The alternating binomial row sum vanishes

Identifier: F:alternating-binomial-row-sum-zero
Proof route: cas-certificate
External status: proved
Axiom footprint: cas.exact-rational-polynomial-normal-form, cas.gamma-functional-equation, cas.hyperterm-specification-denotes-the-summand, cas.telescoped-term-natural-boundary

Recorded description

For every integer n >= 1, the sum over all integers k of (-1)^k C(n,k) is zero. The summand vanishes outside 0 <= k <= n. At n = 0 the sum is 1, which is why the statement starts at 1.

Formal statement

(define-hyperterm F (* (^ -1 k) (gamma (+ n 1)) (/ 1 (gamma (+ k 1))) (/ 1 (gamma (+ (- n k) 1)))))
(assert (forall ((n Int)) (=> (>= n 1) (= (sum ((k Int)) F) 0))))

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

wz-alternating-binomial-row-sum

Kind: witness-replay
Status: checked

Supports: the creative-telescoping certificate, replayed by a checker that shares no code with the search that produced it

Checker command

cargo test -p axeyum-cas --test telescoping_identities

Evidence notes

Certificate: order ZERO -- the summand telescopes in k on its own, with a_0 = n and R(n,k) = -k, so n*F(n,k) = G(n,k+1) - G(n,k) for G = -k*F(n,k). Summing gives n*S(n) = 0, hence S(n) = 0 for every n /= 0; no base case and no induction are needed. The vanishing of the leading coefficient at n = 0 is exactly the reason the identity excludes n = 0, where S(0) = 1 -- the certificate carries its own domain restriction.

cas-certificate-artifact-alternating-binomial-row-sum-zero

Kind: witness-replay
Status: checked

Supports: the serialised certificate itself, re-checked from the committed file with no search run

Checker command

cargo test -p axeyum-cas --test telescoping_certificate_artifacts

Evidence notes

This row exists so the fact rests on an ARTIFACT rather than on re-running the producer. The sweep reads every committed certificate, parses it, and hands it to the independent checker; nothing in that path calls the search. Tamper controls in the same file: a certificate numerator edited by one, a recurrence coefficient edited by one, and a certificate re-pointed at a neighbouring file's summand are each rejected, and a truncated / foreign-format / decimal-bearing file is refused by the reader before the checker sees it.

Provenance

{
  "date": "2026-08-14",
  "established_by": "lane telescoping: axeyum_cas::telescoping::zeilberger (untrusted search) + axeyum_cas::telescoping_check (independent checker)",
  "source": "classical binomial identity, stated by this lane as a HyperTerm specification",
  "prior_art": [
    {
      "who": "Attributed to standard binomial-theorem folklore",
      "what": "the alternating row-sum identity",
      "year": 1991,
      "where": "classical; see Petkovsek-Wilf-Zeilberger, A=B (1996), ch. 4-7",
      "attribution": "standard textbook attribution; this lane did not consult the primary source"
    }
  ]
}