# Verification report

# Written self-audit

Decision: accept the complete counterexample in the stated source scope. This is the originating researcher's self-audit, not independent expert review or formal verification.

1. Source quantifiers: a single constant for all positive epsilon<1/2 and all inputs; arbitrary real lambda. The finite positive coefficients used here are admissible.
2. Negation: the all-N norm lower bound tends to infinity, so it negates that single uniform constant. It does not claim any fixed-epsilon operator is unbounded.
3. Fourier sign: pairing positive and negative t gives -2i sin(t xi); the sine Mellin formula yields +2 Gamma(i mu) sinh(pi mu/2), with a(0)=-i pi. Damped physical-side quadrature independently checks both frequency signs.
4. Gamma amplitude: reflection yields 2pi tanh(pi|mu|/2)/|mu|, at least 1/4 in square on [2pi,4pi]. The common amplitude multiplies the entire aligned sum, so its complex argument is harmless.
5. Normalization: inverse Fourier factor 1/(2pi), triangular mass 2pi, exact input square norm 4pi N/3. Positive disjoint frequency bands avoid sign cancellation.
6. Simultaneous selector: base-16 digits of all target phases are placed consecutively in ONE tau. Multiplying by 16^j exposes the correct digit; earlier digits are integers, the later tail is less than one cell. No independent parameter is assigned to each packet.
7. Boundaries: half-open cells and finite digit sums cover every real x. The selector is measurable and takes at most 16^N values.
8. Huge frequencies: only exact exponent identities and analytic inequalities enter the proof. No floating approximation of exp(16^N) is treated as phase evidence.
9. Packet multiplier error: the relative frequency offsets are at most exp(-16), so the logarithm is real and the elementary Lipschitz estimate is valid. The error sum is bounded independently of N.
10. Test-function domain: smoothing the triangular Fourier transform gives Schwartz approximants with uniform derivative and quadratic-tail bounds. This proves the multiplier identity for sinc-squared packets. Spatial smooth cutoffs recover compactly supported smooth tests, uniformly for the finite selector family (in fact uniformly in lambda).
11. Principal value: both phases are even in t, hence pairing cancels f(x) at zero. Bounded f' controls the local integral; quadratic decay controls infinity. The paired difference identity is legitimate.
12. Transfer: tiny t is bounded with ||f'|| and a=exp(-2*16^N); large t uses b=64. Taylor expansion is used only on the remaining finite annulus. No Taylor approximation is claimed uniformly over the whole line.
13. Constants: near-zero, far-tail, and middle errors each fit N/128. The total is <N/32, leaving output >=N/32 on an interval of length two. The norm ratio is 3N/(2048pi).
14. Threshold: epsilon*(16^N)^3<=2^-16 is exactly epsilon<=2^(-16-12N). Choosing N=floor((log2(1/epsilon)-16)/12) proves the stated log corollary for log2(1/epsilon)>=56.
15. No infinite supremum/limit exchange: the quantitative proof uses an explicit finite selector. Compact-cutoff approximation is controlled in physical space.
16. Prior work: the inherited endpoint reduction and multiplier are explicitly credited. Fixed-exponent and fixed-coefficient theorems are not misrepresented as contradictory or novel.
17. Result scope: one complete frozen-record counterexample, one exact AIM question closure; no second increment for the equivalent Guo even-phase formulation, and no closure of the separate odd-phase question.
18. Verification: 153218 supplementary predicates in each normal and optimized run, identical output bytes. These finite/algebraic/high-precision checks are not a proof of the general analytic theorem; the written argument supplies it.

Known gaps in the accepted theorem: none identified in this self-audit. Remaining limitations: no optimal growth rate, no exhaustive priority certification, no formal proof assistant, no external review. No external-review gate is imposed.


## Supporting check output

```json
{
  "checker_sha256": "e3dde91f099068121ba2987898d9716209ae84f5c570cdc04628c9ab2f465d6a",
  "counts": {
    "derivative_bound": 1,
    "geometric_majorant": 200,
    "infinite_tail": 1,
    "log_interval_length": 200,
    "logarithmic_corollary": 9776,
    "middle_integral": 1,
    "middle_integral_budget": 1,
    "near_zero_rational_majorant": 1,
    "norm_constant": 1,
    "numeric_damped_kernel_sign": 12,
    "numeric_gamma_reflection": 256,
    "numeric_multiplier_lower": 256,
    "numeric_taylor_remainder": 8,
    "one_parameter_same_cell": 64148,
    "packet_error_rational_majorant": 1,
    "phase_distance": 64148,
    "plancherel_norm": 1,
    "selector_range": 13804,
    "small_exponential_argument": 200,
    "threshold": 200,
    "total_budget": 1,
    "triangular_mass": 1
  },
  "floating_huge_frequencies_used": false,
  "numeric_supplements": {
    "damped_kernel_max_absolute_error": "1.1163351e-54",
    "gamma_identity_max_absolute_error": "2.173363e-70"
  },
  "precision_digits": 70,
  "role": "supplementary; not a substitute for the all-N written proof",
  "status": "PASS",
  "total_predicates": 153218
}
```

## Source and novelty

# Source and novelty audit, 28 October 2026

The complete relevant page of the primary AIM workshop PDF was rendered and inspected: Question 12, p. 4, equation (0.17), positive exponent between zero and one half, even absolute-value monomial, no frequency restriction. The live URL returned 404; the exact preserved official download from 6 October 2026 was reused after its SHA-256 was checked. This is recovered primary evidence, not a current successful fetch.

Guo, arXiv:1503.04411, definition (1.1) and Question 1.4 / (1.15), independently verify the operator and uniformity issue. Theorem 1.1 is fixed-exponent boundedness; Theorem 1.3 excludes neighborhoods of zero and one; Theorem 1.6 bounds individual fixed-coefficient multipliers. None asserts the maximal uniform estimate at zero. The odd-phase question (1.16) near one is separate and is not resolved here.

Guo--Hickman--Lie--Roos, arXiv:1610.05203, Theorems 1.1 and 1.2 explicitly allow their constants to depend on the exponent. Ramos, arXiv:2012.08913, Theorem 2 gives local uniformity near exponent two for a family maximizing a linear modulation; its final discussion concerns the odd-phase question near one. These are different claims, not prior proofs of the bound disproved here. All three PDFs and text extractions were saved with transport/hash receipts.

The frozen UnsolvedMath v1.6.0 report already supplies the scaling lambda=mu/epsilon, the logarithmic kernel, its exact Gamma multiplier, and a necessary maximal-operator condition. The paper explicitly cites that contribution. No originality is claimed for those facts or for the elementary Gamma, triangular-transform, and Fourier identities.

Bounded searches covered the exact paper title with counterexample/uniform/zero, Guo Question 1.4, maximal imaginary powers, lacunary phase constructions and logarithmic Carleson operators. Searches and unhelpful results are retained in sources/*.json. No matching quantitative counterexample was located. Search-result material on fixed imaginary powers, maximal truncations of one kernel, weighted spherical means, or maximal bounds with a log(L) domain does not establish the pointwise compact-parameter maximal estimate used here. In particular the Blower 2002 abstract found by search has extra domain assumptions. This is not a certificate of exhaustive literature absence or absolute priority.

Primary references: https://arxiv.org/abs/1503.04411 ; https://doi.org/10.3934/cpaa.2016.15.929 ; https://arxiv.org/abs/1610.05203 ; https://arxiv.org/abs/2012.08913 .


## Written construction

# Quantitative failure of uniformity at the zero fractional exponent

Research proof, 2026-10-28. This file records the argument before publication decisions. The source is Guo's Question 12, equation (0.17), in the AIM workshop list *Carleson theorems and multilinear operators*, and the even-phase part (1.15) of Guo's Question 1.4 (arXiv:1503.04411). The original AIM PDF was recovered from the preserved 2026-10-06 download; its live URL returned 404 on 2026-10-28. The question has no frequency restriction. The odd-phase question near exponent one is outside the present result.

Use the Fourier transform `fhat(xi) = integral f(x) exp(-i x xi) dx`, with inverse factor `1/(2 pi)`. Write

\[
T_{\varepsilon,\lambda}f(x)=\operatorname{p.v.}\int_{\mathbb R} f(x-t)e^{i\lambda |t|^\varepsilon}\,\frac{dt}{t},
\qquad \mathcal C_\varepsilon f(x)=\sup_{\lambda\in\mathbb R}|T_{\varepsilon,\lambda}f(x)|.
\]

**Theorem.** For every integer `N >= 1` and every `0 < epsilon <= 2^(-16-12N)`, the operator norm on `L^2(R)` satisfies

\[
\|\mathcal C_\varepsilon\|_{2\to2}\ge \sqrt{\frac{3N}{2048\pi}}.
\]

The same lower bound is a supremum over compactly supported smooth test functions. Consequently the uniform bound in the question is false. In particular, for `0 < epsilon <= 2^(-56)`,

\[
\|\mathcal C_\varepsilon\|_{2\to2}\ge
\sqrt{\frac{\log_2(1/\varepsilon)}{16384\pi}}.
\]

No optimality of the rate is asserted. The proof uses only finitely many positive modulation parameters for each `N, epsilon`.

## 1. The logarithmic kernel and its exact multiplier

For real `mu`, set

\[
S_\mu f(x)=\int_0^\infty [f(x-t)-f(x+t)]t^{i\mu}\,\frac{dt}{t}.
\]

For a smooth function with bounded derivative and quadratic decay, this paired integral converges absolutely at both endpoints. As a tempered convolution operator its multiplier is

\[
m_\mu(\xi)=a(\mu)\operatorname{sgn}(\xi)|\xi|^{-i\mu},\qquad
a(\mu)=2\Gamma(i\mu)\sinh(\pi\mu/2),
\]

with the continuous value `a(0) = -i pi`. Indeed, for `-1 < Re(s) < 1`, the sine Mellin integral, initially evaluated in `0 < Re(s) < 1` and continued using convergence at zero and integration by parts at infinity, gives

\[
-2i\int_0^\infty t^{s-1}\sin(t\xi)\,dt
=-2i\Gamma(s)\sin(\pi s/2)\operatorname{sgn}(\xi)|\xi|^{-s}.
\]

Putting `s = i mu` gives the formula, including its sign. This identity is valid as a tempered distribution. One can first insert an exponential damping factor, pass to the limit on Schwartz test functions, and then use the convergence of the paired integrals. For the particular packets below an additional direct justification by smooth Fourier approximation is given in Section 6.

The reflection and recurrence identities for Gamma imply

\[
|a(\mu)|^2=\frac{2\pi}{|\mu|}\tanh(\pi |\mu|/2),\qquad |a(\mu)|\le\pi.
\]

For `2 pi <= mu <= 4 pi`, the first identity implies

\[
|a(\mu)|^2\ge \tfrac12\tanh(\pi^2)>\tfrac14,
\qquad |a(\mu)|\ge\tfrac12.
\]

The frozen dataset already supplies the logarithmic scaling reduction and this multiplier formula. They are background, not the claimed new contribution here.

## 2. An explicit packet family

Set `B = 16`, `xi_j = exp(B^j)` for `1 <= j <= N`, and

\[
\phi(x)=\left(\frac{\sin(x/2)}{x/2}\right)^2,\quad \phi(0)=1,
\qquad f_N(x)=\phi(x)\sum_{j=1}^N e^{i\xi_jx}.
\]

The elementary triangular transform identity is

\[
\widehat\phi(\eta)=2\pi(1-|\eta|)_+.
\]

Thus `0 <= phi <= 1`, `phi(x) <= 4/x^2` for nonzero `x`, and `phi(x) >= 1/2` on `[-1,1]`. The last assertion follows from `sin(y)/y >= 1-y^2/6` for `|y| <= 1/2`.

By Plancherel, `||phi||_2^2 = 4 pi/3`. The translated Fourier supports `[xi_j-1, xi_j+1]` are disjoint and positive. Therefore

\[
\|f_N\|_2^2=\frac{4\pi N}{3}.
\]

Fourier inversion gives `||phi'||_infinity <= 1/3`. Consequently

\[
\|f_N'\|_\infty\le N/3+\sum_{j=1}^N\xi_j\le 2N\xi_N.
\]

## 3. One parameter aligns all packet phases

For each `x`, define

\[
\theta_j(x)=\left\{\frac{\xi_jx}{2\pi}\right\},\quad
d_j(x)=\lfloor B\theta_j(x)\rfloor,\quad
\tau(x)=\sum_{j=1}^N d_j(x)B^{-(j+1)},\quad
\mu(x)=2\pi(1+\tau(x)).
\]

Then `0 <= tau < 1/B` and `2 pi <= mu(x) < 2 pi(1+1/B) < 4 pi`. For each `j`, the terms with `k < j` in `B^j tau` are integers, and

\[
\{B^j\tau\}=\frac{d_j}{B}+\sum_{k=j+1}^N d_k B^{j-k-1}
\in[d_j/B,(d_j+1)/B).
\]

Both this fractional part and `theta_j` lie in the same interval of length `1/B`. The integer `B^j` in the factor `mu/(2 pi)` does not affect the phase. It follows that

\[
\operatorname{Re}\exp(i\xi_jx-i\mu(x)B^j)
\ge\cos(2\pi/B)>1/2.
\]

This is one common parameter, not independent parameters for different summands. Its values lie in a fixed finite set of size at most `B^N`; measurability is immediate from the floor functions. Boundary points and finite base expansions cause no exception because all intervals are half-open.

## 4. The multiplier error is uniformly small

For `mu in [2 pi,4 pi]` and `|eta| <= 1`,

\[
|\log(1+\eta/\xi_j)|\le 2|\eta|/\xi_j,
\qquad
|(\xi_j+\eta)^{-i\mu}-\xi_j^{-i\mu}|\le 8\pi/\xi_j.
\]

Since `phihat` is nonnegative with integral `2 pi`, Fourier inversion and `|a(mu)| <= pi` imply

\[
S_\mu f_N(x)
=a(\mu)\phi(x)\sum_{j=1}^N e^{i\xi_jx-i\mu B^j}+E_N(x,\mu),
\qquad
|E_N(x,\mu)|\le8\pi^2\sum_{j=1}^N e^{-B^j}.
\]

The inequality `B^j >= 16j` implies

\[
8\pi^2\sum_{j=1}^N e^{-B^j}
\le\frac{8\pi^2}{e^{16}-1}
<\frac{128}{65535}<\frac1{16}.
\]

Use `pi < 4` and `e > 2` in the strict inequality. On `|x| <= 1`, the common-parameter alignment now gives

\[
|S_{\mu(x)}f_N(x)|\ge \frac12\cdot\frac12\cdot\frac N2-\frac1{16}
\ge\frac N{16}.
\]

In particular the maximal logarithmic operator already fails to be bounded on `L^2`, even with `mu` restricted to the fixed compact interval `[2 pi, 17 pi/8]`.

## 5. Quantitative transfer to a positive exponent

Write `h = B^N`, `a = exp(-2h)` and `b = 64`. Let `g_x(t) = f_N(x-t)-f_N(x+t)`. For `|x| <= 1` and `mu in [2 pi,4 pi]`,

\[
e^{-i\mu/\varepsilon}T_{\varepsilon,\mu/\varepsilon}f_N(x)-S_\mu f_N(x)
=\int_0^\infty \frac{g_x(t)}t
\left[e^{i\mu(t^\varepsilon-1)/\varepsilon}-e^{i\mu\log t}\right]dt.
\]

This is an equality of absolutely convergent paired integrals. Bound its three regions separately.

* For `0 < t < a`, the derivative estimate gives `|g_x(t)| <= 4N exp(h)t`. The difference of phases is at most two. This contribution is at most `8N exp(-h) < N/128`, since `h >= 16` and `e > 2`.
* For `t > b`, both `|x-t|` and `|x+t|` are at least `t/2`, so `|g_x(t)| <= 32N/t^2`. This contribution is at most `32N/b^2 = N/128`.
* On `[a,b]`, `|log t| <= 2h` and `log(b/a) <= 3h`. Taylor's theorem gives

\[
\left|\frac{e^{\varepsilon u}-1}{\varepsilon}-u\right|
\le\frac{\varepsilon u^2}{2}e^{\varepsilon|u|}.
\]

With `M = 4 pi < 16` and `|g_x| <= 2N`, this region contributes at most

\[
12MN\varepsilon e^{2\varepsilon h}h^3.
\]

Our hypothesis is precisely `epsilon h^3 <= 2^-16`. It implies `e^(2 epsilon h) <= 2`, so the last bound is at most `384N/65536 = 3N/512 < N/128`. For example, `2 epsilon h <= 2^-23 < 1/2` and `exp(1/2) < 2` justify the stated exponential bound without an asymptotic argument.

The total error is less than `3N/128 < N/32`. Taking the finite-valued selector `mu(x)` from Section 3 gives

\[
\mathcal C_\varepsilon f_N(x)\ge N/32\quad (|x|\le1).
\]

Consequently `||C_epsilon f_N||_2^2 >= N^2/512`. Dividing by `4 pi N/3` proves the asserted norm lower bound. There is no exchange of an infinite supremum with a limit in this argument.

For the displayed logarithmic consequence put `L = log_2(1/epsilon)` and `N = floor((L-16)/12)`. If `L >= 56`, then `N >= (L-28)/12 >= L/24`, giving the stated constant.

## 6. Test-function and multiplier-domain details

The function `f_N` is smooth and in `L^1 intersect L^2`, with bounded derivative and quadratic decay. To justify the multiplier computation directly, approximate the compactly supported triangular `phihat` by smooth compactly supported functions within a fixed positive-frequency neighborhood after each translation. Equivalently use convolution of `phihat` with a nonnegative smooth approximate identity. In physical space this multiplies `phi` by bounded smooth factors that tend to one, with uniformly bounded first derivatives. The approximants converge in `L^2`, their paired `S_mu` integrals converge pointwise by the derivative bound at zero and quadratic tail domination, and their multiplier expressions converge uniformly by `L^1` convergence of the Fourier transforms on compact supports away from zero. The Schwartz multiplier identity therefore gives exactly the claimed identity for `f_N`. This also rules out any ambiguity at frequency zero.

For the source's Schwartz test-function convention, take a fixed `chi in C_c^infinity`, `0 <= chi <= 1`, with `chi=1` on `[-1,1]`, and set `f_{N,R}(x)=chi(x/R) f_N(x)`. For fixed `N`, `||f_{N,R}||_2 -> ||f_N||_2`. On `|x| <= 1` the paired integrals for `T_{epsilon,lambda}(f_{N,R}-f_N)` have no contribution near zero once `R > 2`, while their tails are dominated by a constant times `N t^-3`; hence they tend to zero uniformly on `[-1,1]` and in `lambda`. Alternatively the relevant finite selector set suffices. Thus the ratio lower bound for `f_N` is a limit of ratios for compactly supported smooth functions. The operator norm on that class is at least the displayed value, whether or not an extremizer attains it.

## 7. Scope and outstanding administrative work

The proof above is an analytic family of counterexamples, not a numerical certificate and not a counterexample to boundedness at any fixed nonzero exponent. It concerns the even-phase uniform estimate near zero only. The logarithmic multiplier and limiting reduction are inherited/classical; the contribution to audit for prior occurrence is the simultaneous digit alignment and explicit positive-exponent lower bound. No absolute priority claim is made. Literature review, independent calculations, paper preparation and publication registration are separate from the mathematical argument.