Purpose of Day 01
Before any new computational or theoretical work is published, the existing content must be audited. Every assertion is classified under exactly one of four epistemic labels. This discipline is the foundation of the 90-day program: readers must be able to distinguish at a glance what has been proved, what has been computed, what is taken from the literature, and what remains heuristic or open.
1. The Four Epistemic Labels
From this day forward every substantive claim on the site carries exactly one of the following labels. The labels are mutually exclusive and exhaustive for the purposes of this project.
Mathematical proof
A statement for which a complete mathematical argument exists. Example: the cycle inequality \(0 < m\ln 2 - k\ln 3 \le k\ln(1+1/(3B))\) is a theorem of elementary analysis.
Machine-checked computation
A finite computational verification that has been executed and is reproducible. Example: “every \(n<2^{36}\) eventually descends” is a computational fact, not a theorem about all positive integers.
Attributed prior result
A result taken from the published literature and correctly attributed. Example: Tao’s 2019 almost-all theorem; Barina’s verification records; the classical Steiner cycle equation.
Heuristic, open, or informal
Probabilistic arguments, geometric-mean heuristics, informal intuition, or statements that remain open. Example: the claim that the average multiplicative factor is \(\sqrt{3}/2\approx 0.866\).
2. Inventory of Current Claims
Every substantive assertion that appears on the public pages as of 2026-10-10 is listed below and assigned exactly one label.
| Claim (paraphrased) | Page | Label | Notes |
|---|---|---|---|
| The map \(T\) is the accelerated Collatz map. | index, all | Kanıtlandı | Definition. |
| Every integer \(2\le n<2^{36}\) eventually falls below its starting value under \(T\). | index, verification | Hesaplandı | Finite computation (157 s). Not a theorem for all \(n\). |
| The Collatz conjecture holds for all positive integers less than \(2^{36}\). | verification | Hesaplandı | Follows by induction from the computation; still computational. |
| Any nontrivial cycle with elements \(>2^{36}\) has ≥190 537 odd terms (length ≥492 531). | index, cycles | Kanıtlandı | Conditional on verification bound \(B\); the inequality is analytic. |
| The cycle inequality \(0<m\ln 2-k\ln 3\le k\ln(1+1/(3B))\). | cycles | Kanıtlandı | Elementary analysis. |
| Table rows for \(B=2^{68}\) and \(B=2^{71}\). | index, cycles | Bilinen literatür | External records (Barina et al.). Not results of this project. |
| Geometric-mean factor ≈ \(\sqrt{3}/2\approx 0.866<1\). | index | Varsayım/Sezgi | Standard probabilistic heuristic. |
| Three negative cycles (smallest |n| = 1, 5, 17). | negatives | Bilinen literatür | Classically known; recovered on the site. |
| Every start in [−200 000, −1] enters one of the three negative cycles. | negatives | Hesaplandı | Finite scan; does not prove uniqueness on all \(\mathbb{Z}^-\). |
| Parity vectors and tail density statistically indistinguishable for positive vs negative maps. | negatives | Varsayım/Sezgi | Empirical / heuristic observation. |
| Parity walks forever above the critical line: measure zero yet nonempty in the 2-adics. | negatives, stopping-times | Bilinen literatür | Standard 2-adic analysis. |
| Whether any positive ordinary integer is divergent remains open. | negatives | Varsayım/Sezgi | Correctly stated as open. |
| Stopping-time density tends to zero (Terras). | stopping-times | Bilinen literatür | Terras 1976. |
| Union-Find clustering partitions long trajectories into families. | families | Hesaplandı | Computational construction. |
| Verification completed in 157 seconds. | verification | Hesaplandı | Timing of a specific run. |
| A genuine proof must exploit positivity. | index, negatives | Varsayım/Sezgi | Meta-mathematical observation, not a theorem. |
3. Critical Corrections Required
Correction A — Table rows \(2^{68}\) and \(2^{71}\)
These rows are Bilinen literatür. They rely on external verification records (Barina et al.). Future versions of the cycles page will separate “this project” from “external records”.
Correction B — Language of “verification”
The phrase “the Collatz conjecture holds for all \(n<2^{36}\)” is true only as a computational statement and must never appear without the label Hesaplandı. The conjecture remains open for the infinite set of positive integers.
4. Label Discipline Going Forward
- Every new page opens with an explicit statement of which results are new computations, theorems, or literature.
- One of the four tags is attached to every numbered claim or highlighted box.
- The verbs “solve”, “prove the conjecture”, or “almost solved” are never used.
- When a computation is reported: hardware, flags, wall-clock time, and content hash whenever feasible.
- When a literature result is cited: precise bibliographic pointer.
5. Summary of Day 01
| Item | Status |
|---|---|
| Four-label system defined | Done |
| Full claim inventory of current site | Done |
| Identification of presentation risks | Done |
| Forward discipline for Days 02–90 | Done |
Next (Day 02): Notation and definitions page — \(T(n)\), Terras map, Syracuse map, stopping time, total stopping time, glide — with every definition labelled and referenced.