The binomial row sum is a power of two

Identifier: F:binomial-row-sum-two-power
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 natural n, the sum of the n-th row of Pascal's triangle is 2^n: sum over all integers k of C(n,k) equals 2^n. The summand vanishes outside 0 <= k <= n, so the sum is finite.

Formal statement

(define-hyperterm F (* (gamma (+ n 1)) (/ 1 (gamma (+ k 1))) (/ 1 (gamma (+ (- n k) 1)))))
(define-hyperterm T (^ 2 n))
(assert (forall ((n Int)) (=> (>= n 0) (= (sum ((k Int)) F) T))))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

0 direct dependencies 0 direct dependents

Evidence

wz-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 1, recurrence coefficients a_0 = 2, a_1 = -1 (i.e. S(n+1) = 2*S(n)), R(n,k) = k / (n - k + 1). Closed form 2^n verified to satisfy the same recurrence (a rational-function identity in its own shift ratio); base case S(0) = 1 = 2^0 checked by exact summation; leading coefficient a_1 = -1 has no integer zero, so the recurrence runs forward from n = 0.

cas-certificate-artifact-binomial-row-sum-two-power

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; the WZ-style proof is Zeilberger's",
      "what": "the row-sum identity and its creative-telescoping proof",
      "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"
    }
  ]
}