Thomson : certificate data for the cap (), part 1 of 2
DefinitionThomsonN7_cap_data_1discrete-geometrythomson-problem
Exact certificate data (part 1 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.
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_core
/-!
# 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_FP0 : Blk := mkBlk 6 4 68 68
0x999999
0x834e6596243a000c6800000000000000008000000000000000080000000000000000800000000000000008000000000000000080000006335b6c7d7800000081174537ad800000000000000008000000000000000080000000000000000800000000000000007e0504a29e6820209800000000000054c590c8c1bae13d40a378000000000000000080000000000000000800000000000000006bd738080ed1544587b6dd88eb2d644b5fa4bbc0c7557bc781f849227714d29be5af8000000000000000080000000000000000ab3e1c6b5b96c2b79875b1a89d9aec3d1a68a79698d018a7cdb78a4e57626513b6c9b44b140f4f32caf47800000000000000009832b121ee8f118796e12dae6c4738bf751fb81cd302742da9d7965f08b7e25f521291611b6e8a4498f5287020f03ecf8b7329
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
def tc_cap_FP1 : Blk := mkBlk 5 4 68 72
0x99999
0x816ef5cb33dab95e4800000000000000008000000000000000080000000000000000800000000000000008d1dd678e5a4e53908eadf2e35567c76e8800000000000000008000000000000000080000000000000000960782704511f26f4a699ca3cd7c3d48d1d2cbf3a38f677a22580000000000000000800000000000000002eaf81c1b2108ed4c3f81c0116bd2fb4008b748b4c26f23a2d1c9c07539489c57908800000000000000008e45393b6833348c97b04350338a3b9db553935bac4b88de7cd773e6960931a0959fb27fa5b16e7df0b57
0x84ab4aaab85b61b7d58072a72790e7b19b03820ee5e2637e1c43cd7e3a33386be20856b580411428e717b62e0d8072a72790e7b19b03831347ed632cd875448083d0496b50e550aa7f9d9aa6c4eead39457fe809f125e1ed5b97820ee5e2637e1c43cd8083d0496b50e550aa8b5b0e6a6d85b782517f962e835d3921743581674ca8067704f0ad7e3a33386be20856b57f9d9aa6c4eead39457f962e835d3921743585c31530d943a4cbb5804e4f9a658071610b80411428e717b62e0d7fe809f125e1ed5b9781674ca8067704f0ad804e4f9a658071610b887816d319f4d69301
def tc_cap_FP2 : Blk := mkBlk 4 4 68 68
0x9999
0x838a15747cfffed76800000000000000008000000000000000080000000000000000cbefc12af13546b43aac55aadc48d1e83a80000000000000000800000000000000005ffba8ca5cd8e3b006f2383da9487cb62f806dc911786f95ebb8000000000000000078c594c633353d4866d9b81eec3226f4fd7f1675c7ccfc56036aa5693224820a4f5c
0xa4a8e402ca6949c6190c855e20c47d3688800d03b7e54a1128a8d833507207fa92ac90c855e20c47d36889bd8c1b6df60aa2187fd8865f01cacbaef84ea0009b40c9efda800d03b7e54a1128a7fd8865f01cacbaef88dec19481b1122eb8840da8d5ad54da088d833507207fa92ac84ea0009b40c9efda8840da8d5ad54da08be9fb2d1ec57db4da
def tc_cap_FP3 : Blk := mkBlk 3 4 68 68
0x999
0x88180c109c4565e57800000000000000008000000000000000096ec34f4f487b59dd8127ad06803074cef800000000000000006d490e554556b09a07d3bfe13ac9a7970f92ba558afd4504c0c
0x92aa736e94fe04fce7ff055373c1fd7f9d79420390146a80a8e7ff055373c1fd7f9d808f73dc51f0999aa806a91133e9c106b679420390146a80a8e806a91133e9c106b69e60dd18e183b5bf2
def tc_cap_FP4 : Blk := mkBlk 2 4 64 64
0x99
0x8b9a0e28d51a3d1080000000000000007a1358bc767e68c984fa41628db5064b
0x91f369db604e90ff824b37d1b06f7011824b37d1b06f70118399615ccf5a1b05
def tc_cap_FP5 : Blk := mkBlk 1 4 40 40
0x9
0xa23610e3f4
0x9dd5c55f70
def tc_cap_FR0 : Blk := mkBlk 6 4 68 68
0x999999
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
def tc_cap_FR1 : Blk := mkBlk 5 4 68 68
0x99999
0x80d61eac675a2d2218000000000000000080000000000000000800000000000000008000000000000000082fcbbd21a1b655c6829906e5c2e5fae768000000000000000080000000000000000800000000000000009e1c153e746f73f7caca31e107713c16c597bf59c3cdc02103a800000000000000008000000000000000065300ce946cc55bb674256b6e8eb5be6219465ad3ed48a4c87d98989a794889288b180000000000000000828889bc41d43b31e836d9edc58355e37a79c1de205afab9b2e76046bd77c6fab626a978a5671a0484932
0x9eba092eab7b7ff6d7f5e69b42d5e566d0783912e3730605bec8ab0f4add89b1497c8b130ff3da54afa427f5e69b42d5e566d08c63913b4f6d5fb8679f505fc37437ee417f2b102f602a3903180957b754ca19dd4e783912e3730605bec79f505fc37437ee41aabf97901ae0e0ad17f40993a372345ff7843c8bcfb776719e28ab0f4add89b1497c7f2b102f602a390317f40993a372345ff7ac83b55a37fec3b6d7be0a260da4f2a8c88b130ff3da54afa4280957b754ca19dd4e843c8bcfb776719e27be0a260da4f2a8c8c451e66886411de72
def tc_cap_FR2 : Blk := mkBlk 4 4 68 68
0x9999
0x80000005e145384b3800000000000000008000000000000000080000000000000000818d3cb62bdcaa47c80c69e5b15ee62f2f80000000000000000800000000000000006e5736f41117ed462772b9b7a088bf704583ce75ba9482f0f638000000000000000085ad3210907903e7a764aaed601687048478f1e8f28b779c9ee8faee4bed8891686c
0x878539ddd12a1a24b800142926ba863edc7ef7cb57705059f64867bc29acc860c5d4800142926ba863edc80f61d34fa66c14e47f83142179b15b42f806172d7bc0e15c6e7ef7cb57705059f647f83142179b15b42f84ca33d0ca23bc653801df4c6d37b47a0e867bc29acc860c5d4806172d7bc0e15c6e801df4c6d37b47a0e8d81a67cb4995241c
def tc_cap_FR3 : Blk := mkBlk 3 4 68 64
0x999
0x815f69bb122bec6be8000000000000000080000000000000000810b9b13d693a6c518062f0a74c8a7aa28800000000000000008ac34516822cfb5f87cf72b2ba802e760987d62ae396aed704d
0xcc378a12a3fbf67d815df3c20176f0385e349d0013fc3ac0815df3c20176f038b4e291a2f2098d09a5bc1ed61b9313115e349d0013fc3ac0a5bc1ed61b931311f0fcff41da5ce741
def tc_cap_FR4 : Blk := mkBlk 2 4 60 56
0x99
0x80001a99891b9638000000000000002e5ac8801241079869594fdeeb4c56
0x92667b46dcf7ac6d9c014e547ff86d9c014e547ff8e071c272fa4bc4
def tc_cap_FR5 : Blk := mkBlk 1 4 60 60
0x9
0xc23cd169ad5e2ab
0xa6238384f90095f
def tc_cap_SA0 : 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,
0x99999999999999999999999999999999999999999999999999999999,
0x8000000105f5486c8f8000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000009dc802a045d212031216a8000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000019c88026d792364440c27b8039385245fc47b27e800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000010b2d7fcc97bfa518e23aee7fd74839675414162580899a0edd57573db18000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000008000000000000000007fffffffffffffb9288010963dad3336d8ce802a71358a5cb8cc66814cd485e9b63efb1881c0b344c818f03156800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000000000800000000000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def tc_cap_SA1 : 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, 76,
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⟩
def tc_cap_SA2 : 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, 76,
0x99999999999999999999999999999999999,
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def tc_cap_SA3 : 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, 76,
0x99999999999999999999999999999999999,
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def tc_cap_SA4 : 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, 76,
0x99999999999999999999,
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def tc_cap_SA5 : TBlk :=
⟨[13], [(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, 68, 72,
0x99999999999999999999999999999999999,
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⟩
def tc_cap_SA6 : TBlk :=
⟨[13, 10], [(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, 68, 72,
0x99999999999999999999999999999999999,
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def tc_cap_SB0 : 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, 72,
0x99999999999999999999999999999999999999999999999999999999,
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⟩
end Cert
end ThomsonN7
end Asm_cap_tc
section Asm_cap_ct
namespace ThomsonN7
namespace Final
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)