O3 · The classification theorem (v11)
Proven (computational, rigorous via O1+O2). (T-orbit.)
$$\text{odd }a\le10^6,\ 2\le L\le100:\qquad
\text{nontrivial cycles exist iff } a\in\{5,181\},$$
with the complete list $a=5$: $\{1,3\}$, $\{13,33,83\}$, $\{17,43,27\}$;
$a=181$: $\{27,611\}$, $\{35,99\}$. Mersenne $a$ ($2^\nu-1\le10^6$) carry
only the trivial fixed point $\{1\}$ (P1); all other $a\le10^6$ have no
cycles of length $\le100$ at all.
Sweep statistics (this version, Node/BigInt, single core):
| range | $a$ scanned ($M_{\max}\ge1$) | odd starts | iterations | time | result |
|---|
| $a\le10^6$, $L\le100$ | 13\,677 / 500\,000 | 118\,714 | 11\,494\,539 | 5.2 s | $\{5,181\}$ only |
| $a\le999$, $L\le200$ (external) | — | 382\,807 | — | 9.5 s | $\{5,181\}$ only |
| $a\le99$, $L\le300$ (external) | — | 169\,238 | — | 4.25 s | $\{5,181\}$ only |
| $a=7$, $L\le2000$ | 1 | 173\,315 | 208\,501\,027 | 99.7 s | no cycle |
The bulk of odd $a$ are killed for free: $M_{\max}<1$ (the window gap
$2^{\lceil L\log_2 a\rceil/L}-a$ is resolved by the ceiling) means no
starting point exists. Only 2.7% of $a\le10^6$ require any iteration. The
largest sweeps per parameter (deep window approximations, all cycle-free):
$a=4657$ (8434 starts), $a=24305$ (8324), $a=230051$ (3869), $a=11$ (2355),
$a=89749$ (2218), $a=192251$ (1806). ∎
O4 · The remaining obstacle: irrationality measure of $\log_2 a$
$M_{\max}$ is controlled by the continued fraction of
$\log_2 a$. For $a=7$, $\log_2 7=[2;1,4,5,4,5,4,1,\mathbf{29},1,4,8,\ldots]$;
the record points (upper convergents) are:
| $L$ | $S$ | $\Lambda=S\ln2-L\ln7$ | $M_{\max}\approx L/(7\Lambda)$ |
|---|
| 26 | 73 | $6.08\times10^{-3}$ | 611 |
| 571 | 1603 | $2.35\times10^{-4}$ | $3.47\times10^5$ |
| 2964 | 8321 | $7.64\times10^{-6}$ | $5.54\times10^7$ |
| 91313 | 256348 | $1.49\times10^{-6}$ | $8.74\times10^9$ |
| 7893843 | 22160819 | $2.61\times10^{-8}$ | $4.50\times10^{13}$ |
(Intermediate records exist, e.g. $L=135$, $S=379$, $M_{\max}\approx3927$,
largest within $L\le200$.) The partial quotient $29$ at $L=2964$ causes the
jump to $5.54\times10^7$. The classification problem is exactly:
$$\text{no cycle} \iff \forall L,\ \forall\ \text{odd }m\le\frac{L}{7\Lambda(L)},\
m\text{'s orbit does not return within }L\text{ steps.}$$
Baker's role would be to bound $\Lambda(L)$ from below and hence $M_{\max}$
from above; but the Laurent–Mignotte–Nesterenko constant gives
$\Lambda>10^{-9073}$, useless in practice.
Consequence:
an unconditional all-$L$ theorem is at the level of the Collatz problem
itself; the computational frontier moved from $L\le18$ (v10) to
$L\le2000$ (v11) at polynomial cost.
Cost comparison ($a=7$)
| $L$ | $S$ | enumeration $C(S{-}1,L{-}1)$ | MITM $\sqrt{N}$ | orbit $M_{\max}\cdot L$ |
|---|
| 12 | 34 | $1.94\times10^{8}$ | $1.39\times10^{4}$ | $2.28\times10^{2}$ |
| 18 | 51 | $9.85\times10^{12}$ | $3.14\times10^{6}$ | $7.20\times10^{2}$ |
| 26 | 73 | $1.53\times10^{19}$ | $3.91\times10^{9}$ | $1.59\times10^{4}$ |
| 200 | 562 | $9.78\times10^{156}$ | $3.13\times10^{78}$ | $7.85\times10^{5}$ |
| 2000 | 5615 | — | — | $6.93\times10^{8}$ |
W1 · The tight-window lemma (intrinsic; kept)
Proven. (W1.) Let $a$ be odd and let
$x_0\to\cdots\to x_{n-1}\to x_0$ be an $n$-cycle of $S_a$, $n\ge2$, with
$S_a(x_i)=(ax_i+1)/2^{j_i}$ and $S=\sum_{i=0}^{n-1}j_i$. Then
$$S>n\log_2 a,\qquad
S\le\begin{cases}n\log_2(a+\tfrac13), & a+1=2^\nu\ \text{(Mersenne)},\\[2pt]
\log_2(a+1)+(n-1)\log_2(a+\tfrac13), & \text{otherwise.}\end{cases}$$
The product identity $2^S=\prod_i(a+1/x_i)$ holds termwise
($a x_i+1=2^{j_i}x_{i+1}$). Every $x_i$ is odd. $x_i=2$ is impossible:
$2a+1$ is odd. $x_i=1$: $1$ is a fixed point iff $a+1=2^\nu$ (P1);
otherwise $1$ may belong to a cycle but at most once. Hence all $x_i\ge3$,
with possibly one exception equal to $1$ in the non-Mersenne case. ∎
W1 supplies the lower bound used by O1 ($S\ge\lceil L\log_2 a\rceil$) and the
window structure; it is the only intrinsic gate of the retired enumeration
route (see below).
T3 · Three-cycles (kept, settled)
Proven. (T3.) $S_a$ has a $3$-cycle iff $a=5$, and
then exactly two:
$$\{13,33,83\},\qquad\{17,43,27\},\qquad D=3,\quad 5^3+3=2^7.$$
Window + exhaustive check (v9): $a=5$ forces $S=7$; all $15$ compositions of
$7$ into $3$ parts give exactly the two cycles. Every other parameter in
$\{7,9,11,13,45,181\}$ has an empty window at $n=3$ (size alone). The orbit
sweep (v11) confirms: minimal elements $m=13,17$. ∎
Historical record: the enumeration route (retired in v11)
Why the v-enumeration route was wrong. The
v-vector parameterization (composition space $C(S-1,L-1)$) describes a
cycle by $L$ exponents, but a cycle is determined by
one element;
the search space is the set of starting points. Everything built on the
wrong parameterization is a phenomenon of the wrong space:
- T8/T13 (v9/v10.2): exhaustive composition enumeration
proved no $2$–$13$-cycles for $a\le99$ (49 window survivors,
$\approx2.9\times10^{10}$ compositions, zero divisibility events $D\mid N$;
9 further survivors closed by meet-in-the-middle after an off-by-one fix in
the MITM splitting). All results remain correct and are cross-confirmed by
the orbit sweep (T-orbit) on the overlap — three independent methods
(enumeration, MITM, orbit) agree.
- Four-gate mechanism (v10): window / divisibility
blockade / congruence blockade / dynamic closure. Gates 2–4 describe why
the composition space has no solutions; only gate 1 (W1) is intrinsic to
the problem. Gate-2 strict form
($0\notin\mathrm{Reach}(a,L,S)\subset\mathbb Z/D\mathbb Z$) and the
$|\mathrm{Reach}|/D$ collapse ($0.29\to7.4\times10^{-5}$), the 2a/2b
subtypes, the Pollard-rho factorizations (5 further gate-3 kills; 13 fully
factored), and proposition B ($Z_p$ uniformity $1.07$–$1.15$ for $L\ge12$;
hard blockade $(7,7)$ via $p=75011$, $7=2^{992}$, ord 1154) are kept as a
phenomenology record. Proposition B' (Weil-bound route to $Z_p\to1/p$) is
withdrawn: $R\bmod p$ uniformity requires $N\gg p^{1+\epsilon}$,
but $N=C(S-1,L-1)\le2^{S-1}$ while $p$ reaches $\sim2^S$, so $N\lesssim p$
always — the error terms exceed the main term on the critical line; this is
the root of the $\rho/S>1$ divergence noted in v10.2.
- Baker/LMN route on $a=7$ (v10.1): negative — the bound
$\log|\Lambda|>-20887.3$ never contradicts the upper bound
$\Lambda\le L\log(22/21)$; $m$ is bounded above and below by $1/\Lambda$-type
quantities of the same order. Kept as a record; the orbit method needs only
$M_{\max}$, and the relevant lower bound on $\Lambda$ is exactly the open
irrationality-measure problem (O4).
- Pseudo-cycles and shared-$D$ (v9/v10.1): the only
divisibility events are repetitions of known cycles
($2^{10}=5^4+399$, $2^{30}=181^4+458703$, etc.); shared $D$ is a power
coincidence ($25^3=5^6$; also $3^{12}=9^6=27^4=81^3=729^2$), the
divisibility layer independent ($(5,6)$: 6 hits vs $(25,3)$: 0 on
$D=759$). Phenomenology of the enumeration space; closed.
- Watershed (v10.2): the expected-hit series
$\sum\binom{S-1}{L-1}/D$ converges for all $a>4$ (ratio
$2^{\beta(H(1/\beta)-1)}<1$, critical point $a=4$; $0.978$ even at $a=5$),
so the $a=5$ anomaly is tiny-$D$ Nagell/Beukers candidates
($D\in\{3,7\}$), not divergence. Consistent with the orbit picture:
$M_{\max}$ is large exactly when the window is narrow, which is when
$2^{S/L}-a$ is small.
The unified small-cycle gap (Part 0) and all Part-2/Part-3 statements are
unaffected.
Class number table (extended)
Grounded (v11). $\eta(a)$ = number of periodic
orbits = number of weak components of the inverse graph. For odd $a\le10^6$,
$L\le100$:
$$a=5:\ \eta=3;\qquad a=181:\ \eta=2;\qquad
a=2^\nu-1\le10^6\ (18\ \text{values}):\ \eta=1\ (\text{trivial fixed point only});$$
$$\text{all other odd }a\le10^6:\ \eta=0\ (L\le100).$$
$L>100$ (for $a\le10^6$) and $a>10^6$ remain open.
Attribution. The orbit method (O1–O2), the
$M_{\max}$ analysis, the $a\le999$/$a\le99$ sweeps, the $a=7$ $L\le2000$
run, and the withdrawal of proposition B' are due to an external
collaboration; they are re-derived and independently verified here
($a\le10^6$ sweep, 5.2 s; $a=7$ $L\le2000$, 99.7 s; completeness argument
restated). The v-enumeration results (T3/T8/T13, four-gate mechanism,
factorizations, pseudo-cycles, watershed) are our own and remain correct on
their stated ranges.
Part 2 · The quadratic family reduction (conjectural core)
Chain.
$$\boxed{
\begin{aligned}
&\text{Core inequality (cycles: proven via P8; general: conj.)} \\
&\quad \Rightarrow k=1: b^2+(2^S-1)=2^N;\\
&S\text{ even} \Rightarrow \text{mod }3\ (\text{proven});\\
&S\ge5\text{ odd} \Rightarrow \text{UP2} \Rightarrow \text{quadratic family};\\
&S=3 \Rightarrow \{b=5,11,181\}\ (\text{Nagell});\\
&\text{discrete-log filter} \Rightarrow \{b=5,181\}\ (\text{proven}).
\end{aligned}
}$$
Conjectural (general solutions).
(Core inequality.) For $D\ne7$, $D\equiv7\pmod8$, every odd-$M_2$ solution of
$a^2+D=2^{M_2}$ satisfies $M_2\le\operatorname{ord}_D(2)+2$.
For cycle candidates, proven via P8. For general
solutions, the mixed-CRT gap remains open. Verified $a\le2\times10^6$.
Conjectural. (Mersenne-gap.)
$b^2+(2^S-1)\ne2^N$ for $S\ge4$, $b\ge5$, $N$ odd. Even $S$: proven (mod $3$).
Odd $S$: reduced (verified $S\le30$). $S=3$: $(5,5),(11,7),(181,15)$.
The quadratic family — positive-sign branch
Positive-sign branch. $u=2^h\ge8$,
$u^2d^2+(u-1)d+1=2^k$; $k>h$, $d=1+u\tau$ ($\tau\ge0$); $N=k-h$,
$B=2u^2+u-1$, $b=B+2u^3\tau$:
$$b^2+D_h=2^M,\quad D_h=(u+1)(3u-1),\quad M=N+3h+2,\quad
b\equiv B\pmod{2u^3},\ b\ge B.$$
Layer 1 — local sieve (odd $M$): odd $h$: $3\mid D_h$
(proven, mod $3$). v11.1 correction: the even-$h$
inert-prime claim is false — counterexamples
$h\in\{64,76,96,160,216,264,288\}$ (verified; the earlier
"$6\le h\le64$ verified" statement was wrong: $h=64$ is a
counterexample). Structure: for even $h$ with $v_2(h)=1$,
$5\mid2^h+1$ always (proven: $2^h\equiv-1\pmod5$ iff $h\equiv2\pmod4$),
so those $h$ have the inert factor $5$; for $v_2(h)\ge2$ all prime factors
of $2^h+1$ are $\equiv1\pmod8$, so an inert factor must divide
$3\cdot2^h-1$, and a counterexample is exactly a $h$ for which all prime
factors of $3\cdot2^h-1$ are $\equiv1$ or $7\pmod8$.
v11.5 — corrected list and structure: the
v11.1 list $\{64,76,96,160,216,264,288\}$ contained an error ($h=160$ is
unresolved, not a counterexample). Full scan $h\le12\,500$
($v_2(h)\ge2$): 13 counterexamples
$64,76,96,216,264,280,288,324,3040,3276,3616,4204,7276$ (density lower
bound $0.42\%$), 2020 hits, 1091 unresolved semiprimes (smallest prime
factor $>10^4$, rho-limited; heuristic $\approx1/4$ of them are
counterexamples, total density $\approx9\%$). The semiprime
counterexamples have a clean structure: small factors are
$\equiv7\pmod8$ P6-type primes ($(2/p)=1$, $\operatorname{ord}_p(2)$
odd, $\langle2\rangle=QR_p$) from congruence classes —
$47\mid3\cdot2^h-1$ iff $h\equiv4\pmod{23}$ (ord 23; $h=96,280$);
$431\mid3\cdot2^h-1$ iff $h\equiv30\pmod{43}$ (ord 43,
$2^{30}\equiv144\equiv3^{-1}$; $h=288,3040$); also $71\,(h=264)$,
$191\,(h=3616)$, $313\,(h=7276)$; $h=3276,4204$ have $3\cdot2^h-1$
prime $\equiv7$. $2^h+1$ contributes nothing (all factors
$\equiv1\pmod8$). $h>12\,500$ requires C/ECM-grade factorization.
Layer 2: $h=4$.
Layer 3 — Pell (odd $M$): $x^2-2y^2=-799$, $y=2^r$,
$x\equiv527\pmod{8192}$. Three candidate orbits $(1,20),(13,22),(49,40)$;
for $0\le n\le160$ per orbit, the odd part $\operatorname{odd}(y_n)>1$, so no
tested $y_n$ is a power of $2$. All-$n$ and orbit completeness open.
Even-$M$: factor-pair problem $AC=D_h$, $A+C=2^{r+1}$;
verified $3\le h\le20$.
Hensel structure. $b_u(T)=u^3T^2+(2u^2+u-1)T+u+1$;
unique $\alpha_u\in\mathbb{Z}_2$; $a_N=\alpha_u\bmod2^N$;
$q_N=b_u(a_N)/2^N$; $b_u'(\alpha_u)^2=1-2u-3u^2$ (exact);
$v_2(b_u(T))=v_2(T-\alpha_u)$; $q_N\equiv\gamma_N\pmod{2^h}$; bit-block
$10^{h-1}$ absent for $u\in\{8,16,32,64,128\}$ over $10u$ bits (all-prefix
claim open).
Part 3 · Frameworks (programmatic)
K-theory decoupling (final)
K-theory is decoupled from the cycle classification.
The natural étale groupoid of the forward map is the Deaconu–Renault groupoid
$G_a=\{(x,m-n,y):S_a^m(x)=S_a^n(y)\}$ on $\mathbb{Z}_2$ (tail equivalence);
the parity-vector map conjugates $S_a$ to the full countable shift, so
$C^*(G_a)\cong O_\infty\otimes K$, and
$$K_0(C^*(G_a))=0.$$
The class number $\eta(a)$ = number of periodic orbits = number of weak
components of the inverse graph is a graph-topological invariant
(verified, Part 1: $\eta(5)=3$, $\eta(181)=2$, $\eta=1$ for the 18 Mersenne
values $\le10^6$, $\eta=0$ otherwise, $L\le100$); it does not arise from
groupoid K-theory. Future direction (separate work):
an AF-inductive-limit construction of the inverse graph truncations, with
$K_0\cong\mathbb{Z}^{\eta(a)}$, would require two proofs — that the
connecting maps kill the boundary pseudo-components, and that each cycle
component contributes exactly $\mathbb{Z}$ (tree part Morita-trivial). This
is not part of the present notes' claims.
Escape rate (finite-time scaling)
Finite-time scaling observed; limit open.
For $a>4$, with $L_a^{\rm known}$ = basins of cycles detected within the stated
search range:
$$\mathcal S_a(N,k)=\#\{1\le n\le N,\ n\ \text{odd},\ n\notin L_a^{\rm known}:\
S_a^j(n)\le N\ \text{for all}\ 0\le j\le k\}.$$
Fitted log-slope $\approx\log(a/4)$ (deviation $<0.004$); normalization,
window, and the limit definition remain open. The exact fitting protocol
(window length, $N$, slope-fit interval) is to be recorded for
reproducibility. The parity-vector conjugacy is classical (Terras, Lagarias);
the survival scaling is a numerical observation.
Spectral statistics
Numerical observation. For $a=3,5,7$,
$\langle r\rangle$ observed at $0.44,0.34,0.44$ (Poisson $0.386$, GOE
$0.536$); no Wigner–Dyson observed. Caution (v9):
per the Gaussian-indistinguishability protocol, raw level statistics require
a null model, sample size, and error bars before any claim; the values above
are reported as observations only, and no spectral claim is made.
Other frameworks
The PLL, EM/RG, resampling/crack, permutation-test/validation,
digital-twin/transport, and kinetic-theory identifications are analogies
(heuristics), not theorems.
Honest status table (v11)
| Statement | Status |
|---|
| P1 fixed points; P2 crack healing ($m=1,3$ / $m\ge5$) | Proven / Conjectural |
| P3 reduction, P4 pinning, P5 gap, P6 parity gate | Proven |
| P7 even-$M_2$; P8 CRT sign consistency; P9 $D=7$ | Proven |
| P10 LTE k-case (cycles) | Proven |
| P11 lock-range measurability ($\mu_{\rm Haar}=0$) | Proven |
| P12 escape rate $v_a=\log(a/4)$ | Finite-time scaling; limit open |
| O1 orbit lemma ($m\le M_{\max}$); O2 orbit algorithm (complete, rigorous) | Proven (v11) |
| T-orbit: odd $a\le10^6$, $L\le100$ → cycle set $\{5,181\}$ (13\,677 parameters scanned, 118\,714 starts, 5.2 s; external $a\le999$/$L\le200$, $a\le99$/$L\le300$ agree) | Proven (computational, rigorous) |
| $a=7$, $L\le2000$ (173\,315 starts, 99.7 s) | Proven (computational, rigorous) |
| T3 three-cycles ($a=5$, $D=3$); T8/T13 enumeration results ($a\le99$, $n\le13$) | Proven (historical; cross-confirmed by T-orbit) |
| Four-gate mechanism; gate-2 Reach; 2a/2b subtypes; factorizations; proposition B | Retired (v11): phenomenology of the enumeration space; records kept |
| Proposition B' (Weil route to $Z_p\to1/p$) | Withdrawn (v11): $N\lesssim p$ always — error terms exceed main term |
| Baker/LMN route on $a=7$ | Negative (v10.1, kept) |
| Pseudo-cycles; shared-$D$ (power coincidence); watershed (convergence for all $a>4$) | Closed (v10.x, kept as record) |
| Core inequality: cycles / general | Proven / Conjectural |
| Family sieve / $h=4$ survivor / Pell $799$ / even-$M$ | Layer-1 inert-prime claim falsified (v11.1); Pell/even-$M$ verified (ranges stated) |
| $K_0(C^*(G_a))=0$ (forward groupoid) | Proven (decoupling) |
| $K_0\cong\mathbb{Z}^{\eta}$ (AF construction) | Future direction (separate work) |
| p-adic uniformity: $S_a$ mod $p^k$ has $L$-step closure rate exactly $1/p^k$ (affine fixed-point argument; 2-adic path decouples from $p$-component) | Proven (v11.6) |
| Convergent discriminants $2^S-7^L$: no gate-3 obstacles ($-7\in\langle2\rangle$ for all checked factors); factors prefer $\equiv1,7\pmod8$ (inert primes only inherited: $5|D(1603,571)$ via $s\equiv l\pmod4$) | Verified (v11.6) |
| log $M(a,L)$ autocorrelation: a=97 ACF 0.55/0.51/0.45 (clustered); a=5,181 ~0.001 (white noise) | Verified (v11.6) |
| a=97 mechanism chain: resonance $\log_2 97\approx33/5$ → E concentrated at L=5 (C=0.969) → D=2,594,335=5·518,867 → 2b blocker Z=0 → empty window | Closed (v11.7) |
| Δ(a) discriminant $|2^S-a^q|$ separates cycle parameters (Δ≤7: a=5,181) from empty ones (Δ≥751) | Verified (v11.7, 11 parameters) |
| Gate-2 precise form: divisibility layer has solutions (13 for a=19, 79 for a=97) but 2-adic exactness excludes all of them (90/92 at step 0) | Verified (v11.8, BigInt) |
The a=97 mechanism chain and the Δ(a) discriminant (v11.6–v11.7)
Complete chain for a=97: the resonance
$\log_2 97\approx 33/5$ (4-digit rational approximation) concentrates
$E(97)=2.74$ in the single window $L=5$ ($C(97)=0.969$; the geometric series
$\sum_k 85\cdot2^{-5k}$ over $L\equiv0\pmod5$ sums to 2.74). The unique window
has $M=170$, $D=2^{33}-97^5=2{,}594{,}335=5\cdot518{,}867$, and $518{,}867$ is a
2b-blocker ($Z=0$, subgroup zero-sum): the window is empty by a single prime.
Not a counting miracle — a resonance located, then blocked by one prime.
Concentration ratio falsified (v11.7):
$C(a)=\max_L E(a,L)/E(a)\ge0.75$ for ALL tested $a$ (5:0.844, 181:0.754,
97:0.969, 45:0.876, 19:0.833, 105:0.885) — concentration is universal, not a
marker of cycles. What differs is whether the single dominant window contains
an integer solution.
The Δ(a) discriminant (v11.7): for the resonance
denominator $q$ (dominant E-window) and $S=\lceil q\log_2 a\rceil$, set
$\Delta(a)=|2^S-a^q|$:
| a | q | Δ(a) | cycles? |
|---|
| 5 | 3 | 3 | yes |
| 181 | 2 | 7 | yes |
| 3 | 2 | 7 | fixed point only (trivial) |
| 19 | 4 | 751 | no (blocked) |
| 11 | 4 | 1743 | no |
| 97 | 5 | 2,594,335 | no (2b blocker 518,867) |
| 45 | 6 | 286,168,967 | no |
| 147 | 5 | 77,991,229 | no |
| 645 | 3 | 99,331 | no |
| 105 | 7 | 27,446,089,703 | no |
| 7 | 26 | 5.7×10¹⁹ | no |
Separation: cycles (excluding the trivial a=3) have $\Delta\le7$; all
cycle-free parameters have $\Delta\ge751$. $\Delta$ is necessary but not
sufficient (19, 11 have small $\Delta$ and are blocked at the divisibility
layer). This is the first quantitative discriminant sharper than $E(a)$:
$E(97)=2.74>E(5)=1.85$ yet 97 is empty — $\Delta$ removes 97 cleanly.
p-adic uniformity theorem (v11.6): for any odd prime
$p$ and odd $a$ with $p\nmid a$, the $L$-step closure rate of $S_a$ on
$\mathbb{Z}/p^k\mathbb{Z}$ is exactly $1/p^k$, independent of $a$. Proof:
on each 2-adic path the $p$-component map is affine $x\mapsto Ax+B$ with
$A-1$ invertible, giving exactly one fixed point. (The "bijection ⟹ uniform"
draft is insufficient; the affine fixed-point argument is the correct one.)
Numerically exact for $p\in\{3,5,7,13,97\}$, $L\le30$.
Convergent factor structure (v11.6): for the upper
convergents (73,26), (1603,571), (8321,2964) of $\log_2 7$: no gate-3
obstacles ($-7\in\langle2\rangle$ for every checked factor, including the
large prime cofactor of (73,26) with $\operatorname{ord}=24{,}080{,}123$);
$m\equiv0\pmod7$ obstruction absent ($D\equiv2^S\pmod7$ always invertible);
prime factors prefer $\equiv1,7\pmod8$ (inert primes only inherited, e.g.
$5|D(1603,571)$ since $5|2^s-7^l \iff s\equiv l\pmod4$, and
$1603\equiv571\equiv3\pmod4$). Weak primitive-factor lemma: primitive
factors of convergent discriminants are $\equiv\pm1\pmod8$.
Gate-2 mechanism: p-layer compatibility vs 2-adic exactness (v11.8)
Identity (Proven): $\Delta(a)\cdot M_{\max}(q)=q\,a^{q-1}+O(M_{\max}^{-1})$,
from $2^S=(a+\varepsilon)^q$, $\varepsilon=1/M_{\max}$; numerically exact to
<0.5% for all tested cases. $\Delta$ small and $M_{\max}$ large are the same
statement. The L=q Diophantine equation
$\Delta\cdot e_q=\sum_{k=0}^{q-1}a^k e_k$ (elementary symmetric functions)
generalizes the L=2 equation (P3) and gives the necessary condition
$\Delta\mid 1+ae_1+\cdots+a^{q-1}e_{q-1}$.
Divisibility layer does NOT close the window (verified,
BigInt): for a=19, $\Delta=751$: 13 odd multisets
$(x_1,\ldots,x_4)\le35$ satisfy $751\mid1+19e_1+361e_2+6859e_3$; for a=97,
$\Delta=518{,}867$: 79 odd multisets $(x_1,\ldots,x_5)\le169$ satisfy
$518867\mid1+97e_1+97^2e_2+97^3e_3+97^4e_4$. The orbit condition rejects all
of them (13/13 and 79/79), and 90 of 92 fail at the first step (the
image of the minimal element is outside the multiset).
Mechanism (v11.8): mod-$\Delta$ orbit compatibility
is equivalent to the product identity mod $\Delta$ — which the divisibility
solutions satisfy by construction. But exact orbits require 2-adic division
(a lossy layer), which mod-$\Delta$ constraints cannot control; the p-layer
compatibility therefore almost always dies at step 0. This is the mirror of
the p-adic uniformity theorem: the p-layer is uniformly compatible, the
2-adic layer is the killer. Gate-2 precise form:
"Reach/divisibility solutions exist (13 and 79) but are excluded by 2-adic
exactness, predominantly at the first step." a=19 and a=97 share the same
mechanism — the $\Delta$-threshold cases are not a different species.
The Δ discriminant, corrected — two levels (v11.9)
Correction (v11.9, external review). The Δ
discriminant must be taken over ALL convergents $p/q$ with $q\ge2$, not
the E-concentration point: $a=11$ has $\Delta_{\min}=|2^7-11^2|=7$
(convergent $7/2\approx3.5$ of $\log_2 11$), not 1743. The complete
non-Mersenne solution set of $|2^p-a^q|\le7$ ($q\ge2$) is
$$|2^p-a^q|\le7\ \Longleftrightarrow\ (a,q,p)\in\{(5,3,7),(5,2,5),(11,2,7),(181,2,15)\},$$
i.e. $\Delta=3$: $5^3+3=2^7$; $\Delta=7$: $5^2+7$, $11^2+7$,
$181^2+7$ — exactly the Ramanujan–Nagell solutions $x^2+7=2^n$ for
$x=5,11,181$ plus the cubic analogue $5^3+3=2^7$. The $q=1$ convergents
are excluded (they encode the Mersenne fixed-point condition
$a=2^k-1$). The discriminant is therefore two-level:
- $\Delta_{\min}(a)\le7$ (necessary: window
$m\le q\cdot a^{q-1}/\Delta$ exists);
- $-a\in\langle2\rangle\pmod\Delta$ (2-subgroup
test): excludes $a=11$ ($-11\equiv3\notin\{1,2,4\}\pmod7$) while
$a=5,181$ pass ($-181\equiv1$).
Both levels verified: candidates $\{5,181\}$ have cycles; all others
fail level1 or level2. For $q\ge3$ the solution set of
$x^q\pm c=2^p$, $c\in\{3,5,7\}$, is closed by classical results
(modular sieving + Zsigmondy; verified for $q=2$: $x^2+5=2^n$ has no
solutions by $\bmod8$; $x^2+3=2^n$ only $(1,2)$; $q=3$:
$x^3+3=2^n$ only $(5,7)$) — turning the "$\Longleftarrow$" half of the
Δ-conjecture into a theorem.
Mod-$a$ structure collapses onto the valuation word (O1,
v11.9). Since $ax+1\equiv1\pmod a$, along a cycle
$x_i\equiv2^{-k_{i-1}}\pmod a$: the entire mod-$a$ pattern is a
function of the word $(k_0,\dots,k_{L-1})$ alone. This is the exact
statement behind "the p-layer is trivial" and complements the p-adic
uniformity theorem.
Wall locality (v11.9). The $L=2964$ wall is
caused by a single large partial quotient (29) of $\log_2 7$ — a LOCAL
continued-fraction feature. The irrationality measure
$\mu(\log_2 7)$ is a GLOBAL approximation quantity and cannot predict
where large partial quotients appear; it is the wrong tool for the wall
(previous framing retracted). The correct complexity statement: the
orbit-method candidate size at length $L$ is controlled by the largest
partial quotient below $L$; walls occur exactly at resonant $a$.
Per-window multiplicities, three gates, and the E-concentration law (v11.10)
Per-window multiplicities (v11.10). The four tiny-Δ
windows (non-Mersenne, $|2^p-a^q|\le7$) are exactly:
| window $(a,q,p,\Delta)$ | equation | minimal words | cycles | count |
|---|
| $(5,2,5,7)$ | $m\cdot7=5+2^{k_0}$, $2^{k_0}\equiv2\bmod7$ | $k_0=1$ ($k_0=4$: $m=3$ not minimal) | $\{1,3\}$ | 1 |
| $(5,3,7,3)$ | $m\cdot3=25+5\cdot2^{k_0}+2^{k_0+k_1}$ | $(1,1)$, $(1,3)$ ($(1,5)$: not minimal; $(1,7)$: 2-adic fail) | $\{13,33,83\}$, $\{17,43,27\}$ | 2 |
| $(11,2,7,7)$ | $2^{k_0}\equiv-11\equiv3\bmod7$ | $3\notin\langle2\rangle=\{1,2,4\}$ | — | 0 |
| $(181,2,15,7)$ | $2^{k_0}\equiv1\bmod7$ | $k_0=3,6$ ($k_0=9,12$: not minimal) | $\{27,611\}$, $\{35,99\}$ | 2 |
Total $1+2+0+2=5$, matching the full classification cycle-by-cycle. The
asymmetry of $a=5$ (three cycles) vs $a=11$ (none) is explained: $5$ has
two tiny-Δ windows (both pass all gates); $11$ has one window, killed at
Gate1.
Three gates (v11.10). Precise failure attribution:
- Gate0 (archimedean/window): $M(a,L)\ge1$, i.e.
$\Delta\lesssim L\cdot a^{L-1}$ (via $\Delta\cdot M_{\max}=q\cdot a^{q-1}$).
- Gate1 (mod-$\Delta$): $\exists$ word with
$\sum a^{L-1-j}2^{K_j}\equiv0\pmod\Delta$; for $L=2$ this is
$-a\in\langle2\rangle\pmod\Delta$.
- Gate2 (2-adic exact lift): $m=\sum\cdots/\Delta$ odd
$\ge1$, $k_j=v_2(ax_j+1)$ exactly, minimality.
Failure loci: $a=45$ → Gate0; $a=11$ → Gate1 (zero divisibility
solutions); $a=19,97$ → Gate2 (13/79 solutions exist, 2-adic lift kills
all); $a=5,181$ → all gates pass.
E-concentration law (v11.10, refined). For $a$
with $E(a)\ge1$, the E-concentration point (dominant convergent)
coincides with the Gate0+Gate1-surviving window. Verified 7/7:
$181(L=2),97(L=5),5(L=3),19(L=4),11(L=4),45(L=6),105(L=7)$ all have
divisibility solutions at their E-peak. $a=7$ is the counterexample
excluded by the refinement: $E(7)=0.076<1$ — no candidate
qualification — which explains why all surviving windows of $a=7$
are empty at Gate1. Mechanism chain closed: tiny $\Delta$ ⇔ Gate0
survives; cycle ⇔ Gate2 also survives; E-concentration is the
Gate0+1 proxy (for $E\ge1$).
Refined ⇔ is a finite task (v11.10). (i) cite
Ramanujan–Nagell / Lebesgue–Nagell: non-Mersenne $|2^p-a^q|\le7$
implies $a\in\{5,11,181\}$; (ii) verify the four windows above gate by
gate (all $L\le3$, finite checks, done). Hence the "⇔" half of the
Δ conjecture is a theorem. The "⇒" half is the honest
Collatz-grade negative; its precise statement: any nontrivial cycle forces
$|2^S-a^L|\le7$ with $(S,L)$ the dominant window.
Open problems (v11)
- Crack-healing $m\ge5$ (reduced to the quadratic family).
- Inert-prime covering — falsified (v11.1):
the claim "every even $h$ has a $\equiv3,5\pmod8$ prime factor of $D_h$"
is false: 23 confirmed counterexamples (lower bound) for $h\le12\,500$;
the $h\le600$ segment is completely closed with 17 elements
($4,64,76,84,96,144,160,196,204,216,228,244,264,280,288,324,580$, zero
undecided, external factordb closure verified here) plus confirmed large
ones $1020,3040,3276,3616,4204,7276$ (all verified via factordb:
$3\cdot2^{3040}-1=47\cdot431\cdot P$, $3\cdot2^{3276}-1$ prime $\equiv7$,
$3\cdot2^{7276}-1=313\cdot P$); the interval $(600,12\,500]$ likely
contains further undiscovered counterexamples — the list is a lower-bound
sample, not a complete set), density
$\gtrsim0.4\%$, $\approx9\%$ heuristic. Reformulation: $v_2(h)=1$
implies the inert factor $5$ (proven); $v_2(h)\ge2$: $2^h+1$ has only
$\equiv1\pmod8$ factors and the question is the prime-factor mod-$8$
distribution of $3\cdot2^h-1$ — counterexamples are exactly the $h$
with all prime factors of $3\cdot2^h-1$ $\equiv1,7\pmod8$; known
semiprime examples have P6-type small factors $\equiv7\pmod8$ from
congruence classes $h\equiv c\pmod{\operatorname{ord}_p(2)}$ (47:
$c=4$, ord 23; 431: $c=30$, ord 43). Three follow-up questions (P6 preference, $c$-value mechanism, density
asymptotics) were answered by the external collaboration and verified here
(v11.5): (i) $4\mid h \Rightarrow$ all prime factors of $2^h+1$
$\equiv1\pmod8$ (proven: $v_2(e)=v_2(h)+1\ge3$); together with
$h\equiv2\pmod4 \Rightarrow 5\mid2^h+1$, only $4\mid h$ and $3\cdot2^h-1$
matter. (ii) Two-level quadratic-reciprocity sieve: $p\equiv7\pmod8$
passes both levels with zero loss ($e$ odd), hence the P6 dominance;
$\gcd(4,e)\mid c \iff 3^{-e/d}\equiv1\pmod p$ (verified 94/94 on
sample); Frobenius/Kummer-tower formulation; global version needs GRH-type
input. (iii) Sieve survival stays $\sim27\%$ up to $p<10^7$ (measured,
$h\le5\cdot10^4$, log-log decay only); density $\sim c\,h^{-\log2}$,
$N(H)\sim c'H^{0.307}$: predicts 18.1 at $H=12\,500$ vs 13 measured (same
order). Open: full determination for $h>12\,500$ (C/ECM factorization;
BigInt rho infeasible beyond $\sim6000$ bits).
- Pell $D=799$ (all $n$; orbit completeness; Carmichael).
- Even-$M$ factor-pair classification for all $h\ge3$.
- Core inequality for general solutions (mixed-CRT branches;
closed for cycles via P8).
- Negative-sign branch of the quadratic family.
- Complete the small-cycle gap proof (record; not needed by
the orbit method).
- Define the escape-rate limit (transient survival, limit
order, normalization; three separate proofs).
- a=97 structure — closed (v11.7): resonance 33/5, window (97,5), D=5·518867, 2b blocker; the Z_{518867}=0 piece is resolved as a 2-adic exclusion event (v11.8): 79 divisibility solutions exist; all fail the exact orbit condition, 78 at step 0; the subgroup zero-sum is the mod-Δ shadow of 2-adic information loss.
- Δ(a) discriminant — corrected (v11.9): two-level form (Δ_min≤7 necessary + −a∈⟨2⟩ mod Δ); a=11 corrected to Δ_min=7; prove the "⟸" half via Lebesgue–Nagell (q=2), generalized Ramanujan–Nagell (q=3), Zsigmondy (q≥4) — solution set of x^q±c=2^p, c∈{3,5,7} is classical; the "⟹" half is Collatz-grade (honest negative, external review). Verify threshold on larger sets. (Δ≤7 vs Δ≥751) on a larger parameter set; prove Δ small ⟺ resonance denominator exists; explain why 19 (Δ=751) is blocked while 5,181 (Δ≤7) are not — the divisibility layer remains the only content layer.
- General cycle classification ($L\ge2$): closed
computationally for odd $a\le10^6$, $L\le100$ (T-orbit) and for $a=7$,
$L\le2000$; open for $a>10^6$ and $L>100$. An unconditional all-$L$ theorem
requires O4 and is at the level of Collatz.
- Future: AF-inductive-limit construction with
$K_0\cong\mathbb{Z}^{\eta(a)}$ (two proofs: connecting maps kill boundary
pseudo-components; each cycle component contributes exactly $\mathbb{Z}$) —
separate from this paper.
- Effective irrationality measure for $\log_2 a$ — reframed (v11.9): $\mu$ is a GLOBAL quantity and cannot locate walls (the L=2964 wall is a local partial quotient 29); drop as wall-tool, keep only for unconditional all-L finiteness; the wall is a complexity theorem about the CF profile.
- Effective irrationality measure for $\log_2 a$ (the only
theoretical input left): an explicit lower bound
$\Lambda(L)\ge cL^{-\kappa}$ for $L\le L_0$ or all $L$ would bound
$M_{\max}$ and close $a=7$ (and every $a$) unconditionally up to a finite
tail; the LMN constant ($10^{-9073}$) is useless — a much sharper
$L$-dependent bound is needed (standard Baker-type applications; external
collaboration proposed). The continued-fraction records of $\log_2 7$
(O4) are the obstruction.
- Deep sweeps beyond v11: $a\le10^6$, $L\le200$
(cost $\sim10\times$); $a=7$, $L\le10^4$ ($M_{\max}=5.54\times10^7$ in
$L\in[2964,91313)$, $\sim40$ h single-core or parallel). Diminishing returns
for the theorem; useful for the record.
Verification records (v11.5)
- Q2 criterion re-checked: $\gcd(4,e)\mid c \iff 3^{-e/d}\equiv1\pmod p$ —
94/94 agreement on primes $p\le500$ (independent).
- Sieve-layer scan: $h\equiv0\pmod4$, $h\le5\cdot10^4$, trial
division over $p\equiv3,5\pmod8$ up to $10^7$ (fast exponentiation):
survival constant $\approx27\%$ across all intervals (23.7\% at
$h\le10^3$, 27.5\% at $(12500,25000]$, 27.1\% at $(25000,50000]$) —
supporting the log-log sieve decay; all 13 known counterexamples survive
(consistency).
- Determination limit: verdict via rho/MR infeasible beyond
$\sim6000$ bits in BigInt (timeouts); full counterexample count for
$h>12\,500$ needs C/ECM.
- v11.5 — list corrected (external re-check, verified here):
$h=160$ and $h=1020$ are genuine counterexamples missed by the original list
(which defaulted undecidable cofactors to non-counterexample):
$3\cdot2^{160}-1=125711\times P_{44}$ ($125711\equiv7$, $P_{44}$ prime
$\equiv1$; $\operatorname{ord}_{125711}(2)=62855$, $c_{125711}=160$ — the
counterexample hits its own arithmetic-progression class), and
$3\cdot2^{1020}-1=33623\times P_{304}$ ($33623\equiv7$, $P_{304}$ prime
$\equiv1$). Both verified independently (divisibility + MR primality).
$h=4$ is the trivial case ($47$ prime $\equiv7$).
- v11.5 — counterexample set enlarged: 7 new elements in
$h\le600$ ($84,144,196,204,228,244,580$) all verified here via factordb
(all-prime-factor mod-8 check; $580$: $383\cdot p_{48}\cdot p_{63}$ all
$\equiv7$); the 5 large elements ($3040,3276,3616,4204,7276$) also verified
via factordb — previous rho verdicts confirmed, none of the earlier list
refuted. Lesson: undecidable cofactors defaulting to non-counterexample
systematically under-counts.
- v11.5 — orbit-sweep cost model (external, measured): JS
110 ns/step, steps/candidate $\approx1.59L^{0.854}$; total work is
$\sum_L M_{\max}(L)\cdot L$ — the $M_{\max}$ plateau does NOT make the sweep
free (L≤2964: 3.33e7 candidates; L≤91313: 1.18e10). C single-core:
L≤20000 $\approx5$h (8-core 37min, sweet spot), L≤91313 $\approx19.5$ days
(8-core 2.4 days, overnight class). $m\equiv1\pmod8$ starts are zero-cost,
$m=1$ fixed point excluded.
- v11.5 — Q3 status: the $N(H)\sim H^{0.307}$ exponent is
neither confirmable nor refutable in the near term: confirming one
counterexample at $h=10^5$ needs full factorization of a 30,103-bit integer
(ECM ~70–80 digits, NFS ~250 digits — hard wall). Moreover the decidable
sample is biased toward small $\omega$ (most CE-prone). Sieve bounds:
$11\le N(2000)\le135$, $N(10^5)\le7145$; sieve survival 28.58% at
$H=10^5$ (matches the 27% measured here). Engineering insight: the sieve
needs no bignum ($p\mid3\cdot2^h-1 \iff h\equiv c_p\pmod{e_p}$, pure 64-bit
arithmetic-progression marking, scalable to $p<10^9$ in C).
— end of v11.10 —