Spectral Shadow

A sharp (L^p) barrier for spectrum that disappears under Loeb pushdown.

Read the theorem note · inspect the claim ledger · reproduce every check

The candidate discovery

Let (W) be a kernel on probability spaces and ((T_Wf)(x)=\int W(x,y)f(y),dy). For (p>2), define

[ a_p=\frac{p}{2(p-1)}, \qquad b_p=\frac{p-2}{2(p-1)}. ]

Then

[ \boxed{ \lVert T_W\rVert_{2\to2} \le \lVert W\rVert_p^{a_p} \lVert T_W\rVert_{\infty\to1}^{b_p}.} ]

Here (\lVert T_W\rVert_{\infty\to1}) uses complex phase tests, matching complex interpolation. The constant is exactly one, and every rank-one rectangular block attains equality. Among homogeneous multiplicative estimates with nonnegative exponents, (b_p) is the largest possible cut-decay exponent; at that endpoint, homogeneity fixes the moment exponent (a_p).

The hyperfinite interpretation is the useful part: if an internal kernel has limited (L^p)-norm and carries an appreciable spectral mode, that mode must leave an appreciable unrestricted internal, hence full-Loeb, bounded phase-cut witness. It cannot hide entirely on Loeb-null microscopic structure. A witness on a prescribed smaller factor needs an additional approximation hypothesis.

At (p=2), this fails as strongly as possible:

[ W_n=n\mathbf 1_{[0,1/n]^2}, \qquad \lVert W_n\rVert_2=1, \qquad \lVert T_{W_n}\rVert_{\infty\to1}=\frac1n, \qquad \lVert T_{W_n}\rVert_{2\to2}=1. ]

Its unit eigenvector (\sqrt n\mathbf 1_{[0,1/n]}) becomes an infinite value on a single Loeb-null cell. Standard part is undefined at that cell; taking it off the exceptional set gives the zero almost-everywhere shadow even though the vector's class in the abstract nonstandard Hilbert hull still has norm one.

What NSA revealed

The moment threshold is really a concentration threshold. If the row and column energy densities

[ r_W(x)^2=\int|W(x,y)|^2dy, \qquad c_W(y)^2=\int|W(x,y)|^2dx ]

are uniformly integrable, then functional-cut convergence to zero forces (L^2)-operator convergence to zero. An (L^p) bound with (p>2) guarantees that uniform integrability because the energy densities are bounded in (L^{p/2}).

So the phase transition is:

• above (2): a bounded moment prevents spectral mass from concentrating on Loeb-null sets;
• at (2): a one-cell spike carries a full eigenvalue while every bounded cut test tends to zero.

The paper proves both the sharp interpolation theorem and this more intrinsic row-energy visibility theorem.

Honest status

This repository presents a candidate discovery and an apparently unrecorded sharp formulation, not certified priority.

Riesz--Thorin interpolation, Loeb measure, nonstandard hulls, and qualitative (L^p)-graphon spectral convergence are prior art. A targeted search found no source stating the exact finite-(p), constant-one modulus together with its full rectangle equality family, row-energy criterion, and explicit (p=2) Loeb-shadow obstruction. The central inequality is elementary enough that an earlier occurrence remains quite plausible.

The claim ledger records the boundary source by source and lists the priority questions that remain open.

Repository map

Path	Role
paper/spectral-shadow.md	Self-contained theorem note and proofs
CLAIM_LEDGER.md	Proved claims, prior art, uncertainty, and forbidden overclaims
SPEC.md	Frozen research contract and verification gates
scripts/verify_extremizers.py	Typed finite-grid norm and equality checks
tests/test_extremizers.py	Deterministic and seeded randomized regressions
SpectralShadow/FiniteSpike.lean	Sorry-free finite one-cell spike certificate
references.bib	Primary-source bibliography
REPRODUCING.md	Exact build and audit commands

Quick verification

Python dependencies are managed with uv:

uv sync --locked --group dev
uv run python scripts/verify_extremizers.py
uv run pytest
uv run mypy
uv run ruff check .
uv run ruff format --check .

Build the finite Lean certificate with:

lake build

These artifacts verify the finite normalizations, equality family, and endpoint spike. They do not formalize Riesz--Thorin, uniform integrability, or Loeb measure; the analytic proof is the primary artifact. The sharp theorem uses complex phase tests; random real-matrix checks use the rigorous (\sqrt2) real-to-complex norm comparison, while positive block extremizers have identical real and complex cut norms.

Citation

If you use the formulation before its priority audit is resolved, cite it as a research note and preserve the qualifier “candidate discovery.” Bibliographic metadata is in CITATION.cff.

License

Code is released under the MIT License. The mathematical text may be read, shared, and cited with attribution under CC BY 4.0.