A C1 counterexample to the homological entropy bound

Source and verification

This article explains family 151 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Topological entropy and homological growth

For a continuous map $f$ of a compact metric space, define the orbit distance

$$d_n(x,y)=\underset{0\le j<n}{\max }d(f^{j}x,f^{j}y).$$

Let $s_f(n,\epsilon )$ be the largest size of a set whose distinct points have $d_n$ distance greater than $\epsilon$. Topological entropy is

$$\begin{array}{c}{h}_{\mathrm{top}}(f)\\ =\underset{\epsilon \downarrow 0}{\lim }\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log s_f(n,\epsilon ).\end{array}$$

It measures exponential growth of distinguishable orbit segments, independently of a chosen invariant probability.

A map of a compact manifold also acts linearly on its finite-dimensional real homology spaces. For this article, homology records persistent cycles, such as the fundamental two-dimensional cycle of a sphere. The spectral radius $\rho (f_{\ast })$ measures the largest modulus of an eigenvalue of the induced action across all degrees.

The general $C^1$ version of Shub’s entropy inequality asks whether $h_{\mathrm{top}}(f)\ge \log \rho (f_{\ast })$.

🏷️ The claimed counterexample

Statement in the manuscript

For some positive integer $q$, there is a noninvertible $C^1$ self-map of

$$M=(\mathbb{R}/100\mathbb{Z})\times (S^2{)}^{q+1}$$

with topological entropy zero and an eigenvalue $2$ on $H_2(M;\mathbb{R})$. (OpenAI, 2026)

Thus its entropy is strictly smaller than $\log 2$. The manifold is compact, smooth, and without boundary, of dimension $2q+3\ge 5$.

Both regularity and invertibility matter. This is not a smooth $C^{\infty }$ counterexample and not a diffeomorphism counterexample. The growing homology occurs in intermediate degree two, rather than in degree one or the top dimension. The source explicitly distinguishes positive results in those other settings.

🏷️ Why a sphere-doubling map is insufficient

On the Riemann sphere $\mathbb{C}\cup \{\infty \}$, $z\mapsto z^2$ doubles the fundamental homology class. But it also has positive topological entropy. Simply taking its product with a zero-entropy map would preserve that positive entropy.

The manuscript modifies the orbit behavior while retaining the homological action. One sphere coordinate $z$ usually follows squaring. A circle coordinate acts as a clock, and additional sphere coordinates store temporary signals, called registers.

Near a special clock position, progress can be arbitrarily slow. The registers prepare corrections for every possible delay, then read and reset them during a completed passage. These corrections force the main coordinate into a region from which its future approaches the simple invariant part of the system. (OpenAI, 2026)

The homology relation at the end is not a dynamical semiconjugacy to $z\mapsto z^2$. A true continuous factor map onto sphere doubling would impose its positive entropy as a lower bound.

🏷️ Infinite prediction and C1 regularity

A signal intended for readout after $m$ steps carries angular oscillation of order $2^m$ and amplitude of order $L^{-m}$, with $L>2$ chosen sufficiently large. The first angular derivative is then on the scale $(2/L{)}^m$.

This is the elementary reason an infinite series of such corrections can have uniformly convergent first derivatives. The actual proof also controls derivatives of the prediction map, and chooses $L$ above that growth bound.

Small prepared signals are amplified before readout. To justify multiplication by the claimed gain, the source proves that each register stays inside its linear regime until it is used. A bound for an already saturated register would not establish the same evolution.

Each register is read at most once in a passage and reset before the next preparation. A finite truncation of the prediction series would miss arbitrarily long waiting times; the infinite construction is essential.

The estimates establish $C^1$ regularity. One cannot silently repeat the same argument for every derivative order: angular derivatives of order $r$ have a factor of size $2^{rm}$, and a single fixed $L$ cannot dominate all such orders.

🏷️ Orbit classification and zero entropy

The manuscript proves that every forward orbit approaches the union of two compact invariant sets. One lies at a fixed clock reading with $z=0$; the other has $z=\infty$ and zero registers. The restrictions to these sets have zero entropy. (OpenAI, 2026)

The entry time is not uniform. Long transient passages therefore cannot be dismissed by claiming that every orbit reaches a simple set after one common time.

The source gives a direct separated-set localization proof. There is also a useful invariant-measure explanation of why the stated orbit property has the needed strength. If every orbit approaches a closed invariant set $A$, then for any invariant probability $\nu$ and any $\delta >0$,

$$1_{\{d(f^{n}x,A)\ge \delta \}}⟶0.$$

Dominated convergence and invariance imply $\nu \{x:d(x,A)\ge \delta \}=0$. Hence every invariant probability is supported on $A$. The variational principle then localizes topological entropy to $A$. Applying this to the claimed union explains the zero-entropy conclusion, once the orbit classification is established.

🏷️ Retaining the homological eigenvalue

Projection onto the main sphere gives a nonzero map on second homology. The construction preserves an intertwining relation with degree-two multiplication there, so the induced homology operator has eigenvalue two.

This is a topological statement about cycle classes. It does not say that an actual observed sphere coordinate evolves forever by the unmodified squaring map. That distinction allows simple eventual orbit dynamics and expanding intermediate homology to coexist in the claimed finite-regularity example.

🏷️ Key steps and proof outlines

The infinite predictor has a summable first derivative

In the construction section, the $\ell$th register receives a series of terms

$$U_{\ell }(t,z)=\frac{\alpha (t)}{10}\sum _{m\ge 1}{L}^{1-m}{\beta }_{\ell }(T_m(t,z)){\Phi }_{2^{m+1}}(z).$$

Here $\alpha$ restricts the preparation interval, ${\beta }_{\ell }$ selects a predicted readout interval, $T_m$ is the real clock coordinate after $m$ iterations of a fixed predictor $G$, and ${\Phi }_k(z)=\chi (z)(z/|z|{)}^k$. The smooth cutoff $\chi$ is supported in a fixed annulus away from zero and infinity, so ${\Phi }_k$ extends smoothly by zero at both poles.

If $\Vert DG\Vert \le B$, then $\Vert D{T}_m\Vert \le B^m$. On the annulus, $\Vert D{\Phi }_k\Vert \le C(1+k)$. Applying the product rule to the $m$th summand gives

$$\Vert F_{\ell ,m}{\Vert }_{\infty }+\Vert D{F}_{\ell ,m}{\Vert }_{\infty }\le C{L}^{1-m}(1+B^m+2^{m+1}).$$

The predictor derivative and angular derivative occur in separate product-rule terms, so their costs add; there is no unnecessary factor $B^m{2}^m$. Choosing $L>\max (B,2)$ makes all three geometric series summable. Uniform convergence of functions and first derivatives in finitely many source charts proves $C^1$ regularity. The real selector ${\beta }_{\ell }$, rather than its periodic extension, is used inside the prediction series so a long delay does not create unintended later-period readouts. (OpenAI, 2026)

Readout timing matches the angular phase exactly

All state coordinates update simultaneously from the old state. A term prepared at time $i$ enters a register at time $i+1$. If read at time $i+m$, it receives exactly $m-1$ intervening gains. While the register remains in its linear region and the gain equals $L$, these cancel its initial factor $L^{1-m}$.

Meanwhile unmodified squaring takes the phase of $z_i$ to its $2^m$th power at time $i+m$; the next main-coordinate square has phase exponent $2^{m+1}$, exactly the exponent stored in the predictor. This explains the formula’s indexing. It is not enough to note that both quantities grow exponentially: an off-by-one delay would produce the wrong correction.

The orbit proof separately establishes agreement of the actual and predicted clock until the intended readout, that the register has not saturated, and that each register is read only once before reset. The gain and phase calculation applies only after these conditions are verified. The reset interval makes both the gain and new input zero, clearing the old state before another preparation.

Long transient times do not themselves contribute invariant entropy

The source’s classification says every forward orbit approaches the union $A$ of two compact invariant sets with zero-entropy restrictions. Uniform entry time is not claimed. For any invariant probability $\nu$, however, the bounded functions $1_{\{d(f^{n}x,A)\ge \delta \}}$ tend pointwise to zero. Invariance makes their integrals independent of $n$, so dominated convergence forces that common integral to vanish. Taking a countable sequence of positive $\delta$ proves $\nu (A)=1$.

The variational principle now bounds the map’s topological entropy by the entropy on $A$, which is zero. This measure argument is a consequence of the full orbit classification, not of a vague claim that most points eventually simplify. Arbitrarily long transient passages can coexist with the conclusion because no invariant probability gives them persistent mass away from $A$.

Homological growth is preserved without a positive-entropy factor

Let $\pi$ project onto the main sphere. The corrections deform the sphere-coordinate map through a homotopy preserving the induced degree-two relation. On second homology, ${\pi }_{\ast }{f}_{\ast }=2{\pi }_{\ast }$, and ${\pi }_{\ast }$ is nonzero. Equivalently, the dual linear action has a nonzero eigenvector of eigenvalue two; a finite-dimensional operator and its dual have the same characteristic polynomial, so $f_{\ast }$ also has eigenvalue two.

This relation holds on homology, not as the pointwise identity $\pi \circ f=(z\mapsto z^2)\circ \pi$. The latter would make squaring a dynamical factor and force its positive entropy. The construction’s temporary corrections alter actual orbits while retaining the topological degree relation. Its differentiability estimate uses one fixed amplification constant and controls only first derivatives, which is why the stated result remains a noninvertible $C^1$ example.

See Also

The standard map concerns entropy for invariant area, rather than topological entropy. Smooth realization obstruction shows a different limitation of entropy as a complete test for smooth dynamics.

🏷️ Manuscripts and verification scope

• A $C^1$ Counterexample to the Entropy Conjecture — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope describes the noninvertible C1 self-map, zero topological entropy, and a nonzero second-homology eigenvector with eigenvalue two. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. A C¹ Counterexample to the Entropy Conjecture.