Thomson : certificate data for the cap (), part 2 of 2
DefinitionThomsonN7_cap_data_2discrete-geometrythomson-problem
Exact certificate data (part 2 of 2) for the cap () in H. Tran's proof of the Thomson problem for seven electrons. Together the two parts contain the typed three-point certificate tc_cap of the cap cell (paper §6.5, §7.2): the PSD blocks , the class minorants and multipliers , the constants and the sum-of-squares blocks, all exact integer data, together with the one-dimensional coercivity certificates of the cap minorants.
These are plain data definitions (integers, integer polynomials and block encodings); their correctness is not assumed anywhere — the solution of the corresponding cell lemma checks every identity and positivity condition in the Lean kernel. The data are split into two files only to respect the platform's file-size limit, and this part imports part 1.
Formalization Note The code is taken verbatim from the source file, restricted to the declarations needed here; the only edits for this Lean/Mathlib version are the lemma renames ite_eq_left→if_pos, ite_eq_right→if_neg and one rewrite of IsHermitian.isSymm.
Definition code
import Definitions.Def_ThomsonN7_cap_data_1
/-!
# Thomson problem for N = 7 (and the D3h stretch target for N = 9)
Minimise the Coulomb energy `E(x) = ∑_{i<j} 1 / ‖x i - x j‖` over configurations of `N` pairwise
distinct points on the unit sphere `S² ⊂ ℝ³`.
* `N = 7`: the unique minimiser (up to `O(3)` and relabelling) is the regular pentagonal bipyramid,
with `E = 1/2 + 5√2 + 5/(2 sin(π/5)) + 5/(2 sin(2π/5)) = 14.452977414…`.
* `N = 9`: the (numerically known) minimiser is the D3h tricapped trigonal prism, which has one free
parameter `z` (the height of the two triangular faces); numerically `z ≈ 0.70365`,
`E ≈ 25.759986531`.
-/
open Real
namespace ThomsonN7
/-! ## Auxiliary facts (already proved; they show the definitions are well-posed) -/
/- BEGIN M0 -/
namespace Base
open Finset
/-! ### Invariance -/
/-! ### Gram values of the pentagonal bipyramid -/
end Base
/- END M0 -/
/- BEGIN M1 -/
namespace ThreePoint
open Finset Matrix
open scoped RealInnerProductSpace
/-! # Bachoc–Vallentin three-point positivity on `S²` (any block size, any `k`)
`Q3 k u v t = ((1-u²)(1-v²))^{k/2} T_k((t-uv)/√((1-u²)(1-v²)))` via the Chebyshev recursion.
-/
/-! ### Addition theorem (de Moivre) -/
/-! ### Combinatorics of ordered triple sums -/
section Comb
variable {n : ℕ}
end Comb
section Comb2
variable {n : ℕ}
end Comb2
section Comb3
variable {n : ℕ}
end Comb3
section Comb4
variable {n : ℕ}
end Comb4
section Final
variable {n : ℕ}
end Final
/-! ### The kernels at the diagonal `(1,1,1)` -/
section FinalZ
variable {n : ℕ}
end FinalZ
/- BEGIN M5 -/
section Critical
end Critical
section Critical
end Critical
section Critical
end Critical
section CriticalZ
variable {n : ℕ}
end CriticalZ
section CriticalZsym
variable {n : ℕ}
section Perm
variable (τ : Fin n → Fin n → ℝ) (hτ : ∀ i j, τ j i = τ i j) (f : ℝ → ℝ → ℝ → ℝ)
include hτ
end Perm
end CriticalZsym
section CriticalFinal
variable {n : ℕ}
end CriticalFinal
/- END M5 -/
end ThreePoint
/- BEGIN CERT3 -/
/-!
# Kronecker-substitution checking of polynomial identities
A polynomial `e : Ex` in three variables `u, v, t` with integer coefficients is given as an
expression tree. To prove `∀ u v t : ℝ, e(u,v,t) = 0` it suffices to check, with kernel
(GMP) integer arithmetic, that
* every exponent of every variable is `< D`,
* the `ℓ¹`-norm of the (uncollected) coefficients is `< 2^w`,
* the value of `e` at the integers `u = 2^w`, `v = 2^(w D)`, `t = 2^(w D²)` is `0`.
-/
namespace Kron
namespace Ex
end Ex
namespace Ex
end Ex
open Ex
end Kron
/-! ## Builders for `Ex` with evaluation lemmas -/
namespace Kron
namespace Ex
variable (u v t : ℝ)
/-! ### Substitution of variables by variables or `1` -/
end Ex
end Kron
/- BEGIN CERT1 -/
section Cert1Block
open Kron Kron.Ex
namespace Cert
open ThreePoint
/-! ## `Q_k` as a polynomial expression -/
/-! ## Quadratic forms: `L D Lᵀ` plus a diagonally dominant remainder -/
/-! ## Integer data of a positive semidefinite block -/
namespace Blk
end Blk
end Cert
end Cert1Block
/- BEGIN CERT3FILE -/
section Cert3Block
open Kron Kron.Ex
namespace Cert
open ThreePoint
/-! ## Helpers: zero-skipping scaling, monomial-preserving substitution -/
/-! ## The `F`-blocks: `Λ · Fp(u,v)` -/
/-! ## Block matrices and their quadratic forms -/
/-! ## Symmetrisation and the `F`-part of the identity -/
/-! ## SOS blocks with multipliers -/
/-! ## The certificate and its soundness -/
open scoped RealInnerProductSpace
namespace Cert3
end Cert3
namespace Cert3
end Cert3
namespace Cert3
end Cert3
end Cert
end Cert3Block
/- END CERT3 -/
/- BEGIN M2 -/
namespace M2
/-! ## The number field `ℚ(√2, 2 sin (π/5))` -/
namespace K8
/-! ### Rational enclosures and positivity of elements of `K8` -/
end K8
namespace K8
end K8
/-! ### Polynomials with `K8` coefficients: dense lists, lowest degree first -/
namespace Pl
end Pl
open Pl
end M2
/- END M2 -/
/- BEGIN M3 -/
namespace M3
open scoped InnerProductSpace
end M3
/- BEGIN P3ext -/
/-! ## P3: Bregman (Taylor) lower bounds for the Coulomb pair potential (agent7) -/
namespace Base
end Base
/- END P3ext -/
/- BEGIN P1 -/
/-! ## P1: the bipyramid is a critical point of the Coulomb energy on the constraint set (agent7) -/
namespace Reg
open Base
end Reg
/- END P1 -/
/- BEGIN GAUGE -/
/-! ## G: gauge (Procrustes) lemma `exists_gauge` (agent7) -/
namespace Reg
open scoped RealInnerProductSpace
end Reg
/- END GAUGE -/
/- BEGIN P2 -/
/- BEGIN RegB (P2: exact energy identity; needs M0 = namespace Base and the Challenge preamble) -/
namespace RegB
open Base Finset
section vectors
variable (y : Fin 7 → R3)
end vectors
end RegB
/- END RegB -/
/- END P2 -/
/- BEGIN GV -/
namespace GV
open Base
open scoped InnerProductSpace
end GV
/- END GV -/
/- BEGIN TWOREGIME -/
namespace TwoRegime
open scoped InnerProductSpace
end TwoRegime
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
/- END TWOREGIME -/
/- BEGIN TR_D -/
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
/- END TR_D -/
/- BEGIN TR_E -/
namespace ThreePoint
open Finset Matrix
open scoped RealInnerProductSpace
section FinalCut
variable {n : ℕ}
end FinalCut
end ThreePoint
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
/- END TR_E -/
/- BEGIN TR_G -/
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
/- BEGIN TR_G3 -/
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
/- END TR_G3 -/
/- BEGIN TR_G4 -/
namespace TwoRegime
open scoped InnerProductSpace
open Base
end TwoRegime
/- END TR_G4 -/
/- END TR_G -/
/- BEGIN INTERFACES -/
namespace Interfaces
open Finset Base
end Interfaces
/- END INTERFACES -/
/- BEGIN GLUE -/
namespace Glue
open Finset Base Interfaces
end Glue
/- END GLUE -/
/- BEGIN TR_F -/
namespace EPBounds
open Real
end EPBounds
/- END TR_F -/
end ThomsonN7
section RegLocalSection
/- BEGIN REGLOCAL (agent7): perturbative strict local minimality of the pentagonal bipyramid.
Splice AFTER Level1d (needs GAUGE = exists_gauge, P1/P3ext, Base, TwoRegime.LocalMinAt).
Pieces in order: Loc1 (H, B3) | QCore1 (certificate defs/lemmas) | Q2 (expansion) | QCore2 (core) | Loc2 (glue). -/
open Real
open scoped RealInnerProductSpace
namespace ThomsonN7
/- BEGIN LOC1 (agent7): exact chart identity (H) and cubic Bregman lower bound (B3) in h = y - P.
Splice AFTER the GAUGE block (needs P1, P3ext); compiles against Level1b.lean lines 1-3641. -/
namespace Reg
open Base
end Reg
/- END LOC1 -/
end ThomsonN7
/- BEGIN QCORE1 -/
namespace ThomsonN7
namespace Reg
end Reg
end ThomsonN7
/- END QCORE1 -/
/- BEGIN Q2 -/
open Real
open scoped RealInnerProductSpace
namespace ThomsonN7
namespace Reg
open Base
end Reg
end ThomsonN7
/- END Q2 -/
/- BEGIN QCORE2 -/
namespace ThomsonN7
namespace Reg
end Reg
end ThomsonN7
/- END QCORE2 -/
/- BEGIN LOC2 -/
open Real
open scoped RealInnerProductSpace
namespace ThomsonN7
namespace Reg
open Base
/-! ### The atom box for the penalised Hessian certificate -/
end Reg
end ThomsonN7
/- END LOC2 -/
namespace ThomsonN7
namespace Reg
end Reg
end ThomsonN7
/- END REGLOCAL -/
end RegLocalSection
/- BEGIN CASE1 -/
namespace ThomsonN7
namespace CutOneD
end CutOneD
end ThomsonN7
namespace ThomsonN7
namespace Case1Data
open ThomsonN7 ThomsonN7.Cert
end Case1Data
end ThomsonN7
namespace ThomsonN7
namespace Case1
open scoped InnerProductSpace
open ThomsonN7.Cert ThomsonN7.Cert.Cert3
/-! ### Chunked evaluation of the Kronecker check for `Case1Data.cf`
The identity expression `Case1Data.cf.idE` has about 147 000 nodes. Evaluating `chk` on it in one
kernel call is correct but keeps every intermediate kernel term alive until the end of the
declaration. Here the same check is split into separate declarations, one per piece of the
expression (`hE`, four `F`-blocks, eight `S`-blocks). For every expression `e`, `c1Stat w D e`
computes, in one structural pass, the tuple `(ℓ¹-norm, deg u, deg v, deg t, Kronecker value)`
(`c1Stat_eq`). The 13 pieces are evaluated by `decide +kernel` against exact literals, in 13
separate declarations; the statistics of `Case1Data.cf.idE` follow by the composition lemma
`c1Stat_idE` and closed arithmetic on the literals, and `chk` is then decided by its own definition
(`Nat.log2 ℓ¹ + 1 = 179`, `max deg + 1 = 11`, Kronecker value `0`). The statement of `cf_ok` is
unchanged; the literals are data found by a program outside Lean and are only ever checked. -/
/-! The 13 pieces, each evaluated in its own declaration (`w = 179`, `D = 11`). -/
end Case1
end ThomsonN7
/- END CASE1 -/
/- BEGIN CASE2RED -/
namespace ThomsonN7
namespace Case1
section Case2Red
open scoped InnerProductSpace
open Base
end Case2Red
end Case1
end ThomsonN7
/- END CASE2RED -/
section Asm_Typed
open Finset Matrix
open scoped RealInnerProductSpace
namespace ThomsonN7
namespace ThreePoint
/-! # Typed (root-dependent) three-point bound
Every root `i` carries its own kernel `s i`; every pair `{i, j}` carries its own minorant `H i j`.
-/
section TypedRoot
end TypedRoot
section TypedComb
variable {n : ℕ}
end TypedComb
end ThreePoint
end ThomsonN7
end Asm_Typed
section Asm_Typed2
open Finset Matrix
open scoped RealInnerProductSpace
namespace ThomsonN7
namespace ThreePoint
/-! # Typed three-point bound with free pair shares
Each triple `{i, j, l}` receives a share `W i j l` of the pair function of `{i, j}` (and
similarly for the constant), so that the total over the triples containing a pair is prescribed.
-/
section TypedComb2
variable {n : ℕ}
end TypedComb2
end ThreePoint
end ThomsonN7
end Asm_Typed2
section Asm_Typed7
open Finset Matrix
open scoped RealInnerProductSpace
namespace ThomsonN7
namespace ThreePoint
section Sym
end Sym
/-! # The `n = 7` typed instance: two poles (indices `0, 1`) and five ring points (`2..6`) -/
section Typed7
end Typed7
end ThreePoint
end ThomsonN7
end Asm_Typed7
section Asm_T4
namespace ThomsonN7
namespace M6
/-! ### Integer interval arithmetic (exact, no rounding) -/
/-! ### Homogeneous Gram determinants and their interval enclosures -/
/-! ### Boxes, bisection and the two refutation checkers -/
end M6
end ThomsonN7
open scoped InnerProductSpace
namespace ThomsonN7
namespace T4
open M6
/-! ### Combinatorics of the ring: a 2-colouring of `K₅` without monochromatic triangle is a pentagon -/
/-! ### The ring colouring of a configuration -/
/-! ### The relabelling of the bipyramid -/
/-! ### Nominal Gram entries of the relabelled bipyramid -/
/-! ### Ring rigidity -/
end T4
end ThomsonN7
end Asm_T4
section Asm_Glue1
/-!
# Glue for the near-antipodal case (Case 2)
Abstract, numerics-independent assembly:
* `Glue.exists_minpair_perm`: every 7-configuration is a relabelling of one whose pair `(0,1)` has the
smallest inner product;
* `Glue.RootedClaim`, `Glue.case2_of_rooted`, `Glue.seven_of_rooted`: the rooted claim implies both
Challenge statements (through `Case1.seven_of_case2`);
* `Glue.rooted_of_cap_slabs`: a cap certificate plus finitely many slab certificates give the rooted
claim;
* `Glue.slab_of_typed`, `Glue.cap_of_typed`: a typed pair-minorant bound gives the slab / cap claims
(with the equality analysis through `M3.contact_rigidity`).
-/
namespace ThomsonN7
namespace Glue
section Rooted
open scoped InnerProductSpace
open Base
end Rooted
section Typed
open scoped InnerProductSpace
open Base
end Typed
end Glue
end ThomsonN7
end Asm_Glue1
section Asm_Glue2
/-!
# Glue for the near-sharp (tube) cap certificate, route S2
A typed pair-minorant bound `e <= sum_{i<j} H_cls(<y_i, y_j>)` which is only *near*-sharp
(`E(P) <= e + delta`) still gives the cap claim, provided
* the pair slacks `phi - H_cls` control the distance to the nodes (`slack <= delta` puts the inner
product within `tau` of a node of its class);
* the tube rigidity `TubeRigid tau` (closeness of all 21 inner products to the class nodes gives a
relabelling of the pentagonal bipyramid within `tau` in every Gram entry);
* the local statement `LocalGramA` holds on the window `tau` (Regime B).
The proof is: `E(y) <= E(P)` implies `sum of slacks <= delta`, hence every slack `<= delta`, hence the
tube, hence the window, hence the local statement.
-/
namespace ThomsonN7
namespace Glue
section Tube
open scoped InnerProductSpace
open Base
end Tube
end Glue
end ThomsonN7
end Asm_Glue2
section Asm_Glue2b
namespace ThomsonN7
namespace Glue
section TubeFromT4
open scoped RealInnerProductSpace
end TubeFromT4
end Glue
end ThomsonN7
end Asm_Glue2b
section Asm_Glue3
/-!
# Glue3: the typed three-point bound (`Typed7`) feeds the cap / slab glue (`Glue1`, `Glue2`)
`ThreePoint.typed7_bound` bounds `e` by the typed double sum `∑ i<j, H7 HA HB HC i j ⟪x i, x j⟫`.
Here we identify `H7` with the class selector `Glue.cls3` (poles are the indices `0, 1`) and
restate the bound in exactly the shape needed by `Glue.cap_of_typed`, `Glue.cap_of_typed_tube`
and `Glue.slab_of_typed`.
-/
namespace ThomsonN7
namespace Glue
section Bridge
open scoped InnerProductSpace
open Base
end Bridge
section Assembly
open scoped InnerProductSpace
open Base
end Assembly
end Glue
end ThomsonN7
end Asm_Glue3
section Asm_Glue4
/-!
# Glue4: the final interface (`CapSpec`, `SlabSpec`) and the assembly `seven_of_specs`
The near-sharp cap and the margin slabs are certified by *typed* three-point data
(`Typed7`): PSD blocks `FP FR`, class minorants `HA HB HC`, multipliers `ψBa ψCb`, constants
`cal cbe`, together with one-dimensional facts on the minorants. `CapSpec a0` (resp.
`SlabSpec lo hi`) is the proposition "such data exist"; every certificate producer has to prove
one of these propositions and nothing else. `seven_of_specs` is the assembly: a cap spec at
`a 0` and slab specs on `[a k, a (k+1)]` for `k < K`, with `a K ≥ -9/10`, give both Challenge
statements for `N = 7`.
The tube-rigidity hypothesis of the near-sharp cap and the local (Regime B) statement are
discharged here once and for all (`Glue.tubeRigid_of_le`, `Glue.localGramA_of_le`).
-/
namespace ThomsonN7
namespace Glue
section Final
open scoped InnerProductSpace
open Base
end Final
end Glue
end ThomsonN7
end Asm_Glue4
section Asm_EPEnc
/-! agent9: 40-digit enclosure of the minimal energy `E(P)` (for the near-sharp cap route). -/
namespace ThomsonN7
namespace Glue
section EPEnc
open Real
end EPEnc
end Glue
end ThomsonN7
end Asm_EPEnc
section Asm_Coerce
/-!
# One-dimensional facts for the typed cap / slab certificates (CoerceCert)
The typed three-point bound uses class minorants `H_cls ≤ phi` of the pair potential
`phi t = (√(2 - 2t))⁻¹`. For a polynomial `H = Q / Dq` write `y = √(2 - 2t) / 2 ∈ (0, 1]`, so
`t = 1 - 2 y²`, `phi = 1 / (2 y)` and
`phi t - H t = F(y) / (2 y Dq)`, `F(y) = Dq - 2 y Q(1 - 2 y²)`,
a *polynomial* in `y` (no square roots, no sign case split). Hence
* `H ≤ phi` on a `y`-interval is the polynomial inequality `F ≥ 0` (`PlainCert`);
* exact double contact at a rational node `y = p / q` is the factorisation `F = (q y - p)² G`
(`Contact1`; two nodes: `Contact2`; simple contact at the boundary `y = 1`: `ContactA`), and
`G ≥ g0 > 0` gives *coercivity*: `phi - H ≤ δ` forces `y` (hence `t`) close to the node.
Positivity of a polynomial on a rational `y`-interval is certified by exact Bernstein pieces
(`BPiece`), checked by list arithmetic in the kernel (no SDP data, no rounding).
-/
namespace ThomsonN7
namespace Glue
namespace Coerce
open CutOneD Base
/-! ## A. Polynomial helpers and Bernstein pieces -/
/-! ## B. The substitution `y = √(2 - 2t) / 2` -/
/-! ## C. Plain certificates: `H ≤ phi` on a `y`-interval -/
/-! ## D. Contact certificates with coercivity -/
/-! ## E. Shapes of the hypotheses `hA hB hC` of `cap_of_typed_tube7` and closeness of the nodes -/
end Coerce
end Glue
end ThomsonN7
end Asm_Coerce
section Asm_Coerce2
/-!
# Relaxed contact certificates (Coerce2)
`Coerce.Contact1` / `Contact2` need an *exact* double root of `F` at a rational node, which integer
SDP data cannot provide. Here the factorisation is relaxed to
`s · F = (q y - p)² · G + R` (resp. `(q1 y - p1)² (q2 y - p2)² · G + R`)
with integers `s > 0`, polynomials `G, R` with `G ≥ g0 > 0` and `R ≥ 0` on `[0, 1]` (both by exact
Bernstein chains). Then `F ≥ 0` on `[0, 1]` (so `H ≤ phi`), and `F ≤ 2 Dq δ` forces
`g0 (q y - p)² ≤ 2 s Dq δ`, i.e. the same coercivity as in the exact case with `Dq` replaced by
`s Dq`.
-/
namespace ThomsonN7
namespace Glue
namespace Coerce
open CutOneD Base
/-! ## A. One node -/
/-! ## B. Two nodes -/
end Coerce
end Glue
end ThomsonN7
end Asm_Coerce2
section Asm_CertF
/-! # Fast (linear-traversal) checking of sum-of-squares blocks
The blocks `⟨d, l, Δ⟩` of `Cert1` are read by random access (`List.getD`), which costs `O(r³)`
kernel steps per block. Here the same quadratic form and the same positivity check are
implemented by traversing the coefficient lists once, and proved equivalent to the random-access
versions. Also: packed integer data (one natural-number literal per array, decoded in the kernel).
-/
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
/-! ## Packed integer data -/
/-! ## The quadratic form of a block, by list traversal -/
/-! ## The positivity check, by list traversal -/
end Cert
end ThomsonN7
end Asm_CertF
section Asm_CertT
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
/-! ## Multiplier atoms of the typed certificates -/
/-! ## Kronecker values of the quadratic form of a block -/
/-! ## Size bounds of packed blocks -/
/-! ### Degrees -/
/-! ## Analytic bounds of typed blocks -/
/-! ## Sums of expressions -/
/-! ## The hybrid check -/
end Cert
end ThomsonN7
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
/-! # Flat (Nat-only inner loops) evaluation of the quadratic form of a packed block -/
end Cert
end ThomsonN7
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
open scoped RealInnerProductSpace
/-! ## One-variable polynomials, marginals, kernel sums -/
/-! ## The three type slacks, scaled to integer polynomials -/
section RealSide
variable (K : ℕ) (m : ℕ → ℕ) (Lam : ℕ) (blP blR : ℕ → Blk)
end RealSide
/-! ## The certificate -/
namespace TCert
variable (cf : TCert)
/-! ### The real data of a certificate -/
end TCert
end Cert
end ThomsonN7
end Asm_CertT
section Asm_SlabHead
/-!
# One-dimensional facts for the typed slab certificates: `H ≤ φ` on an interval
`φ(t) = (√(2 - 2t))⁻¹`. For `t < 1` put `y = √((1 - t)/2) > 0`, so that `t = 1 - 2 y²` and
`φ(t) = 1/(2y)`. A polynomial `H(t) = Q(t)/Lam` satisfies `H ≤ φ` iff `q(y) = Lam - 2 y Q(1 - 2y²) ≥ 0`.
A certificate for `q ≥ 0` on `{ν₂ y² ≤ μ₂}` (i.e. `t ≥ lo`) or `{ν₂ y² ≤ μ₂, μ₁ ≤ ν₁ y²}`
(i.e. `lo ≤ t ≤ hi`) is an exact polynomial identity
`m · Lam · q(y) = Σ_T mult_T(y) · Σ (weight · square)`
where `mult_T` is a product of a subset of the atoms `y`, `μ₂ - ν₂ y²`, `ν₁ y² - μ₁`, and all
weights are natural numbers. The identity is checked by kernel evaluation of integer polynomials.
-/
namespace ThomsonN7
namespace SlabOneD
/-! ## Fast check by evaluation at a large integer point (Kronecker substitution)
If an integer polynomial `R` satisfies `R(X) = 0` and all its coefficients have absolute value
`< X`, then `R = 0`. The coefficient bound is obtained from `ℓ¹`-norm majorants along the same
expression tree as the identity, so the kernel only evaluates a few hundred big-integer operations
instead of expanding all polynomial products. -/
end SlabOneD
end ThomsonN7
end Asm_SlabHead
section Asm_Bridge
/-!
# Bridge: the certificate checker `Cert.TCert.sound_lo` feeds `Glue.SlabSpec` / `Glue.CapSpec`
`TCert.sound_lo` bounds `ef` by the typed double sum `∑ i<j, H7 HAf HBf HCf i j ⟪x i, x j⟫`.
Here we restate it with the class selector `Glue.cls3` (via `Glue3.sum_H7_eq_sum_cls3`) and, for a
certificate whose four boolean checks hold, produce the final interface of `Glue4`. The remaining
inputs are the one-dimensional facts about the minorants (`HAf ≤ phi`, ...) and the numerical
comparison of `ef` with the energy `E(P)`.
-/
namespace ThomsonN7
namespace Glue
section Bridge
open scoped InnerProductSpace
open Base
end Bridge
end Glue
end ThomsonN7
end Asm_Bridge
section Asm_Bridge1D
/-!
# Bridge1D: agent2's one-dimensional slab facts (`SlabOneD`) feed `Bridge`
`SlabOneD.peval Q t / Lam` is the polynomial `polyR Lam Q t` of `CertT`; a generated slab
`SlabOneD.Slab_x.HA_le_phi` therefore gives `HAf ≤ phi` on the slab of a certificate whose
`HA`, `Lam` are the data of `Slab_x.certHA`.
-/
namespace ThomsonN7
namespace Glue
section Bridge1D
open scoped InnerProductSpace
open Base
end Bridge1D
end Glue
end ThomsonN7
end Asm_Bridge1D
section Asm_Bridge2
/-!
# Bridge2: near-sharp cap specification from a certificate and its contact factorisations
`Bridge.capSpec_of_tcert` asks for coercive one-dimensional facts about the three minorants
`HAf, HBf, HCf` of a cap certificate. `Coerce` produces exactly these facts from an exact
factorisation of `F = Dq - 2 y Q(1 - 2 y²)` at rational `y`-nodes (`ContactA`, `Contact1`,
`Contact2`). Here the two are glued: the contact data are attached to a certificate `cf` by the
equalities `c.Q = cf.HX`, `c.Dq = cf.Lam`.
-/
namespace ThomsonN7
namespace Glue
section Bridge2
open scoped InnerProductSpace
open Base
end Bridge2
end Glue
end ThomsonN7
end Asm_Bridge2
section Asm_Bridge3
/-!
# Bridge3: near-sharp cap specification with relaxed contacts
Same as `Bridge2.capSpec_of_contacts`, but the pole--ring and ring--ring minorants come with the
*relaxed* factorisations `s F = (q y - p)² G + R` (`Coerce.Contact1R`, `Coerce.Contact2R`), which
integer SDP data can satisfy; the pole--pole minorant keeps the exact simple contact at `y = 1`.
-/
namespace ThomsonN7
namespace Glue
section Bridge3
open scoped InnerProductSpace
open Base
end Bridge3
end Glue
end ThomsonN7
end Asm_Bridge3
section Asm_cap_tc
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
def tc_cap_SB1 : TBlk :=
⟨[10, 4], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0), (0, 0, 4), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0)], 35, 4, 72, 72,
0x99999999999999999999999999999999999,
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def tc_cap_SB2 : TBlk :=
⟨[11, 6], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0), (0, 0, 4), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0)], 35, 4, 72, 72,
0x99999999999999999999999999999999999,
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def tc_cap_SB3 : TBlk :=
⟨[12, 8], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0), (0, 0, 4), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0)], 35, 4, 72, 72,
0x99999999999999999999999999999999999,
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def tc_cap_SB4 : TBlk :=
⟨[3], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0)], 20, 4, 72, 68,
0x99999999999999999999,
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def tc_cap_SG0 : TBlk :=
⟨[], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0), (0, 0, 4), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0), (0, 0, 5), (0, 1, 4), (0, 2, 3), (0, 3, 2), (0, 4, 1), (0, 5, 0), (1, 0, 4), (1, 1, 3), (1, 2, 2), (1, 3, 1), (1, 4, 0), (2, 0, 3), (2, 1, 2), (2, 2, 1), (2, 3, 0), (3, 0, 2), (3, 1, 1), (3, 2, 0), (4, 0, 1), (4, 1, 0), (5, 0, 0)], 56, 4, 72, 76,
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def tc_cap_SG1 : TBlk :=
⟨[10, 4], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0), (0, 0, 4), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0)], 35, 4, 72, 72,
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⟩
def tc_cap_SG2 : TBlk :=
⟨[11, 6], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0), (0, 0, 4), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0)], 35, 4, 72, 72,
0x99999999999999999999999999999999999,
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⟩
def tc_cap_SG3 : TBlk :=
⟨[12, 8], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0), (0, 0, 4), (0, 1, 3), (0, 2, 2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0)], 35, 4, 72, 72,
0x99999999999999999999999999999999999,
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def tc_cap_SG4 : TBlk :=
⟨[3], [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 0, 2), (0, 1, 1), (0, 2, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0), (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0), (3, 0, 0)], 20, 4, 72, 72,
0x99999999999999999999,
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def tc_cap : TCert where
dd := 5
Lam := 87112285931760246646623899502532662132736
an := -1
ad := 1
bn := -99
bd := 100
e := 1259031901072922459286254611941137639131858
cal := 19127427689929623000262339271757972228010
cbe := 16215043688115070926263752263445740533402
HA := [170717757480552085088979247788979045424340, 56191821671676908679482209660775856271648, -220309737546812222628554408661585740914768, -77479581194809140134631691140886025002102, -44807708364056816690875659833292417362464, -233648679192088102608262280903944296522196, -101078167729932081174450918159852766529868, -66934474842571285191037482939485041242696, -224432908240006636817633580419638758821368, -216062636282412238476404472408487364331876, -74466642474068062185006448695369901556726]
HB := [61597688107009154933968282397669419072688, 30798844053504577477837635728002975243864, 23084009998671252465573998689971353965276, 19238310478029424323344171619969680680508, 16835936748528215710789925485357870404654, 16263191929301309062843297932274020659776, 14672295122868470394680686460290544912008, -12644532183649820551035083229779674511842, -24752329292309356219373268859975129186766, 68561193987253405316099134272501091166204, 66446725468145523634310745021115587937566]
HC := [61594484493696253203633737325808935281570, 30808343403275704592719602158382344123874, 22758145382235113847716537355248743285798, 19694456291593967907480642660091266215732, 22195366481427575273352526265412332639396, 14849570979845450533510314797337514508754, -14421706729754048369243870323955118931358, -13911379967526343876638099783403961110224, 40092006734241262637000245067100303166362, 59918385972582782963955451852797591146636, 22690058154622992236519099732025979254482]
ψBa := [1337210185706445623479231266058894184986, 1956802171773322209165946620013038810122, 1238209245396996757745314187269414830452, 1601347522618807261286063417528769765280, 4406799829556204198791926232478824781922, -16112932419692785765710664809270740986740, -40097479791795464097822501401984837340802, 47768851310778224551852030163272903240220, 72392307838636285058775346679559434119532, -17595841901142585351683228038099041985914, -35840271479052094940221890379439263600918]
ψCb := [-16215043688115070926263758455895228124450, 4723783828725235634330206518590377074992, 12344504738551974509636123811021192424450, 3338726022713904035812737344624377474490, -937010120224116522682777948762060314088, 2710132300291695349115028737790296459892, 2312861661064001423369501654143476522580, -2522298722742911061062278442666295714092, 565681642402936970483039944464258648414, 4636724880466302696690146829544874878682, 2271611859189418467658065772811752777544]
FP := [tc_cap_FP0, tc_cap_FP1, tc_cap_FP2, tc_cap_FP3, tc_cap_FP4, tc_cap_FP5]
FR := [tc_cap_FR0, tc_cap_FR1, tc_cap_FR2, tc_cap_FR3, tc_cap_FR4, tc_cap_FR5]
mA := 2
mB := 2
mG := 2
SA := [tc_cap_SA0, tc_cap_SA1, tc_cap_SA2, tc_cap_SA3, tc_cap_SA4, tc_cap_SA5, tc_cap_SA6]
SB := [tc_cap_SB0, tc_cap_SB1, tc_cap_SB2, tc_cap_SB3, tc_cap_SB4]
SG := [tc_cap_SG0, tc_cap_SG1, tc_cap_SG2, tc_cap_SG3, tc_cap_SG4]
end Cert
end ThomsonN7
end Asm_cap_tc
section Asm_cap_ct
namespace ThomsonN7
namespace Final
/-- Contact data of the pole--pole minorant (boundary node `y = 1`). -/
def capA : Glue.Coerce.ContactA where
Dq := Cert.tc_cap.Lam
Q := Cert.tc_cap.HA
G := [87112285931760246646623899502532662132736, 2151734199360815430923414870928107483308888, 2151734199360815430923414870928107483308888, -27944443296251496900472893114914500939203904, -27944443296251496900472893114914500939203904, 159201582776544735021594855797065455719683808, 159201582776544735021594855797065455719683808, -586573914610842804085002223160528587775467392, -586573914610842804085002223160528587775467392, 1450037954435626738735731572631618042357026048, 1450037954435626738735731572631618042357026048, -2441384163564702410646849165275238940292955136, -2441384163564702410646849165275238940292955136, 2760663489507302453260323117454585217533232128, 2760663489507302453260323117454585217533232128, -1994958844910011448073037772515726056596251648, -1994958844910011448073037772515726056596251648, 831278874700755697619411007458761294628540416, 831278874700755697619411007458761294628540416, -152507683786891391354893206928117558388174848, -152507683786891391354893206928117558388174848]
g0n := 1741757945213855404949078441533013189155331532714531
g0d := 500000000000
ya1 := 199499
ya2 := 200000
pieces := [
{ u1 := 199499, u2 := 200000, w1 := 1, w2 := 1, Lam := 992554853457087130445962004320801543609802955047510001, β := [18263655791605636450998848759089208378317409172476729496381213723597986948594168067017071381090156586936506404033275564569962093490038472072472576000000000000, 2085136902531408818913515919867434547315309307641564030707628895766793596924423702975547672793076972024233049702228662674574015605411049723904000000000000, 111265533930447848523501852066023208461084552563500667691048800534516841326940013583647414038010991803730502923382991703275241114428526592000000000000, 3702495673753447773739822209238980686613794199407859210490708864746484110475087821946160493713375230896164748804208056313667176230912000000000000, 86379038420910523266932272383460301281545975888593748762307415737586508652132729540631751729827922709758185234673616419026726400000000000000, 1504552220456233976428589736519845879607486077607103681584843326735353937480716010446662674653740197459556607140607051418112000000000000, 20328835555486871241320394917138682751857690187151847161172791363606551751842027936882827017672366520320981944521311232000000000000, 218415495700693313533455749672409155713015595930244749537465291747725816256749348118469804485614460526117181707264000000000000, 1896780278576304742307353219996050135244792564196656338983457333570207380867552325593824495419489964933082112000000000000, 13454509238563133780545883785982315266811698341827645136439922451352312811040077781800754924080615607296000000000000, 78423017685477330603857011479264524592912705214550413778003177299413686687200228863134474502890624000000000000, 376445861679566553733757114961148166053453467522031622038624307811614179478475207320460137600000000000000, 1486105303393733522460450053298175239158979563128324072002504322353892558258361516917056000000000000, 4800163092974866370282467233577033996015188093988202915983857393814822536539693760000000000000, 12565477921662182591985262771106657905314349822056037367518901124436573537968000000000000, 26253533962748251411870231821014233760602679139477354548361790297441392000000000000, 42763287408077259875507286314148910279512481109363631618334127712000000000000, 52345256801739913889601019077447301238713446284274745973568000000000000, 45305506991313448251410722700770366460340775668258616400000000000, 24725315449732370881841499696642092045526378377141876000000, 6399835911555275647070089742207047615572668467285469] }]
/-- Relaxed contact data `capB` (generated). -/
def capB : Glue.Coerce.Contact1R where
Dq := Cert.tc_cap.Lam
Q := Cert.tc_cap.HB
p := 99516432383215
q := 140737488355328
s := 1606938044258990275541962092341162602522202993782792835301376
G := [14134776518227130154875496557487337897079831320175669235330531433379027882, -50918894616535578276190759481441371896738273580868608708937669524005112785, -172289935731956639809401918348431941786595725611878676361441375742698348210, 501348494809364628159500136630701490776548008629326947984048554572056657759, 1762607553133398842365923693184858731757468887334252364945271801185428015508, -2591044328314967441363161999039709810278816379028228089090345376131241790493, -10853795165892652797666957515999250755729042362407275115861346468471900406326, 6796447812768495020034612552473033956575272156112229305982218854661094838034, 40930847677341756070161837039348991957366755079442704792390994061415493891255, -5488041710531617006406480477014579134875341368207263456108474937735052012156, -97384221390489970130009314570910447846531069375704101233130772762240528175182, -16941780046440544984835808671286331601411710206517363441730557715211395687667, 146849852556144408596767462008194320666294248333237922024597483469304242069304, 55743644427074680324508253312476287434218135355293033545059965880137672485631, -136032869182544843674399713277811504583525329751881677362476789127977734656393, -71261131755857544896082019885684217958335263324944472037901708249287142147353, 70508820366710786442552016091722572890509935533119030106621930744448932956699, 44337087032234619766781304483864230340924711121610056896529511223589238472704, -15613421139216925700704711253632563898977704548944635898377321063239510917120, -11040355965061700033886132041039380201344775924060030095845468820413456842752]
R := [2484733551080715115810433578000602460623352581434881638516233804382449856395007019286, -6192000245156207830648096646606126400209510705172072278160054509979121284227082461967, 3885337784451458141838923813647037813284813678104279042497929951062523912908785963762, 17672654475726993555996892097, -6451323497313032668239841300, -2842118242697367497212757315, -4329807621513977222900940042, 3711213739786526486521761966, 4483194863417362760707691369, 9857214568157920645161508348, -22227887901471970286787380210, 15641151576488477283247704147, 2050535454027668521435083208, -13336225251668464287565997023, -1126166202796860805514550871, 12118484763222459473638897465, 1116707439931749251440842117, -8896753369993911989350432768, 3330278956850433979731410944]
g0n := 2857243563662596013557082535148112696054254694783109386710762956990844
g0d := 1
pieces := [
{ u1 := 0, u2 := 1, w1 := 1, w2 := 2, Lam := 1, β := [14131919274663467558861939474952189784383777065480886125943820670422037038, 486094037820676188960562940566741839909845254907405064076927515952032294659, 7660862641942574352509297341186976482449329938265777520262375859961790925522, 73031765391794265485908844919971271077771509717941692670141685867740814912775, 468086596029634004030564045067606056031135798987563071094397593187651012325684, 2119118763391933611163020825224213446341867583492305000014973627099807751874715, 6920736750297930669048207424863894342893498899556155153032674111797973390517614, 16330049364723679565273174132597323942042509802110202809119126089834210015021818, 27401422433383591180137188153169321798276715526295797943814775528747264011752183, 31804399991935374645932174969833507023564795766753162940470267779567379972944478, 26373567220941112477122476913584749108153087178729191256132820713009723178726822, 25224639503563176193227038281405721564610742425359796557591271429970926412877785, 45790849243134849914540757108246202102771979261658010304812872997073138312806792, 77772372939460992872639621739203764109991077455598851083662890945515333258383551, 88570650695888733364670973024023674196229211415689954664273952760659084464843211, 66312820464866884793003748491268398220893773151735260828470991217625446952012785, 32568697934385739977108924825874802138970195546454866624873982530847292294915531, 10140886465618085926122542681176871639526564225729617746726695513502810512375434, 1819720765192362165895486373931311702761058421363639642059055405769393554757268, 143616199150221355613905378359007505423619067709420097565128949116554626237160] },
{ u1 := 1, u2 := 2, w1 := 3, w2 := 4, Lam := 65536, β := [17952024893777669451738172294875938177952383463677512195641118639569328279645, 568335227647236617275204227325100550453021252088708284636779911013103323250448, 8132379214235987009697252454329796688357711575055364974657781365161788489338254, 69263860817309941395320531069120541543231039215412190712385516594412980325820662, 388674609908818018109345899617196924950784184071572855093154343845109227083039166, 1499331357104930628234329901813857241647552584723151365855416253125559356663407406, 4016490826708350834526699399929658256416112415473409290548615856897941241482503602, 7393759130807471217982017746421765499995956479499501233867568120756603622342695030, 9500485713214401862434010674577255806063453111560078999694235790345741541407779880, 11839711435875668465249097909021265223378734239428807572104565071395772610120658310, 27335595848703688252440481523112096121538120788841854573664362908668425064709117110, 71328037304783662533464401686121384428059527774498504335363057038236562115550622346, 132072557847641459755111051459395243981544572746609485613299429815302069988017777442, 166096061618492915202919642651959699512681390089860509037461639350093923096469085962, 145565174713580157896186773416255479731834555706742835443922580768956659515353460262, 92079441118434372915761399218814397179919889016566192349650602936211111992027023762, 43842798249991994639751508069415793717986757173314445157006300408021026563855213723, 15822076339662536028996902335360225382739319020627410619372697797065510519074912690, 3839524924977140669697597079132442909587152262092347358187082040213027281826790372, 442180252092255952272112589999185338547154480766999252952009100000366196558535160] },
{ u1 := 3, u2 := 4, w1 := 13, w2 := 16, Lam := 4294967296, β := [55272531511531994034014073749898167318394310095874906619001137500045774569816895, 4291009262351254265806937736457215167850671393585029289258337552451091763675907432, 157811799569644877441712682168014812511794409273101115546706865806435327704401899100, 3651935116010587798564244317219410769196230760940391409126448934706600277019031689252, 59564086799432988894056504264247764581915575063136678475832046996670801190101262174184, 726832264200191083130716089260444724056317499782330411997747888699636621837148063329252, 6873939846673856866708829895453166378111850094208746983079684316170307002734387882629836, 51500392918760334750992168991081544267435700105434107041454377239535887480587944720204468, 309800686023731034249457284207150519224720520806682881132247824413578445856134616045762702, 1507452985254562068261230607913308949791788215913727755346695799329987490078453266468879620, 5948951535444894600093517174912173568244899706211806137812890275658385741031644743679918180, 19009067066021112737777899017746589926568383893187660828709075619254656541940905989196043820, 48890209029040032508932675855636164029972664026722699285105190677409434638404487228683746160, 100120577674197585779551135879818639348944511961969024383551415089737144300163059447646780652, 160507635063151241972814180670612820034870960049928768806715286118259382686807548729282680548, 196354103711677267159665460449977695895965914784753472293872980156548536764184380943856232764, 176298644504791787226435838451518119844178307382571616240401237903646799033348279280590183259, 109109961467184696319059220902816051164748236366865535212464083572065866166269219821916125124, 41500786507798370942894300538193338653657204872540708384619555537510277312877927988120159504, 7331167362107652464148316841299256591181490580390318339716455738646528952701126485204075200] },
{ u1 := 13, u2 := 16, w1 := 7, w2 := 8, Lam := 281474976710656, β := [916395920263456558018539605162407073897686322548789792464556967330816119087640810650509400, 7036325858056081866628677363537399740641738910291828960671693494907936934445693405329638724, 24505143813550750337675530512309611277421057412842634386883283280806241539170512585654452194, 51718590778279847639825421038895566160651661910880646252001174492705775558283289180444387867, 77197414465535234921806726885205686569321376596022204319117312477373894057701713793427937400, 97892211723214413531980200702727996687031913272262909522876556215564873107514894511418094448, 129481469459717784644389819936798437160466046439419421106519031959426662505485659917078670720, 180647176582376596643330881905563775706674105497420128626072721521446487677004520873166336428, 226938313484122489559909711232848421781087441440476421488458894121699047135686603472392083176, 231144224913519949352825249310978576735707176305008285906090460887037850267944633658457541224, 185038936608913590206734104150323729920077156934363081399363807015971982110412018726912046196, 115994464408331440329942865810002625126846555646495500213351655375948753608070214244680933306, 56974641229481256475291215052332704915919530198870842228953083909508475295745999558304496744, 21866633369012764463742816164646391411717371386240555214188661682473922235669844205566281488, 6496410291726647684879277469308709253321878695807206669280849989315872347727588774049588624, 1466328527677615435874596165952390889194319696856030137684135315842096583777588449127213724, 243312790085218922627270331053744524919412036306844801253745183247160599672534237051526368, 28011644406068240533752013055227404816693038468482689283572675575885914356671421341569108, 2000118694152818999010816206368567545786240618879008138306939974327261502080957657878426, 66754875179485885984503043533907369543275062451234954360000292349882125120676127162211] },
{ u1 := 7, u2 := 8, w1 := 1, w2 := 1, Lam := 1, β := [34178496091896773624065558289360573206156831975032296632320149683139648061786177107052032, 115514188228984096134774865604105761163548028213848592653636908135425256307045221783526464, 181927902584358344938544975280065592570690929291734756600306935109459070440162140150214664, 177625432096718817968741546570954119544985052301038256680492575224347983684479071414831931, 120630734861299524627802391498767391532604153918488173558383961935514254075137481712021417, 60590222673137921535115391649212396786254445308941784961420139346120472314234277883449320, 23355575189314231729856980538244128847746162777642195414229554239138441814664040509778952, 7073371763168570056588483557476548816441329607263786256375230287959767734418813956209789, 1708474227096897682150641043495044032214990279965989058181130013116874677000383405537090, 332042781360734604569537835719470682505054324015014603641014388898097717973615328150823, 52135640590182554109516493257152171040291678845934488346835867051174818350747044060231, 6611402820117230567856938779702535925332256475299229144983563142767486935670203787339, 673945437757513923627317849881922522392186682630278611528477988788771200657299181051, 54690073654643791479671037644198374297205875924766617334176507105001808038447579634, 3476614119664941611067415251551730191514082638045111867515311611278104092509698981, 168831611300987924891280213210361181170976462406256422246458085147349409405895178, 6020969097859289879453320756842154540580445908186015208641971694349419198250805, 147714937489680610269088458129625430235281813791262064865287072619730524867312, 2207129871492300597976651217562789383213366122498110825708599460462413805082, 14918324792914275449878887122090553515071010956242854644204844009915327495] }]
piecesR := [
{ u1 := 0, u2 := 1, w1 := 1, w2 := 2, Lam := 1, β := [2484733551080715115810433578000602460623352581434881638516233804382449856395007019286, 83258407593749536338527512161415562182231182226483666708424362447789073545993170232329, 1314014262710538042775788987565407446107653229540401384356991164893831712362932278059916, 12976223297192377876676791383470905311260385865692636946274856052668467176133710110321441, 89833035463798741379435865651922747005356314716405785768963640633099111157874444000669450, 462869250039900285876800295064270616327406114992444765879890009026821378328432816252075761, 1839127908843168117530648848167246796915730011103500676680043586334142853376996585013881072, 5760109039767943178289781880524163562814481516092001628325991386627664206870327720804799974, 14411368086723273290847330864505282318979987749094166156403723584205816473515368481629747549, 29007896748647338513835513820120244934121147907776330130011175574869620609428782367912185128, 47067424839463723445731404940351734007453642389874206736801636759954732754549704876117774666, 61389242796492560764904383491530194405856988311283057649395654727271233283548081029792465779, 63852307946322331868430304738205055454050967804793348600837945902986297501595194384867724946, 52215293278349331457037833635531752181593500323197312733948313861484648386468036558023090493, 32811140161478891773057467587460387935071950988466893081613073032652150241529986431895334235, 15276762207091327367197827849027384955938274725952156742626969905673308855506942380209775673, 4959000997283169860982766297041254197472174234072799448480117763352049682928795245174254435, 1000676127333539917520526136108242848000083254874258970895913139145195479807938582140261232, 94389632923199271323858687258604002040820701167593785933794971877020839398573659451155468] },
{ u1 := 1, u2 := 2, w1 := 1, w2 := 1, Lam := 1, β := [94389632923199271323858687258604002040820701167593785933794971877020839398573659451155468, 698337265284046966308930234546629188734689366142429175912396354641179629366387287980537192, 2389120669862479775684201133767538093716326179852246191120225095067914954175609244816600095, 4997240897408567026003567195932520540914882509069943713187424435861905766710240483645629623, 7112336887271990920071815954723226897513470446261360508014335374749321650618442214817325660, 7242387548487254964312943278241720365866159374337630730150863853845157630933449547745944396, 5387530497501632427887947273728014093605424571275761995901886788951667675085515422051708743, 2924465347671861324362026027053153045411445077765471044460442829202812691163989013219021494, 1130813986818888171019552646834058652817858420681817261496277142443321886902606067629884039, 297514965744316467140606136671697837473782927031086707946035983415162231088654622500665057, 56177195271762267499877022781008770656305258721544445371605018583388755986949172116720835, 18032683187338768950800343834454143485008519942952943913369364699480865732934755751188765, 12103613567521260013700177402339254401806785056076095276115538307404110899254261725796492, 6113176220988571497563127373043730698925308480588620528934581694866556951484527551377247, 1995328960161873229582564172840653615263851925664219355324446643159538993757927903475712, 427979510958539569554891172173463398390034038443841744327418685225708927423505596684651, 59189884087154212063598966435663254820618046815521864162365494994686144707371025829152, 4831883929788046919015635840806550226791483306178696696268718997850887487479560841698, 178071090375965427001260745041513873698655554367088402864880399216006492277020413213] }]
/-- Relaxed contact data `capC` (generated). -/
def capC : Glue.Coerce.Contact2R where
Dq := Cert.tc_cap.Lam
Q := Cert.tc_cap.HC
p1 := 165446840199891
q1 := 281474976710656
p2 := 33462326346837
q2 := 35184372088832
s := 1606938044258990275541962092341162602522202993782792835301376
G := [4567192616659064565207375596310372691788896649, -2775389871167025506860849556654292053975904899, -66228926333360093084332320304769771251338071338, -36726124390425997450955554565051486066099711117, 494665945336253261825265300485632868509824819266, 746670369464906674358226701398524568306100741190, -1557158647455653546733274145731967487559542514424, -3958414052429971936879442411320465006369102663443, 1348314034767786567242672473151556382108991627670, 9504848098033785643805071680678742839312008710545, 3286319531038044742724682758587946451627954390713, -11160291266192137611279427994422809748720553769953, -8808837467377752997053670096398694124071288296300, 5413661212846226151217187387779474180235010418411, 7641996013102661822220482199571929829645914567751, 254518863789669326790426949796210761430801838259, -2343200557714410128443987374743392955400770533225, -761352013425342678636808048659483943127926964224]
R := [86036297928008664663301821971718389582051380377417714927289896840010541410073458497175, -459265148251999805426709238832364872244299307012798382938706274059679093199282624967389, 905380580979978180821054897484156713429957185035899054596526188506012676713540159046122, -780652673716676891023246367769807062784152589765867954560258570530629024085532690677779, 248661618204893321077691124073155564541360238164940687684576576497501583886760687973886, 140906252676326453266363396631423581129677713892120219258, 55550233974717637208410520790195988649967834017961016312, -146874521796170757723273058069032699992073851681459001421, 131820989072328559011450333864198777157585148182018730794, -123109836167601679081831341242139876377551511668793827441, 112929332898039660386151501723630848955328453692389924711, -80379704012661063564538613839861464042677899165842564927, 66764796020773841333814666282129259679339038687276659180, -73154337718168017425760870990448392638597716288657841291, 87224962417001896758614461256920399146120688054642541977, -113425800692339586522746674879411498931028004286506071379, 132404009946224287341997585443563606607841805025740319561, -121969925840651043028138299907914365629379233425790074880, 71312198047128090362867710259955497476032603327505104896, -14756845640053692606753727029465806114945415034768982016, -2304121637324211010276510100909859696940442515033554944]
g0n := 43785061443414541663713289352661989633065319
g0d := 1
pieces := [
{ u1 := 0, u2 := 1, w1 := 1, w2 := 2, Lam := 1, β := [4523407555215650023543662306957710702155831330, 151020467006165075293623668879907871819322360321, 2305692307826608703503872788867287504994205215414, 21251556047821735094803389998947741410213046232423, 131467798482176764863584989879799282469379046044910, 574054101221860061884931553027749328652941417507598, 1812207867001194649789451723956990189028966685493000, 4176057086483186710551271343013225061427481282378909, 7072186026080310452542709088351766669286055322260442, 9099307603750829918736732006427652051028739750646945, 10167233033250908733423048990365431388618044140612585, 12433946609696456139724106311457412046055762028456717, 16338931336537573985010208794044028235323450960780604, 17693116408086934011888539976809471515805464248070063, 13289537891283720640790922528778680562791741226738639, 6324422735913430746356746023241612613165143233412437, 1707124851221935365180213313647507219646503073166167, 198692430936421254466381418148481213739530651593890] },
{ u1 := 1, u2 := 2, w1 := 5, w2 := 8, Lam := 65536, β := [99346215468210627233190709074240606869765325796945, 6737303463575967949640996957662944364283549619574158, 213855989965988164871144837807100711011812148323144514, 4218559019953976948670918543739057094714147993346734994, 57902449608678597557802814426382308777182465147237033334, 586660212381863020298637035028171935358773605702876011946, 4541895722099056747011914232458997589777160179742052376154, 27434716318152760480201664298154877995137954750941411035130, 130846182411136224766960951110798194761628436007398188449960, 495353983319525321154116368397716807075771238071595338648798, 1487899598449051586315389219157101419579141133618869767183078, 3524164185660527559299810840464155671102616561741575871584630, 6498223792498544029462971856063033451913196731344822154282154, 9127481937139030978503280482627164867875125530512793903794990, 9426148356935422164919892945672892121382555416156884610576766, 6740547892570835454022456804399768117863811972556711311590014, 2978403448668305315900367552234099785198863223217845223514343, 612012228964954402889443957667623168505389851039759811564860] },
{ u1 := 5, u2 := 8, w1 := 3, w2 := 4, Lam := 4294967296, β := [306006114482477201444721978833811584252694925519879905782430, 2223700497533807108659906087940697147096950510620113174786967, 7443472176065685250121221894452315130912322244331566233361020, 15187587357271234035360971688201378118353100079450710015724596, 21059226162648814471460648200662355707028932911547039066115284, 20938680938702360023798940232637340394965763877853257008706992, 15352162628558101411767060690441774259952126971391635530278732, 8419285477841815538389868885762070401224503560129313073148180, 3489017059227094869924712381798939720191520306147653057073136, 1122383622446082001645005897490711949623302225483258775010534, 307962369762692808840788701742812127325721701049294018201908, 87114614591036390390965916545618980789452997424469644568508, 27210919252201585804919627685360623631557732106435202438636, 7727148429013174044366669752025769986059191413583073379976, 1643402295884330742900853781320576468838771477140364450052, 233582972455481958080858385540212372001829393422085725820, 19659627939128749388293715566280702904902103481253281458, 741379510379956590032012563146623074376965606351343563] },
{ u1 := 3, u2 := 4, w1 := 1, w2 := 1, Lam := 1, β := [2965518041519826360128050252586492297507862425405374252, 18150727090249036703338925154197073888323142442665240255, 49342338020592213410485853912551318900983582123993184237, 77947568808081939023019463486113798719397999981913565868, 79839508495673029754881059895375284262987361147205840748, 56040479161935893813920868457292252418187683789547883337, 27661547581486147044618987325345957614898866938110816932, 9634998530999056243842760910873767263892299166395935064, 2323559395602675450134286203506674881208497768894172158, 372496057682978482588659389165079627541754983037427544, 40567118226501389017865054147513541997479480987546392, 5828510308844370725291729444758127556514879349689531, 1770462344908530987580140313099086018472827593991009, 419176386683585440941467863818123972944085251669040, 58029124504406274924736889316829749606956331868888, 4400145941367846154332953897485462031766460664140, 148275639595516875377201165468761338530424694327, 533030744327825138800318053686777258164516212] }]
piecesR := [
{ u1 := 0, u2 := 1, w1 := 1, w2 := 2, Lam := 1, β := [86036297928008664663301821971718389582051380377417714927289896840010541410073458497175, 2982186768868346781105363640036370711037755908083910214152889599540742563203655714919611, 48840891372690570718715488520360267650505632605387023847666009372646218606796628868138340, 502326723940633692435978226962393974993902169116821964217723451939761465249162539051524537, 3637109772401313718720080131699320694811828748322303848268371579071120712541720820494962536, 19696835534349044770689492638436902371549279119211239784614319025425780989695764155968598426, 82734397764658652550265910003067547651907722147064956028910691999883379641838880727972689348, 275830656621846461280950104395501401681265559780120992720856536821519397927143105966040347835, 740758527285607466744893447616541184474587118817389153681662481033980248226189134124051056808, 1616890992744424347044334327793641894486925358020143166816297664968391320199411125285646216903, 2881355451507024619409941428589425923901899005482215317574188660077112491211815328725464291345, 4194599677846114127559644197915591793566225157214766222918445692822686351006569393687809354753, 4973047705261700351558736355152977901063161427394473838784511548230433236179616957982506987074, 4768087898573309508681684370660513593526448397533156529096729872220657651411791613540716050789, 3654166716463762242073053454445119493854362970720128343151123492716334780372055276529667822079, 2199197261755810871881634122670267940306504439289181594005880971260854392481868905872523223837, 1012350270741211112932205535103710577678295927005635158965098990796487875650648121395989612703, 342456166122160308245220294967305566826922254581438351757118377943280300123646017247327429856, 79796721841083803012052093944650757704512752702393208376933799668815644850365649937128663672, 11374087966835019905938647557335903608488039705370237653316701402429640994998753394640799664, 743058671114155033531070503899314692124790931282370072902173026047122864101704771628215536] },
{ u1 := 1, u2 := 2, w1 := 5, w2 := 8, Lam := 65536, β := [46441166944634689595691906493707168257799433205148129556385814127945179006356548226763471, 3222355698609091471326859704703728874718938357343533248973993737490812776260810648346247142, 105477428841119864949528139587907883299880024398780345353930237871091524677174820000049157158, 2164036190696284918798693853172887250888698822689515728396317504885560460774753392014348514742, 31182436502446230811387376512033614350859102389175936062913125096534747497278267124198636689298, 335089077686070965725493224871465158838056041911455852765835857448138013801128705248281437144366, 2782900003888438879083418750318758061740807518758491178372665254622682324781057227705055475666766, 18262803703914697376303853113345510415834413959701361878362372146740972905068124868633244024016558, 96009858571063939108000161903333574991515146773711164463857840142634675124807346980791944795274304, 407433330663200984522347203804001142294929859478547746404676951690101856980737747563601328529189794, 1399662807871202570315064162682114411086680621586429350944302204050489149514770655127420214592321650, 3887307108160595371361950249554889525568418135661320470985941807918625891296740603991866379859549666, 8683803601693700755441952711898636046072338321028526728593377808233844160775495368233261027378110318, 15470319581732163752072201375568797520064240197045016600903985965605096499495550624121979800887853050, 21744343061691077337041839229705333453197964070517625363835699683721077560092060395356127499799107002, 23893884683118700999242118500452649652517735920574960222447684093904918606786431900970390022674498362, 20577441647241510673479061842162792698906985686607824469157443420947807612172668433151970283431408881, 14302292262082932864287460822845300507822541064390592013441740895849908505809503924398269900402360016, 8356340341271917324005277904701307075095573269663909826434797294151304618944838624529744628593314400, 3845034174265543108352222794915469186447460710128665384144631944162244705531492639947309666873207552, 979251749209816661835120286996863973976318996386482561098072597219861180148581801880256148613730560] },
{ u1 := 5, u2 := 8, w1 := 3, w2 := 4, Lam := 4294967296, β := [61203234325613541364695017937303998373519937274155160068629537326241323759286362617516009288358160, 743435414729077938749872509381646319164466156717020028354511753504545886994290672356906477408012256, 4585743443303887307956843746338847141929962212949877582304810548138346493363323533585617463112511064, 18036467060955614740603578366941635869517064994828958104381882121203045037197820053279222337782225768, 49437881422871117883021900574904400916076481782879499094025206416498310643672343332276589775783645009, 99564740340115177424289723042701962614298847218615841826242163395865372184858225640972969901865055580, 152755011552818094500315203134143222350796277032089232139091847793543836434980572891898037284368758196, 183166735826890862297016206603340136202356123638696654003028583488956797637090805945600873342703661404, 174784178268426405333603602890844508422698323180857471744275117132126720192571783920867184994208540440, 134373327925205979681341133249997576428052364393308423608930432232597651899976072676042209451480038988, 83871740451720840153807391662202984719382969275496329141422790581898796271253345969582965496801800804, 42657666335078337470376870856036155105836021135407130656821756895733166490415002032375118333850076892, 17676305451181652016675119645463297635899171130969744951438016666449655933003063814462040150045748082, 5943072654768531862459063217411231648904043526575187162467553741738532734033007970791693922223775380, 1607232877061961953411042368906946394311356648585035424162338884463934070090367862078945380356199100, 344561927042685654064208744763355237394089388583899177532010444132438771209246117790474919819590740, 57229113390724003907133694710936344887895359084802411568408435526975176894775842253901604512563200, 7102079214991039643736779192025030449897656436017161179893390381917017348234646672701988639086820, 619856488318853727625376406752527223951109070939056476007462687370158403992192708243087936808276, 33941295965792164215213970801339288181667411763098549045841539618677504550694197417092928224524, 877276925236497067815142109315274171050467462655242478454293614327649652088331291111455651477] },
{ u1 := 3, u2 := 4, w1 := 1, w2 := 1, Lam := 1, β := [224582892860543249360676379984710187788919670439742074484299165267878310934612810524532646778112, 1196500451097450230636451568484938373207080703757823978330628943422814372774763523174042295302144, 2965538566769765653153169248084906081477855385326461096821392153693144794778719412310083072577248, 4548401686609202672677279336297656467237900767246340119130490847727653020416907551717393453918832, 4842274412659264432771595030682754684728280954797254626764702989270330586051107713665204909801285, 3802860760592677327653879963897224008316061425051799877357655267993678397889688181678024705952803, 2284604626993565209155040551144587737620589225147557015478398293838376180201215400878778071275284, 1073993504949438658676918321153035846217872977435421571714663936331869686674803231632957703147368, 400647994830270206294101235732247955080112431263781445842410540085569304053663533496982219539239, 119537572141685387111173946611664655128720062723333034010659488041126793262936098472994067812380, 28615191997924662892281523810777326780180343008374529408735739350469615955982003183193191012416, 5493592633623238476185696644630452879514631371607755718031765780242639242522944147127597741666, 843757713632767937807785416590884907791202959036024548653487305357800935466497710997422174520, 103561977692479910221939540920767094373674279719372514239880019460589632549000760341316000628, 10240717670785999292538336492273652795112475779147348411798839060546167291890695060225967513, 841902832642803194391824138654169067244008982057671505222842917729625538825570667048139263, 61082147943852217994406397578453616512329624157675012377217072550214533380703101867938992, 4048410821414696087837994052756572616346779639507880487337475522085683427047685909404232, 225825338747919848289490663540047183551088536298342419826851867842667359128774970136384, 8669546158778441510953005038810518991836489733071305246949113834212562796096837567497, 160675144203470112092236926981670206464477174297470789620651248595383936371792470405] }]
end Final
end ThomsonN7
end Asm_cap_ct
section Asm_cap_capspec
namespace ThomsonN7
namespace Final
end Final
end ThomsonN7
end Asm_cap_capspec
section Asm_s99_98_tc
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
end Cert
end ThomsonN7
end Asm_s99_98_tc
section Asm_s99_98_1d
namespace ThomsonN7
namespace SlabOneD
namespace Slab_s99_98
end Slab_s99_98
end SlabOneD
end ThomsonN7
end Asm_s99_98_1d
section Asm_s99_98_spec
namespace ThomsonN7
namespace Final
end Final
end ThomsonN7
end Asm_s99_98_spec
section Asm_s98_96_tc
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
end Cert
end ThomsonN7
end Asm_s98_96_tc
section Asm_s98_96_1d
namespace ThomsonN7
namespace SlabOneD
namespace Slab_s98_96
end Slab_s98_96
end SlabOneD
end ThomsonN7
end Asm_s98_96_1d
section Asm_s98_96_spec
namespace ThomsonN7
namespace Final
end Final
end ThomsonN7
end Asm_s98_96_spec
section Asm_s96_94_tc
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
end Cert
end ThomsonN7
end Asm_s96_94_tc
section Asm_s96_94_1d
namespace ThomsonN7
namespace SlabOneD
namespace Slab_s96_94
end Slab_s96_94
end SlabOneD
end ThomsonN7
end Asm_s96_94_1d
section Asm_s96_94_spec
namespace ThomsonN7
namespace Final
end Final
end ThomsonN7
end Asm_s96_94_spec
section Asm_s94_93_tc
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
end Cert
end ThomsonN7
end Asm_s94_93_tc
section Asm_s94_93_1d
namespace ThomsonN7
namespace SlabOneD
namespace Slab_s94_93
end Slab_s94_93
end SlabOneD
end ThomsonN7
end Asm_s94_93_1d
section Asm_s94_93_spec
namespace ThomsonN7
namespace Final
end Final
end ThomsonN7
end Asm_s94_93_spec
section Asm_s93_90_tc
namespace ThomsonN7
open Kron Kron.Ex
namespace Cert
open ThreePoint
end Cert
end ThomsonN7
end Asm_s93_90_tc
section Asm_s93_90_1d
namespace ThomsonN7
namespace SlabOneD
namespace Slab_s93_90
end Slab_s93_90
end SlabOneD
end ThomsonN7
end Asm_s93_90_1d
section Asm_s93_90_spec
namespace ThomsonN7
namespace Final
end Final
end ThomsonN7
end Asm_s93_90_spec
section Asm_Final
namespace ThomsonN7
namespace Final
end Final
end ThomsonN7
end Asm_Final
namespace ThomsonN7
end ThomsonN7
Source
H. Tran, Thomson problem N = 7 Lean proof package (snapshot 2026-09-27), https://github.com/huwngtran/thomson-n7-lean @ 25f2fa53119273458cfbb3c4230904bffdd61e53, file formal/lean/ThomsonN7/Solution.lean (certificate data of cell `cap`; paper/PAPER.md §5.2 / §6.5)