changjonathanc

A Short Proof of Graffiti.pc Conjecture 143

Status: Proof candidate pending independent expert review. No claim of publication priority is made.

Abstract

Graffiti.pc Conjecture 143 asserts that if $G$ is a finite simple connected graph that is not a tree, then

changjonathanc

Let me think about this problem. Codeforces contest 2237 problem H — I don't know this specific problem, but let me derive.

Slime occupies a connected set of m vertices in a tree. A move: remove one occupied vertex and add an unoccupied vertex, keeping connectivity and size m. Food appears sequentially at vertices; slime must occupy the vertex to eat. f = min moves to eat all foods in order, starting from initial set. Queries are online: each query appends a food vertex and asks total minimal moves.

Key: online with q up to 1e5, sum n 1e5. We need incremental computation.

Classic result: a connected set of size m in a tree is like a "subtree of size m". Moving it toward a target vertex v: cost to reach v = distance from v to the set (dist to nearest occupied vertex). Actually each move can shift the slime one step: remove a leaf of the occupied subtree, add adjacent vertex toward v. So cost to make v occupied = dist(v, S) where S is current set. But then subsequent foods depend on the resulting shape of S,

changjonathanc

Below is a consolidated reference to the current Model‑Context Protocol (MCP) features that govern how I think, reason, and format replies—excluding the tool catalog itself.

---

1. Instruction‑layer hierarchy

• System → Developer → User: Every response is shaped by this ordered stack. Higher‑layer rules always override lower‑layer rules.
• Guarantees that user‑level preferences are honored unless they conflict with developer or system‑level safety or policy constraints.

2. Multi‑channel response model

changjonathanc

#!/usr/bin/env python3

import argparse
import base64
import hashlib
import json
import os
import secrets
import sys
import time

changjonathanc

import asyncio
import logging
import os
import pty
import pwd
import subprocess
from contextlib import asynccontextmanager

import uvicorn
from fastapi import FastAPI, HTTPException, Request, Response

changjonathanc

import torch

def precompute_freqs_cis(dim: int, end: int, theta: float = 10000.0, dtype=torch.float32, mod=True):
    freqs = 1.0 / (theta ** (torch.arange(0, dim, 2)[: (dim // 2)].float() / dim))
    t = torch.arange(end, device=freqs.device)  # type: ignore
    angles = torch.outer(t, freqs) # type: ignore
    if mod:
        angles = angles % (2 * torch.pi)
    # Now convert angles to the desired lower precision
    angles = angles.to(dtype)

changjonathanc

"""Generates a document causal attention mask based on a document ID tensor"""

from typing import List, Union

import torch
from torch import Tensor
from torch.nn.attention.flex_attention import _mask_mod_signature, or_masks
from attn_gym.masks import causal_mask

changjonathanc

Run huggingface transformers on M1 Macs, on GPU

• Requirement: macOS 12 Monterey

• First, install a conda env with M1 support

wget https://github.com/conda-forge/miniforge/releases/latest/download/Miniforge3-MacOSX-arm64.sh
chmod +x ~/Downloads/Miniforge3-MacOSX-arm64.sh
sh ~/Downloads/Miniforge3-MacOSX-arm64.sh

changjonathanc

from transformers import AutoTokenizer, AutoModelWithLMHead
  
tokenizer = AutoTokenizer.from_pretrained("gpt2")

# model = AutoModelWithLMHead.from_pretrained("gpt2")

import re

nums = []

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