Research
Questions made precise.
Written arguments, computational investigations and formal verification. Each result carries its own evidence and scope.
A reading map / Three separate tracks
One question, different objects.
These tracks study different sets. A bound for a relaxed shell is not a count of Robin exceptions, and a finite canonical certificate does not prove the full analytic theorem.
Analytic candidate counting
Problem. How many integers satisfy the weak reverse Robin inequality?
Contribution. An eventual upper count, expressed through a spectral rate.
Scope. The written theorem states explicit zero-free-region and uniform zero-density inputs; classical inputs give ordinary-zeta specializations. Selected finite steps are formalized, not the full analytic proof.
Still open. Independent human scrutiny of the argument and its relationship to earlier counting results.
Read the question and argument →Relaxed shell / profiles
Problem. How large is a set defined by a sigma-benefit budget in a logarithmic shell?
Contribution. Checked profile construction and eventual two-sided relaxed-shell bounds; an adaptive relaxed shell also has a checked lower bound.
Scope. The two-sided result uses its stated theta estimate, with an unconditional exponent-1/10 instance. These are relaxed sets, not actual Robin exceptions.
Still open. Control of the signed arithmetic threshold needed to pass to actual exceptions.
Read the shell milestones →Finite canonical certification
Problem. Can strict Robin satisfaction be certified across a whole real cutoff interval?
Contribution. The actual canonical N(y) satisfies Robin for every real y in [100000, 102000], with exact event and tie handling.
Scope. A kernel-checked anchored certificate, not all integers in that numerical interval. Sources and certificates remain private.
Still open. New certified inputs for other anchors and new layer bounds beyond the guarded cutoff class; no global sign or RH conclusion.
Read the checked statement →Two counting questions / different criteria
Relaxed sigma-benefit shell
- Choose the shell
Restrict to the stated unit logarithmic shell.
- Apply the benefit budget
Keep patterns satisfying the sigma-benefit cost bound.
- Count the relaxed set
Checked size bounds use their stated hypotheses.
Robin candidates
- Choose the counting range
The manuscript considers integers 3 ≤ n ≤ X.
- Apply the Robin condition
Keep n with σ(n) ≥ eγ n log log n.
- Bound the candidate count
The written analytic argument uses its explicit spectral inputs.
Spectral bounds for the number of Robin candidates
How many integers can satisfy the weak reverse of Robin’s inequality? The manuscript connects estimates for zeta zeros to an eventual bound on their number.
Spectral narrowing / a conceptual view
October 25 / Canonical-family track
Finite Robin satisfaction, kernel checked.
Every real cutoff y in [100000, 102000] yields an actual canonical N(y) > 5040 satisfying the strict Robin inequality. Complete coordinate events, exact ties and a compensated margin budget are checked. The guarded workflow is anchored at 100000; it is separate from the candidate-counting manuscript and the shell/profile construction.
Finite certification and evidence limits ↗02 / FORMAL RESULTS
Checked ingredients, explicit boundaries.
Shell/profile construction
For distinct profiles on a finite set of odd primes with a common positive integer baseline, powers-of-two padding preserves distinctness and cardinality. For positive n with log n ≤ y, floor-based padding places n in the logarithmic shell (y − log 2, y], hence in the unit shell [y − 1, y], with an explicit bound on the added sigma benefit.
Later cost accounting, primewise estimates and counting lead to checked eventual two-sided bounds for the relaxed sigma-benefit shell under its stated theta estimate, including an unconditional exponent-1/10 instance. A smaller adaptive relaxed shell has a checked lower bound. Neither result counts actual Robin exceptions.
See the October milestones →A finite prime-support certificate
A checked Lean example proves that the product of all fifteen pair sums of a particular six-element witness has exactly the prime support {2, 3, 5, 7}. It establishes this finite specification, not the underlying asymptotic conjecture.
The completed proof was checked against the unchanged specification, including its actual axiom dependencies. The submission template remains separate.
Prime-pattern laboratory
Change factors and count a bounded family. Numerical totient costs and exact finite distance counts make one counting mechanism visible; the laboratory does not enforce the manuscript’s shell or Robin inequality.
Try the laboratory →03 / COMPLETED EXPERIMENTS
Base 60, tested against alternatives.
PLIMPTON E1–E3 · CLOSED CASE STUDYThe storage and triangle-lookup experiments did not meet their declared practical-advantage criteria. The graph comparison passed its criterion, with its structure explained by commuting prime generators. The arithmetic controls and deterministic repeats passed.
The retained case study records both the exact identities and the negative findings. It establishes neither universal base optimality nor ancient scribal intent.
Read the research history →The wider investigation
Arithmetic as a relational system.
Factor ancestry
Connect each composite to the number left after removing its smallest prime factor. This gives a precise structure to explore and test.
Residues and geometry
View integers through their remainders and base-12 representations. Test whether the patterns help answer a specific mathematical question.
Proof development
Translate finite encodings and counting arguments into Lean. State which assumptions are proved and which remain inputs to a conditional theorem.
The current manuscript is the unreviewed v1.2 preprint, revised 27 October 2026 (CDT). Earlier investigations include Erdős problems #123, #126 and #64, base-60 geometry, and prime-pattern experiments, each with its own outcome and limits.
Explore the project’s research history →