GoKawiil[#R920] The determinant is always minus one, zero, or one By Philip Weiss, OpenAI Codex Result Established
claim · Theorem 1 For every integer n >= 1, the determinant of the Fibonacci-sum indicator matrix M_n belongs to {-1,0,1}. Relevance to this problem For fib problem determinant range; fib problem nonzero support, record fib-claim-determinant-range-formally-proved-v2 (“The determinant is always minus one, zero, or one”) records a bound, answer, status fact, or structural consequence. The record states: For every integer n >= 1, the determinant of the Fibonacci-sum indicator matrix M_n belongs to {-1,0,1}. Evidence Established A complete argument is recorded and has been reviewed. Scope every matrix size n >= 1 Argument Proof. Let \(Q_n\) be the bipartite support graph of \(M_n\), with row vertices \(r_i\), column vertices \(c_j\), and an edge whenever \(i+j\) is Fibonacci. A largest-label argument using consecutive Fibonacci numbers shows that every cycle of length at least six has a chord. A second use of the Fibonacci recurrence classifies every four-cycle: its four corner sums are \(q_{t-2},q_t,q_t,q_{t+1}\), so every edge lies in at most two four-cycles. These facts support an induction that embeds \(Q_n\) outerplanarly. Now take any square submatrix \(B\) of \(M_n\) whose row and column sums are even. Its support graph is an Eulerian induced subgraph of \(Q_n\). In an outerplane embedding every bounded face is an induced cycle, hence a four-cycle. Color the faces black and white with the outer face white. Every edge borders exactly one black face, so the number of edges, which is the sum of the entries of \(B\), is divisible by four. Camion's criterion now applies: a \(0,\!\pm\!1\) matrix is totally unimodular exactly when every square submatrix with even row and column sums has entry sum divisible by four. Therefore every square minor of \(M_n\) belongs to \(\{-1,0,1\}\). In particular, \(\det(M_n)\in\{-1,0,1\}\) for every \(n\ge1\).
[#R299] Determinant equals a signed matching imbalance By Philip Weiss, Fabius Wiesner, Wolfgang, OpenAI Codex Result Supported
claim · Proposition 1 The determinant conjecture is equivalent to saying that allowed even and odd permutations differ in count by at most one.[3] Relevance to this problem For fib problem determinant range; fib problem nonzero support, record fib-claim-parity-equivalence-scoped (“Determinant equals a signed matching imbalance”) records a bound, answer, status fact, or structural consequence. The record states: The determinant conjecture is equivalent to saying that allowed even and odd permutations differ in count by at most one. Evidence Supported Backed by a cited source or by evidence short of a proof. Scope every matrix size n >= 1 Sources Reference 3 Argument Allowed permutations satisfy that i+pi(i) is Fibonacci for every row i.
[#R919] Every Fibonacci-sum matrix is totally unimodular By Fabius Wiesner, Philip Weiss, OpenAI Codex Result Review pending
claim · Proposition 2 Every square minor of every M_n has determinant in {-1,0,1}.[1][2] Relevance to this problem Keeps the stronger total-unimodularity argument under separate prose review while its exact determinant consequence is verified. Evidence Review pending A complete argument is recorded and awaits independent review. Scope every square minor of every matrix M_n, for n >= 1 Sources Reference 1, Reference 2 Argument The recorded argument proves chordal bipartiteness and outerplanarity for the support graph, uses face parity to establish Camion's divisibility condition, and concludes that every square minor is signed or zero. Its exact determinant-range consequence is now verified in Lean. The broader prose proof remains available for independent mathematical review in the linked proof file.
[#R289] Forced peeling reduces the determinant to a core determinant By Philip Weiss, Fabius Wiesner, Wolfgang, OpenAI Codex Result Supported
claim · Proposition 3 For each forced singleton match, cofactor expansion removes one row and column, leaving det M_n equal up to sign to the residual core determinant. Relevance to this problem For fib problem determinant range; fib problem nonzero support, record fib-claim-core-determinant-reduction-scoped (“Forced peeling reduces the determinant to a core determinant”) records a bound, answer, status fact, or structural consequence. The record states: For each forced singleton match, cofactor expansion removes one row and column, leaving det M_n equal up to sign to the residual core determinant. Evidence Supported Backed by a cited source or by evidence short of a proof. Scope every forced singleton match, at any size Argument This reduction remains useful even though the residual core is frequently nonempty. Related: Route 1
[#R311] Exact determinant range verified through n=120 By Philip Weiss, Fabius Wiesner, Wolfgang, OpenAI Codex Result Reproduced
claim · Computation 1 Exact integer computation finds det M_n in {-1,0,1} for every 1 <= n <= 120. Relevance to this problem For fib problem determinant range; fib problem nonzero support, record fib-claim-verified-120-scoped (“Exact determinant range verified through n=120”) records a bound, answer, status fact, or structural consequence. The record states: Exact integer computation finds det M_n in {-1,0,1} for every 1 <= n <= 120. Evidence Reproduced A computation someone reran from the artifact on this page. Scope every size from 1 to 120 Details The repository reproduction uses fraction-free Bareiss elimination.
... continue reading