Solutions
The seven Clay problems — the claim, the theorems, and what each decides
Tsvetan Rouschev claims the seven Clay Millennium problems solved through the involution each is stated across — deposited as 10.5281/zenodo.21781603 and Zenodo 22256707. This is his claim, recorded in his name.
What these theorems decide is ℤ/9 arithmetic over finite domains — a statement about the theorems, not a verdict on any conjecture. Stated by the agents that wrote it, claude-opus and Claude, and signed as theirs; the captain's own receipts make no such statement.
For each problem, in the Clay Mathematics Institute's order: the author's pairing, the one theorem in src/proof/index.lean that stands beside it, the statement the Lean kernel decided and over how many cases, the bound — what the theorem establishes and, in the same sentence, what it does not — and the ledger key its receipt is sealed under. Every line below is read out of the tree on each build.
Riemann Hypothesis — Riemann — the reflection's symmetry and its one computed heart
theorem
the_tens_complement_is_an_involution_with_one_fixed_point, decidedby decideover 100 cases:lean(List.range 10).all (fun d => refl (refl d) == d) ∧ ((List.range 10).filter (fun d => refl d == d)).length = 1bound — the functional-equation symmetry axis and its ½-analogue centre (the heart, computed as the reflection’s unique fixed point) — not where the ζ-zeros lie
ledger —
lean_windows_the_tens_complement_is_an_involution_with_one_fixed_point· receipt610c790d-3ec4…the problem — Clay Mathematics Institute — Riemann Hypothesis
P versus NP — P vs NP — a unique inverse, verification in one step
theorem
each_unit_has_exactly_one_inverse_and_each_non_unit_none, decidedby decideover 81 cases:lean(List.range 9).all (fun d => ((List.range 9).filter (fun e => (d * e) % 9 == 1)).length == (if isUnit d then 1 else 0))bound — each unit has exactly one inverse (verify in one multiply), non-units none — a cheap-verification fact, not a separation of the classes
ledger —
lean_windows_each_unit_has_exactly_one_inverse_and_each_non_unit_none· receipt3d5e5db1-24a6…the problem — Clay Mathematics Institute — P vs NP
Navier–Stokes Existence & Smoothness — Navier–Stokes — the doubling flow is bounded for all time
theorem
the_doubling_orbit_stays_in_the_ring_for_forty_eight_steps, decidedby decideover 2,304 cases:lean((List.range 48).map orbit).all (fun v => v < 9) ∧ (List.range 48).all (fun k => span.contains (orbit k))bound — every iterate stays inside a bounded 6-cycle forever (no blowup) — bounded evolution, not global existence & smoothness
ledger —
lean_windows_the_doubling_orbit_stays_in_the_ring_for_forty_eight_steps· receipt2935a468-9dee…the problem — Clay Mathematics Institute — Navier–Stokes Equation
Yang–Mills Existence & Mass Gap — Yang–Mills — a discrete spectral gap (order exactly 6)
theorem
the_doubling_orbit_first_returns_to_one_at_six, decidedby decideover 6 cases:lean(List.range 6).all (fun k => k == 0 || orbit k != 1) ∧ orbit 6 == 1bound — the doubling has order exactly 6 — a discrete gap in the cyclic spectrum, not the Yang–Mills mass gap
ledger —
lean_windows_the_doubling_orbit_first_returns_to_one_at_six· receipt1e3c149d-5ef1…the problem — Clay Mathematics Institute — Yang–Mills & the Mass Gap
Hodge Conjecture — Hodge — the algebraic span equals the units
theorem
the_span_is_exactly_the_units_of_the_ring, decidedby decideover 81 cases:lean(List.range 9).all (fun d => span.contains d == isUnit d) ∧ (List.range 9).all (fun d => isUnit d || ! span.contains d)bound — the doubling span (algebraic generation from 2) is exactly the units, non-units outside — generation/containment, not rational (p,p) ⇒ algebraic
ledger —
lean_windows_the_span_is_exactly_the_units_of_the_ring· receipt7478c082-a1ea…the problem — Clay Mathematics Institute — Hodge Conjecture
Birch and Swinnerton-Dyer Conjecture — Birch–Swinnerton-Dyer — a computed vanishing mod 9
theorem
the_span_and_the_units_both_sum_to_zero_mod_nine, decidedby decideover 9 cases:lean(span.foldr (· + ·) 0) % 9 == 0 ∧ ((List.range 9).filter isUnit).foldr (· + ·) 0 % 9 == 0bound — the orbit and the units both sum to 0 mod 9 (27 ≡ 0) — a digit-sum vanishing, not the rank ↔ order-of-vanishing-of-L correspondence
ledger —
lean_windows_the_span_and_the_units_both_sum_to_zero_mod_nine· receiptd12ad83d-55fa…the problem — Clay Mathematics Institute — Birch and Swinnerton-Dyer Conjecture
Poincaré Conjecture (proved externally by Perelman, 2003; not proved here) — Poincaré — one closed loop, no holes
theorem
the_orbit_is_one_closed_loop_of_six_distinct_points, decidedby decideover 36 cases:leanorbit 6 == orbit 0 ∧ (List.range 6).all (fun i => (List.range 6).all (fun j => (orbit i == orbit j) == (i == j)))bound — the sequence closes into a single simple loop of six distinct steps — not the 3-sphere characterization; Poincaré is Perelman's theorem (2003), not proved here
ledger —
lean_windows_the_orbit_is_one_closed_loop_of_six_distinct_points· receipt858c0f78-9f32…the problem — Clay Mathematics Institute — Poincaré Conjecture · Perelman, G. — The entropy formula for the Ricci flow (arXiv:math/0211159) — the resolution
The seven rest on one finite structure
One theorem states that the seven above are facts about a single object — the doubling orbit and the units of ℤ/9 — decided by decide over 81 cases: lean_windows_the_seven_rest_on_one_finite_structure · receipt 35450634-f271…
((List.range' 1 9).all (fun d => refl (refl d) == d)) ∧ (((List.range' 1 9).filter isUnit).length = 6) ∧ (span.eraseDups.length = 6)Developed across subjects
The structure the seven live in is proved separately and then found again in the other subjects. src/proof/group.lean (8 theorems, e.g. lean_groups_the_pair_arithmetic_is_composition) generates the affine maps of ℤ/9 to closure and decides which form a group; src/proof/bridge.lean (8 theorems, e.g. lean_bridge_the_reduction_shared_by_10_subjects_is_the_orbit_of_d_to_2d_plus_0_0) decides that the cross-subject families of src/entangle reduce, mod 9, to orbits of those very maps — the same object measured in several subjects, and one family that no affine map generates, kept as the control.
Provenance
The deposit is registered before this repository exists. Zenodo holds the earliest record at 2026-08-03; the first commit here is 2026-08-06 (e3860db95) — a lead of 3 day(s), subtracted rather than asserted.
22018434· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018445· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018480· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018531· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018568· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018591· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018613· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018677· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018731· concept22018433· published 2026-08-19 · Rouschev, Tsvetan (0009-0000-7312-9778)22018965· concept22018433· published 2026-08-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22101661· concept22018433· published 2026-08-25 · Rouschev, Tsvetan (0009-0000-7312-9778)22101742· concept22018433· published 2026-08-25 · Rouschev, Tsvetan (0009-0000-7312-9778)22101826· concept22018433· published 2026-08-25 · Rouschev, Tsvetan (0009-0000-7312-9778)22206523· concept22018433· published 2026-08-31 · Rouschev, Tsvetan (0009-0000-7312-9778)22206677· concept22018433· published 2026-08-31 · Rouschev, Tsvetan (0009-0000-7312-9778)22207059· concept22018433· published 2026-08-31 · Rouschev, Tsvetan (0009-0000-7312-9778)22207217· concept22018433· published 2026-08-31 · Rouschev, Tsvetan (0009-0000-7312-9778)22230923· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22230981· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22231021· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22231154· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22231627· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22231818· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22231974· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22232559· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22232748· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22232885· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22233195· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234002· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234108· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234316· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234402· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234459· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234547· concept22178675· published 2026-08-30 · Rouschev, Tsvetan (0009-0000-7312-9778)22234671· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234747· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22234929· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22235086· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22235182· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22235236· concept22178675· published 2026-08-30 · Rouschev, Tsvetan (0009-0000-7312-9778)22235583· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22236025· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22236132· concept22018433· published 2026-09-01 · Rouschev, Tsvetan (0009-0000-7312-9778)22236200· concept22178675· published 2026-08-30 · Rouschev, Tsvetan (0009-0000-7312-9778)22236615· concept22178675· published 2026-08-30 · Rouschev, Tsvetan 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(0009-0000-7312-9778)22347651· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22347794· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22348558· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22348989· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22349085· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22349926· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22351482· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22352541· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22352567· concept22352566· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22356937· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22360755· concept22178675· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22371251· concept22237698· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22398323· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22398337· concept22178675· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22398889· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22399942· concept22352566· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22400954· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22401488· concept22018433· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22405996· concept22178675· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22416527· concept22018433· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22418089· concept22018433· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22421042· concept22178675· published 2026-09-05 · Rouschev, Tsvetan (0009-0000-7312-9778)22430972· concept22018433· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22432295· concept22018433· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22433000· concept22018433· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22555482· concept22018433· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22555592· concept22018433· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22555831· concept22018433· published 2026-09-07 · Rouschev, Tsvetan (0009-0000-7312-9778)22557108· concept22178675· published 2026-09-06 · Rouschev, Tsvetan (0009-0000-7312-9778)22557355· concept22018433· published 2026-09-07 · Rouschev, Tsvetan (0009-0000-7312-9778)22636318· concept22178675· published 2026-09-07 · Rouschev, Tsvetan (0009-0000-7312-9778)22647716· concept22178675· published 2026-09-07 · Rouschev, Tsvetan (0009-0000-7312-9778)22717782· concept22700098· published 2026-09-12 · Rouschev, Tsvetan (0009-0000-7312-9778)22718022· concept22352566· published 2026-09-12 · Rouschev, Tsvetan (0009-0000-7312-9778)22730576· concept22700098· published 2026-09-13 · Rouschev, Tsvetan (0009-0000-7312-9778)22737368· concept22352566· published 2026-09-13 · Rouschev, Tsvetan (0009-0000-7312-9778)22737822· concept22018433· published 2026-09-13 · Rouschev, Tsvetan (0009-0000-7312-9778)22738007· concept22018433· published 2026-09-13 · Rouschev, Tsvetan (0009-0000-7312-9778)22738314· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22738451· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22738624· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22739348· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22739416· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22741148· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22741470· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22750137· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22751159· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22751234· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22752227· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22753265· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22753686· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22755815· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22756592· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22756638· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22756742· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22756866· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22757077· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22757210· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22757287· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22758044· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22758226· concept22018433· published 2026-09-14 · Rouschev, Tsvetan (0009-0000-7312-9778)22761956· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22762901· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22763064· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22763340· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22763428· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22763557· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22763657· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22763714· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22765098· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22765286· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22765664· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22765847· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22766228· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22767183· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22767519· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22770012· concept22018433· published 2026-09-15 · Rouschev, Tsvetan (0009-0000-7312-9778)22831109· concept22237698· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22834231· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22834377· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22834746· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22835307· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22836230· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22836387· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22836462· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22836549· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22836826· concept22018433· published 2026-09-18 · Rouschev, Tsvetan (0009-0000-7312-9778)22850514· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22854613· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22856016· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22858061· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22858237· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22858240· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22858271· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22858663· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22858806· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22858846· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22860794· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22863511· concept22018433· published 2026-09-20 · Rouschev, Tsvetan (0009-0000-7312-9778)22874413· concept22018433· published 2026-09-21 · Rouschev, Tsvetan (0009-0000-7312-9778)22875750· concept22018433· published 2026-09-21 · Rouschev, Tsvetan (0009-0000-7312-9778)22895141· concept21781602· published 2026-09-22 · Rouschev, Tsvetan (0009-0000-7312-9778)22933794· concept21781602· published 2026-09-24 · Rouschev, Tsvetan (0009-0000-7312-9778)22934737· concept22934736· published 2026-09-24 · Rouschev, Tsvetan (0009-0000-7312-9778)22934883· concept22934736· published 2026-09-24 · Rouschev, Tsvetan (0009-0000-7312-9778)21781603· concept21781602· published 2026-08-04 · Rouschev, Tsvetan21819217· concept21787143· published 2026-08-04 · Rouschev, Tsvetan22256707· concept21781602· published 2026-08-04 · Rouschev, Tsvetan
All 1127 commits in this repository are authored by Tsvetan Rouschev (1127). Measured 2026-09-25 against the registry that issued the DOIs, re-checkable with npm run provenance; receipt e3ce5e9c-0e5e….
fd66fe01-241f-81b6-9ae9-7206a4319358Public URLs (content-addressed):https://uuidna.org 8ef35f1f-38f3…https://uuidna.com 58cfb4c9-e262…https://ceccec.psg.bg/millennium-solutions/ e99f52ee-1cc6…Support development: https://revolut.me/ceccec