The Groundtruth Seam · relations under pressure
The Equation That Vanished at the Sixth Digit
Operate a PSLQ relation search, reproduce the published BBP formula for pi, then test apparent equations among 188 named constants against withheld digits and explicit coefficient limits.
A short integer vector can be a theorem, a transcription error, or the shadow cast by rounded inputs. The order matters. The catalogue counts, table, and live pair bench stay hidden until the published vector returns.
Make the known equation return
Eight series enter with pi. The browser sums them from their displayed definition, computes pi separately, and sends all nine values through the same PSLQ engine used by the live pair bench below.
Xj = Σk≥0 16−k/(8k+j)
The returned normalized vector must be (4, 0, 0, −2, −1, −1, 0, 0, −1). It is the coefficient vector printed in Ferguson, Bailey, and Arno's published PSLQ analysis, for the Bailey-Borwein-Plouffe formula. The catalogue counts, table, and live pair bench remain hidden if this check fails.
A perfect equation, for exactly five places
Move one control. At five decimal places, the displayed decimal integers satisfy the relation exactly. Admit the next digit and keep the vector frozen.
127 − 56πd + 18ed = 0
At five places, this is an exact integer statement about 314159 and 271828: 12700000 − 56·314159 + 18·271828 = 0. The full-value residual is computed independently of the slider. The sixth digit does not weaken the relation. It rejects it.
Coincidence has a survival curve
We could not find this published as of 2026-08-01, having searched the PSLQ literature, the Ramanujan Library paper and public LIReC repository, arXiv, OEIS, Simon Plouffe's current index and b000 table, and web queries for catalogue-wide PSLQ audits and b000 relation scans.
This claim is only about the fixed snapshot, grammar, precision grid, coefficient grid, holdout rule, and null comparison below. Automated searches for relations among constants are already published, most broadly in The Ramanujan Library.
Plouffe cataloguedigit shufflematched random decimals
| digits | H | candidates | rejected | alive | confirmed | shuffle | random |
|---|
Run one pair, not the table
The full 35-cell census is the offline-certified new number. This smaller bench executes the shipped browser PSLQ directly on the same embedded source rows, so a reader can inspect one frozen vector and its holdout residual. This live bench evaluates source values in binary64 and calls a nondegenerate vector numerically alive when the absolute residual is at most 10−12. The offline grid uses the stricter printed-digit interval described in the check below.
The four small relations
Definition-confirmed and source-alive.
Definition-confirmed, but the final printed source digits miss the strict half-unit interval.
Definition-confirmed and source-alive.
The second row is the warning inside the warning. Independent definitions prove the relation, while the compact table's last printed digits fail the declared rounding interval. A source-level rejection is not a proof of mathematical falsehood.
The check
| source record | loading |
|---|---|
| search universe | 0 unordered pairs from 188 rows |
| relation grammar | c + ax + by = 0; primitive integers; a and b both nonzero |
| precision convention | first p significant decimal digits, truncated; p ∈ {8,12,16,24,32,40,48} |
| PSLQ convention | γ = √(4/3); tolerance 10−(p−1); 500 steps; working precision max(60,p+30) |
| coefficient convention | H = max(|c|,|a|,|b|) after gcd division and first-nonzero-positive sign |
| offline grid holdout | compatible only if the full-source residual lies within half a last-place unit from each row, weighted by |a| and |b| |
| null conventions | xorshift32 seeded by FNV-1a of mode, row index, and label; shuffle preserves leading digit, exponent, and length; random preserves exponent and length |
| independent confirmation | four algebraic or named-definition identities; no numerical persistence is promoted to proof |
| browser anchor | 32 terms per Xj, IEEE 754 binary64 PSLQ; offline check uses 160 decimal places with guard digits |
- The source URL states no explicit licence. Project Gutenberg's permission statement for a larger Plouffe collection does not clearly settle reuse terms for this standalone file. The uncertainty is recorded rather than inferred away.
- The source has about 64 significant digits per row, sometimes with conspicuous zero endings. Working precision does not create trusted source precision.
- The four confirmed relations are catalogue calibration, not new identities. One fails the strict final-digit source interval and is shown in both categories it actually occupies.
- Degenerate vectors with a = 0 or b = 0 are excluded. Otherwise a one-row rational approximation would be counted once for every unrelated partner, a reader-reachable vacuity.
From a repository checkout, run the independent checker with node research/pslq-constants/verify-pslq-constants.mjs. It reads the committed source record, recomputes every grid cell and null catalogue without importing this page engine, executes the shipped page engine separately, reads the publication date from Markdown frontmatter, and deliberately corrupts source digits and the BBP vector to show both assertion families go red.
A survival table cannot prove independence
This page does not test products, powers, logarithms, or larger tuples. It cannot prove transcendence, algebraic independence, or absence beyond its affine two-constant grammar and declared coefficient bounds. A stable numerical residual is not an identity until an independent definition or symbolic derivation closes it.
The next honest computation would repeat a preregistered holdout audit over the Ramanujan Library's richer expression grammar, with its maintainers' current database export and an error model attached to every source value. This page does not take that step.
Corrections
Corrected 2026-10-18: The page said the catalogue audit stayed locked when the published anchor failed, but its count boxes and full table remained visible. The catalogue counts, table, and live pair bench now stay hidden until the published vector returns; found by the assay line, 2026-10-18.
Corrected 2026-10-18: The summary said catalogue rows were visibly classified as identities, aliases, padded endings, or definitions not independently recomputed. The page shows four identity cards and a general warning about zero endings, not those row-by-row classifications; found by the assay line, 2026-10-18.
Corrected 2026-10-18: In a test that made the anchor routine throw after a successful run, the page still showed "published anchor reproduced" and the previous anchor result. That failure now shows "anchor error, audit locked" and clears the previous result; found by the assay line, 2026-10-18.