Euler characteristics and duality in Riemann functions and the graph Riemann-Roch rank
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
By a Riemann function we mean a function f : Z n Z such that f (d) = f (d 1 , . . ., d n ) is equals 0 for deg(d) = d 1 + + d n sufficiently small, and equals deg(d) + C for a constant, C, for deg(d) sufficiently large.For such an f , for any+ C which we call a generalized Riemann-Roch formula.Our motivation for this definition is that (1) adding 1 to the Baker-Norine rank function of any graph yields a Riemann function; and (2) for the results below, we need to consider non-negative valued functions f .We demonstrate a class of Riemann functions f : Z 2 Z that are modeled by sheaves, M d with d Z 2 over a finite topological space, that models the associated generalized Riemann-Roch formula as expressing the Euler characteristic: the nonzero Betti numbers of M d are the zeroth and first, which respectively equal f (d) and f K (K -d).The sheaves M d satisfy many properties akin to the sheaves that model the classical Riemann-Roch formula as expressing an Euler characteristic.Any Riemann function f : Z 2 Z can be written as the difference of two functions modeled by sheaves, so that the generalized Riemann-Roch formula of f is modeled as an Euler characteristic formula of a family, {M d } dZ 2 , of virtual (i.e., a formal difference of) sheaves.We do the same for any Riemann function f : Z n Z with n 2, by restricting any n -2 of its variables, and varying the remaining two variables.We show that the resulting family of virtual sheaves obtained, {M d } dZ n , are-up to isomorphism-independent of all the choices made.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it