Encyclopedia Masses Masses Mass Genesis T10 Hypercube Substrate Hypercube Edge Connectivity Three In

ARTICLE 4 claims 3 theorems 1 model

Masses Mass Genesis T10 Hypercube Substrate Hypercube Edge Connectivity Three In

A machine-checked proof shows that a three-dimensional hypercube graph cannot be split into two nonempty parts by cutting fewer than three edges.

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The three-dimensional wall

A hypercube graph is a network whose vertices are strings of bits, with an edge between two vertices when they differ in exactly one bit. The three-dimensional hypercube is the familiar cube: eight vertices, each connected to three neighbors. Its edge connectivity is three, meaning that any way of dividing the eight vertices into two nonempty groups must cut at least three edges. The declaration hypercubeEdgeConnectivity_three_inhabited is a machine-checked proof of exactly this fact, established in the framework's library of formal theorems.

The proof is not a numerical check but a general argument. It shows that for any nonempty proper subset of the cube's vertices, the number of edges crossing from the subset to its complement is at least three. This is the classical edge-connectivity property of the cube, proved in full generality for the three-dimensional case. The same library carries the general statement for any dimension d, and the three-dimensional instance is discharged by a bipartite handshaking argument, not by exhaustive computation.

In Recognition Science, this graph appears as the substrate on which particle masses are modeled. The framework derives the substrate graph from ledger atomicity: legal posts to a discrete record of events induce a graph on parity patterns, and that graph is the hypercube. The three-dimensional case matters because mass-genesis modules use d = 3, giving a wall of four edges. The theorem establishes that this wall is real: no smaller cut can separate the substrate's states.

What the declaration does not claim is equally precise. It does not derive the dimension d itself; lower bounds on d come from measured lepton ratios, not from this theorem. It does not predict any particular mass value, since the wall is an upper bound on what the substrate can support, not a specific outcome. And it does not settle whether d greater than 3 is legitimate here; that question remains open in the framework.

THEOREM hypercubeEdgeConnectivity_three_inhabited · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean
hypercubeEdgeConnectivity_three_inhabited · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean:983 
theorem hypercubeEdgeConnectivity_three_inhabited :
    HypercubeEdgeConnectivity 3 :=
  hypercubeEdgeConnectivity_three
THEOREM hypercubeEdgeConnectivity_three · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean
hypercubeEdgeConnectivity_three · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean:859 
theorem hypercubeEdgeConnectivity_three : HypercubeEdgeConnectivity 3 := by
  intro A hne hneU
  have hpos : 0 < A.card := Finset.card_pos.mpr hne
  have hlt := card_lt_univ (d := 3) hneU
  by_cases hbig : 5 ≤ A.card
  · have hAc : (Aᶜ).card = 8 - A.card := by
      simpa [Patterns.card_pattern] using Finset.card_compl A
    have hAcpos : 0 < (Aᶜ).card := by omega
    have hle : (Aᶜ).card ≤ 3 := by omega
    have hge := hypercubeCut_ge_when_small (d := 3) Aᶜ hAcpos hle
    have : 3 ≤ (Aᶜ).card * (3 + 1 - (Aᶜ).card) := by
      interval_cases (Aᶜ).card <;> norm_num
    have := le_trans this hge
    simpa [hypercubeCut_compl A] using this
  · push_neg at hbig
    by_cases h4 : A.card = 4
    · have hsum := degree_sum_eq_internal_plus_cut (d := 3) A
      have hin := hypercubeInternalDir_le_parts (d := 3) A
      set ae := (A.filter (fun p => hammingWeight p % 2 = 0)).card with hae
      set ao := (A.filter (fun p => hammingWeight p % 2 = 1)).card with hao
      have hsumparts : ae + ao = 4 := by
        have hsplit :
            A =
              A.filter (fun p => hammingWeight p % 2 = 0) ∪
                A.filter (fun p => hammingWeight p % 2 = 1) := by
          ext p; simp; constructor
          · intro hp; by_cases h : hammingWeight p % 2 = 0
            · exact Or.inl ⟨hp, h⟩
            · exact Or.inr ⟨hp, by omega⟩
          · intro h; rcases h with ⟨hp, _⟩ | ⟨hp, _⟩ <;> exact hp
        have hdisj :
            Disjoint (A.filter (fun p => hammingWeight p % 2 = 0))
              (A.filter (fun p => hammingWeight p % 2 = 1)) := by
          refine Finset.disjoint_left.mpr ?_
          intro p hp0 hp1
          have h0 := (Finset.mem_filter.mp hp0).2
          have h1 := (Finset.mem_filter.mp hp1).2
          omega
        have hcu := Finset.card_union_of_disjoint hdisj
        have : ae + ao = A.card := by
          rw [hae, hao, ← hcu, ← hsplit]
        omega
      have hprod : ae * ao ≤ 4 := by
        have : ae ≤ 4 := by omega
        interval_cases ae <;> omega
      have hin' : hypercubeInternalDir A ≤ 2 * (ae * ao) := by
        simpa [Nat.mul_assoc] using hin
      have hle8 : hypercubeInternalDir A ≤ 8 :=
        Nat.le_trans hin' (Nat.mul_le_mul_left 2 hprod)
      have h12 : hypercubeCut A + hypercubeInternalDir A = 12 := by
        have : 3 * A.card = 12 := by simp [h4]
        omega
      have : 4 ≤ hypercubeCut A := by omega
      exact Nat.le_trans (by decide : 3 ≤ 4) this
    · have hle : A.card ≤ 3 := by omega
      have hge := hypercubeCut_ge_when_small (d := 3) A hpos hle
      have : 3 ≤ A.card * (3 + 1 - A.card) := by
        interval_cases A.card <;> norm_num
      exact le_trans this hge
MODEL T10HypercubeSubstrateCert · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean
T10HypercubeSubstrateCert · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean:989 
structure T10HypercubeSubstrateCert : Prop where
  oneBitDiff_iff_adj :
    ∀ {d : Nat} (p q : Pattern d), OneBitDiff p q ↔ HypercubeAdj p q
  legal_tick_is_hypercube_edge :
    ∀ {d : Nat} {L L' : LedgerState d},
      LegalAtomicTick L L' → HypercubeAdj (parity d L) (parity d L')
  post_realizes_flip :
    ∀ {d : Nat} (L : LedgerState d) (k : Fin d) (side : Side),
      HypercubeAdj (parity d L) (parity d (post L k side)) ∧
        parity d (post L k side) = flipAt (parity d L) k
  bipartite :
    ∀ {d : Nat} {p q : Pattern d},
      HypercubeAdj p q → hammingWeight p % 2 ≠ hammingWeight q % 2
  xor_preserves_adj :
    ∀ {d : Nat} (c : Pattern d) {p q : Pattern d},
      HypercubeAdj p q → HypercubeAdj (xorBy c p) (xorBy c q)
  xor_transitive :
    ∀ {d : Nat} (p q : Pattern d), ∃ c : Pattern d, xorBy c p = q
  evenClass_attains_maxcut :
    ∀ d : Nat, 0 < d → hypercubeCut (evenClass d) = d * 2 ^ (d - 1)
  singleton_cut_eq_degree :
    ∀ {d : Nat} (p : Pattern d), hypercubeCut ({p} : Finset (Pattern d)) = d
  connectivity_three : HypercubeEdgeConnectivity 3
  wall_three : hypercubeCut (evenClass 3) = 4 * 3
  muon_forces_d : ∀ d : Nat,
    (206768 : ℚ) / 1000 ≤ (2 : ℚ) ^ (d - 1) → 9 ≤ d
  tau_forces_d : ∀ d : Nat,
    (347723 : ℚ) / 100 ≤ (2 : ℚ) ^ (d - 1) → 13 ≤ d
  muon_forces_cells : ∀ N : Nat,
    (206768 : ℚ) / 1000 ≤ (2 : ℚ) ^ (3 * N - 1) → 3 ≤ N
  tau_forces_cells : ∀ N : Nat,
    (347723 : ℚ) / 100 ≤ (2 : ℚ) ^ (3 * N - 1) → 5 ≤ N
  posting_witness :
    let L := zeroLedger 3
    let L' := post L (0 : Fin 3) Side.debit
    LegalAtomicTick L L' ∧
      HypercubeAdj (parity 3 L) (parity 3 L') ∧
      parity 3 L' = flipAt (parity 3 L) 0
  scope_does_not_derive_d : True
  scope_wall_is_capacity_not_prediction : True
  scope_d_gt_three_is_open : True
THEOREM muon_ratio_forces_accounts · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean
muon_ratio_forces_accounts · IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean:938 
theorem muon_ratio_forces_accounts {d : Nat}
    (h : (206768 : ℚ) / 1000 ≤ (2 : ℚ) ^ (d - 1)) : 9 ≤ d := by
  by_contra hlt; push_neg at hlt; interval_cases d <;> norm_num at h

What this page does not claim

The theorem does not derive the dimension d from ledger atomicity. It does not predict any particular particle mass value. It does not establish whether d greater than 3 is legitimate in the framework.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/MassGenesis/T10HypercubeSubstrate.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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