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.

refinementPreprint

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

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

alignmentPreprint

Repetend Rigidity at the Golden Scale

A smooth formula admits a period that long division cannot supply. The golden-scale condition forces a one-digit repeating tail.

September 5, 2026 · 15 min

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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.

Collision CapacityPreprint

The Weight of a Carry

Carry weights leave a trace in the collision measurements. The full sequence can recover their source. A finite stretch cannot.

August 18, 2026 · 8 min

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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.

Collision CapacityPreprint

The Bias Beneath the Secondary Term

Growing the observation window adds a bias of its own. Remove it, and the arithmetic fluctuations approach a stationary clock law.

August 9, 2026 · 9 min

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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.

Collision CapacityPreprint

The Clocks Beneath Collision Energy

Remainder clocks keep different times. Their weights add to one, and their changing positions determine the collision deficit.

July 31, 2026 · 11 min

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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.

collisionPreprint

Magnitude Decorrelation in the Collision Spectrum

Magnitude correlation tends to zero as the base grows. The squared weights retain an excess fixed by the cubic law.

July 22, 2026 · 8 min

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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.

boundaryPreprint

The Secondary Term of the Cubic Law

The cubic law leaves a growing shortfall. Its first correction comes from rounding, and survives passage to the finite collision table.

July 1, 2026 · 11 min

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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.

boundaryPreprint

Digit Collisions and the Cubic Law

Collision energy grows like the cube of the prime base. Its leading term divides one to two between self-terms and interactions.

June 4, 2026 · 13 min

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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.

boundaryPreprint

The Orbit's Edge

Multiplication closes a remainder orbit. Addition exposes its boundary, where the orbit's size and its arithmetic depth come apart.

May 30, 2026 · 11 min

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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.

boundaryPreprint

Carry Boundaries and Bernoulli Spectra

Change the digit rule and its boundary changes. A classical sawtooth handles the counting, leaving that boundary to carry the difference.

May 29, 2026 · 12 min

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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 collision count can change while the structure beneath it holds. An essay on what survives, and why the work takes its name from that.

April 15, 2026 · 4 min

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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.

collisionPreprint

The Collision Spectrum

Two factors lie behind each collision coefficient. The trilogy's third part separates the finite digit boundary from the L-value.

March 31, 2026 · 14 min

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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.

collisionPreprint

The Collision Transform

A finite table chooses the L-functions in the trilogy's second part. Convergence leaves their zeros an exact cancellation test.

March 30, 2026 · 12 min

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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.

collisionPreprint

The Collision Invariant

The first part of the trilogy gathers the finite arithmetic. Collision counts grow, but a few final digits keep their deviation fixed.

March 29, 2026 · 20 min

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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 digit boundary stays fixed as the transform moves through the critical strip. Its active channels inherit their L-functions' zeros.

January 24, 2026 · 15 min

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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 entries in base five determine a sum of fourth powers of L-values. The digits supply an exact answer to an analytic question.

March 10, 2025 · 17 min

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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

Collision weights and prime sums overlap less than shuffled pairings predict in the tests. Whether that persists remains open.

January 9, 2025 · 14 min

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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.

collisionResearch note

The Double Transversality

Mix centered collision signals from two coprime bases. Averaging recovers each one intact. Their energies add without a cross-term.

October 3, 2024 · 11 min

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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.

collisionResearch note

The General Neutrality Theorem

Several remainder conditions narrow the prime list. Whole balanced groups survive, keeping every allowed channel free of drift.

July 21, 2024 · 15 min

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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.

collisionResearch note

The Neutrality Theorem

Sort primes by their remainders on division by three. Each group cancels its own drift. Neither needs the other to balance.

April 23, 2024 · 16 min

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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.

collisionResearch note

The Collision Transform and the Critical Strip

Give large primes more weight. A small digit table selects the L-functions whose zeros can obstruct convergence of the sum.

January 13, 2024 · 21 min

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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.

collisionResearch note

The Collision Periodic Table

Past one hundred, a prime's last two digits fix its collision deviation. Forty small integers give every possible answer in decimal.

December 2, 2023 · 17 min

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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.

collisionResearch note

The Centered Collision Sum

Subtract the average for each last-digit family. The shared bias disappears, but the finer differences between its primes survive.

October 6, 2023 · 13 min

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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.

collisionResearch note

The Collision Fluctuation Sum

The collision sum drifts at a rate fixed by forty integers and a small correction. In decimal, the coefficient is minus nine-tenths.

April 28, 2023 · 13 min

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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.

collisionResearch note

Silent Primes

Shift a repeating decimal one place. Exactly seven primes above ten give no matches. Seventy-three is the last of them.

January 21, 2023 · 12 min

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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.

collisionResearch note

The Character Structure of the Collision Fluctuation

A collision sum drifts downward. Primes ending in one, three, seven, and nine contribute differently. Four weights separate them.

October 14, 2022 · 12 min

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

spectralResearch note

Bin Derangements and the Gate Width Theorem

Every prime greater than ten has exactly nine multipliers that change every first digit. The same nine fractions supply the list.

July 4, 2022 · 16 min

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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.

spectralResearch note

Phase-Filtered Ramanujan Sums and the Spectral Gate

At thirteen and fifty-three, the larger digit bins sit in the same places. Ramanujan's weights recover their different returns.

April 17, 2022 · 14 min

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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.

spectralResearch note

The Autocorrelation Formula

Shifting a decimal means multiplying its remainders. A line through a frequency table turns that multiplication into a match count.

January 26, 2022 · 14 min

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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.

spectralResearch note

The Spectral Power of the Digit Function

At thirteen, two pairs of neighboring remainders make the spectrum rise and fall. The other digit bins add a flat background.

October 26, 2021 · 14 min

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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.

spectralResearch note

The Spectral Structure of Fractional Fields

Six binary fractions retain only four independent directions. Complementary digit patterns let us see why two directions disappear.

July 13, 2021 · 13 min

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

spectralResearch note

The Cross-Alignment Matrix

Compare every fraction with every other. A grid keeps the location of each match, revealing arrangements that a single average loses.

April 11, 2021 · 17 min

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

spectralResearch note

The Coherence Decomposition

Seven and seventy-seven have low alignment for different reasons. How much of the reference score is already shared by other rows?

January 12, 2021 · 16 min

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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

Ratios of shrinking alignment gaps approach musical intervals. Fifths, octaves, and a major scale share one prime-number lattice.

October 31, 2020 · 15 min

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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

Sixths and twelfths sit across an alignment gap that no denominator can fill. Pair each fraction with its complement to see why.

October 16, 2020 · 15 min

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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

Fifty-three has four repeating cycles and two alignment limits. Which cycle a denominator selects decides where its score goes.

July 7, 2020 · 16 min

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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

For some primes, one digit identifies its remainder. Keep the repeating tails in step, and they match at every place or at none.

April 3, 2020 · 13 min

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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 fractions over twelve score seven-elevenths, just above a golden threshold. The crossing points back to the prime three.

January 19, 2020 · 11 min

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