Petty's Notebook

Cyan arcs subdivide one fixed interval. Gold paths overlap, leaving a small surviving segment. Finer readings share the same whole; their signed overlap determines what survives as a correction.

refinementPreprint

Unity, Refinement, and Signed Capture

Finer clocks reveal more of the same whole. Signed readings turn that detail into an improving fit, with a cost that stays bounded across refinement levels.

October 4, 2026 · 20 min

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$Only the small gold loop closes. The unfinished blue paths stand for longer repeating blocks excluded by the golden-scale condition. With rational weight on terminating fractions, the tail must repeat a single digit.$

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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The branching blue paths refine a single gold carry boundary. Each split conserves the parent's boundary mass. Dividing those shares by their periods produces the collision clock weights, linking the small local construction to the full 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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The returning blue arcs stand for remainder clocks; the sloping gold boundary for an observation cutoff that grows with the measurement. Unfinished cycles introduce a bias even though every complete cycle balances. 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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The overlapping circles stand for remainder clocks with different periods. Their positions change while their assigned weights stay fixed and add to one. The collision deficit comes from their weighted ages, not from a loss of total mass.

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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The two chains occupy the same spectral channels, but their larger loops no longer rise together. They picture fading magnitude correlation. The heavier loops still matter to the squared weights, where the cubic law leaves a persistent excess.

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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The narrow stepped band marks a shortfall beneath the cubic leading term. Its small offsets stand for rounding. Their accumulated contribution grows, and the paper shows that its leading coefficient survives the passage 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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The gold diagonal separates self-comparisons from two mirrored fields of interactions. Repeated points along each ray stand for scaled copies of one reduced pair. Counting those copies as the window grows exposes the arithmetic behind the cubic energy law.

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 closed blue loop stands for a remainder orbit returning under multiplication. Its offset gold edge asks a different question, where addition crosses the orbit's boundary. A larger orbit need not carry a deeper arithmetic spectrum.

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 the places where the digit condition changes along the blue contour. Those boundaries carry the chosen rule's arithmetic. The floor function supplies a common sawtooth response, so changing the rule changes the boundary factor rather than the whole construction.

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 larger outline fades while a small gold framework remains. It pictures the distinction behind the name nfield. Counts and measurements can change while the finite structure that determines them survives.

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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The paths share a finite source but carry unequal weights. Each collision coefficient combines a digit-boundary factor with an L-value. The differently weighted loops recall that the strength of a spectral channel 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 finite gold junction separates the incoming signal into selected character paths. It fixes which L-functions enter and with what weights. If the prime sum converges past one of their zeros, the weighted contributions there must 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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The compact lattice represents the finite collision table. The long outgoing path stands for increasing denominators, whose deviations keep returning to values already held in that lattice.

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 forms change around an unchanged gold junction. That junction stands for the finite digit boundary, which stays fixed as the analytic parameter moves through the critical strip. Each active channel carries the zeros of its corresponding 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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A small finite grid sends weighted paths into a much larger field. The relation is exact in the paper. At base five, twenty collision entries determine a fourth moment of L-function values, linking a finite digit count to 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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The strongest blue and gold concentrations occur at different places along shared channels. They picture the measured mismatch between collision weights and prime-sum strength. The reduced overlap is a finite observation, not a proved permanent separation.

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 families cross without losing their separate paths. They represent centered collision signals from two coprime bases. Averaging recovers either signal from the mixture, and their energies add without a cross-term.

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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Each ring keeps its balance around the shared axis. Their repetition stands for whole centered groups surviving several simultaneous remainder conditions. A narrower selection of primes still contains the arithmetic needed to 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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The traces meet the raw table's shared level of minus one-half. Centering removes that level. Even after the primes separate by their remainder modulo three, each group retains 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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The light converges on a boundary between finite digit arithmetic and the infinite prime sum. Giving larger primes more weight tests how far that sum can converge. Zeros of the L-functions selected by the table can obstruct the passage.

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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The filaments gather into a compact grid, the finite table behind an unbounded set of denominators. In decimal, the last two digits choose one of forty entries. Larger denominators keep returning to the same collision deviation.

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 separate while finer threads remain visible. The image follows family centering, which removes the shared level of each last-digit group without erasing its internal differences. Those surviving differences occupy 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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The diverging and leveling trails contrast a prime sum with its bias still present and the same sum after centering. Forty finite table entries and the constructive-mean correction fix the decimal drift at minus nine-tenths. Removing it leaves a convergent sum.

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 breaks in the gold path stand for primes whose repeating decimal has no matches after a one-place shift. The path continues, but the list of silences does not. In decimal there are seven such primes above 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 traceable branches carry contributions from primes ending in one, three, seven, and nine. Their gold phase marks evoke the character weights that recombine those streams, exposing differences concealed by the total without losing any of the original sums.

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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$The nine gold routes stand for the fixed family of multipliers that move every remainder out of its digit bin. The blue field can grow with the prime, but the same nine rational fractions supply the decimal zero set.$

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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The narrow and broad enclosures recall digit bins with the same coarse arrangement but different internal returns. The surviving gold crossings stand for what remains after phase cancellation. A prime's last digit fixes the bin pattern, not which multipliers preserve a digit.

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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The repeating loops and the gold frequency line connect two ways of counting digit matches. Shifting a word multiplies its remainders. A corresponding line through the cross-spectrum recovers the total agreement across the remainder cycles.

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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The spreading fans picture phase sums reinforcing and cancelling as frequency changes. At thirteen, just two neighboring pairs of remainders produce the changing part of the spectrum. The singleton bins contribute the flat background.

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 overlapping loops picture complementary digit rows whose sums repeat the same pattern. Some apparent directions are dependent. The smaller gold structure recalls the independent directions left after those relations are taken into account.

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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$The gold diagonal records each fraction agreeing with itself. Mirrored lights away from it stand for matches between different fractions. Keeping their positions preserves the arrangement that a single average loses.$

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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$A gold strand stands apart from the blue field of shared agreement. It marks the distinction between alignment with one reference fraction and agreement already present among the other rows. Subtracting the background reveals the focused part.$

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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The lattice records multiplication along prime directions. Two gold routes nearly meet without closing, recalling the gap between twelve fifths and seven octaves. The near miss comes from tuning arithmetic, not a golden spiral.

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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The separated layers stand for the three alignment tiers. Their empty space matters as much as their occupied levels. No denominator fills 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 looped families represent the repeating cycles at fifty-three. The two gold levels picture the distinct alignment limits reached when factors from the base select different cycles.

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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The divided fan pictures a digit map that keeps every nonzero remainder distinct. Separate loops retain separate identities, so synchronized repeating tails agree at every position or at 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 winding families recall the residue classes behind the alignment count. Their gold crossing marks the point where that count passes the golden threshold, selecting three among odd prime cores.

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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