The Kumar-Courtade problem. The discrete cube Omega_n = {-1,+1}^n carries
uniform measure. A sign function u : Omega_n -> [-1,+1] is the conditional
mean of a Boolean label U, so u(x) = +1 forces U = +1, u(x) = -1 forces
U = -1, and u(x) = 0 leaves U fair. Boolean labels f : Omega_n ->
{-1,+1} are the special case u in {-1,+1}. Every coordinate passes
independently through a binary symmetric channel B_i = X_i Z_i, where Z_i
is a fresh sign independent of everything with E Z_i = rho in [-1,+1].
The receiver sees B = (B_1,...,B_n) and forms the posterior mean
T_rho u(B) = E[u(X) | B]. The information carried by U about B is
I(U; B). The conjecture of Kumar and Courtade, from 2013 and 2014,
asserts that at every rho and every n a single coordinate u(x) = x_i
carries at least as much information about B as any other Boolean u.
./courtade-kumar-2014.pdf
Entropy and information. Put L = log 2 with log = log_e. A sign S equal
to +1 with probability (1+r)/2 has entropy
H(r) = -(1+r)/2 log((1+r)/2) - (1-r)/2 log((1-r)/2),
so H(+1) = H(-1) = 0 and H(0) = L. The information of S about its own
sign, when only its mean r is known, is
psi(r) = L - H(r),
which rises on [0,1] from 0 to L with psi'(r) = atanh(r) = V(r) and
psi''(r) = 1/(1-r^2). For a soft field u and any sign U with conditional
mean u(X), the mutual information between U and X is
I(u) = H(E u) - E H(u) = E psi(u) - psi(E u).
Under the channel the conditional mean of U given Y is T_rho u(Y), so
I(U;Y) = I(T_rho u). The quantity to compare with a coordinate is
therefore I(T_rho u).
Sharp contraction theorem. For every finite n >= 0, every
u : Omega_n -> [-1,+1], and every rho in [-1,+1],
I(T_rho u) <= psi( |rho| psi^{-1}( I(u) ) ).
The symbol psi^{-1} : [0,L] -> [0,1] is the increasing inverse. Equality
holds at every n >= 1 for every soft coordinate u(x) = a x_i with
|a| <= 1. The parameter r = psi^{-1}(I(u)) is the amplitude of the fair
soft coordinate carrying the same information as u; the theorem says
this effective amplitude shrinks under noise at least as fast as |rho|.
Courtade-Kumar corollary. For every Boolean f : Omega_n -> {-1,+1},
write m = E f and r_m = psi^{-1}(H(m)). Then
I(f(X); Y) <= psi(|rho| r_m) <= psi(|rho|).
For signed coordinates m = 0 and both inequalities are equalities at
every rho. For 0 < |m| < 1 and 0 < |rho| <= 1, the refined upper curve
psi(|rho| r_m) sits strictly below the universal psi(|rho|). In bits,
the universal inequality reads I_2(f(X);Y) <= 1 - h_2(eps) with
eps = (1-|rho|)/2, which is the original Kumar-Courtade statement.
Prior work. Samorodnitsky proved the Boolean conjecture in a
dimension-independent high-noise range and Ordentlich, Shayevitz and
Weinstein obtained asymptotic high-noise bounds for balanced Boolean
functions.
https://doi.org/10.1109/TIT.2016.2584625
https://doi.org/10.1109/ISIT.2016.7541349
Yu enlarged the proved range for balanced functions by local expansion
and Phi-stability.
./yu-2023.pdf
Nearby formulations. Pichler, Piantanida and Matz prove that dictator
functions maximize mutual information when the observation is also
reduced to one Boolean output. Li and Medard identify lexicographic
optimizers at low noise and relate high-noise optimizers to first-level
Fourier weight, in dimension-dependent ranges. Barnes and Ozgur show the
balanced conjecture is equivalent to a symmetrized Li-Medard moment
inequality. Kindler, O'Donnell and Witmer prove fixed-mean Gaussian and
spherical analogues. Javanmard and Woodruff extend the coordinatewise
sum bound to arbitrary output bias and refine high-noise entropy
estimates. These fixed-mean questions are distinct from attainability of
the refined upper curve in the corollary.
The differential approach used below originates with Chen, Gohari and
Nair, who reduce the global statement to a dimension-free local
entropy-production inequality and integrate along the noise semigroup.
./chen-gohari-nair-2025.pdf
In 2026 several independent proofs of the full Boolean conjecture
appeared: Chen, Gohari, Javanmard, Lin, Mirrokni, Nair and Woodruff via
a computer-assisted Bellman inequality; Ky and Tran via entropy
production and spectral estimates; Mahdavifar and Beirami via entropy
flow and pivotal sets; and a finite-noise Boolean bound retaining the
output bias is due to Kramer and Saglam.
./chen-et-al-2026.pdf
./kramer-saglam-2026.pdf
From information loss to a production inequality. The proof turns the
contraction into a differential inequality. For an interior soft field
g : Omega_n -> (-1,1) define
m = E g, e = E H(g), I(g) = H(m) - e,
D(g) = E[ V(g) N g ],
where N = (1/2) sum_i (Id - flip_i) is the coordinate-summed flip
generator and flip_i flips coordinate i. The noise semigroup is
P_t = T_{e^{-t}} = e^{-tN},
because a Walsh character chi_A(x) = prod_{i in A} x_i satisfies
N chi_A = |A| chi_A, so P_t chi_A = e^{-t|A|} chi_A. The mean is conserved
under P_t and differentiation gives
(d/dt) I(P_t g) = -D(P_t g) for every t >= 0,
so D is the instantaneous entropy-production rate of the flow. For the
soft coordinate g(x) = r x_i one computes D = r V(r). This pins the
coordinate rate F : [0,L) -> [0,infinity) by
F(psi(r)) = r V(r) for 0 <= r < 1,
with F(0) = 0 by continuity.
The linear baseline. Two classical objects determine the shape of the
rate. The scalar curve r V(r) is the one-bit entropy deficit of Mrs.
Gerber's lemma (Wyner and Ziv, 1973), which gives the entropy of a noisy
one-bit channel. The linear part 2I reflects the cube logarithmic
Sobolev inequality: the hypercontractive framework of Bonami, Gross and
Beckner yields D(g) >= 2 I(g) for every soft field, and the modified
logarithmic Sobolev inequality of Bobkov and Tetali places the same
estimate in the discrete functional-inequality setting. The surplus
S(I) = F(I) - 2I is what those linear estimates miss, and the local
joining machinery supplies it.
Reduction of the theorem to a single inequality. Suppose every interior
g on every cube satisfies
D(g) >= F(I(g)).
Let I(s) = I(P_s g). For s > 0 every value of P_s g lies in (-1,1) and
I(s) lies in (0,L). The rate identity then gives
I'(s) = -D(P_s g) <= -F(I(s)).
Setting r(s) = psi^{-1}(I(s)) turns this into r'(s) <= -r(s), because
F(psi(r)) = r V(r) and psi'(r) = V(r). Integrating from a small s = eps
to s = t, and letting eps -> 0, gives
psi^{-1}( I(T_{e^{-t}} g) ) <= e^{-t} psi^{-1}( I(g) ).
Taking rho = e^{-t} in (0,1], negative rho via T_{-rho} u(x) =
T_rho u(-x), and boundary-valued inputs by approximation, yields the
sharp contraction theorem. For Boolean f one has H(f) = 0 everywhere,
so I(f) = H(m) and the CK corollary follows directly.
Scalar facts about the rate. The whole proof rests on the following
elementary properties of F. Integrating psi'' = 1/(1-r^2) term by term
gives
psi(r) = sum_{n>=1} r^{2n} / (2n (2n-1)),
so psi(r) >= r^2/2 and psi(r) <= L r^2 on [-1,1], and psi(r)/r^2 is
strictly increasing. The rate is then smooth on (0,L) with
F(0) = 0, F'(0) = 2, F''(0) = 4/3,
F''(I) >= 4/3, F'''(I) >= 0 for all I in [0,L).
To see this, write r = tanh(v). Then
F'(I) = 1 + sinh(2v)/(2v),
F''(psi(r)) = cosh^2(v) A(v), A(v) = 2 int_0^1 (1-s^2) cosh(2 v s) ds.
The integral A(v) is positive, strictly increasing in v, and has
A(0) = 4/3, which gives both F'' >= 4/3 and F''' > 0 for I > 0.
The same substitution yields three further bounds. First, the
log-derivative of the rate curvature satisfies
(d/dI) log F''(I) > 2 on (0,L),
equivalently I |-> e^{-2I} F''(I) is strictly increasing on [0,L).
Second, the rescaled curvature
P(h) = h^2 F''(L-h), 0 < h <= L,
satisfies
1/2 <= P(h) <= M := 4L^2/3, P(L) = M.
Third, the numerical constants obey
2/3 < L < 25/36 < 7/10, 1/2 < M < 2/3, 2 > 3M.
The margin 2 > 3M is the numerical coincidence that closes the thinning
estimate below.
Coordinate split and joining cost. The single inequality D(g) >= F(I(g))
is proved by induction on n. Fix an interior soft field g on Omega_n and
split along the last coordinate:
g_+ = g restricted to the half x_n = +1,
g_- = g restricted to the half x_n = -1.
Both live on Omega_{n-1}. Write
a = E g_+, b = E g_-, m = (a+b)/2, delta = (a-b)/2,
e_+ = E H(g_+), e_- = E H(g_-), e = (e_+ + e_-)/2,
I_+ = H(a) - e_+, I_- = H(b) - e_-.
The parent information is I(g) = H(m) - e. The generator N splits as
N' + (1/2)(Id - flip_n), where N' acts on the first n-1 coordinates.
Computing D(g) = E[V(g) N g] gives the exact identity
D(g) = (D(g_+) + D(g_-))/2 + E[ c(g_+(X'), g_-(X')) ],
where X' is uniform on Omega_{n-1} and
c(a,b) = (a-b)/4 * (V(a) - V(b))
is the joining cost of the two faces.
Properties of the joining cost. The function c on [-1,+1]^2 is symmetric,
nonnegative, jointly convex, and vanishes at the equal corners (1,1) and
(-1,-1). At the mixed corners (+1,-1) and (-1,+1) it takes the value
+infinity. Convexity follows from
c(a,b) = (1/8)[ J(1+a,1+b) + J(1-a,1-b) ], J(s,t) = (s-t) log(s/t),
and J is the perspective of p |-> p log p, hence convex.
The local joining problem. Induction reduces the global question to a
local one at each coordinate split. Given child means a = m + delta and
b = m - delta, what is the minimum joining cost over all joint laws
(A,B) on [-1,+1]^2 with E A = a, E B = b, and average entropy
E[H(A)+H(B)]/2 at most e? Write
K(m, delta, e) = inf { E c(A,B) : E A = a, E B = b,
E H(A) + E H(B) <= 2e }.
The infimum runs over laws on [-1,+1]^2, not only the physical faces of
the cube. This relaxation is legitimate: only a lower bound on the
actual joining cost is needed.
Pricing entropy. Attach a price p >= 0 to entropy and minimize
E[c(A,B)] + p [E H(A) + E H(B)]
subject only to E A = a and E B = b. Standard convex duality then inverts
the budget: for every e in (0, e_max] some price p >= 0 makes the priced
optimizer hit the budget e exactly, where e_max = (H(a)+H(b))/2.
The three-point optimizer. At every fixed p > 0 the priced minimizer is
a three-point law
lambda * delta_{(x,y)} + pi_+ * delta_{(1,1)} + pi_- * delta_{(-1,-1)},
with -1 < y < x < 1. The equal corners contribute neither cost nor
entropy, since H(+-1) = 0, so the priced value is
lambda [ c(x,y) + p (H(x) + H(y)) ].
Change of variables. Set
z = log((1+x)/(1+y)), w = log((1-y)/(1-x)),
u = 1/(e^z - 1), v = 1/(e^w - 1), n = 1 + u + v.
Then
x = (1+u-v)/n, y = (u-v-1)/n, x - y = 2/n,
lambda = delta n, pi_+ = (1+b-2 delta u)/2,
pi_- = (1-a-2 delta v)/2,
and pi_+ >= 0, pi_- >= 0 become floors
z >= z_0 := log((1+a)/(1+b)), w >= w_0 := log((1-b)/(1-a)).
The priced objective equals 2 delta F_p(z,w) with
F_p(z,w) = (z+w)/8 + p H_norm(z,w),
H_norm(z,w) = (H(x) + H(y)) / (x - y).
Convexity. The normalized entropy has the double-series representation
H_norm(z,w) = (q(z)+q(w))/2
+ sum_{i,j>=1} e^{-iz-jw} / max(i,j),
q(t) = t/(e^t - 1) - log(1 - e^{-t}),
which is a sum of convex terms with positive coefficients, so H_norm is
strictly convex and smooth on (0,infinity)^2. Hence F_p is strictly
convex, and its minimization over the floored quadrant has a unique
solution at every p >= 0.
Support certificate. At the minimizer (z*,w*), with u, v evaluated
there, define
gamma_+ = (F_p)_z / (2u(u+1)), gamma_- = (F_p)_w / (2v(v+1)).
Both multipliers are nonnegative; a derivative vanishes exactly when its
floor is active. The function
Q(z,w) = F_p(z,w) + gamma_+(1+2u) + gamma_-(1+2v)
has zero gradient at (z*,w*) and is strictly convex on the whole
quadrant; let kappa > 0 be its minimum. Multiplying Q >= kappa by
x - y > 0 and using (1+2u)(x-y) = 2+x+y and the analogous identity for
v gives the pointwise bound
c(x,y) + p[H(x)+H(y)] - kappa(x-y)
+ gamma_+(2+x+y) + gamma_-(2-x-y) >= 0.
Averaging against any law with the prescribed means,
E c(A,B) + p[EH(A) + EH(B)] >= 2 delta kappa + 2 gamma_+ + 2 gamma_-,
and the three-point law attains equality. At every p the unique
optimizer is exactly this three-point law, with the corner masses
satisfying the complementarity gamma_+ pi_+ = gamma_- pi_- = 0.
Coverage: from price to budget. Let e(p) be the average entropy of the
unique priced optimizer. It is continuous in p >= 0, since the
minimizers lie in a common compact subset of the floored quadrant
because F_p >= (z+w)/8. At p = 0 the optimizer is the deterministic pair
(a,b), so e(0) = e_max. As p -> infinity, comparison with a reflected
core (r,-r) of mass delta/r and r near one shows e(p) -> 0. The
intermediate value theorem then gives K(m,delta,e) = 2 delta F_p(z*,w*)
at the price p = p(e) with e(p) = e, and the same affine certificate
bounds every law feasible at budget e from below by that value.
The two canonical geometries. By symmetry assume m >= 0, so
z_0 <= w_0. At any strict minimizer z <= w. If both w > w_0 and z < w,
then (F_p)_w = 0 while strict convexity and symmetry give
(F_p)_z < 0, contradicting constrained optimality. Hence either z = w
or w = w_0.
Reflected geometry (z = w). Writing x = -y = r with 0 <= r < 1 gives
lambda = delta/r, e = (delta/r) H(r), K = delta V(r),
0 <= m <= 1 - delta/r.
As r ranges in [delta/(1-m), 1), the budget e ranges over (0, e_0] and
the cost K over [K_0, infinity), where
e_0 = (1-m) H(delta/(1-m)), K_0 = (1-m) V(delta/(1-m)).
One-corner geometry (w = w_0, so pi_- = 0). Put k = lambda. Then
x = 1 - (1-a)/k, y = 1 - (1-b)/k,
e = (k/2)[H(x)+H(y)], K = k c(x,y), 1 - m <= k <= 1.
As k ranges in [1-m, 1], the budget e ranges over [e_0, e_max] and the
cost K over [K_0, c(a,b)].
Threshold and direction. The two geometries meet at k = 1 - m,
r = delta/(1-m), where e and K both equal e_0 and K_0. The one-corner
entropy is strictly increasing in k, with derivative
d/dk [(k/2)(H(x)+H(y))] = (1/2)[log(2/(1+x)) + log(2/(1+y))] > 0,
and the reflected entropy k H(delta/k) also increases strictly in k.
Hence e <= e_0 gives the reflected geometry with both equal corners,
and e >= e_0 gives the one-corner geometry with the positive corner
only. At e -> 0 the reflected radius r -> 1 and K -> infinity; at
e = e_max the law is deterministic with K = c(a,b). Smaller budget
forces a higher minimum cost.
Reflected-core motion. Fix a reflected core (r,-r) with mass k.
Increasing the parent mean m while keeping (r,k) fixed shrinks the
negative-corner mass pi_-; at m = 1 - k the negative corner disappears.
Throughout this motion the core (r,-r), its mass k, the average entropy
e = k H(r), and the cost delta V(r) stay fixed, and only the parent mean
m grows. The reflected optimizer at (m, delta, e) and the one-corner
optimizer at (1-k, delta, e) therefore have the same cost and the same
entropy. The comparison below exploits this: the payment demanded by
the strict profile increases when m grows at fixed cost and entropy.
The variance-refined one-bit inequality. The capacity comparison pays the
interior core of the canonical optimizer. Consider a single fair sign X
in {-1,+1} and a soft value
g(X) = m + delta X, |m| + delta < 1,
so g takes the two values m+delta and m-delta with probability 1/2 each
and has mean m. Write
J = (1/2)[psi(m+delta) + psi(m-delta)] - psi(m),
D = (delta/2)[V(m+delta) - V(m-delta)],
alpha = 1 - m^2, beta = alpha - delta^2 > 0.
The quantity J is I(g) = H(m) - (1/2)[H(m+delta) + H(m-delta)]. The
quantity D is the Dirichlet form of g, and it also equals the joining
cost c(m+delta, m-delta) of the deterministic pair.
Claim. D >= alpha F(J/alpha).
Proof. By reflection assume m >= 0. The case delta = 0 is trivial.
Set
w_+ = (1+m)/2, w_- = (1-m)/2,
r_+ = delta/(1+m), r_- = delta/(1-m),
i_+ = psi(r_+), i_- = psi(r_-).
The two posterior representations of the one-bit channel give
J = w_+ i_+ + w_- i_-, D = w_+ F(i_+) + w_- F(i_-).
Let
phi(s) = F(s)/s, phi(0) = 2,
the secant slope from the origin. Since F is strictly convex with
F(0) = 0 the slope is strictly increasing, and since F''' >= 0 the
slope is also convex, because
phi''(s) = (1/s^3) int_0^s t^2 F'''(t) dt >= 0.
Introduce size-biased weights px = w_+ i_+ / J, py = w_- i_- / J. These
sum to one, with mean information
s_* = (w_+ i_+^2 + w_- i_-^2) / J.
Convexity of phi and Jensen's inequality give
D/J = px phi(i_+) + py phi(i_-) >= phi(s_*).
It remains to show s_* >= J / alpha. The quotient psi(r)/r^2 increases
in r, so the ratio q = i_+ / i_- satisfies
q <= q_0 := ((1-m)/(1+m))^2.
The ratio
R(q) = (w_+ q^2 + w_-) / (w_+ q + w_-)^2
is strictly decreasing on [0,1], because
R'(q) = 2 w_+ w_- (q-1) / (w_+ q + w_-)^3 <= 0.
Substituting q_0 gives exactly
R(q_0) = (1 + 3m^2)/(1 - m^2) >= 1/(1-m^2) = 1/alpha.
Hence
s_* = R(q) J >= R(q_0) J >= J/alpha.
Monotonicity of phi then gives
D/J >= phi(s_*) >= phi(J/alpha) = alpha F(J/alpha)/J,
so D >= alpha F(J/alpha). This also shows s_* < L, so J/alpha < L: the
rate is always evaluated at a valid argument in [0,L).
Since F is convex with F(0) = 0, the ordinary one-bit bound follows:
c(m+delta, m-delta) = D >= alpha F(J/alpha) >= F(J).
The variance-refined form is what lets the child channels at m+delta
and m-delta pool the surplus without loss. The multiplier
alpha = 1 - m^2 is exactly the variance of the sign X.
The same estimate can be derived by differentiating at correlation one
the pointwise comparison of Kramer and Saglam;
./kramer-saglam-2026.pdf
the direct proof above, due to the size-biased argument, is the one used
below.
The reserve profile. Let S(I) = F(I) - 2I on [0,L) and S(I) = 0 for
I < 0. Write, for v, e > 0,
R(v,e) = v S(L - e/v),
a perspective of the surplus. For |m| < 1 and 0 <= I < H(m), put
alpha = 1 - m^2 and define
B_m(I) = 2I + R(alpha, H(m) - I).
This is the reserve profile. Equivalently
B_m(I) = inf { 2u + alpha F(v/alpha) : u+v = I,
0 <= u <= c_m, 0 <= v < alpha L },
c_m = H(m) - alpha L >= 0,
and F' >= 2 makes the optimum u = min(I, c_m). The construction therefore
allocates information between two capacities: the linear capacity c_m is
paid at rate 2, and the remaining capacity alpha L uses the exact
coordinate rate F. The quantity alpha = 1 - m^2 is a variance weight.
The same two coefficients alpha and c_m occur in the finite-noise Boolean
bound of Kramer and Saglam.
./kramer-saglam-2026.pdf
Child pooling. Consider a physical state (m, delta, e) with
|m| + delta < 1 and 0 < e <= h_bar := (H(m+delta)+H(m-delta))/2, and put
alpha = 1 - m^2, beta = alpha - delta^2,
J = H(m) - h_bar, j = h_bar - e,
a = m + delta, b = m - delta.
Average child information is j and average variance capacity is
(1-a^2 + 1-b^2)/2 = beta. Since R is a convex perspective,
(B_a(I_a) + B_b(I_b))/2 >= 2j + R(beta, e).
The parent reserve is B_m(I) = 2(J+j) + R(alpha, e), so the payment the
reserve needs from a join is at most
W(m, delta, e) = 2J + R(alpha, e) - R(beta, e).
Proving W <= K(m, delta, e) is enough: the canonical cost problem pays
the interior core through the one-bit inequality, and the next two
estimates show that the payment survives adding either common output.
A logarithmic bound for the corner geometry. For -1 < m < 1 and
0 <= delta < 1 - |m|, put
A = 1 - m, kappa = (1/2) log( 1 / (1 - delta^2/(1+m)^2) ).
Then
A^2 log(alpha/beta) <= 3 kappa.
The proof is elementary. The case delta = 0 is immediate. Otherwise, set
h = (1-m)/(1+m) > 0, t = delta^2/(1+m)^2 < min(1, h^2), f(t) = -log(1-t).
The ratio f(t/h)/f(t) decreases in t for h >= 1 and increases for h < 1;
its derivative is (1-t)/(h-t). For h >= 1 the ratio is at most 1/h, so
A^2 f(t/h)/f(t) <= A^2/h = 1 - m^2 <= 1. For 0 < h < 1 it suffices to
bound the ratio at t = h^2. The function G(h) = (2-h)f(h) - (2+h)log(1+h)
has G(0) = 0 and G'(h) > 0, so f(h^2)/f(h) >= 2h/(2+h) and
A^2 f(h)/f(h^2) <= 2h(2+h)/(1+h)^2 <= 3/2. Both cases give the bound.
Common-output contraction. For every physical state (m, delta, e) and
0 < tau <= 1,
W(1 - tau + tau m, tau delta, tau e) <= tau W(m, delta, e),
with the symmetric statement for the negative common output; there is no
sign restriction on the original mean. The positive-corner ray scales
A = 1 - m, delta, and e by the same factor. Hold delta/A and e/A fixed,
put eta = 1 + delta^2/A^2, and write the entropy intercept
gamma(h) = S(L-h) + h S'(L-h) = int_h^L P(u)/u du for 0 < h < L,
with gamma(h) = 0 for h >= L, where P(u) = u^2 F''(L-u). Since P <= M,
gamma(e/alpha) - gamma(e/beta) <= M log(alpha/beta).
Differentiation of the two perspectives in W gives the Euler identity
A (dW/dA) - W = 2 kappa + A^2 { eta gamma(e/beta) - gamma(e/alpha) }.
Because eta >= 1 and A^2 log(alpha/beta) <= 3 kappa,
A (dW/dA) - W >= 2 kappa - M A^2 log(alpha/beta) >= (2-3M) kappa >= 0.
So W/A is nondecreasing; comparing A and tau A proves the contraction.
The intermediate states stay physical by entropy concavity, and W is even
in m by reflection.
Local joining for the linear reserve. Let (A,B) have any joint law on
[-1,+1]^2 with positive average entropies. Put
a = E A, b = E B, m = (a+b)/2, e = (e_a+e_b)/2,
I = H(m) - e, I_a = H(a) - e_a, I_b = H(b) - e_b.
Then
E c(A,B) >= B_m(I) - (B_a(I_a) + B_b(I_b))/2.
First pay the deterministic capacity state: for any interior pair (x,y),
apply the one-bit inequality at (m_0, delta_0, e_0), and use
[L - e_0/alpha_0]_+ <= J_0/alpha_0 < L
to obtain
W(m_0, delta_0, e_0) <= alpha_0 F(J_0/alpha_0) <= c(x,y).
When delta > 0, the exact canonical optimizer is a core (x,y) of mass k
plus two corner masses. Starting from the core and adding the positive
corner, then the negative corner, the common-output contraction and the
capacity bound give
W(m, delta, e) <= (k+pi_+) (k/(k+pi_+)) W(m_0, delta_0, e_0)
<= k c(x,y) = K(m, delta, e).
For delta = 0, W = K = 0 directly. Combining with the child pooling bound
finishes the local joining inequality.
The pooling identity is exact. If e < beta L its minimum over the split
is at e_a/(1-a^2) = e_b/(1-b^2) = e/beta, and if e >= beta L both child
entropies can be allocated in [(1-a^2)L, H(a)] and [(1-b^2)L, H(b)] with
the prescribed average. The perspective R(beta, e) is the exact
common-budget value of the child reserve, not an approximation.
Thinning: adding a common output. The canonical optimizer with one
positive common corner has cost linear in the mass of its interior core.
The key estimate is that the rate increment paid by adding a common
output is at most the core's increment multiplied by that mass. Take an
interior ordered pair a_c > b_c with nonnegative center, and write
x_c = (a_c+b_c)/2 >= 0, d_c = (a_c-b_c)/2 > 0,
h_c = (H(a_c)+H(b_c))/2, J_c = H(x_c) - h_c.
For 0 < lambda <= 1, put mass lambda on (a_c,b_c) and mass 1-lambda on the
common output (1,1). The mixture has means x+d and x-d:
x = 1 - lambda(1-x_c), d = lambda d_c, e = lambda h_c.
Define
h_bar = (H(x+d)+H(x-d))/2, nu = h_bar - e,
J = H(x) - h_bar, I = nu + J = H(x) - e.
The average child information is nu and J is the entropy gained by
averaging the two child means. Since 0 < d_c < 1 - x_c, the mixture has
x >= 0 and 0 < d < 1 - x. Concavity gives nu >= 0 and J > 0, and e > 0
gives I < L.
Chain rule reading. Let X be a fair sign and B in {0,1} an independent
flag with Pr(B = 1) = lambda. Given X and B, draw a sign U with
E[U | X, B] = (1 - B) + B (x_c + d_c X).
Then B = 0 selects the common output and B = 1 selects the core, so
H(U) = H(x), H(U|X) = h_bar, H(U|X,B) = e. The chain rule reads
I = I(U; X, B), nu = I(U; B | X), J = I(U; X), I = nu + J,
so nu is the information about the flag that remains after the input is
revealed.
Payment theorem. For this core and mass,
F(nu + J) - F(nu) <= lambda F(J_c).
The proof shows that (F(nu+J) - F(nu))/lambda increases with lambda,
matching the mixture cost lambda c(a_c,b_c).
Path derivatives. For the state above, set
p = 1 + x, sigma = 1 - x, b = log(1+x),
q = e + psi(x) = L - I, s = e + L - h_bar = L - nu,
kappa = -(1/2) log( 1 - d^2/p^2 ) > 0,
so s - q = J and 0 < q < s <= L. Let Delta(lambda) = F(I) - F(nu) and
A = log(2/(1+x)). The entropy identity
t h_bin'(t) - h_bin(t) = log(1-t)
for h_bin(t) = H(1-2t), applied to I and to the two child expressions,
averaged, gives the two exact derivatives
lambda I' = I - A, lambda nu' = nu - A - kappa.
Substitution yields the Euler form
lambda Delta' - Delta = int_nu^I (u-A) F''(u) du + F'(nu) kappa.
The last term is at least 2 kappa; convexity makes the integrand
nonnegative where u >= A, and on the remaining part the curvature bound
F''(u) <= M/(L-u)^2 with h = L-u gives
lambda Delta' - Delta >= 2 kappa - M int_q^s (h-b)_+/h^2 dh.
It therefore suffices to bound this integral by 3 kappa. The strict
margin 2 > 3M then makes (Delta/lambda)' nonnegative.
A channel reading of kappa. In the channel above, the posterior of the
fair input at the positive output is
Pr(X = z | U = 1) = (1/2)(1 + z d/p),
so
kappa = (1/2) sum_{z=+-1} log( (1/2) / Pr(X=z | U=1) )
is the relative entropy from the uniform prior to this posterior. It is
an auxiliary path quantity, distinct from the canonical cost K.
An interval estimate with a logarithmic cutoff. Let 0 <= x < 1 and put
p = 1+x, sigma = 1-x, gamma = sigma/p, beta = log(1+x)/L.
Suppose 0 <= z < gamma and sigma - p z <= u <= 1 - z. For beta > 0 put
k_beta(v) = (v-beta)_+/v^2 on v > 0 with k_beta(0) = 0, and for beta = 0
put k_0(v) = 1/v. Then
int_u^{u+z} k_beta(v) dv <= -(3/2) log(1 - gamma z),
and for x > 0 the endpoint z = gamma is also allowed.
Proof sketch. First bound the cutoff from below: beta >= b_* := 4x/(3+x),
so 1 - beta <= 3 sigma/(3+x) <= (3/2) gamma. For this, the function
R(x) = (3+x) log(1+x)/x has R(x) >= R(1) = 4L, obtained by
differentiating once.
For an admissible interval with z < gamma, the quantity
a = (2-u)/(1+z) lies in [1,p]. The admissible family
u_a(s) = 2 - a(1+s), v_a(s) = u_a(s) + s, 0 <= s <= z
grows the interval from length zero to the target, so it is enough to
prove the derivative bound
D_a(s) := a k_beta(u_a(s)) - (a-1) k_beta(v_a(s)) <= (3 gamma)/(2(1-gamma s)).
The function k_beta increases then decreases, with minimum on [u,1] at an
endpoint. If k_beta(u) <= 1 - beta, then D_a <= 1 - beta and the bound is
immediate. Otherwise the algebra reduces to a cubic polynomial
P(t,u) >= 0 in t = b_*/u, 0 <= t <= 1/(1+u),
P(t,u) = 2 + C_1 t + C_2 t^2 + C_3 t^3,
C_1 = 3u^2 - u - 8, C_2 = 12 - 8u + 8u^2 - 6u^3,
C_3 = u(3-u)(2 - 3u^2),
which is nonnegative because 8(C_2 - u) - C_1^2 >= 4 and C_3 t >= -u.
Integrating from 0 to z gives the claimed interval bound.
The entropy window on the path. To apply the interval estimate to
[q, s], one needs a lower bound on q and an upper bound on the length
J = s - q. Both follow from the entropy of the oriented core. Write
h_bin(t) = H(1-2t), sigma_c = 1 - x_c, t = (1 - d_c/sigma_c)/2 in (0,1/2),
and note that the two negative-output probabilities at the core are
sigma_c t and sigma_c(1-t). Concavity and h_bin(0) = 0 give
h_bin(sigma_c t) >= sigma_c h_bin(t), h_bin(sigma_c(1-t)) >= sigma_c h_bin(t).
Averaging and multiplying by lambda gives
e >= sigma H(d/sigma),
which together with H(r) >= L(1-r^2) gives
q >= e >= L(sigma - d^2/sigma).
Expanding the binary entropies in J gives the posterior identity
J = (p/2) psi(d/p) + (sigma/2) psi(d/sigma),
and psi(r) <= L r^2 gives J <= L d^2/(sigma p) =: L z with 0 < z < sigma/p.
Together, these estimates say
q >= L(sigma - p z), q + J = s <= L,
L(sigma - p z) + L z = L(sigma - x z) <= L.
So [q, s] can be enlarged to an interval of length L z inside
[L(sigma - p z), L], and the integrand is nonnegative. Rescaling h = L v
with beta = b/L turns the integrand into k_beta(v) dv, and the lower
endpoint and length satisfy the interval estimate. Hence
int_q^s (h-b)_+/h^2 dh <= -(3/2) log(1 - gamma z)
= -(3/2) log(1 - d^2/p^2)
= 3 kappa.
End of the payment proof. Combining the derivative reduction with the
window bound gives
lambda Delta' - Delta >= 2 kappa - 3 M kappa = (2 - 3M) kappa >= 0.
So (Delta/lambda)' >= 0, and comparing with the endpoint lambda = 1
where nu = 0, I = J_c, and Delta = F(J_c), proves the payment inequality.
No limit as lambda -> 0 is needed.
The canonical high-entropy value. For x >= 0, d > 0, x+d < 1, define
h_bar(x,d) = (H(x+d)+H(x-d))/2, e_0(x,d) = (1-x) H(d/(1-x)).
For every e_0(x,d) <= e <= h_bar(x,d),
K(x,d,e) >= J_0(x,d,e) := F(H(x)-e) - F(h_bar(x,d)-e).
Proof. The canonical classification represents every state in this
closed high-entropy branch by the thinning path with a nonnegative core
center, and K(x,d,e) = lambda c(a_c,b_c). With alpha_c = 1 - x_c^2, the
one-bit inequality and convexity of F give
c(a_c,b_c) >= alpha_c F(J_c/alpha_c) >= F(J_c).
The payment theorem then gives
J_0(x,d,e) = F(nu+J) - F(nu) <= lambda F(J_c) <= lambda c(a_c,b_c)
= K(x,d,e).
The canonical optimizer is used only to evaluate a lower bound on the
cost. For an original pair law with means x+-d and individual entropies
e_a, e_b, the value e in the bound is the actual average, and the
original child information values remain H(x+d)-e_a and H(x-d)-e_b. The
average of these is h_bar(x,d)-e, so convexity of F gives
(F(H(x+d)-e_a) + F(H(x-d)-e_b))/2 >= F(h_bar(x,d)-e).
No individual entropy of the canonical law is substituted for either
original child entropy.
The hybrid profile. The reserve bound alone does not imply the sharp
rate F(I) at every state, so the proof takes the maximum
B_hat_m(I) = max{B_m(I), F(I)}.
Closure under a maximum is a feature of the local-production framework of
Chen, Gohari and Nair;
./chen-gohari-nair-2025.pdf
F alone does not satisfy a joining inequality for every law, so the proof
compares the two branches. Above half information the reserve B_m
dominates F. Below half information the demand from F increases with the
common mean of the child states, so every reflected core can be paid at
its one-corner wall by the thinning theorem.
The reserve above half information. For |m| < 1 and L/2 <= I < H(m),
B_m(I) >= F(I),
with strict inequality when m is not zero. Consequently a state with
F(I) > B_m(I) must have I < L/2. The proof sets S = F - 2I,
k_0 = L - 1/2, Xi(I) = (I - k_0) S'(I) - S(I),
so Xi(I) = int_0^I (s - k_0) F''(s) ds. The exact balance
int_0^{L/2} (s - k_0) e^{2s} ds = [(1/2) e^{2s}(s-L)]_0^{L/2} = 0,
together with strict monotonicity of e^{-2s} F''(s) and the sign of
s - k_0, gives Xi(L/2) > 0; since Xi' = (I - k_0) F''(I) > 0 on
[L/2, L), it follows that Xi > 0 throughout that range.
For the mean-dependent comparison, set alpha = 1 - m^2 and
c_m = H(m) - alpha L, with 0 <= c_m <= m^2 k_0. For I >= L/2 the argument
s = L - (H(m) - I)/alpha = (I - c_m)/alpha
lies in (0, L) with no clipping. The tangent inequality for convex S and
S' >= 0 then give
B_m(I) - F(I) = alpha S(s) - S(I) >= m^2 Xi(I) >= 0,
with strict inequality at nonzero m. At mean zero the two profiles agree.
The demand below half information. Let m, d > 0 with m+d < 1 and
0 < e <= h_bar(m,d). Put I = H(m) - e, j = h_bar(m,d) - e, and J = I - j.
If I <= L/2, then at fixed d and e,
partial_m {F(I) - F(j)} > 0.
The proof compares rate slopes with entropy slopes. First
F''(u)/F'(u) < 2 for 0 < u <= L/2,
because setting h = L-u and using both bounds of the rescaled curvature
gives 2 h^2 F'(u) >= (4 - L^{-1}) h^2 + h > (4/3) L^2 >= h^2 F''(u). Since
0 <= j < I <= L/2, this yields log(F'(I)/F'(j)) < 2J. For the entropy
slopes, put A = -partial_m h_bar(m,d) = (V(m+d)+V(m-d))/2 > 0. With
v_+- = V(m+-d) and G(v) = cosh^2(v) - v^2, whose derivative
G'(v) = sinh(2v) - 2v is positive, one has
partial_d { log(A/V(m)) - 2J } = (G(v_+) - G(v_-))/(v_+ + v_-) > 0,
so log(A/V(m)) >= 2J. Finally I_m = -V(m) and j_m = -A, so
partial_m {F(I) - F(j)} = -F'(I) V(m) + F'(j) A > 0.
Local inequality for the hybrid profile. Let (A,B) have a joint law on
[-1,+1]^2 with positive average entropies. Put
a = E A, b = E B, m = (a+b)/2, e = (e_a+e_b)/2,
I = H(m) - e, I_a = H(a) - e_a, I_b = H(b) - e_b.
Then
E c(A,B) >= B_hat_m(I) - (B_hat_a(I_a) + B_hat_b(I_b))/2.
If the parent maximum is B_m(I), the local reserve inequality applies
and the child terms increase to the B_hat values. It remains to
suppose F(I) > B_m(I). Put d = |a-b|/2 and
h_bar(m,d) = (H(m+d)+H(m-d))/2, j = h_bar(m,d) - e = (I_a+I_b)/2.
Since B_hat >= F, convexity at the original child states gives
(B_hat_a(I_a) + B_hat_b(I_b))/2 >= (F(I_a)+F(I_b))/2 >= F(j).
For d = 0, one has I = j and nonnegativity of cost proves the claim.
Otherwise reflect to m > 0, which is forced in the strict-parent case
because B_0 = F. Put
J_0(x,d,e) = F(H(x)-e) - F(h_bar(x,d)-e).
The actual law is feasible in the common-budget cost problem, so it is
enough to prove K(m,d,e) >= J_0(m,d,e).
Use the exact optimizer. In the one-corner geometry, the entropy is at
least the transition value, and the canonical high-entropy value
comparison gives K(m,d,e) >= J_0(m,d,e). In the reflected geometry write
d = k r, e = k H(r), ell = 1 - k, 0 < r < 1, 0 < k <= 1.
Then 0 < m <= ell, and the cost is d V(r). Hold d, e, k, r fixed and
increase the common mean from m to ell. The law with mass k on (r,-r) and
masses (1-k+x)/2, (1-k-x)/2 at the equal corners keeps every x in [m,ell]
physical at the same entropy and the same cost, and its means stay
interior because ell + d = 1 - k(1-r) < 1.
The reserve lemma gives H(m) - e < L/2, and H(x) - e decreases along the
interval, so the wall monotonicity applies throughout:
J_0(m,d,e) <= J_0(ell,d,e).
At x = ell the negative-corner mass vanishes and the state is at the
one-corner transition, so the canonical high-entropy value comparison
gives J_0(ell,d,e) <= d V(r) = K(m,d,e). This proves K(m,d,e) >=
J_0(m,d,e). A zero-length interval requires only the last step, and the
case k = 1 would force m = 0 and is already covered by the reserve
branch. The child terms in the pooling inequality always use the
original e_a and e_b, not the entropies of a replacement optimizer.
Dimension induction. The production bound
D(g) >= B_hat_m(I(g)) >= F(I(g))
holds for every interior g on every cube by induction on n. At n = 0 a
constant field has D = I = 0 and B_hat_m(0) = 0. For a coordinate split,
the exact energy decomposition reads
D(g) = (D(g_+) + D(g_-))/2 + E c(g_+, g_-).
Each within-face production has its face probability 1/2, and symmetry
of c combines the two orientations of every joining edge into the last
expectation. The induction hypothesis bounds the two within-face
productions, and the local inequality for B_hat supplies the remaining
difference on the joining edges. The sum is the parent value
B_hat_m(I(g)). Every face is interior-valued, so all child entropies are
positive and every local use is physical.
Sharp interpolation under noise. With the production bound proved, the
main theorem follows by integrating along the flow. Fix a soft field u
with 0 < rho < 1. Write rho = e^{-t} with t > 0. If u is constant, both
sides of the contraction inequality are zero. Otherwise every value of
g_s = T_{e^{-s}} u lies in (-1,1) for s > 0, because the noise kernel
gives strictly positive weight to every input and a nonconstant
[-1,1]-valued input cannot average to either endpoint. The mean stays
fixed, and by the flow identity and the production bound
(d/ds) I(g_s) = -D(g_s) <= -F(I(g_s)) for s > 0.
For nonconstant u the field g_s is nonconstant at every finite s: the
Walsh characters chi_A form a basis and T_{e^{-s}} chi_A = e^{-s|A|} chi_A
with all eigenvalues positive. Strict convexity of psi gives
0 < I(g_s) < L, so r(s) = psi^{-1}(I(g_s)) lies in (0,1). The identity
F(psi(r)) = r V(r) and psi'(r) = V(r) turn the inequality into
V(r(s)) r'(s) <= -r(s) V(r(s)), r'(s) <= -r(s).
Thus for 0 < eps < t, r(t) <= e^{-(t-eps)} r(eps). On a finite cube the
noise operator and the information functional converge as s -> 0, and
psi^{-1} is continuous including at L, so
psi^{-1}( I(T_{e^{-t}} u) ) <= e^{-t} psi^{-1}( I(u) ).
Applying the increasing function psi proves the claim for 0 < rho < 1.
The positive-time argument requires no finiteness of the initial entropy
production. At rho = 0 the field is constant; at rho = 1 the assertion
is an identity. For negative correlation T_{-rho} u(x) = T_rho u(-x),
and the sign flip preserves the uniform measure and the information.
This proves the full stated range.
Sharpness. For u(x) = a x_i with |a| <= 1, the initial information is
psi(|a|), and after noise it is psi(|rho a|), so every soft coordinate
attains equality. No soft combination of coordinates does better at the
same initial information.
The Courtade-Kumar corollary. Since f is Boolean, H(f(x)) = 0 at every
input, so I(f) = H(m) and I(f(X);Y) = I(T_rho f). The first inequality
of the corollary is the sharp contraction theorem. The second follows
from r_m <= 1 and monotonicity of psi. If 0 < |m| < 1, then
0 < H(m) < L and 0 < r_m < 1, which gives the stated strict comparison
with the unbiased bound. For a signed coordinate f, T_rho f = rho f, so
its information is exactly psi(|rho|). In bits, dividing by L and
substituting eps = (1-|rho|)/2 gives
I_2(f(X);Y) <= 1 - h_2(eps),
which is the original Kumar-Courtade form.
Comparison with the mean-dependent profile. For 0 < m < 1 and
0 <= I < H(m), put h = H(m) - I. The mean-dependent profile of Chen,
Gohari, Javanmard, Lin, Mirrokni, Nair and Woodruff, converted to natural
logarithms and to the generator N, is
C_m(I) = max{ F(L - h) - m V(s), F(I) }, m H(s) = h s, m <= s < 1.
The equation for s has a unique solution because H(s)/s decreases
strictly from H(m)/m to zero on [m,1). The F(I) branch of C_m yields the
sharp contraction theorem by the same inverse-entropy integration above,
with the initial information unchanged.
At every fixed 0 < m < 1,
B_m(H(m)-h) ~ ((1-m^2)/2) log(1/h),
C_m(H(m)-h) ~ ((1-m)/2) log(1/h) as h -> 0.
The verification uses H(r) = (eps/2) log(2 exp(1)/eps) + O(eps^2) and
V(r) = (1/2) log(2/eps) + O(eps) at r = 1 - eps, giving F(L-h) ~
(1/2) log(1/h) and, from m H(s) = h s, s -> 1 and V(s) ~ (1/2) log(1/h);
the F(H(m)-h) term remains bounded. The first equivalent follows from
B_m(H(m)-h) = 2(H(m)-h) + (1-m^2) S(L - h/(1-m^2)) and S(u) = F(u) - 2u
for positive u. Hence C_m does not dominate B_m: the ratio
B_m(H(m)-h)/C_m(H(m)-h) tends to 1+m > 1 as h -> 0. This is a comparison
of two explicit production profiles.
./chen-et-al-2026.pdf
Origin of the differential method. The local-to-global program is due to
Chen, Gohari and Nair. The energy decomposition and the induction that
bounds each coordinate split derive from that work, and the October
2026 revision notes that a proposed auxiliary inequality fails; the
comparisons above are proved directly. The symmetric fixed-entropy
optimizer is a predecessor of the arbitrary-mean joining problem treated
above. The two-capacity profile B_m uses the same coefficients alpha and
c_m that appear in the finite-noise Boolean bound of Kramer and Saglam.
./chen-gohari-nair-2025.pdf
./kramer-saglam-2026.pdf