A Glossary for the Notebook

Most of the words in these articles are ordinary arithmetic. Some are standard terms that number theory has used for a century or more, some I borrowed from other fields, and some are mine, coined along the way for things I kept meeting. Each entry says where its word comes from.

Arithmetic marks the words of school arithmetic. Number theory marks standard terms any number theorist would recognize. Wider mathematics covers standard tools from algebra, analysis and statistics, music covers the one tuning term, and mathematics and philosophy covers unity, a word much older than these articles. Borrowed marks a word I took from physics, chemistry, geometry or signal processing and put to a new use. Coined here marks a word I made up, or an ordinary word I use in a particular sense. Each entry ends with the article where the word first does real work.

Long division and its digits

Remainder (arithmetic). The small number left at the bottom of each step of long division. It is the only thing the next step uses, so when a remainder comes back, every digit after it comes back too. Long Division and Euclid’s Lemma

Terminating and repeating decimals (arithmetic). A fraction in lowest terms stops when its denominator has no prime factors except those of the base, the twos and fives in decimal. Otherwise its digits eventually repeat forever. Geometries Hidden in the Number System

Repetend (arithmetic). The block of digits a fraction repeats, like 142857 in one seventh. Silent Primes

Period (number theory). The length of the repetend. For a denominator sharing no factor with the base, it equals the multiplicative order of the base. Geometries Hidden in the Number System

Multiplicative order (number theory). The number of times you multiply by the base before a remainder returns to where it started. Gauss studied it in 1801, and it is why a repeating decimal repeats. Foundational Tables of Multiplication

Cosets (number theory). When multiplying by the base doesn’t reach every remainder, the remainders fall into separate loops of equal length. At fifty-three in decimal there are four loops of thirteen. The Alignment Limit for All Primes

Pipes and cumulative closure (coined here). My notation for a repeating tail, as in 1/12 = 0.08|333|. Copies of the shortest block go between the pipes until their digit sum is a multiple of one less than the base. Geometries Hidden in the Number System

Open and closed cycles (coined here). Labels from my 2011 worksheet for the part of an expansion before repetition starts and the repeating block itself, each compressed to a single digit. Long Division and Euclid’s Lemma

Unzipping (coined here). My word for what long division does to a fraction, opening a compact number into a sequence of digits and remainders that can be drawn and compared. The Circle of Nine

Digit function (coined here). The rule that tells which digit a remainder writes. In base $b$ with denominator $p$, remainder $r$ writes $\lfloor br/p\rfloor$. The Spectral Power of the Digit Function

Digit bin (coined here). The remainders that write the same digit. Because the digit depends only on size, each bin is a run of neighboring remainders. Digit-Partitioning Primes and the Alignment Formula

Digit-partitioning prime (coined here). A prime small enough that every nonzero remainder gets a digit of its own, which in base $b$ means $p\le b+1$. In decimal there are three, 3, 7 and 11. Digit-Partitioning Primes and the Alignment Formula

Fractional field (coined here). All the proper fractions of one denominator, read together as a table with one row per numerator. The Spectral Structure of Fractional Fields

Supported part and rough part (number theory). Split a denominator as $n=tm$, where $m$ holds the primes the base supplies, the twos and fives in decimal, and $t$ holds the rest. Number theorists call $t$ the rough part, and supported part is my name for $m$. A fraction terminates exactly when its numerator cancels the whole rough part. The Three-Tier Theorem

Alignment

Alignment (coined here). The share of a fractional field whose repeating tails match a chosen reference tail, with terminating rows counted as matches. The twelfths score 7/11, just above one over the golden ratio. Three and the Golden Ratio

Tiers (coined here). The three levels that alignment scores gather on. No denominator scores between 0.600 and 0.636. The Three-Tier Theorem

Alignment deficit (coined here). How far a family of alignment scores falls short of its limit. Ratios of those deficits land on musical intervals. Primes and the Major Scale

Pairwise and focused alignment (coined here). Pairwise alignment averages the agreement over every pair of rows in a field. Focused alignment is what is left of a reference score after that shared background is taken out. The Coherence Decomposition

Cross-alignment (coined here). Agreement between every pair of fractions in a field, kept in a grid instead of averaged away. The Cross-Alignment Matrix

Collisions

Collision (coined here). An ordinary word in a particular sense. It is a place where a repeating decimal agrees with itself shifted one place, the same as a remainder writing the same digit before and after one step of division. Across a field, it counts the fractions that begin with a doubled digit, like 0.33… or 0.77…. The Collision Invariant

Collision invariant (borrowed). A term from physics for a quantity that survives every collision, like the total momentum of two colliding particles. Here it is the finite structure that stays fixed while collision counts grow. The Structure That Survives

Silent prime (coined here). A prime whose repeating decimal never agrees with itself one place over. In decimal there are exactly seven past ten, and 73 is the last. Silent Primes

Collision deviation (coined here). A prime’s collision count minus a tenth of the prime, rounded down. In decimal it lies between minus nine and plus eight, and it depends only on the prime’s last two digits. The Collision Periodic Table

Collision table (coined here). The finite list of possible deviations, forty entries in decimal, one for each ending a prime past a hundred can have. The Collision Periodic Table

Reflection identity (coined here). Entries of the collision table for an ending and its reflection always add to minus one. The Collision Invariant

Half-group law (coined here). At every interior step of the collision count, exactly half of the multipliers sharing no factor with the modulus cross a boundary, though which half crosses keeps changing. It holds for every modulus from three up. The Collision Invariant

Bin derangement (borrowed). A derangement, in the classical counting problem that goes back to Montmort, is a shuffle that leaves nothing in place. A bin derangement is a multiplier that moves every remainder out of its own digit bin. Bin Derangements and the Gate Width Theorem

Gate width (coined here). The number of bin derangements, which is always one less than the base for primes larger than the base. In decimal it is nine, and the nine are fixed fractions known before the prime is chosen. Bin Derangements and the Gate Width Theorem

Spectral gate (coined here). A number between zero and one that measures how much of a multiplier’s digit-keeping survives phase cancellation. It is zero exactly when the multiplier is a bin derangement. Phase-Filtered Ramanujan Sums and the Spectral Gate

Constructive mean (coined here). Test every multiplier except one, ignore those that keep no digits, and average the rest. It is the reference level that decimal’s count is compared with. The Collision Fluctuation Sum

Centering (wider mathematics). Subtracting an average, a standard step in statistics. Here it is done family by family in the collision table, so no last-digit family leans before any prime arrives. The Centered Collision Sum

Collision sum (coined here). The deviations added over the primes, each divided by a power of its prime. The Collision Fluctuation Sum

Neutrality (coined here). The property that a whole class of primes, sorted by remainder, cancels its own drift in the collision sum. When each class does it with no help from the others, I call it independent neutrality. The Neutrality Theorem

Transversality (borrowed). A word from geometry, where transversal things cross cleanly without lining up. Here it means collision tables from two bases with no common factor don’t interfere. Each can be recovered from a mixture, and their energies add. The Double Transversality

Collision transform (coined here). The collision sum read as a function of its exponent and rewritten through the table’s characters as a weighted sum of logarithms of $L$-functions. The Collision Transform and the Critical Strip

Collision zero ledger (coined here). The weighted count of zeros shared by the active $L$-functions at one point. If the collision sum converges there, the ledger has to balance to zero. The Collision Transform and the Critical Strip

Analytic collision transform (coined here). The collision coefficient extended to a function of a complex variable. Across the critical strip it stays a fixed multiple of an $L$-function, with the original coefficient at zero. The Analytic Collision Transform

Characters and spectra

Modular arithmetic (number theory). Arithmetic that keeps only remainders, so that numbers leaving the same remainder count as the same. A circle of nine is arithmetic modulo nine. The Circle of Nine

Dirichlet character (number theory). A pattern that gives each remainder an arrow and turns the arrows consistently under multiplication. Any table indexed by remainders can be rewritten as a mix of these patterns. The Character Structure of the Collision Fluctuation

Primitive, odd and even characters (number theory). A primitive character needs its full modulus and can’t be read off a smaller one. An odd character flips sign when a remainder is reflected, and an even character doesn’t. The Collision Spectrum

Conductor (number theory). The least modulus a character actually needs. Sorting energy by conductor shows which resolution of the digits it lives at. The Orbit’s Edge

Dirichlet $L$-function (number theory). The sum of a character’s values divided by powers of the integers. Its value at one is the end of a walk whose steps shrink like $1/n$ and turn with the character. The Collision Spectrum

Critical strip and critical line (number theory). The band of complex numbers with real part between zero and one, and the line down its middle at one half. The Riemann Hypothesis, and its generalization to $L$-functions, puts every nontrivial zero on that line. The Collision Transform and the Critical Strip

Ramanujan sum (number theory). A sum of evenly spaced arrows, introduced by Ramanujan in 1918. At a prime it takes only two values. Phase-Filtered Ramanujan Sums and the Spectral Gate

Gauss and Jacobi sums (number theory). Classical sums that measure how a character interacts with addition. The Orbit’s Edge

Prime race (number theory). The way some classes of primes stay slightly ahead of others, first noticed by Chebyshev in 1853. The Character Structure of the Collision Fluctuation

Active character (coined here). A character whose coefficient in the centered collision table is not zero. Only active characters, and their $L$-functions, can affect the collision sum. The Collision Transform and the Critical Strip

Collision spectrum (coined here). The list of character coefficients of a collision table. Each one splits into a digit-boundary factor and an $L$-value. The Collision Spectrum

Spectral power (borrowed). The power spectrum of signal processing, taken over the digit bins. For each frequency, add the arrows inside every bin, square the length of each bin’s total and add the squares. A bin with one remainder contributes one at every frequency. The Spectral Power of the Digit Function

Cross-spectrum (borrowed). A name from signal processing for a Fourier transform taken in two directions at once, here over the table of remainder pairs that share a digit. Unlike the spectral power, it keeps enough to rebuild that whole table. The Autocorrelation Formula

Avoidance (coined here). The tendency, at base five, for collision weight and prime weight to sit on different characters. It is a measured pattern, and whether it lasts is open. The Spectral Repulsion

Effective support (wider mathematics). The number of equal weights with the same entropy as a given list, a way of counting how many places a weight is really spread over. Statisticians call it the effective number, or perplexity. The Spectral Repulsion

Boundaries and energy

Carry (arithmetic). The digit moved into the next column when a sum overflows. The Weight of a Carry

Floor and fractional part (arithmetic). The floor of a number is the whole number at or below it, and the fractional part is what lies above the floor. Carry Boundaries and Bernoulli Spectra

Sawtooth (wider mathematics). The fractional part with one half taken away. It climbs steadily and drops back at every whole number. Digit Collisions and the Cubic Law

Generalized Bernoulli number (number theory). A classical constant attached to each character. Here it carries the floor function’s share of every collision coefficient. Carry Boundaries and Bernoulli Spectra

Dedekind sum (number theory). A classical sum comparing two sawtooths running at different speeds. Digit Collisions and the Cubic Law

Parseval’s identity (wider mathematics). The total of the squares stays the same whichever coordinates you measure in, so a table’s energy can be computed address by address or pattern by pattern. The Collision Spectrum

Carry boundary (coined here). The places where a digit rule switches between yes and no. For any finite rule about leading digits, the character coefficients come out as one fixed factor from the floor function times a sum over those places. Carry Boundaries and Bernoulli Spectra

Remainder orbit (number theory). The cycle of remainders that repeated multiplication by the base visits. The Orbit’s Edge

Collision energy (borrowed). Energy is signal processing’s word for a sum of squares. Collision energy is the sum of the squares of a centered collision table. Digit Collisions and the Cubic Law

Cubic law (coined here). In a prime base the collision energy grows like the cube of the base, with coefficient exactly one. Digit Collisions and the Cubic Law

Sampling defect (coined here). The difference between a smooth curve’s average and the average of the evenly spaced readings the digit table actually takes. The Secondary Term of the Cubic Law

Clocks and capacity

Remainder clock (coined here). A clock with $k$ marks whose hand shows the remainder of a counting number on division by $k$. Each period gets one. The Clocks Beneath Collision Energy

Clock weight (coined here). The fixed share of the whole each clock carries. The period-two clock gets a quarter, period three a sixth, and all the shares add to exactly one. The Weight of a Carry

Capacity and deficit (coined here). The continuous collision energy at resolution $N$ is its capacity, and the deficit is how far it falls short of $N$. The deficit is two thirds of the weighted average position of the clock hands. The Clocks Beneath Collision Energy

Profinite integers (number theory). Every way of setting all the remainder clocks consistently at once. Adding one advances every hand, the classical adding machine. The Clocks Beneath Collision Energy

Drag (coined here). The steady downward pull on the clock readings, because each newly admitted clock starts at the bottom of its cycle. The Bias Beneath the Secondary Term

Mediant and the Stern-Brocot tree (number theory). The mediant of two fractions adds their tops and adds their bottoms. Splitting gaps at mediants, again and again, produces every fraction in lowest terms exactly once. The Weight of a Carry

Unity slice (coined here). The collision measure restricted to pairs whose first entry is one. It gives the integer $m$ the weight $1/(m(m+1))$. Unity, Refinement, and Signed Capture

Nyman-Beurling criterion (number theory). The classical theorem that the Riemann Hypothesis is equivalent to approximating the constant one, as closely as you like, by sums of fractional-part functions. Unity, Refinement, and Signed Capture

Spine (coined here). The doubling blocks of addresses, one, then two and three, then four through seven, from which the constant one is built. Unity, Refinement, and Signed Capture

Signed capture (coined here). What a new reading actually adds to an approximation, with its sign and its overlaps with other readings counted. Unity, Refinement, and Signed Capture

Linear algebra and statistics

Gram matrix (wider mathematics). A table of the inner products between a set of rows, which records every overlap among them. The Cross-Alignment Matrix

Rank and kernel (wider mathematics). The rank counts how many independent directions a table really has. The kernel collects the combinations of rows that cancel to nothing. The Spectral Structure of Fractional Fields

Correlation (wider mathematics). A measure of whether two quantities rise and fall together. It can miss a total relation, since a number and its square have none. Magnitude Decorrelation in the Collision Spectrum

From the early drawings

Casting out nines and the digit sum (arithmetic). Adding a number’s digits, and adding again until one digit is left, gives its remainder on division by nine. Bookkeepers used it to check their sums. The Circle of Nine

Nines’ complement (arithmetic). Pairing each digit with the one that adds to nine. When the decimals repeat from the start, the fractions $k/n$ and $(n-k)/n$ have complementary digits. Geometries Hidden in the Number System

Fibonacci numbers (arithmetic). 1, 1, 2, 3, 5, 8, 13 and on, each the sum of the two before it. The Circle of Nine

Golden ratio (wider mathematics). About 1.618, the one way to cut a whole into two pieces that repeats its own proportion. The Golden Ratio

Golden scale (coined here). Evaluating a curve at powers of the golden ratio, the test that repetend rigidity applies. Repetend Rigidity at the Golden Scale

Pythagorean comma (music). The small gap between twelve pure fifths and seven octaves, about a quarter of a semitone. Primes and the Major Scale

Spokes (coined here). The positions on a circle of nine, one for each remainder on division by nine. Adding nine moves a number one ring outward on its spoke. The Circle of Nine

Polarity (borrowed). I borrowed it in 2009 from chemistry’s picture of charge and incomplete valence shells. It began as a plus or minus on each position and was later defined from a number’s digit sum and prefix. The Circle of Nine

Neutral seam (coined here). Where the circle of nine closes, between eight, nine and one, the region my early drawings treated as neutral. The Circle of Nine

Field glyphs (coined here). My 2010 drawings of a denominator’s fractions traced around the circle of nine. Geometries Hidden in the Number System

Radial palindrome and torsion (borrowed). Torsion is the engineer’s word for twisting. In 2010 I used it for the twist between neighboring spokes, and radial palindrome for the mirrored runs along a single spoke. Their exact versions measure the change in polarity from address to address. Arithmetic Foundations

Unity (mathematics and philosophy). An old word. Mathematics uses it for one, as in roots of unity, and Greek mathematics treated the unit as the source of number rather than a number among others. In these articles unity is the whole that every share is a share of, both the total of the weights and the target the parts have to account for. The number one is how that whole gets written down. The Golden Ratio