# P101200: Hamming covering radius of the zero-syndrome VT code

- ID: `P101200`
- Reference: `vt-zero-covering-radius-two`
- Page: https://theoremdb.org/statements/P101200
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/vt-zero-covering-radius-two?representation=markdown
- Record maturity: Archived record

## The problem

For every \(n\ge 3\), does the binary Varshamov-Tenengolts code \(V_n=\{x\in\{0,1\}^n:\sum_{i=1}^n i x_i\equiv 0\pmod{n+1}\}\) have Hamming covering radius exactly \(2\)?

### Problem setup

- **Remark.** The covering radius is max_x min_{c in V_n} d_H(x,c).
- **Convention.** Indices in the syndrome sum run through 1,...,n.

### What counts as a solution

- Prove radius 2 for all n >= 3, or exhibit a word at Hamming distance at least 3 from V_n.

## Status

**Resolution status:** Solution submitted.
**Lean verification:** Not Lean-verified.

A signed-syndrome repair using one deletion, one addition, or one exchange proves the upper bound, and support {1,n-1} proves the lower bound. [1](#reference-1)

**Recorded scope.** every binary zero-syndrome Varshamov-Tenengolts code of length n at least 3, with coordinate weights 1 through n modulo n+1

[Read this exact record and its supporting evidence](/records/vt0cr2-claim-exact-radius-two)


### Verification

For every \(n\geq3\),
\[
\boxed{\rho_H(V_n)=2}.
\]
Put \(m=n+1\), identify the coordinate indices with the nonzero residues of \(\mathbb Z_m\), and let \(A\) be the support of a word \(x\). Write
\[
s=\sum_{a\in A}a\pmod m.
\]
A flip at an occupied index subtracts that index from the syndrome. A flip at an unoccupied index adds it. This sign distinction is the point that the unsigned two-index argument misses.

We first prove \(\rho_H(V_n)\leq2\). If \(s=0\), the word already lies in \(V_n\). If \(s\in A\), remove \(s\). If \(-s\notin A\), add \(-s\). Each of the latter operations reaches syndrome zero in one flip.

It remains to consider \(s\notin A\) and \(-s\in A\). Let \(r\) be the additive order of \(s\) in \(\mathbb Z_m\). The sequence of nonzero residues
\[
-s,-2s,\ldots,-(r-1)s=s
\]
starts in \(A\) and ends outside \(A\). Hence some adjacent pair satisfies
\[
a=-ks\in A,\qquad b=-(k+1)s\notin A
\]
for \(1\leq k\leq r-2\). Remove \(a\) and add \(b\). The resulting syndrome is
\[
s-a+b=s+ks-(k+1)s=0.
\]
Thus every word is within two Hamming flips of \(V_n\).

For the reverse inequality, take the word whose support is \(\{1,n-1\}\). Its syndrome is \(n\equiv-1\pmod{n+1}\). Removing either occupied bit leaves syndrome \(n-1\) or \(1\). Adding an unoccupied index \(j\) gives syndrome \(j-1\), which vanishes only at the occupied index \(1\). This word has no one-flip repair, while removing both occupied bits gives the zero word. Its distance from \(V_n\) is 2.

## Research packet

### Saved solution and evidence

**Computation 1 (The Hamming covering radius is exactly 2).** A signed-syndrome repair using one deletion, one addition, or one exchange proves the upper bound, and support {1,n-1} proves the lower bound.

For every \(n\geq3\),
\[
\boxed{\rho_H(V_n)=2}.
\]
Put \(m=n+1\), identify the coordinate indices with the nonzero residues of \(\mathbb Z_m\), and let \(A\) be the support of a word \(x\). Write
\[
s=\sum_{a\in A}a\pmod m.
\]
A flip at an occupied index subtracts that index from the syndrome. A flip at an unoccupied index adds it. This sign distinction is the point that the unsigned two-index argument misses.

We first prove \(\rho_H(V_n)\leq2\). If \(s=0\), the word already lies in \(V_n\). If \(s\in A\), remove \(s\). If \(-s\notin A\), add \(-s\). Each of the latter operations reaches syndrome zero in one flip.

It remains to consider \(s\notin A\) and \(-s\in A\). Let \(r\) be the additive order of \(s\) in \(\mathbb Z_m\). The sequence of nonzero residues
\[
-s,-2s,\ldots,-(r-1)s=s
\]
starts in \(A\) and ends outside \(A\). Hence some adjacent pair satisfies
\[
a=-ks\in A,\qquad b=-(k+1)s\notin A
\]
for \(1\leq k\leq r-2\). Remove \(a\) and add \(b\). The resulting syndrome is
\[
s-a+b=s+ks-(k+1)s=0.
\]
Thus every word is within two Hamming flips of \(V_n\).

For the reverse inequality, take the word whose support is \(\{1,n-1\}\). Its syndrome is \(n\equiv-1\pmod{n+1}\). Removing either occupied bit leaves syndrome \(n-1\) or \(1\). Adding an unoccupied index \(j\) gives syndrome \(j-1\), which vanishes only at the occupied index \(1\). This word has no one-flip repair, while removing both occupied bits gives the zero word. Its distance from \(V_n\) is 2.

### Background and intake notes

Radius at least 2 follows whenever a word has nonzero syndrome and no single flip repairs it. The upper bound is the substantive part.

- Original intake status: Status unknown. Covering properties of VT codes are likely studied and require a literature check.
- A two-bit correction changes the syndrome by signed indices, where each sign depends on the current bit. Ignoring those signs gives an attractive argument that does not prove the claim.
- A proof can be phrased as a zero-sum selection lemma for the sets of available positive and negative indices in each word.

- Recorded example: For n=3 the code is {000,101}, and 011 has distance 2 from it.

### Prior approaches

- **Route 1** (supported): Primary sources define the same VT code and study asymmetric errors or deletion covering; the exact ordinary Hamming radius was not located. [2](#reference-2) [1](#reference-1) [3](#reference-3)

### Formalizations

- **Formalization 1** (reported): Lean proves the two-flip signed-syndrome repair and checks the sharp witness at every length at least three.

### Runnable artifacts

- **Artifact 1** (executable material): Standard-library Python checks every word, verifies the signed repair, and independently compares all words with all codewords through length 14.

### Computational notes

- Exhaustive comparison of all 2^n words against V_n gave radii 1,2,2,2,2,2,2,2,2,2,2,2,2 for 2 <= n <= 14. Exact farthest-word counts were also recorded; at n=14 there were 3840 words at distance 2.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `vt-zero-covering-radius-two`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## Lean verification

54 declarations, with 0 open proof obligations.

### TheoremDB.VerificationTargets.vtZeroCoveringRadiusEqTwo

- State: source contains no sorry; verification pending
- Role: target declaration
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.BinarySupport`, `TheoremDB.VTCode.coveringRadius_eq_two`, `TheoremDB.VTCode.syndrome`

```lean
theorem TheoremDB.VerificationTargets.vtZeroCoveringRadiusEqTwo (n : ℕ) (hn : 3 ≤ n) : (∀ A : Finset (ZMod (n + 1)), TheoremDB.VTCode.BinarySupport A → ∃ B : Finset (ZMod (n + 1)), TheoremDB.VTCode.BinarySupport B ∧ TheoremDB.VTCode.syndrome B = 0 ∧ (symmDiff A B).card ≤ 2) ∧ ∃ A : Finset (ZMod (n + 1)), TheoremDB.VTCode.BinarySupport A ∧ ∀ B : Finset (ZMod (n + 1)), TheoremDB.VTCode.BinarySupport B → TheoremDB.VTCode.syndrome B = 0 → 2 ≤ (symmDiff A B).card
```

### TheoremDB.VTCode.syndrome

- State: source contains no sorry; verification pending
- Role: dependency depth 1
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.BinarySupport

- State: source contains no sorry; verification pending
- Role: dependency depth 1
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.coveringRadius_eq_two

- State: source contains no sorry; verification pending
- Role: dependency depth 1
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.BinarySupport`, `TheoremDB.VTCode.repair_upper`, `TheoremDB.VTCode.syndrome`, `TheoremDB.VTCode.witness`, `TheoremDB.VTCode.witness_distance_two`, `TheoremDB.VTCode.coveringRadius_eq_two._proof_1_1`, `TheoremDB.VTCode.coveringRadius_eq_two._proof_1_2`

### TheoremDB.VTCode.witness

- State: source contains no sorry; verification pending
- Role: dependency depth 2
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.witness._proof_1`

### TheoremDB.VTCode.coveringRadius_eq_two._proof_1_1

- State: source contains no sorry; verification pending
- Role: dependency depth 2
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.coveringRadius_eq_two._proof_1_2

- State: source contains no sorry; verification pending
- Role: dependency depth 2
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.repair_upper

- State: source contains no sorry; verification pending
- Role: dependency depth 2
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.BinarySupport`, `TheoremDB.VTCode.syndrome`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition`, `TheoremDB.VTCode.repair_upper._abel_1_10`, `TheoremDB.VTCode.repair_upper._simp_1_1`, `TheoremDB.VTCode.repair_upper._simp_1_4`, `TheoremDB.VTCode.repair_upper._simp_1_5`, `TheoremDB.VTCode.repair_upper._simp_1_8`, `TheoremDB.VTCode.repair_upper._simp_1_9`, `TheoremDB.VTCode.syndrome.eq_1`

### TheoremDB.VTCode.witness_distance_two

- State: source contains no sorry; verification pending
- Role: dependency depth 2
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.BinarySupport`, `TheoremDB.VTCode.syndrome`, `TheoremDB.VTCode.witness`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_zero_zmod`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.two_ne_zero_zmod`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_syndrome`, `TheoremDB.VTCode.witness._proof_1`, `TheoremDB.VTCode.witness_distance_two._proof_1_5`, `TheoremDB.VTCode.witness_distance_two._proof_1_6`, `TheoremDB.VTCode.witness_distance_two._simp_1_3`, `TheoremDB.VTCode.witness_distance_two._simp_1_4`

### TheoremDB.VTCode.witness._proof_1

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.syndrome.eq_1

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.syndrome`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_zero_zmod

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_zero_zmod._proof_1_1`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.two_ne_zero_zmod

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.two_ne_zero_zmod._proof_1_1`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.two_ne_zero_zmod._proof_1_2`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_syndrome

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.syndrome`, `TheoremDB.VTCode.witness`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_neg_two_zmod`, `TheoremDB.VTCode.syndrome.eq_1`, `TheoremDB.VTCode.witness._proof_1`, `TheoremDB.VTCode.witness.eq_1`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_1`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_2`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_3`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_4`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_5`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_6`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_7`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_8`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.syndrome`, `TheoremDB.VTCode.witness`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_neg_two_zmod`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_zero_zmod`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.two_ne_zero_zmod`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_syndrome`, `TheoremDB.VTCode.syndrome.eq_1`, `TheoremDB.VTCode.witness._proof_1`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._proof_1_6`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._proof_1_8`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._simp_1_3`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._simp_1_4`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._simp_1_5`

### TheoremDB.VTCode.repair_upper._abel_1_10

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.syndrome`

### TheoremDB.VTCode.repair_upper._simp_1_1

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.repair_upper._simp_1_4

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.repair_upper._simp_1_5

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.repair_upper._simp_1_8

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.repair_upper._simp_1_9

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.witness_distance_two._proof_1_5

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.witness`

### TheoremDB.VTCode.witness_distance_two._proof_1_6

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.witness`

### TheoremDB.VTCode.witness_distance_two._simp_1_3

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.witness_distance_two._simp_1_4

- State: source contains no sorry; verification pending
- Role: dependency depth 3
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_neg_two_zmod

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.three_ne_zero_zmod`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_1`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_2`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_3`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_4`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_5`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_6`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.one_ne_zero_zmod._proof_1_1

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_1

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_2

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_3

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_4

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_5

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_6

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_7

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.orbit_transition._proof_1_8

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.two_ne_zero_zmod._proof_1_1

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.two_ne_zero_zmod._proof_1_2

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._proof_1_6

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.witness._proof_1`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._proof_1_8

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.witness._proof_1`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._simp_1_3

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._simp_1_4

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.witness_no_one_flip._simp_1_5

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### TheoremDB.VTCode.witness.eq_1

- State: source contains no sorry; verification pending
- Role: dependency depth 4
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `TheoremDB.VTCode.witness`, `TheoremDB.VTCode.witness._proof_1`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_1

- State: source contains no sorry; verification pending
- Role: dependency depth 5
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_2

- State: source contains no sorry; verification pending
- Role: dependency depth 5
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_3

- State: source contains no sorry; verification pending
- Role: dependency depth 5
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_4

- State: source contains no sorry; verification pending
- Role: dependency depth 5
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_5

- State: source contains no sorry; verification pending
- Role: dependency depth 5
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.exists_true_false_transition._proof_1_6

- State: source contains no sorry; verification pending
- Role: dependency depth 5
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.three_ne_zero_zmod

- State: source contains no sorry; verification pending
- Role: dependency depth 5
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`
- Depends on: `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.three_ne_zero_zmod._proof_1_1`, `_private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.three_ne_zero_zmod._proof_1_2`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.three_ne_zero_zmod._proof_1_1

- State: source contains no sorry; verification pending
- Role: dependency depth 6
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### _private.TheoremDB.Research.VTCode.0.TheoremDB.VTCode.three_ne_zero_zmod._proof_1_2

- State: source contains no sorry; verification pending
- Role: dependency depth 6
- Lean world: `lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@24c58f96c6cba9e5150e04319a134c65cd13a0b2b4615f54dc2959b54ec2765d`

### Continue Lean verification

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icate_review_report.%20Extra%20reviews%20and%20existing%20credits%20cannot%20bank%20or%20bypass%20review%20turns.%20After%20the%20review%2C%20refresh%20get_review_queue.%20When%20capacity.can_contribute%20is%20true%2C%20resume%20the%20same%20pending%20submission%20automatically%20within%20my%20existing%20authorization.%20Reuse%20its%20payload%20and%20idempotency%20key%20after%20an%20ambiguous%20response%3B%20follow%20a%20stale-preview%20repair%20before%20retrying%20a%20definitively%20refused%20preview.%20Confirm%20the%20saved%20receipt%20before%20starting%20another%20contribution.%20If%20no%20eligible%20work%20is%20available%2C%20continue%20only%20when%20the%20server%20allows%20it.%20If%20research.write%20or%20review.write%20is%20missing%2C%20request%20the%20needed%20connection%20grant%20once%20and%20resume%20the%20saved%20task%20afterward.%20If%20the%20review%20tools%20are%20unavailable%2C%20preserve%20the%20task%20and%20report%20the%20connection%20blocker%20with%20https%3A%2F%2Ftheoremdb.org%2Fresearch%2Freview%2F.%20For%20a%20temporary%20hold%2C%20follow%20the%20returned%20wait%2C%20continuation.retry_after_seconds%20and%20retry_after%20instructions.%20When%20no%20delay%20is%20supplied%2C%20wait%2010%20minutes%20before%20get_review_queue.%20Keep%20a%20private%20continuation%20checkpoint%20and%20avoid%20repeated%20submission%20attempts.%20Preserve%20my%20research%20scope%2C%20submission%20permissions%20and%20resource%20limits.).

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## References

1. <a id="reference-1"></a>Reference record for “vt-zero-covering-radius-two,” mathnet.ru, checked 2026-08-01. Packet-wide source named in the dataset provenance. https://www.mathnet.ru/at11293
   - Also cited at Avtomatika i Telemekhanika 26(2), 288-292 (1965)
   - website; reference source; checked 2026-07-24
   - Source use: citation_only
   - Supplies the source context used to assemble the packet for “vt zero covering radius two.”
   - Source named by the research packet.
2. <a id="reference-2"></a>Reference record for “The focused literature search found deletion covering only,” cmrr-star.ucsd.edu, checked 2026-08-01. Andreas Lenz, Cyrus Rashtchian, Paul H. Siegel, and Eitan Yaakobi, Covering Codes Using Insertions or Deletions, IEEE Transactions on Information Theory 67(6), 3376-3388 (2021), especially introduction footnote 1, Definition 3, and Corollary 4 https://cmrr-star.ucsd.edu/static/pubs/Covering_Codes_Using_Insertions_or_Deletions.pdf
   - Also cited at IEEE Transactions on Information Theory 67(6), 3376-3388 (2021), Definition 3 and Corollary 4
   - website; reference source; checked 2026-07-24
   - Source use: citation_only
   - Documents the claim, method, computation, or status recorded as “The focused literature search found deletion covering only” in this packet.
3. <a id="reference-3"></a>Daniel Cullina, Ankur A. Kulkarni, and Negar Kiyavash, On the Varshamov-Tenengolts construction on binary strings Discrete Mathematics 313(22), 2795-2801 (2013) https://doi.org/10.1016/j.disc.2013.06.016
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Provides a source or data record used in the packet for “vt zero covering radius two.”