The Ramanujan Machine publishes conjectures checked by digit agreement and argued from collision probability. This page re-decides their complete zeta(3) result sheet with certificates: rigorous enclosures whose every ingredient is proved inside the audit — including the two rows the Machine itself marks "new and unproven". All five survive. None was taken on faith, and a refutation would have been a discovery.
Local working document. SURVIVES means the claimed form lies inside a rigorous enclosure of the stated width — consistency proved to that slack, equality not proved and not claimed. Verdicts are unconditional: no convergence theorem, no special-function identity, no external constant is consumed.
| row | the Machine says | claimed form | evaluator | certified enclosure | width | verdict |
|---|---|---|---|---|---|---|
| rm-z3-pos | known | 5/(2 zeta(3)) | positive CF · proved seed | [2.079768431451768, 2.079768431451769] | 8.88e-16 | SURVIVES |
| rm-z3-inv | known | 1/zeta(3) | minus-CF · tail band | [0.831907372580699, 0.831907372580719] | 2.01e-14 | SURVIVES |
| rm-z3-apery | known (Apery) | 6/zeta(3) | minus-CF · tail band | [4.991444235484244, 4.991444235484246] | 1.78e-15 | SURVIVES |
| rm-z3-new1 | new and unproven | 8/(7 zeta(3)) | minus-CF · tail band | [0.950751282949380, 0.950751282949380] | 2.22e-16 | SURVIVES |
| rm-z3-new2 | new and unproven | 12/(7 zeta(3)) | minus-CF · tail band | [1.426126924424070, 1.426126924424070] | 8.88e-16 | SURVIVES |
Transcribed from rm_zeta3.pdf (sha256 6429e280dad89fd3…, re-hashed during this build — a drifted source refuses the audit). The slow row is honest: rm-z3-inv's continued fraction genuinely converges slowly (§4), and its certificate says so rather than borrowing precision it does not have.
A minus continued fraction b0 − a1/(b1 − a2/(b2 − …)) offers no free enclosure: the subtractions could take a tail anywhere, so the positive-CF seed argument does not apply. The certificate replaces it with a TAIL BAND: two polynomials L(n), U(n) and a base index N0, with four families of polynomial inequalities proved for every n ≥ N0 by shift-and-check — substitute n = N0 + m, expand exactly over BigInt rationals, and require every coefficient nonnegative:
Every depth-k truncation then lives inside the band by finite induction, so one backward interval pass from the proved seed encloses EVERY deep convergent at once. Convergence is proved inside the same certificate: deepening the truncation lowers the terminal, increasing maps propagate the drop, so the convergents decrease monotonically and are bounded below by the enclosure — no external convergence theorem is consumed. For the Apéry row the checker reports, verbatim:
a(n) > 0 for all n >= 1
L(n) > 0 for all n >= 52
(T) b(n) − L(n) >= 0 for all n >= 52 (shift-and-check coefficient positivity)
(T) U(n) − b(n) >= 0 for all n >= 52 (shift-and-check coefficient positivity)
(I−) band invariance from below for all n >= 52 (shift-and-check coefficient positivity)
(I+) band invariance from above for all n >= 52 (shift-and-check coefficient positivity)
The checker can refuse, and the battery proves it does: a band that excludes the tail fails (T), a band that merely touches optimality fails (I−), a negative partial numerator fails (P) — three distinct red controls, firing for three distinct reasons.
The claimed forms all speak ζ(3), so the audit needs the constant — without borrowing it. The bracket is computed from the defining series Σ 1/k³ alone: the first K = 6000 terms summed EXACTLY over BigInt, and the tail bracketed by convexity — the tangent-line bound gives tail ≤ 2/(2K+1)², the midpoint Taylor remainder gives tail ≥ 2/(2K+1)² − 2/(2K−1)⁴. Both ends are exact rationals; the bracket is 9.65e-17 wide and costs 7 ms. Elementary calculus, no cited theorem.
Every final comparison — form bracket against CF enclosure — is made in exact rational arithmetic (enclosure endpoints convert losslessly, every double being m·2^e). The verdict never touches floating point.
The row rm-z3-inv (1/ζ(3), with a_n = n⁶ and b_n = n³+(n+1)³) hides a trap: its tail recurrence is EXACTLY solved by s_n = n³ —
(re-proved during this build: the polynomial identity cancels monomial by monomial). That solution is spurious — following it gives CF value 0, not 1/ζ(3) — and it sits directly against the true branch: the fixed-point equation c² − 2c + 1 = 0 has a DOUBLE root. Two consequences. A naive cubic band [αn³, βn³] provably cannot work for this row (the invariance margin vanishes at the double root), so the certificate uses L(n) = n³ + 2n², which excludes the spurious branch — and the exponent 2 is sharp. And the continued fraction genuinely converges slowly: the per-level contraction is only 1 − O(1/n), so the certified enclosure at depth 10⁷ is honestly 2.01e-14 wide, while every fast row of the sheet reaches machine precision by depth 80. An audit that cannot say "this one is slow" would have nothing to say at all.
The corpus certifies against BYTES: each transcription carries the sha256 of its source PDF and re-hashes it at certify time. When the pinned sheet was re-read against the transcription, the audit was found to be running on FOUR rows of a five-row table — the missing row (5/(2ζ(3))) is a positive CF sitting in a minus-CF table, and the first transcription pass walked past it. It is row one of §1 now, surviving at 8.88e-16. A transcription error the instrument caught is instrument yield; a transcription error the instrument would have missed is the reason the pins exist.
make test re-runs everything above inside the cf battery: Brouncker and Euler calibrations, Apéry's proved identity as the anchor (an evaluator that refuted it would be broken), the transcription float-guard over all ten corpus rows, seven red controls, and the zeta(3) bracket cross-checked at two depths. The certificates themselves are four shift-and-check positivity facts plus one outward-rounded backward pass — small enough to re-verify by hand, which is the point.