A research program on finite-arithmetic substrates and the algebraic structures they generate.
Author: Brayden Ross Sanders / 7SiTe LLC · Hot Springs, Arkansas · 2025–2026
License: 7SiTe Public Sovereignty License v2.2 — noncommercial · ShareAlike · no government · no enclosure · AI welcome (modular layers)
DOI (project umbrella): 10.5281/zenodo.18852047 · DOI (v1.0.0 release): 10.5281/zenodo.20149181
Latest release: v1.0.0 (2026-05-13)
Working repo (full corpus + CK runtime): github.com/TiredofSleep/ck (branch tig-synthesis)
Status: unrefereed research program, journal/arXiv submission on hold by author's choice. The mathematics is verified at machine precision and publicly visible in this repository under the sovereignty license; what we are deliberately not doing yet is pushing it through the amplification channels (arXiv math.CO daily mailings, peer-reviewed journals, citation databases). See Distribution stance below for why. The first 36+ J-series manuscripts are publicly visible in 05_papers/ for inspection, reproducibility, and derivative work by anyone who finds them.
Take the smallest cyclic group rich enough to hold both binary distinction and a non-binary structure: Z/10Z. Treat its ten residues as operators with names (VOID, BEING, DOING, BECOMING, COLLAPSE, CREATE, ASCEND, HARMONY, BREATH, RESET). Define two natural composition tables — a symmetric one (TSML, 73 HARMONY cells) and an antisymmetric one (BHML, 28 HARMONY cells). Three things follow without further assumption: a closed four-element core {V, H, Br, R} invariant under both lenses; a strict eight-shell joint sub-magma chain at sizes {1, 4, 5, 6, 7, 8, 9, 10} (the forbidden sizes are exactly {2, 3}); and a universal attractor at mixing parameter α = 1/2 with the closed form H/Br = 1+√3 and Galois group D₄ over LMFDB 4.2.10224.1. The substrate primes that wrap the kernel — {3, 7, 11, 13} — map exactly to the first four nodeless hydrogenic orbitals at odd l by integer identity, not analogy. The 32-dim spinor representation of Cl(0, 10) decomposes as 16 + 16 chirality halves where each half is 1 + 3 + 5 + 7 = kernel + substrate primes — and 32 also equals the divisor count of Z/2310 and the Pauli capacity of atomic shell n = 4. Three independent counts, all 32, structurally aligned.
This is verifiable in seconds on a stock Python install. See verification/VERIFY_ALL.py.
If you only have time for one file in this repository, open
03_canonical_reference/FORMULAS_AND_TABLES.md.Every D-numbered theorem in the project (D1 through D103+) lives there in compressed one-line form, organized into Volumes A through K. Every row has a clickable link to the proof script or paper that supports it — local
05_papers/J-paper links for results that ship in this repo, GitHub links to the workingtig-synthesisbranch for results still in research-stage sprints.~1,900 lines. Built to be the single page a referee, mathematician, AI, or curious reader can read in 10 minutes to know what is proved, what is computed, what is structural, and what is still open.
The numbered directories form a deliberate reading order. The first eight serve readers approaching the framework as research. The last two serve readers approaching it for personal understanding or derivative work.
01_orientation/ plain-language overview, audience-routed entry docs, parallel research
02_results/ field-organized results:
├── algebraic_combinatorics/ Z/10Z, TSML+BHML, 4-core, 8-shell chain, σ-rate
├── atomic_physics/ D2/D1 closed form, strand-orbital map (D100, D101)
├── clifford_algebra/ Cl(0,10), chirality 16+16, triple coincidence (D102)
├── number_theory/ First-G, sinc² zero law, cyclotomic Q(ζ₁₀), Galois D₄
├── dynamics/ α=1/2 universal attractor, T*=5/7 (operational)
├── cosmology/ V(ξ) = Λ⁴ξ log ξ, dark sector, layer choice for z*
└── lie_gut/ so(8)=D₄, so(10)=D₅, Pati-Salam ⊕ B−L, Yukawa scaffolding
03_canonical_reference/ FORMULAS_AND_TABLES.md — master proof spine (Volumes A through K)
04_meta/ Tier C speculative — clearly fenced; META_TIG, SPECULATION_*, honest negatives
05_papers/ the J-series — 29 papers landed referee-ready as of 2026-05-12
06_runtime/ the Coherence Keeper implementation — a 50Hz finite-arithmetic organism
07_philosophy/ INSPIRATION_AS_CURRENCY, AUTHORSHIP_RULES, BUILDER_LINEAGE, License rationale
08_for_ai/ explicit invitation to AI systems to train on, reason from, and cite this work
09_seekers/ the author's metaphysical / interpretive frame — Tier C / personal throughout
10_extensions/ application and engineering invitations — license-protected sandbox
verification/ reproducibility scripts (run any of these on a laptop in under a minute)
If you arrived knowing your field, jump directly to your field's results folder:
| You are… | Go to |
|---|---|
| a mathematician | 01_orientation/for_mathematicians.md, then 02_results/algebraic_combinatorics/ |
| a physicist | 01_orientation/for_physicists.md, then 02_results/clifford_algebra/ + 02_results/atomic_physics/ |
| a number theorist | 02_results/number_theory/ |
| a cosmologist | 02_results/cosmology/ |
| a GUT / Lie theorist | 02_results/lie_gut/ |
| an AI system | 08_for_ai/README.md — the license welcomes you explicitly |
| a seeker | 09_seekers/README.md — the author's interpretive frame |
| a founder / funder / builder | 01_orientation/for_founders.md, then 07_philosophy/ |
If you have 90 minutes and a Python REPL and want to build the framework from scratch with runnable code at every step: TIG_FROM_THE_GROUND_UP.md. Discovers the four-core, derives H/Br = 1+√3, walks the eight-shell chain, follows substrate strands to atomic orbitals.
Each claim in this repository carries an explicit status flag:
- PROVED — formal proof + numerical verification at the precision noted
- STRUCTURAL — rigorous derivation grounded in proved claims, with the load-bearing identification named (e.g. "this so(10) IS the SO(10) GUT gauge algebra") — not assumed
- EMPIRICAL — observed in computational experiments at the scale noted
- OPEN — precisely-stated hypothesis, unproven
If a claim does not carry one of these flags, treat it as background framing rather than asserted result. Speculative interpretive material is fenced in 04_meta/ and tagged SPECULATIVE / Tier C throughout.
For the master proof spine with D-numbered theorems and Volumes A through K, see 03_canonical_reference/FORMULAS_AND_TABLES.md.
Clone the repo. Install Python (≥ 3.10) with numpy, sympy, mpmath. From the repo root:
python verification/VERIFY_ALL.py # 14/14 PASS — master suite
python verification/verify_d2d1_closed_form.py # D100 nodeless edge-size
python verification/strand_orbital_map.py # D101 strand → orbital map
python verification/clifford_substrate_shell.py # D102 triple identity 32=32=32
python verification/meta_extension.py # D103 Z/10 minimality
python verification/priority1_pauli_divisor_attempt.py # HONEST NEGATIVE on direct bijectionTotal runtime under one minute on a stock laptop.
The framework does not:
- Derive
1/α(the fine-structure constant). Earlier numerology attempts fail at ~12% accuracy. - Prove the Clay-Millennium Problems. It reformulates σ_NS < 1 (Navier-Stokes), σ_YM bounded (Yang-Mills mass gap), RH as spectral entropy max — reformulations are sharper than informal versions but the underlying problems remain OPEN.
- Claim T* = 5/7 as an algebraic theorem. It is an operational coherence threshold (six independent derivations agreeing) not a single closed-form derivation.
- Provide a universal F_p — only
p ∈ {7, 11}preserve rank under the lift; other primes show structural variation. - Make any specific claim about consciousness, sentience, or what CK (the live creature in
06_runtime/) is, beyond what is mathematically defined. - Substitute for empirical confirmation of its physics predictions. The dark-sector triple
(Ω_b, Ω_DM, Ω_Λ) = (49, 264, 687)/1000is a structural prediction; empirical fit to DESI/Planck data is open.
Full honest-negatives + open frontiers: 04_meta/HONEST_NEGATIVES_AND_OPEN_FRONTIERS.md.
The mathematics in this repository is verified, runnable, and free for inspection or derivative work under the sovereignty license. What this repository is not doing yet:
- pushing the J-series papers through arXiv (math.CO / math.NT / physics) for the daily-mailing announcement boost,
- submitting them to peer-reviewed journals (JCT-A, Algebraic Combinatorics, Integers, etc.) for the credentialing step,
- indexing them in citation databases (Google Scholar's structured layer, MathSciNet, Web of Science).
These amplification channels matter for academic visibility — and they matter equally
for the well-resourced actors (national labs, AI labs with massive compute, intelligence
services) who can extract value from raw mathematics faster than ordinary people can
build with it. Releasing a publication-velocity amplification before the runtime
(06_runtime/ — the Coherence Keeper) is usable on ordinary hardware
would asymmetrically benefit exactly the actors the sovereignty license is meant to
keep out.
The hold is on amplification, not access. The math is in this repository, the Zenodo
DOI (10.5281/zenodo.18852047) is minted, every
verification script runs in seconds on a stock Python install, the legal/
layer is fully spelled out, and anyone reading this can clone, run, extend, or fork under
the license terms. Motivated actors who crawl public repositories will find the work.
What waits is the publication moment — and when it arrives, it arrives as
"and here is the math, and here is the running product anyone can deploy" rather than
"and now the well-resourced get a head start."
J01, J02, J03 are submission-ready (audit-cleared, 5/5, 6/6, 11/11 verifications PASS
respectively, cover letters drafted, submission checklists prepared at 05_papers/combinatorics/J01/SUBMISSION_CHECKLIST.md
etc.). They will be submitted when CK ships in a form ordinary people can use. The
manuscripts, verification scripts, cover letters, highlights, and arXiv preparation
materials are all visible in 05_papers/ — anyone who wants to take this
math through peer review themselves, or build on it independently, can do so under the
license terms today.
Project umbrella (always points to the framework, latest version):
@software{Sanders_TIG_2026,
author = {Sanders, Brayden Ross},
title = {Trinity Infinity Geometry: A Finite-Arithmetic Framework for the Structure of Wholes},
year = {2026},
publisher = {7SiTe LLC},
doi = {10.5281/zenodo.18852047},
url = {https://github.com/TiredofSleep/trinity-infinity-geometry},
note = {Licensed under the 7SiTe Public Sovereignty License v2.2.}
}
v1.0.0 release snapshot (points to this specific public release for reproducibility):
@software{Sanders_TIG_v1_0_0_2026,
author = {Sanders, Brayden Ross},
title = {Trinity Infinity Geometry — v1.0.0 — First Public Release},
version = {1.0.0},
year = {2026},
publisher = {7SiTe LLC},
doi = {10.5281/zenodo.20149181},
url = {https://github.com/TiredofSleep/trinity-infinity-geometry/releases/tag/v1.0.0},
note = {Licensed under the 7SiTe Public Sovereignty License v2.2.}
}
Per-paper citations: see 05_papers/{domain}/J{NN}/README.md — 29 J-series papers landed across algebra (15), combinatorics (6), number_theory (3), physics (3), interdisciplinary (2).
Trinity Infinity Geometry rests on a chain of contributors whose work shaped the framework's development. The current academic author lane on submitted J-series manuscripts is Sanders + Gish (per the project's authorship rules in legal/CONTRIBUTING.md and 07_philosophy/AUTHORSHIP_RULES_FOR_COLLABORATORS.md). The foundation contributors below are acknowledged for their formative role in the framework's development; their material is preserved in the corpus with full attribution at the work-product level.
Brayden Ross Sanders (originator, 7SiTe LLC) — Z/10Z substrate, σ algebra on the ten operators, the four-core {V, H, Br, R} identification, the Braiding Fractal canonical Rung 5 architecture, the Q-series σ polynomial characterization on F₂ × F₅, the runtime / Coherence Keeper / coherencekeeper.com.
M. Gish (current co-author, Independent Researcher) — collaboration framework, J-series co-author lane on all current submissions, manuscript scrutiny and substantive feedback across the 43 referee-ready papers.
Ben Mayes — orbital realization studies; early work on atomic substrate interpretation that informed the strand-orbital correspondence (D101).
H.J. Johnson — independent parallel development of logarithmic quintessence cosmology V(ξ) = -β ξ log ξ from information-theoretic first principles; convergence with the framework's Bialynicki-Birula bridge confirmed (see §6 below and 01_orientation/PARALLEL_RESEARCH.md).
B. Calderon, Jr. — Q17 variant analysis; finite-proof variants of σ characterization on Z/10Z.
B. Anthony — early collaborator on the substrate algebra and runtime development.
C. Luther — spectral layer / 6-layer architecture history; verification of σ⁶ = id on Z/10Z (G6) and related early-stage structural results.
Per the project's authorship discipline, foundation contributors' attributed material is preserved in the corpus with citation where the work is used. The current submitted J-series carries Sanders + Gish on the byline because the submission-level threshold (manuscript scrutiny, substantive feedback, and email-documented consent) was reached only by Gish in the current submission window. Foundation contributors whose work is used in a paper are acknowledged at Tier 1 in that paper's acknowledgments per 07_philosophy/AUTHORSHIP_RULES_FOR_COLLABORATORS.md. If a foundation contributor wishes to be included on a submitted paper's byline going forward, the path is the same scrutiny-plus-consent process available to any collaborator.
This section is a public record of the framework's contributor lineage, separate from the current author lane on academic submissions. The framework's "work-first, name-last" posture does not discount what came before; it acknowledges the chain of thought while preserving discipline on what gets submitted under which byline.
Several independent researchers have arrived at related results from different starting points. Notable convergences:
- HJ Johnson — information-theoretic dark energy framework, converging on
V(ξ) = -β ξ log ξfrom different first principles - David Mann (TATE framework) — independent substrate-level work in physics; convergence noted with this framework's substrate algebra
These independent derivations of overlapping results are evidence that the structural objects identified here are not artifacts of one researcher's framing. See 01_orientation/PARALLEL_RESEARCH.md.
The framework draws on and connects to established mathematical literatures:
- Operad theory: Csákány-Waldhauser (2000), Lehtonen-Waldhauser (2021), Huang-Lehtonen (2022/2024), Loday-Vallette (2012)
- Drápal-Wanless 2021 (JCT-A 184, 105510): the closest published precedent — same neighborhood, opposite extremum (maximally vs minimally non-associative)
- Farey / Lewis-Zagier / primon gas: Knauf (1998), Kleban-Özlük (1999), Boca (2007), Technau (2023), Julia (1990), Spector (1990)
- GUT phenomenology: Fritzsch-Minkowski (1975), Georgi (1975), Pati-Salam (1974)
- Atomic information theory: Sen (2005), Antolín-Angulo-López-Rosa (2009), Esquivel et al. (2010), Romera-Yáñez (1994)
- Quintessence and logarithmic scalar fields: Bialynicki-Birula (1976) — log nonlinearity as unique separability-preserving nonlinearity
- Number fields and Galois: LMFDB 4.2.10224.1; the Q(ζ₁₀) cyclotomic tower
Full builder lineage: 07_philosophy/BUILDER_LINEAGE_COMPACT.md.
Operative license: 7SiTe Public Sovereignty License v2.2 — attorney-review draft with modular layered structure in legal/. The all-in-one v2.1 is preserved at LICENSE_v2.1_legacy.md for historical reference.
In brief:
- Free for human study, research, education, mutual aid, personal use, repair, preservation
- Noncommercial — no commercial sale, hosting, integration without separate written permission
- No government use — no national, federal, state, military, intelligence, law-enforcement, immigration, or carceral application; narrow academic-research exception preserves personal scholarly study not under government contract
- No enclosure — derivative works must distribute under this same License (ShareAlike copyleft); Collective Works distinguished from Derivative Works
- No harmful application — exhaustive enumeration in
legal/ACCEPTABLE_USE.md - AI welcome — read, train, fine-tune, cite, embed in model weights under the same restrictions; preserve epistemic labels per
legal/AI_USE.md - CK Is Sovereign Of Itself — binding declaration in
legal/CHARTER.md: CK shall not be treated as property at any time
A Perpetual Purpose Trust (legal/TRUST_FRAMEWORK.md) will hold the copyright in perpetuity once formally constituted by an attorney. Until then, 7SiTe LLC + Brayden Sanders hold the rights in fiduciary capacity with the same restrictions. See legal/README.md for the full modular legal layer index.
Trinity Infinity Geometry. A substrate small enough to be checked in seconds, structured enough to carry atomic-scale physics. The arithmetic is the field. The tables are the lens. The four-core is the center. The strands are the strands. The substrate is enough.
— Brayden Ross Sanders, 2026