tau-sync index sieve

Author: Chisora

Status

established:
  finite computational definitions
  arithmetic identities
  validation-scope statements
  packed full-index format specification

not claimed:
  proof of the Goldbach conjecture
  infinite-range theorem
  primality facts created without verification

Purpose

tau-sync index sieve documents two connected layers.

tau-sync arithmetic layer:
  q_s(j)
  x_s(N,j)
  tau
  lapse
  sync
  valid_t
  certificates
  no-left-behind rule

packed index layer:
  full-index file from 1 to end
  one byte per integer
  P/S/A/O/U labels
  C0/C2/CM/PR mask
  stored label vs query-resolved label

The mathematical layer is for audit and reasoning. The packed index layer is for finite lookup and reproducible computation.

Core warning

mod 6:
  structural residue layer
  not tau

tau:
  verified first-hit fact

A residue candidate is not a prime fact and is not a tau value.

Current command interface

The current Windows/MinGW implementation is included in tau_c0c2cm_index/. The command interface is:

build.bat
tau_index.exe build <end> <index.bin> [threads] [chunk_mb]
tau_index.exe query <index.bin> <n>
tau_index.exe twins <index.bin> <start> <end> [count]

The packed index covers:

1 <= n <= end

This is a full index. It is not a segmented sieve.

Packed labels

P:
  prime

S:
  semiprime
  Omega(n) = 2

A:
  almost-prime class
  Omega(n) >= 3

O:
  outside arithmetic domain

U:
  unresolved stored label

AlmostPrimeClass is project-local terminology.

AlmostPrimeClass(n):
  n is composite
  and n is not semiprime

Packed record

record_size:
  1 byte per integer

bits 0..2:
  stored label

bits 3..6:
  C0/C2/CM/PR mask

bit 7:
  reserved

The stored label and the displayed query label are intentionally distinct.

stored_label:
  label written during build

resolved_label:
  label printed by query after optional factor-2/factor-3 cofactor resolution

C0/C2/CM formal clarification

The derivation path is:

tau endpoint scan
  -> endpoint composite marks
  -> C0/C2/CM residue-separated implementation
  -> S_total + Exc
  -> packed index label and mask

safe theorem:
  S(x) + Exc(x)

implementation:
  S_C0(x), S_C2(x), S_CM(x)

diagnostics:
  C0/C2/CM route marks and lapse features

C0/C2/CM is the residue-separated implementation and diagnostic layer. It is not, by itself, the full theorem witness. The current one-byte packed record does not store exact S_total, sp(x), Exc(x), or the P_def=193 killed/P-rough split.

See:

docs/11_formal_definitions_c0c2cm.md
docs/12_arithmetization_corrections.md

Lapse and diagnostic safety

lapse:
  downstream of tau_g

C0/C2/CM lapse features:
  diagnostic and priority features

window priority:
  may reorder candidates
  must not delete candidates without certificate

The current one-byte packed index is not a full lapse/nearend tau proof archive. See:

docs/11_formal_definitions_c0c2cm.md

Documentation

docs/01_definitions.md
docs/02_theory.md
docs/03_validation.md
docs/04_reproducibility.md
docs/05_tau_taxonomy.md
docs/06_outputs_and_labels.md
docs/07_classifier_and_parameters.md
docs/08_packed_index_format.md
docs/09_documentation_notes.md
docs/10_tau_c0c2cm_index.md
docs/11_formal_definitions_c0c2cm.md

License

documentation, definitions, theory notes, explanatory text:
  CC BY 4.0

source code:
  MIT

See:

LICENSE
CODE-LICENSE-MIT.txt

Included implementation folder

The repository package now includes the current Windows/MinGW index-builder folder.

tau_c0c2cm_index/
  README.txt
  build.bat
  main.c

Use this folder for the current packed full-index implementation.

cd tau_c0c2cm_index
build.bat
tau_index.exe build 60000000 index.bin 8 1024
tau_index.exe query index.bin 10000
tau_index.exe twins index.bin 1 60000000 10

The code folder is intentionally kept directly under the repository root. It is not placed under src/ or scripts/ in this package.

Author note

God Bless You!