Two papers on capacity-allocation bounds in coordinated systems, and two standalone mathematics papers.
Author. Riley Rook rileyrook@gmail.com
The Tangent Space Cannot See the Discriminant (the Residue Ledger
project). Standalone mathematics paper (draft v0.15, not yet submitted),
independent of the capacity-allocation pair. A local rigidity theorem
for rational Weil classes on polarized abelian 2n-folds of Weil type
(kernel of the infinitesimal Hodge obstruction = the n²-dimensional
Weil period domain; rank n(n+1); local Hodge locus smooth, reduced,
and equal to the Weil locus), certified by sparse exact SymPy
computations in dimensions four and six. In dimension six the
certificates run on both a split (solved) and a nonsplit (open)
discriminant class and return identical infinitesimal data, and an
explicit real conjugacy theorem upgrades this to every order, while
the arithmetic quotients ARE separated — by their rational boundary
(Witt indices 3 vs 2; totally degenerate cusps exist only on the
split side; local difference exactly at the primes 2 and 3; both
genera of class number one, with a dyadic type invariant separating
the boundary data and predicting cusp spectra (2,2,1) vs (1,1), with the corank-one
counts 2 vs 1 proved unconditionally): the tangent
cannot see the discriminant, the marked real-analytic germ cannot
see it, the rational boundary can. Includes a K3 period-loop
instrument and an interpretive companion essay. See
residue-ledger/README.md.
The Prime Torus Cannot See the Critical Line (draft v0.1, not yet
submitted). Standalone mathematics paper, sibling to residue-ledger:
the same blindness-and-coupling architecture at the number-theoretic
substrate. The Möbius lift of the integers' free multiplicative
structure to the infinite prime torus is critical at exponent 1/2
unconditionally (energy ζ(2σ)/ζ(4σ)); unimodular twists are unitary
rotations, so all unmarked torus statistics are twist-blind while
Helson zeta functions realize essentially arbitrary zero sets — and
symmetrically, the exact ζ functional-equation data (gamma factor,
trivial zeros, counting law TO ALL ORDERS) provably contains both
RH-true and RH-false spectra (witnesses: Nakamura's exact-gamma-factor
functions times an explicit FE-symmetric quartic; distributional
realization in Burnol's property-S class). The Riemann Hypothesis is
repositioned as a marked restriction statement (Littlewood's
criterion, decomposed as free field → Archimedean height cutoff →
aligned phase), and the finite Möbius inverse defect gets an exact
leakage ledger: polynomial windows redistribute the boundary defect
into the generalized von Mangoldt tower, with Selberg's Λ₂ appearing
at degree two — the taper ladder meets the sieve parity barrier. All
finite identities machine-certified in exact arithmetic. Makes NO
claim of progress on RH: the results are no-go statements, a ledger,
and an assembled architecture, with everything classical cited as
such. See prime-torus/README.md.
Capacity-Allocation Predicts a Coordination Ceiling at k ≈ 7 ± 2. Workshop-shaped (~8 pp excluding references). Concentrated empirical core: a single equation k_max = B / [(1−φ)c + φσ], a structural argument for the substrate-invariant ceiling k̄ ∈ [4, 10] from K log₂(K) entropy combined with working-memory-class B, eight substrates clustering in the band, and an own-collected pre-registered cross-architecture probing experiment across four open-weight LLMs. The experiment returns a mixed verdict (one cross-architecture confirmation of the concept-probe middle peak in 4/4 models; a scope-bounded falsification of a literature-anchored magnitude band that sharpens what the framework actually commits to; one substantive structural finding about hidden-state language preservation), and is reported honestly rather than tidied into a clean confirmation.
The experiment artifact (pre-registration commit 708f13f,
ADDENDA 001/002/003, source, configs, processed data, results,
plots) lives at convergence/experiment/. The heavy caches
(downloaded raw data, intermediate activations, virtualenv) are
not included; the README and requirements*.txt are sufficient
to reproduce.
A Capacity-Allocation Framework for Coordinated Systems. Long-form companion (arXiv preprint or journal venue). Develops four extensions of the bound: three-mechanism taxonomy (mechanism-1 / mechanism-2 / mechanism-3, with mechanism-1 as the terminal-depth special case of mechanism-3); projection-class taxonomy (P_3-rich / P_3-chunked / P_3-recognition); per-direction multi-channel generalization with the φ-bimodality theorem; cross-depth propagation form with three continuity constraints; the architectural lever; a 19-row cross-mechanism table.
The convergence paper stands alone and is the recommended entry point. The framework paper assumes the empirical convergence is real and develops the structural framework that organizes it. Each paper cross-references the other where appropriate but does not require it.
Both papers are authored in Markdown. To build PDFs:
./build.sh # builds both papers
./convergence/build.sh # builds the convergence paper only
./framework/build.sh # builds the framework paper only
Requires pandoc and a LaTeX engine that handles Unicode well
(xelatex or lualatex). On macOS:
brew install pandoc
brew install --cask mactex # or basictex + missing-package install
On Debian/Ubuntu:
sudo apt-get install pandoc texlive-xetex texlive-fonts-recommended \
texlive-latex-recommended texlive-latex-extra
PDF outputs land at convergence/paper.pdf and
framework/paper.pdf (gitignored; rebuild on demand).
These papers are part of the surrounding repository and inherit its license (Apache 2.0).
v0 drafts, 2026-05-16.