Numbers are weird and perfect uniformity is (almost uniformly) a trap. Do not use BIDDER to generate secrets.
Arithmetic Congruence Monoids encoded as Champernowne reals, and the BIDDER block generator that falls out of them. Named for George Bidder's logarithms, which everyone seems to forget about.
For each positive integer n, the multiplicative monoid nZ+ has irreducible elements — n-primes (multiples of n not divisible by n²).
The first few n-primes for small n:
| n | first n-primes |
|---|---|
| 2 | 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, … |
| 3 | 3, 6, 12, 15, 21, 24, 30, 33, 39, 42, … |
| 4 | 4, 8, 12, 20, 24, 28, 36, 40, 44, 52, … |
| 5 | 5, 10, 15, 20, 30, 35, 40, 45, 55, 60, … |
| 10 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 110, … |
Concatenating those digits as a sequence in order gives the Champernowne real C(n):
| n | C(n) |
|---|---|
| 2 | 0.2610141822263034… |
| 3 | 0.3612152124303339… |
| 4 | 0.4812202428364044… |
| 5 | 0.5101520303540455… |
| 10 | 0.1020304050607080… |
Mathematical definitions in this README and the core theory docs are written in BQN as executable notation. Vocabulary at BQN-AGENT.md; the prose carries the meaning if you skip the block.
NPn2 ← {(0≠𝕨|·)⊸/ 𝕨×1+↕𝕩×𝕨} # n-primes for n >= 2
Digits10 ← {𝕩<10 ? ⟨𝕩⟩ ; (𝕊⌊𝕩÷10)∾⟨10|𝕩⟩}
ChamDigits10 ← {⥊ Digits10¨ 𝕩} # exact digit concatenation
LeadingInt10 ← {⊑ Digits10 𝕩} # leading digit of an integer
ChamDigits10 (5↑ 3 NPn2 5) # ⟨3,6,1,2,1,5,2,1⟩ — digits of C(3)
LeadingInt10 3 # 3 — the leading digit of n is the
# leading digit of C(n)- ACM-CHAMPERNOWNE.md — the construction and proofs
- BLOCK-UNIFORMITY.md — exact leading-digit uniformity, integer and sieved
The construction is the deletion rule. The consequences are larger than the construction. Below: a bilingual algebra that organises the substrate's residuals (the open heart, potentially the most important feature); a keyed permutation that mates with the substrate as a block-structured generator; and the art that falls out when exact data has room to move.
Our constructive normals leave algebra in their residue.
The single deletion rule M_n = {1} ∪ nZ_{>0} does three things at
once: it discards unique factorisation, exposes the arithmetic
progression Hardy inverts, and supplies the generating function
ζ_{M_n}(s) = 1 + n^{-s} ζ(s) whose Mercator log is a bilingual
program. Bilingual because the K-index it runs on positions an
atom in the substrate lattice and a position in the base-b digit
stream of C_b(n) simultaneously, polylog-bignum work to convert
in either direction.
What that buys: the substrate organises its own residuals as
algebra. offset(n) indexed by ord(b, n), β(n) for the
O(b^{-k}) tails, the off-spike denominator process between
consecutive boundary spikes, the (n−1)/n² smooth-block density,
the α_n = (n−1)/n mult-table asymptote, the lucky-cancellation
locus — distinct objects, all referring back to the same deletion
rule. Other constructive normals — classical Champernowne (1933),
Stoneham (1973), Becher–Figueira — have residues. None of them
expose those residues as algebra in the way this substrate does.
The open commitment is absolute normality of every C_b(n)
across every base, not just the base of concatenation that
Copeland–Erdős (1946) settles. We submit it will remain open. The
algebra of residuals would either close it or record exactly where
closing fails — independent of whether absolute normality itself
ever falls.
- THE-OPEN-HEART.md — the manifesto: closure refused, log identity recovered at higher altitude, UFD traded for K-indexed access
- arguments/THE-WHOLE-MACHINE.md — the eleven parts of the BIDDER UTM, named once each
- algebra/Q-FORMULAS.md — the master expansion
Q_n(m) = Σ_{j=1}^{ν_n(m)} (−1)^{j−1} τ_j(m/n^j) / jand its row specialisations - algebra/FINITE-RANK-EXPANSION.md — the rank lemma: Mercator truncates at
j = ν_n(m)by integer divisibility - algebra/WITHIN-ROW-PARITY.md — first closed-form deliverable: autocorrelation factored as algebraic
V× combinatorialD - experiments/acm/cf/DENOMINATOR-PROCESS.md — the CF coordinate of the cost ledger; three-population decomposition
Not that kind of crypto. A block-structured randomization tool that mates smoothly with a small, well-understood block cipher.
BIDDER exists to keep two guarantees separable. An algebraic substrate — the ACM-Champernowne digit block [b^(d-1), b^d - 1] — gives exact leading-digit uniformity by a counting argument from positional notation: across the block there are exactly b^(d-1) integers with each leading digit, with no error term. A keyed permutation — Speck32/64 in cycle-walking mode for operating blocks well below 2^32, with a Feistel fallback when the cycle-walking ratio gets bad — provides the disorder. Neither piece is asked to do the other's job.
Proved:
- The algebraic substrate is exact, not merely tested. The integer block lemma in core/BLOCK-UNIFORMITY.md is a counting argument from positional notation; there is no hidden block size where it stops working.
- At
N = period, the MC estimate equals the left-endpoint Riemann sumRfor any key and any integrand. This is the structural permutation-invariance theorem in generator/RIEMANN-SUM.md. - The root entry points are stable:
BIDDER.md documents
bidder.cipher(period, key)andbidder.sawtooth(n, count), with core/API.md as the detailed cipher-path reference.
Measured:
- Full-period digit counts are exactly
period / (b - 1)for every digit, every key, across the tested bases and digit classes. - All
ddigit positions of each permuted index are independently uniform, with pairwise joints exact on the tested ranges. - SHA-256 rekeying across period boundaries shows no detectable seam
for
d ≥ 3. - Stratified sampling at block boundaries is totalizing for smooth functions.
- The theory front is now executable. Three red-team test files
under
tests/theory/attack the structural, quadrature, and statistical layers independently. The organizing document is RED-TEAM-THEORY.md — it decomposes the total errorE_N − I = (E_N − R) + (R − I)into four layers, names what would falsify each claim, and tracks the measured coupling gap between the cipher backend and the ideal without-replacement null.
Not claimed:
-
The PRP properties anyone would want for adversarial use — distinguishing advantage in this exact composition, key independence beyond the structural
E_P = Rresult, side-channel behavior, robustness under chosen input — are inherited from Speck and have not been independently verified for BIDDER. -
The Feistel fallback at small block sizes shows a measurable variance gap at intermediate
N: therandom.shufflenull matches the finite-population correction formula, but the Feistel-keyed permutations run roughly1.5×to2.5×worse. This is a backend property, not an algebra failure. -
BIDDER.md — root API reference:
bidder.cipherandbidder.sawtoothin three layers (natural language, Python, BQN) -
core/API.md — detailed cipher-path reference
-
generator/BIDDER.md — cipher design doc, observed properties, known limitations, open questions
Uniform numbers, given too much room, make a star.
Art folders coordinate experiments on exact data. The next time someone says some of science is an art you should take it seriously. The below are a random set of examples.
corona_attempt.png— the sun above. An instructive failure: binary run lengths on polar axes collapse toward the center because short runs dominate exponentially. The "sun" is a failure mode caused by structure of structure.dense_bloom.png— decimal block structure rendered as a bloom.escalating_bidder_mul.png— 1, then 5, then 10 multiplications in a sea of additions. Each burst deepens the Benford scar. The additive staircase is the Champernowne sawtooth; the multiplicative kick is ε doing its work.art_groove.png— four Benford demos rendered as vinyl records. Groove eccentricity is mantissa non-uniformity; a perfect circle is Benford equilibrium. The BS(1,2) walk's record has one bright scratch (the initial delta) and then a machined surface.butterfly.png— a keyed permutation of 20,000 integers, rotated 45 degrees and cropped to an oval. Colored by output mod 9 on a CMB scale. The density variations are the Feistel round function's fingerprint, almost but not quite uniform.
