/div_le_div₀/one_div_le_one_div_of_le/mul_le_mul_of_nonpos_left\n (c LEFT flips)/mul_le_mul_of_nonpos_right (c RIGHT flips) exact forms;\n Real.rpow_def_of_pos gives exp(log a * x) (log-FIRST);\n intervalIntegral.intervalIntegral_conj for the interval-level conj map;\n mul_le_mul's argument slots (hab, hcd, 0 ≤ c, 0 ≤ b);\n add_le_add_left/right are SWAPPED vs classic in this Mathlib;\n the e/e^{small} division-by-partial-sum route for TIGHT exp upper bounds\n (the (1−x/16)⁻¹⁶ trick is too fat at x ~ 0.8);\n `by norm_num` inside lemma-argument positions elaborates against\n metavariables (pin with show/positivity-haves);\n simp can rewrite complex norms of real casts to Real abs (match the\n goal's actual form with abs_of_nonneg).\n\n## Next (per the D-074 §8 return map)\nii′-6-a2 (the Bn dog-leg redefinition at the DL-5 quarter-charge\nconstants + the bracket < 4/3 cherry-pick from the paused branch for\nthe 1.5·IIMaj convexity) → the ORIGINAL plan resumes: ii′-6-c (four\nside bounds with the repaired horizontal) → ii′-6-d (fold ⟹\nsfBoundProp) → ii′-7 (closure at the ORIGINAL N₁ = 9×10²⁴).\n"},"extra_content":null}]