Petty's Notebook

One line, divided again and again, each arc a finer look at the same whole. Two gold readings cross over it, and only the short bright stretch where they meet adds anything new.

CapacityPreprint

Unity, Refinement, and Signed Capture

One sawtooth clock already matches the flat line on half the integers. More clocks keep closing the gap, and the hard part is telling what each new reading actually buys.

October 4, 2026 · 20 min

ReadArticle ReadPaper

$Only the small gold loop closes. The open blue paths are longer repeating blocks the golden scale turns away, and with rational weight on terminating fractions the tail has to repeat a single digit.$

AlignmentPreprint

Repetend Rigidity at the Golden Scale

Change the base and one fifth can repeat every one, two or four digits, never three. That missing three keeps the golden crossing tied to a single repeating digit.

September 5, 2026 · 15 min

ReadArticle ReadPaper

Blue paths branch off one gold carry boundary, and every split keeps the parent's boundary mass. Divide those shares by their periods and out come the clock weights, from one small local rule to the whole capacity sequence.

CapacityPreprint

The Weight of a Carry

In 1861 a Paris clockmaker found good fractions by splitting gaps at their mediants. The same splitting, made by a carry, sets the weight of every remainder clock.

August 18, 2026 · 8 min

ReadArticle ReadPaper

Blue arcs come back to the line one cycle at a time while a sloping gold edge cuts across them as the cutoff grows. Every finished cycle balances, the unfinished ones under the edge leave a bias, and the paper separates that bias from the arithmetic fluctuation.

CapacityPreprint

The Bias Beneath the Secondary Term

The clock model behaves perfectly on average and fails at every integer we count with. The fault lies in how the readings are taken, and once that bias is removed the integers follow the clocks after all.

August 9, 2026 · 9 min

ReadArticle ReadPaper

Circles of different sizes overlap along one line, clocks with different periods. Their hands move while their weights stay fixed and add to one, and the collision deficit comes from the clocks' weighted ages, with no mass ever lost.

CapacityPreprint

The Clocks Beneath Collision Energy

Raise the resolution from five to six and the collision deficit goes down. The reason is a set of remainder clocks with fixed weights adding to one, and three of them reset at six.

July 31, 2026 · 11 min

ReadArticle ReadPaper

Two chains run along the same channels, and their big loops no longer rise together. The plain correlation fades to zero, yet the heavy loops still count in the squared weights, where the cubic law keeps a factor of three.

CollisionsPreprint

Magnitude Decorrelation in the Collision Spectrum

Correlation can miss a relation completely, since a number and its square have none. Two magnitudes in the collision spectrum drift to zero correlation, yet breaking their pairs still costs a factor of three.

July 22, 2026 · 8 min

ReadArticle ReadPaper

A thin stepped band runs beneath the curve of the cube. Its small steps are rounding, and they add up to a shortfall whose leading coefficient carries through to the finite digit table.

BoundariesPreprint

The Secondary Term of the Cubic Law

Legendre fitted a constant to his prime tables and it was wrong. The cubic law's next term carries a pi and has to survive the same danger, a finite table that can invent patterns of its own.

July 1, 2026 · 11 min

ReadArticle ReadPaper

A gold line splits self-comparisons from two mirrored fields of pairs. Points repeat along each ray as scaled copies of one reduced pair, and counting those copies as the window grows brings out the cube.

BoundariesPreprint

Digit Collisions and the Cubic Law

Prime reciprocals look so random that their digits were once proposed as codes. Measure how far their agreements stray from fair and the total grows exactly like the cube of the base, with coefficient one.

June 4, 2026 · 13 min

ReadArticle ReadPaper

The blue loop is a remainder orbit closing under multiplication. The gold edge beside it asks where addition crosses the orbit's boundary, and a longer orbit can still have a shallow edge.

BoundariesPreprint

The Orbit's Edge

In binary, one over 243 repeats every 162 digits, a long and busy cycle. Its edge is plain, switching off at every multiple of three, while a small two-digit rule in base fifteen has a deeper one.

May 30, 2026 · 11 min

ReadArticle ReadPaper

Gold joins mark where the digit rule switches on and off along the blue contour, and the rule's arithmetic lives at those joins. The floor function gives every rule the same sawtooth response, so changing the rule changes only the boundary factor.

BoundariesPreprint

Carry Boundaries and Bernoulli Spectra

Ask any question about the first two digits of a fraction and it reduces to a small table of carries. The floor function supplies the same factor every time, and the question only gets to choose its edges.

May 29, 2026 · 12 min

ReadArticle ReadPaper

The large outline fades while a small gold frame holds. Counts and measurements can change while the finite structure that fixes them survives, and that difference is behind the name nfield.

Essay

The Structure That Survives

A count can change while the structure beneath it holds. A short essay on that difference, and on why the work takes its name from it.

April 15, 2026 · 4 min

ReadArticle ReadPaper

Every path starts at one gold source, and the circles they reach differ in size. Each collision coefficient multiplies a digit-boundary factor by an L-value, so a channel's strength depends on both.

CollisionsPreprint

The Collision Spectrum

Eight L-values stand behind the base-five collision table, infinite sums no one can finish. Twenty fractions from long division still fix an exact average of their squared sizes, 6π²/25.

March 31, 2026 · 14 min

ReadArticle ReadPaper

A single signal comes in and the gold junction splits it among a few character paths. The junction decides which L-functions take part and how much each one counts, and if the prime sum converges past one of their zeros, the shares meeting there have to cancel.

CollisionsPreprint

The Collision Transform

The collision scores come from a forty-number table that favors some endings before any prime arrives. Remove those preferences exactly and the primes decide the rest, down to which L-function zeros could stop the sum.

March 30, 2026 · 12 min

ReadArticle ReadPaper

A long path runs out of a compact lattice. The denominators keep growing along it, and their collision deviations keep landing on values the lattice already holds.

CollisionsPreprint

The Collision Invariant

Line up a prime's repeating decimal with itself shifted one place and count the matches. The count looks as if it should depend on everything about the prime, yet in decimal its deviation is fixed by the last two digits.

March 29, 2026 · 20 min

ReadArticle ReadPaper

The blue petals change shape around a gold junction that never moves. The junction is the finite digit boundary, fixed while the analytic parameter travels through the critical strip, and each active channel carries the zeros of its own L-function.

AvoidanceResearch note

The Analytic Collision Transform

The base-five collision coefficient carries the size of an L-value. Let the exponent move and the coefficient becomes a function whose digit boundary never changes, while its channels inherit the zeros of their L-functions.

January 24, 2026 · 15 min

ReadArticle ReadPaper

From a small circle, weighted paths fan out into a much wider field. At base five, twenty collision entries fix a fourth moment of L-function values exactly, a finite digit count reaching an analytic quantity.

AvoidanceResearch note

The Collision Spectrum and the L-Function Landscape

Twenty fractions from long division in base five fix an exact average of fourth powers of L-values, 192π⁴/625. An infinite analytic quantity comes straight out of a finite digit count.

March 10, 2025 · 17 min

ReadArticle ReadPaper

Blue and gold rise at different places around the same ring. Collision weight and prime-sum strength mostly peak apart, a mismatch measured in the finite tables that has yet to be proved permanent.

AvoidanceResearch note

The Spectral Repulsion

At base five the collision weights and the prime weights very nearly avoid each other. Shuffled pairings overlap far more in every test I ran, though whether the avoidance lasts is still open.

January 9, 2025 · 14 min

ReadArticle ReadPaper

Blue and gold strands pass through each other and come out whole. They are centered collision signals from two coprime bases. Averaging pulls either one back out of the mix, and their energies simply add.

CollisionsResearch note

The Double Transversality

Read collision tables in two bases at once and the weights seem to swallow each other. Over a whole table nothing is lost, since tables from coprime bases don't interfere and each can be pulled back out intact.

October 3, 2024 · 11 min

ReadArticle ReadPaper

Ring after ring stays balanced on one axis. Each is a whole centered group, and the stack keeps its balance under several remainder conditions at once, so even a narrow selection of primes can cancel its own drift.

CollisionsResearch note

The General Neutrality Theorem

Keep only the primes ending in 01 or 51 and the collision sum starts to tilt. The general theorem finds exactly which selections stay balanced, and it comes down to keeping whole centered groups.

July 21, 2024 · 15 min

ReadArticle ReadPaper

Three traces come down to the line at minus one half, the level every family of the raw table shares. Centering removes it, and even split by remainder mod three, each group of primes keeps enough of the table to cancel its own drift.

CollisionsResearch note

The Neutrality Theorem

Two sums growing in opposite directions can add up to one that converges. Split the collision sum by remainder mod three and each half settles on its own, so nothing is cancelling by accident.

April 23, 2024 · 16 min

ReadArticle ReadPaper

Light runs toward a bright edge where finite digit arithmetic meets the infinite sum over primes. Weighting the large primes more pushes against that edge, and zeros of the L-functions the table selects can stop the sum there.

CollisionsResearch note

The Collision Transform and the Critical Strip

Count the fractions of each prime that start with a doubled base-three digit and add the counts over the primes. How heavily the large primes can be weighted before the sum stops settling comes down to the zeros of one L-function.

January 13, 2024 · 21 min

ReadArticle ReadPaper

Filaments from far away gather into a small grid. However large the denominator, its last two digits pick one of forty entries, and its collision deviation keeps coming back to the same table.

CollisionsResearch note

The Collision Periodic Table

No one will ever write out the fractions of a thousand-digit prime, yet how many begin with a doubled digit is known exactly. It is a tenth of the prime, rounded down, plus a correction read from its last two digits.

December 2, 2023 · 17 min

ReadArticle ReadPaper

Broad streams pull apart while finer threads stay visible. Family centering removes each last-digit group's shared level and keeps the differences inside it, and those differences ride on the finer character channels.

CollisionsResearch note

The Centered Collision Sum

The decimal collision table leans by last digit, with primes ending in 1 averaging minus 1.7 and those ending in 9 plus 0.7. Take out each family's own lean and the sum settles, with the finer differences intact.

October 6, 2023 · 13 min

ReadArticle ReadPaper

One trail climbs away and the other levels off, the same prime sum before and after centering. Forty table entries and one mean correction fix the decimal drift at minus nine tenths, and removing it leaves a sum that converges.

CollisionsResearch note

The Collision Fluctuation Sum

All the primes below ten million add up to only about 3.04 in reciprocals. The weighted collision sum drifts like minus nine tenths of log log x, at a rate fixed before any prime is added.

April 28, 2023 · 13 min

ReadArticle ReadPaper

The gold path breaks where a prime's repeating decimal never agrees with itself one place over. The path goes on and the breaks stop. In decimal there are seven such primes past ten, and seventy-three is the last.

CollisionsResearch note

Silent Primes

Shift a prime's repeating decimal one place and count the matches. Exactly seven primes past ten give none, 11, 13, 19, 23, 37, 41 and 73, and nine subtractions prove the list is complete.

January 21, 2023 · 12 min

ReadArticle ReadPaper

Four strands leave one knot, carrying the primes that end in 1, 3, 7 and 9. The gold marks along them are character weights, which recombine the strands to show differences the total hides, with every original sum still there.

CollisionsResearch note

The Character Structure of the Collision Fluctuation

Chebyshev noticed in 1853 that some kinds of primes run ahead of others. The collision sum hides a race like that by last digit, and four character weights pull the runners apart.

October 14, 2022 · 12 min

ReadArticle ReadPaper

$Nine gold routes cross a blue field that grows with the prime. They are the multipliers that move every remainder out of its own digit bin, and in decimal they always sit at the same nine fractions, whatever the prime.$

SpectraResearch note

Bin Derangements and the Gate Width Theorem

Multiply every remainder of thirteen by one number and ask whether any keeps its first digit. For exactly nine multipliers none does, and the count stays nine at twenty-nine, eighty-three and a thousand and nine.

July 4, 2022 · 16 min

ReadArticle ReadPaper

A narrow ring and a broad one share an outline and return differently inside. After the phases cancel, the gold crossings survive. A prime's last digit sets the shape of its bins and leaves open which multipliers keep a digit in place.

SpectraResearch note

Phase-Filtered Ramanujan Sums and the Spectral Gate

Ramanujan's 1918 sums take just two values at a prime. They turn up exactly in long division, counting how many remainders keep their digit when every remainder is multiplied by the same number.

April 17, 2022 · 14 min

ReadArticle ReadPaper

Two linked rings of remainders and one gold line through both. Sliding a digit word multiplies its remainders, and reading along that line through the cross-spectrum gives back the total agreement.

SpectraResearch note

The Autocorrelation Formula

Long division runs on two kinds of arithmetic at once, size for the digits and multiplication for the remainders. Sliding a decimal against itself is where they meet, and one frequency table per prime answers it exactly.

January 26, 2022 · 14 min

ReadArticle ReadPaper

Fans of phase lines open and close as the frequency changes, adding up in some places and cancelling in others. At thirteen only two neighboring pairs of remainders move the spectrum, and the single bins lay down its flat floor.

SpectraResearch note

The Spectral Power of the Digit Function

Change the base and the fractions of thirteen sort their remainders into different bins. The frequency spectrum of those bins has an exact formula, and at thirteen just two neighboring pairs of remainders make it rise and fall.

October 26, 2021 · 14 min

ReadArticle ReadPaper

The loops overlap because complementary digit rows repeat the same sums, so many of their directions depend on one another. The small gold frame inside holds the independent directions that remain.

SpectraResearch note

The Spectral Structure of Fractional Fields

Write the fractions of 127 in binary and you get 126 different rows. Every agreement between them fits into just eight independent directions, and a plain fact about long division accounts for the other 118.

July 13, 2021 · 13 min

ReadArticle ReadPaper

$Down the gold diagonal every fraction agrees with itself. The mirrored lights on either side are matches between different fractions, and keeping them in place keeps the arrangement a single average would throw away.$

SpectraResearch note

The Cross-Alignment Matrix

The mileage chart at the back of a road atlas is enough to redraw the map. Keep every agreement between fractions in one grid and the same idea redraws their structure, which a single average throws away.

April 11, 2021 · 17 min

ReadArticle ReadPaper

$A blue field of waves is the agreement the rows already share. Take it away and the gold strand remains, the agreement that comes from one chosen fraction.$

SpectraResearch note

The Coherence Decomposition

Grading tests against an answer key mixes two things, agreement with the key and agreement the students already share. The same split runs through every alignment score, and the golden crossing needs both halves.

January 12, 2021 · 16 min

ReadArticle ReadPaper

Steps across the lattice multiply by primes. Two gold routes come within a hair of meeting and never close, the old gap between twelve fifths and seven octaves.

AlignmentPreprint

Primes and the Major Scale

As a musician I kept hearing that the golden ratio explains music, and the explanations never held up. Repeating decimals led somewhere better, to octaves, fifths and a major scale built on one lattice of prime ratios.

October 31, 2020 · 15 min

ReadArticle ReadPaper

Three floors with dark air between them. Alignment scores gather on the three tiers, and no denominator lands in the gap between the scores at six and twelve.

AlignmentResearch note

The Three-Tier Theorem

Batting averages land on almost every value, and alignment is an average too. Yet no denominator scores between 0.600 and 0.636, and the golden line runs right through that gap.

October 16, 2020 · 15 min

ReadArticle ReadPaper

Four looping families hold the cycles of fifty-three. The two gold rims are the two limits the count can settle on, depending on which cycle the factors of the base pick out.

AlignmentResearch note

The Alignment Limit for All Primes

Halving a third parks a 1 in front of the repeating 3s and changes nothing that lasts. At fifty-three the same move picks between two different limits, depending on which cycle the remainders follow.

July 7, 2020 · 16 min

ReadArticle ReadPaper

Each wedge of the disk holds one nonzero remainder with a digit of its own, so the loops never share. Two repeating tails then agree in every place or in none.

AlignmentResearch note

Digit-Partitioning Primes and the Alignment Formula

Stack the elevenths and every column uses every digit, the nine times table running down the page. For primes like eleven a digit names its remainder, and repeating tails match at every place or at none.

April 3, 2020 · 13 min

ReadArticle ReadPaper

Three arms wind in toward one center, like the remainder classes behind the alignment count. Where the gold arm crosses, the count passes one over the golden ratio, and among the odd primes only three gets there.

AlignmentResearch note

Three and the Golden Ratio

The golden ratio gets credit for a lot it never did. When it turned up in my table of twelfths I checked, and it really is there, at a crossing only the prime three can reach.

January 19, 2020 · 11 min

ReadArticle ReadPaper