Premise: what follows consists of original ideas and observations reorganized, presented, and elaborated by ChatGPT/Codex from author-directed research notes.
Status
This is not a proof of the Collatz conjecture. The conjecture
remains open. The core contribution is a precise equivalent
reformulation: a three-branch affine-residual dynamics on positive
integers in which every finite operational prefix simultaneously
gives an affine map, a single congruence cylinder modulo a power of
2, a density, a descent threshold, and a cycle
candidate.
The first part below is the stable core. The later extension sections collect theorem-level developments from the research notes: certificate graphs, shielding, defects, pressure bounds, automata, active budgets, tail-deficit formulas, and extremal arithmetic gates.
Classical Collatz and Finite Descent
The classical Collatz map is
Col(n) = 3n+1 if n is odd,
Col(n) = n/2 if n is even.
Its conjecture is equivalent to finite descent: every
n>1 eventually reaches a value smaller than itself.
Indeed, if this holds for every smaller value, then once the orbit
of n drops to m<n, strong induction
brings m, and therefore n, to
1.
Conversely, any nontrivial cycle or divergent orbit would produce a seed with infinite stopping time. For a cycle, take the minimum element of the cycle. For a divergent orbit, take the smallest divergent seed. If either ever fell below itself, it would produce a smaller counterexample.
Goal: for every n>1, prove that some k≥1 has Col^k(n)<n.
From Syracuse to a Modified Odd Map
It is standard to pass from Collatz to the Syracuse map on positive odd integers:
Syr(x) = (3x+1) / 2^ν₂(3x+1).
A classical Collatz orbit reaches 1 if and only if
its odd subsequence under Syr reaches 1:
the omitted even steps are only divisions by powers of
2.
The affine-residual normalization begins by splitting odd integers
into three cases. If x=8k+1, then
3x+1 = 24k+4 = 4(6k+1),
so the next Syracuse odd value is 6k+1. If
x=4k+3, then
3x+1 = 12k+10 = 2(6k+5),
so the next Syracuse odd value is 6k+5. Finally, if
x=8k+5, then
3x+1 = 24k+16 = 8(3k+2).
In this last class, replace the Syracuse jump by the strictly descending step
8k+5 -> 2k+1 = (8k+5-1)/4.
This replacement preserves the next Syracuse node, because
3(2k+1)+1 = 6k+4 = 2(3k+2).
Thus 8k+5 and 2k+1 have the same next
odd Syracuse successor after removing powers of 2.
Repeating the replacement, if necessary, creates a finite strictly
descending chain before the same Syracuse jump.
Equivalence Theorem
Define a modified odd map S* as follows: on
8k+1 and 4k+3 it follows the Syracuse
jump above, while on 8k+5 it sends
8k+5 to 2k+1.
The modified odd map is equivalent to Syracuse, hence to
Collatz. Every S*-orbit is obtained from the
corresponding Syracuse orbit by inserting finite descending chains,
and every Syracuse step appears after finitely many
S* steps. Therefore convergence to 1,
nontrivial cycles, and divergent behavior are preserved.
Proof. The only changed case is
x=8k+5. There S*(x)=(x-1)/4<x, and
Syr(S*(x))=Syr(x). Since repeated replacements strictly
decrease the positive odd integer, they cannot continue forever
before the next Syracuse jump. Collapsing these inserted descending
chains recovers the Syracuse orbit; expanding each affected
Syracuse arrow recovers the modified orbit. The two dynamics
therefore have the same convergence and obstruction behavior.
The Three-Branch Map on Positive Integers
Use the bijection from positive odd integers to positive integers
φ(o) = (o+1)/2, so o = 2n-1.
Conjugating S* by φ gives the
three-branch map
T(n) = (3n+1)/4 if n ≡ 1 mod 4,
T(n) = 3n/2 if n ≡ 0 mod 2,
T(n) = (n+1)/4 if n ≡ 3 mod 4.
Since φ is a bijection and
T=φ∘S*∘φ^{-1}, this three-branch map is not merely
analogous to Collatz: it is an equivalent reformulation.
The three cases are just the three odd classes above translated by
o=2n-1:
n=4k+1 corresponds to o=8k+1 and maps to 3k+1,
n=2k corresponds to o=4k-1 and maps to 3k,
n=4k+3 corresponds to o=8k+5 and maps to k+1.
We denote the branches by
O1(x)=(3x+1)/4, E(x)=3x/2, O3(x)=(x+1)/4.
Equivalently, in the notation of the base affine-residual paper,
these are b,r,v. The central branch E is
the only branch with principal multiplier greater than
1; the two odd branches are descending to first order.
Operational Words and Affine Maps
A finite operational word is a string
w = s1 s2 ... sk, with si ∈ {O1,E,O3}.
On the cylinder of seeds whose first |w| branches are
exactly w, the iterate has an exact affine form:
T^|w|(x) = (A_w x + B_w) / D_w.
Starting from A=1, B=0,
D=1, appending a letter on the right gives
O1: A -> 3A, B -> 3B + D, D -> 4D
E: A -> 3A, B -> 3B, D -> 2D
O3: A -> A, B -> B + D, D -> 4D.
Hence
A_w = 3^(#O1(w)+#E(w)),
D_w = 4^(#O1(w)+#O3(w)) 2^(#E(w)),
λ_w = A_w/D_w.
The multiplier λ_w records the expanding or
contracting principal part; the affine term B_w
records the order of the letters.
Residual Cylinders and Homogeneity
Let E_w be the set of positive integers whose first
|w| branches are w. The word is realized
exactly when the affine numerator is divisible by the denominator:
E_w = {x : A_w x + B_w ≡ 0 mod D_w}.
Since A_w is odd, it is invertible modulo the power of
2 given by D_w. Thus every finite word
gives one and only one residue class:
E_w = {x : x ≡ a_w mod D_w}.
Therefore the natural density of the cylinder is
δ(E_w)=1/D_w.
This is the main homogeneity of the normalization: the same number
D_w is the affine denominator, the modulus of the
residual cylinder, and the reciprocal of its natural density. With
fresh seeds, the symbolic weights are
P(O1)=1/4, P(E)=1/2, P(O3)=1/4.
Average Drift and the Terras-Korec Mechanism
The principal multiplier of a word is
λ_w = (3/4)^#O1(w) (3/2)^#E(w) (1/4)^#O3(w).
Under the natural cylinder weights, the mean logarithmic multiplier is
μ = (1/4)log(3/4) + (1/2)log(3/2) + (1/4)log(1/4)
= (1/4)log(27/64) < 0.
Thus the typical geometric multiplier is contracting. For
s>0, set
Λ(s)=(1/4)(3/4)^s+(1/2)(3/2)^s+(1/4)(1/4)^s.
Since Λ(0)=1 and Λ'(0)=μ<0, some
positive s has Λ(s)<1. Hence for words
of length k,
P(λ_w ≥ 1) ≤ Λ(s)^k.
Noncontracting prefixes are exponentially rare. This recovers the mechanism behind Terras/Korec almost-everywhere finite stopping: density zero of possible exceptions is natural, but density zero is still not emptiness.
Affine Centers and 2-Adic Balls
If D_w≠A_w, define
p_w = B_w/(D_w-A_w).
This is the fixed point of the affine map associated with
w. Moreover,
T^|w|(x)-x = (A_w/D_w - 1)(x-p_w).
Therefore, when D_w>A_w, the word is contracting and
T^|w|(x)<x ⇔ x>p_w.
The same p_w is simultaneously the affine fixed point,
the candidate cycle value, the exact descent threshold in the
contracting case, and the 2-adic center of the
cylinder:
E_w closure = p_w + D_w Z_2.
Indeed, modulo D_w, D_w-A_w≡-A_w, so
p_w gives the same residue class as the cylinder
congruence.
Exact Cycle Criterion
If a finite word w generates a positive cycle, then
for some x>0
(A_w x + B_w)/D_w = x,
hence
(D_w-A_w)x = B_w.
Thus w generates a positive cycle if and only if
D_w>A_w,
D_w-A_w divides B_w,
x_w=B_w/(D_w-A_w) is a positive integer in E_w.
The candidate is unique. The trivial cycle corresponds to
O1, where A=3, B=1,
D=4, and x=1.
Cyclic rotations sharpen the loop obstruction. If w
generates a cycle, then each cyclic rotation generates the same
cycle starting from another point. Since A_w and
D_w depend only on symbol counts, the common modulus
M_w = D_w-A_w
must divide the affine numerator B of every cyclic
rotation. Equivalently, a cycle must solve the local linear system
O1: 4x_{i+1}-3x_i=1,
E: 2x_{i+1}-3x_i=0,
O3: 4x_{i+1}-x_i=1.
The absence of nontrivial loops can therefore be attacked as a finite simultaneous-divisibility problem over primitive cyclic words.
Exact Stopping Criterion
For a seed x, let w_k be the word traced
by its first k steps. The seed has not descended by
time k exactly when
A_{w_k}x+B_{w_k} ≥ D_{w_k}x.
If D_{w_k}≤A_{w_k}, this inequality is automatic. If
D_{w_k}>A_{w_k}, it is equivalent to
x ≤ B_{w_k}/(D_{w_k}-A_{w_k}) = p_{w_k}.
Therefore a seed has infinite stopping time if and only if for
every prefix w_k, either the prefix is not contracting
or the seed lies at or below its affine descent threshold.
For uniform cylinder certification, let m_w be the
least positive element of E_w. A contracting word kills
its whole cylinder precisely when
p_w < m_w.
This is the exact descent certificate used by the later graph and shielding theory.
Infinite Branches and Residual Minima
Let E_k be the set of seeds not yet certified as
descending in the first k steps. The Terras-type result
says δ(E_k)->0. The full conjecture would require
⋂_{k≥1} E_k = {1}.
Every infinite word gives a compatible sequence of congruences
x≡a_{w_k} mod D_{w_k}, hence a unique
2-adic candidate. But a 2-adic candidate
is an ordinary positive integer only if the least positive residues
of the finite cylinders stabilize to a finite value.
This produces the residual-minimum machine. Let
m_w=min(E_w∩N_{>0}) and
q_w=T^|w|(m_w). For any seed in the same cylinder,
x = m_w + tD_w,
T^|w|(x)=q_w+tA_w.
To append a letter, choose the least nonnegative lift
t that puts q_w+tA_w in the branch class
of the next symbol. Then
m_{ws}=m_w+tD_w.
Thus an infinite word represents a positive integer exactly when
these residual jumps are eventually zero. One possible divergence
program is to show that every surviving infinite branch except
O1^∞ has infinitely many nonzero residual jumps, or
else eventually contains a certifying contracting prefix.
2-Adic Inputs and 3-Adic Outputs
Input cylinders are governed by powers of 2, but their
images satisfy congruences modulo powers of 3. If
y=T^|w|(x), then
x = (D_w y - B_w)/A_w.
Since A_w is a power of 3, outputs satisfy
y ≡ B_w/D_w mod A_w.
Put ρ_w=B_w/D_w∈Z_3. The three branches evolve this
3-adic offset by
O1: ρ -> (3/4)ρ + 1/4,
E: ρ -> (3/2)ρ,
O3: ρ -> (1/4)ρ + 1/4.
This explains why a purely 2-adic cylinder argument
reaches the Terras/Korec barrier but does not cross it. To go
beyond density-zero statements, one needs statistical transport
between the 2-adic input cylinders and the
3-adic output constraints.
Core Established Facts
The stable core of the reformulation proves the following facts:
the three-branch map is equivalent to Collatz; every finite word
gives an exact affine map; every word gives one residual cylinder
modulo D_w; D_w is both cylinder modulus
and reciprocal density; the natural symbol weights are
1/4,1/2,1/4; the mean log multiplier is negative; the
center p_w unifies fixed point, descent threshold,
cycle candidate, and 2-adic center; loops reduce to
divisibility plus itinerary validity; infinite branches reduce to
2-adic candidates plus stabilization of least positive
residues.
Selected Extensions
The remaining notes contain many candidate lemmas, experimental reductions, and proof attempts. The extensions below are the theorem-level pieces that seem worth preserving in a public draft: they have clean hypotheses, short proofs or proof skeletons, and a structural role in the program. They still do not constitute a proof of Collatz.
Certificate Graphs, Ranks, and Prefix Codes
No closed certifying walk. For a cyclic word
w, build the directed graph on cyclic positions with an
arrow i -> j whenever the factor w[i,j) is
nontrivially certifying. If w were a nontrivial positive
cycle, this graph would be acyclic.
Proof. A certifying arrow sends the cycle value at
its initial vertex to a strictly smaller value at its terminal
vertex. A directed cycle of such arrows would give a strict chain
returning to its starting value, hence x < x.
This single observation yields several hard constraints: every
hypothetical nontrivial cycle has a critical rotation with no
outgoing certifying proper prefix; at most n(n-1)/2 of
its n(n-1) proper cyclic factors can be certifying; and
there is a rank function R on the cyclic positions such
that every certifying interval points downward in rank.
Minimal certificates form a prefix code. If
M is the set of certifying words with no certifying
proper prefix, then no element of M is a proper prefix
of another. Therefore the associated cylinders are disjoint and
Σ_{m∈M} 1/D_m ≤ 1.
This reframes finite stopping as a completeness problem for a
prefix code in the natural 2-adic itinerary measure.
Escape, Shielding, and Local Defects
One-letter escape. Odd extensions preserve
certification: if w is strictly certifying, then
wO1 and wO3 are strictly certifying, with
the usual caveat around the trivial fixed point. Thus the only
one-letter extension that can destroy certification is
E.
The even escape has an exact test. Let A=A_w,
D=D_w, let m be the least positive seed in
the cylinder of w, let y=F_w(m), and set
g=m-y>0. If t∈{0,1} is chosen so that
y+tA is even, then
wE is certifying ⇔ 3g > m + t(3A - 2D).
Proof. The lifted seed is m+tD and the
lifted image is 3(y+tA)/2. Subtracting these two
quantities gives the displayed inequality.
Exact shielding. For an odd-only tail
z, write F_z(x)=λ_zx+β_z, and let
Y*_{z,r} be the least value after z that
is divisible by 2^r. Then zE^r is
nontrivially certifying exactly when
λ_z(3/2)^r < 1,
[1 - λ_z(3/2)^r]Y*_{z,r} > β_z.
The proof changes variables from the input x to the
post-tail value y=F_z(x). The descent inequality becomes
a linear inequality in y, so it is necessary and
sufficient to test the least admissible y.
Define σ(z) as the largest even burst absorbed by
z. Two explicit shields are solid:
σ(O3)=3,
σ(O1^h)=floor(h log(4/3)/log(3/2)).
Terminal suffixes are monotone: σ(az)≥σ(z) for
odd-only a,z. Hence a shielding dictionary can be built
from odd suffixes. Using only (O3,3), every nontrivial
positive cycle must contain O1E or E^4;
otherwise every even burst is shielded and the cyclic word
decomposes into certifying blocks.
For each r≥1, the unshielded-tail set
U_r={z: zE^r is not certifying} is finite. One explicit
universal bound is
|z| < ceil((r log(3/2) - log(1 - 2^-r)) / log(4/3)).
Consequently every nontrivial positive cycle contains at least one
local defect zE^r with z∈U_r. Dually, a
forward compensation theory studies E^a z, produces
finite forward-defect sets, and shows that the rotation at the
minimum must begin with E^a z where this first forward
block is not certifying.
Defects, Arcs, and Feedback
Defect-feedback theorem. Let H(W) be
a directed auxiliary graph whose arrows represent factors that
would be certifying unless they cross a defect set D.
If W is a nontrivial positive cycle and every clean
arrow is genuinely certifying, then the clean graph is acyclic.
Equivalently, D intersects every directed circuit in
H(W).
Proof. A clean directed circuit would be a closed
certifying walk. The no-return theorem then gives the contradiction.
In the vertex-local case, this says that D is an
ordinary feedback vertex set. Therefore
|D|≥τ(H(W)), and with weights
Σ_{d∈D}ω(d)≥τ_ω(H(W)).
A key boundary result prevents overclaiming: in the pure interval model, one defect already linearizes every clean interval graph. Quantitative lower bounds require a stricter endpoint-local or vertex-local model.
In the endpoint-local circulant graph C_n^(L), with
arrows i -> i+ℓ for 1≤ℓ≤L, the feedback
sets are exactly the sets containing L consecutive
vertices. Hence
τ(C_n^(L)) = L.
More generally, for C_n(S), every certified length
s∈S forces defects in every orbit of
i -> i+s, so |D|≥gcd(n,s). If
[1,L]⊂S, a surviving cycle must contain a wall of
L consecutive defects; if S⊂[1,M], a wall
of M defects is sufficient to cut all endpoint-local
arcs.
There is also an exact arc-cover transfer. Given a map
Γ:E(H)->2^X assigning to each arc the defects that can
explain its failure, a defect set D⊂X is a
Γ-feedback set if and only if the covered arcs
B_Γ(D) contain a feedback arc set of H.
If each defect covers at most Δ relevant arcs, then
τ_Γ(H) ≥ ceil(τ_arc(H)/Δ).
For translational templates in a circulant graph, the same counting
becomes geometric. If U_Γ(σ) is the shadow of a cyclic
template σ, then every feedback set must satisfy
|D| · |U_Γ(σ)| ≥ n.
The exact condition for covering all translates is
D - U_Γ(σ)=Z/nZ. Thus the true obstruction is not only
the size of the shadow, but its cyclic covering number.
Pressure, Automata, and Thin Survivors
Pressure-dimension criterion. For a language
L, define
Z_n(s;L)=Σ_{w∈L_n}D_w^-s,
P_L(s)=limsup n^-1 log Z_n(s;L).
If P_L(s)<0, then the associated
2-adic survivor has zero s-dimensional
Hausdorff measure and Hausdorff dimension at most s.
The proof is the direct cylinder cover: at level n,
the survivor is covered by cylinders of diameter D_w^-1.
First-passage languages admit a uniform Chernoff bound. If every
word in F_τ(n) has its length n-1 prefix
multiplier at least τ, then for t≥0 and
s>0,
Σ_{v∈F_τ(n)}D_v^-t ≤ τ^-s(2^-t+2·4^-t)Φ_t(s)^(n-1),
Φ_t(s)=2^-t(3/2)^s+4^-t(3/4)^s+4^-t(1/4)^s.
In particular,
|F_τ(n)| ≤ C_τ ρ_top^n, ρ_top≈2.4986862226<3,
Σ_{v∈F_τ(n)}1/D_v ≤ C*_τ ρ_wt^n, ρ_wt≈0.9536553832<1.
A stronger dimensional statement is also available: there is an
explicit universal threshold t_fp≈0.9499843135<1 such
that every survivor covered at all depths by weak first-passage
containers has Hausdorff dimension at most t_fp.
Finite-state containers make this computable. For a finite automaton
with edge labels in {E,O1,O3}, form the weighted matrix
M_t with edge weight δ(label)^-t, where
δ(E)=2 and δ(O1)=δ(O3)=4. The pressure is
the logarithm of the maximum spectral radius among relevant
strongly connected components. Thus ρ_rel(t)<1
implies a Hausdorff bound dim_H≤t.
Finite forbidden dictionaries give a concrete special case: avoiding a finite set of certifying factors is a subshift of finite type recognized by a sliding-window automaton. Adding forbidden certificates can only decrease pressure.
For the run-limited language R_N, where no more than
N odd letters occur consecutively, the weighted spectral
radius ρ_N(t) is the unique positive root of
x^(N+1)=e_t(x^N+o_t x^(N-1)+...+o_t^N),
e_t=2^-t, o_t=2·4^-t.
It satisfies the exact gap identity
(e_t+o_t)-ρ_N(t)=e_t(o_t/ρ_N(t))^(N+1).
Hence every uniform bound on odd-run length automatically yields
reduced entropy, exponentially small natural cylinder mass, and a
Hausdorff dimension bound below 1.
Lassos, Complexity, and Active Budgets
Finite-state symbolic evidence must be filtered arithmetically. A
lasso u v^∞ always determines a rational
2-adic point
x_{u v∞}=(D_u p_v - B_u)/A_u,
p_v=B_v/(D_v-A_v).
It is realized by a positive integer if and only if
p_v is a positive integer and the displayed preimage is
a positive integer. Expansive periods cannot be natural, since then
D_v-A_v<0 and p_v≤0. Thus an eventually
periodic natural orbit must enter a positive cycle.
Morse-Hedlund gives the symbolic sieve: if a natural itinerary
ω has factor complexity p_ω(n)≤n for some
n, then ω is eventually periodic and is
governed by the lasso criterion. Therefore any aperiodic natural
survivor must satisfy
p_ω(n) ≥ n+1 for every n≥1.
There are stronger restrictions when the orbit grows slowly. Let
P_n be the number of active branches
E or O1 among the first n
steps. Decomposing the prefix into O3-runs after the
j-th active step gives the local active budget
n ≤ P_n+(P_n+1)a_x + (β/2)P_n(P_n+1),
β=log_4(3/2), a_x=log_4(3x/2).
Hence
P_n ≥ sqrt(2n/β)-O_x(1).
If a natural orbit is aperiodic and polynomially bounded, its
ordinary factor complexity must satisfy
p_ω(L)≥exp(c sqrt L) along all large scales. With
bounded O3-runs this improves to exponential
complexity. Thus very low-complexity symbolic containers cannot
contain polynomially bounded aperiodic natural survivors.
Tail Deficit and Extremal Gates
For odd tails, a useful order-sensitive coordinate is the deficit
mass. If Z is an odd-only word and
η_Z its multiplier, define
Δ(Z)=Σ_{j:s_j=O3} Π_{i>j}λ_{s_i}.
Then
1-F_Z(1)=Δ(Z)/2,
1-p_Z=Δ(Z)/(2(1-η_Z)).
The local exchange O1O3 -> O3O1 decreases
Δ by exactly λ_B/4, where
B is the suffix after the exchanged pair. Therefore,
at fixed counts (h,ℓ), the extremal orders are
O1^h O3^ℓ and O3^ℓ O1^h.
For blocks U=E^aZ, the rough floor-certification test
has the exact deficit-mass form:
U certifies by the rough test ⇔
Δ(Z) > 2((3^a-1)η_Z-(2^a-1)).
This recovers the universal residual strip and refines it by an
order constraint: a noncertified odd tail inside the strip must have
exceptionally small deficit mass, forcing the O3
letters far to the left and a long terminal run of O1.
The extremal tail
X_{a,h,ℓ}=E^a O3^ℓ O1^h is especially explicit. Its
cylinder has least positive seed
m_X=2^a ρ_{a,h,ℓ},
ρ_{a,h,ℓ}=res_{4^(h+ℓ)}(3^-(a+1)(1+2·4^ℓ)).
Its affine center is
p_X = [1 - (3/4)^h(2+4^-ℓ)/3] / [1 - (3/2)^a(3/4)^h4^-ℓ].
Therefore exact certification is the concrete inequality
m_X>p_X. If such an extremal block is contracting
and still not certified, then for fixed a,ℓ the
parameter h is unique:
h = floor(H_-(a,ℓ))+1,
H_+(a,ℓ)-H_-(a,ℓ) ≤ 1.
Noncertified extremal blocks also pass through finite arithmetic
gates. A small inverse L must satisfy
3^(a+1)L≡1 mod 4^ℓ. Since the subgroup generated by
3 modulo 4^ℓ is exactly the set of units
congruent to 1 or 3 mod 8 for
ℓ≥2, this becomes a discrete-log gate:
a+1 ≡ ι_ℓ(L) mod ord_{4^ℓ}(3),
ord_{4^ℓ}(3)=2^(2ℓ-2) for ℓ≥2.
In the high-threshold regime, the same obstruction can be converted
into a continued-fraction gate. If a non-initial upper convergent
P/Q of log_2 3 is compatible, then
Q≤(5/2)(a-2), the dilation lies in a finite interval,
and the next partial quotient must exhibit an exponential spike:
a_next(P/Q)+2 > (2 log 2 / 5)(T2^a - 1).
Finally, the autonomous residual-strip program reduces a remaining zero-margin case to
(2^a-1)/(3^a-1) ≤ 3^h/4^(h+q) < (2/3)^a.
The notes prove the reduction to this strip, classify the cases
a≤6, and reduce a≥7 to convergents of
log_2 3. The final infinite exclusion is kept as an
open residual-strip sublemma, not as a proved theorem.
Affine Geometry and Quasi-Neutral Cycles
The affine-center geometry supplies restrictions independent of
congruence arithmetic. If a cycle is decomposed into blocks
U_i with centers p_i and multipliers
λ_i, then each block acts as a homothety
x -> p_i+λ_i(x-p_i). The center p_i never
lies between the incoming and outgoing boundary points. For a
contracting block it lies on the side toward which the point moves;
for an expanding block it lies on the opposite side.
For mixed decompositions with total multiplier Λ<1,
the barycentric formula becomes signed. In every rotation,
x_j = c_j + B_j(c_j-e_j),
where c_j is a convex barycenter of contracting centers,
e_j is a convex barycenter of expanding centers, and
B_j is the positive mass of the expanding part. Thus if
c_j≤1, a nontrivial positive cycle requires
e_j<c_j and
B_j > (1-c_j)/(c_j-e_j).
Expanding blocks are therefore not just allowed noise; they must provide signed leverage in every critical rotation.
A complementary cycle tradeoff says that if w is a
positive cycle word with total multiplier Λ<1, minimum
value m, and o(w) offset-producing letters
O1,O3, then
1-Λ ≤ o(w)/(4m).
The proof chooses a cyclic first-passage rotation and bounds its
affine offset by o(w)/4. Large-minimum cycles are forced
to be quasi-neutral, which places their return words in the thin
first-passage containers described above.
What Is Not Claimed
The framework does not prove Collatz, does not prove the absence of nontrivial loops, does not prove the absence of divergent orbits, and does not yet reproduce Tao's almost-everywhere theorem. It also does not turn every pressure or Hausdorff-dimension estimate into an emptiness statement for positive integers.
Some names that occur in the wider notes should remain outside a public claim for now: the global residual-strip sublemma, the general Lemma M, Lemma ML, Lemma BLD, the Exchange Lemma, count-sensitive shielding, and finite defect-divisibility obstructions. The notes also explicitly record a false Lemma G; it should be mentioned only as a failed attempt that led to residual-affine compensation, not as a result.
Research Program
The clean program has three layers.
- First, keep the core affine-residual framework as the stable basis: words, affine coefficients, congruence cylinders, centers, cycle criteria, and stopping certificates.
- Second, develop the certificate theory separately: certifying graphs, shielding dictionaries, defects, feedback sets, and finite-state containers.
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Third, attack the global gap: either cover all cycle languages by
certificates and defects, close the residual strip via a
Diophantine estimate for
Θ, or prove a genuine3-adic transport theorem for live cylinders.
Conclusion
The affine-residual reformulation organizes Collatz into the chain:
word -> affine map -> congruence cylinder -> 2-adic ball -> center/threshold -> certificate.
Its conceptual strength is that descent becomes a property of cylinders and words, then of languages, graphs, and containers. The strongest later developments do not replace the core paper; they suggest a second layer of certificate combinatorics and Diophantine gates around the remaining exceptional cases.
Generative AI Use Statement
During the preparation of this draft, the author used OpenAI ChatGPT/Codex, based on a GPT-5 system available in June 2026, for author-directed assisted mathematical exploration, generation of candidate formulations, structural organization, terminology normalization, and expository revision. Some lemmas, conjectures, and draft arguments emerged from interactions with the system and were subsequently reworked within the formalism of the manuscript.
The AI system is not an author of the manuscript and is not cited as a primary mathematical source. Before any submission, the author must independently verify every statement, proof, reference, and calculation, and assumes full responsibility for the final text.