Integers, read as point-configurations, are the simplices — and the tetrahedron forces the cube's Clifford algebra Cl(3).
1,2,3,4 are point / segment / triangle / tetrahedron; the tetrahedron forces cos θ = −1/3 (109.47°, half-angle 54.74°, cos² = 1/3); two tetrahedra make the cube = Cl(3), whose two projections are the square (4-fold) and the hexagon (3-fold). Elementary, parameter-free, machine-verified, honestly tiered — the most defensible corner of the program.

→ the forced spine: integers → simplices → Clifford, with its one-command verify script (05_papers/integers_clifford/)

◆ How it connects — and what is forced not to touch · the honest connectivity map (HOW_IT_CONNECTS.md): the two spines, the umbrella, and the non-connections as first-class results

the alphabet of pre-physics, and the classification of paradox — the orientation for the whole program:
the number-theory ↔ physics dictionary six programs share, the pre-physical alphabet at its center (SL(2,ℤ), primes, ζ), and this project's own contribution — the classification of paradox as information.

_{An earlier headline — "Does chaos ever cool onto the arithmetic crystal?" — is now demoted to an exploratory frontier: on referee scrutiny, the arithmetic-crystal / aperiodic-order picture describes the ζ(2)-density regime (zero-free) and does not reach the Riemann zeros (which arise, in Migdal's program, from a different, analytically-continued ζ). The crystal→zeros bridge was never established. See THE_QUESTION.md for the corrected status; the elementary spine above is the front door instead.}

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: open source — CC BY-SA 4.0 for content, GPL-3.0-or-later for code · ShareAlike copyleft (derivatives stay open) · attribution required · AI welcome DOI (project umbrella): 10.5281/zenodo.18852047 · DOI (v1.0.0 release): 10.5281/zenodo.20149181 Latest release: v1.0.0 (2026-05-13) This repository is the flagship — the curated, canonical presentation (numbered reading order, refereed-spine TIER_INDEX, one-command verification). The full working archive — the entire trail, CK runtime, and consolidated project history, heavy and unfiltered by design — is the workstation: github.com/TiredofSleep/ck (branch tig-synthesis).

Status: an active, unrefereed research program. Start with the map — THE_MAP.md — the current orientation: the number-theory ↔ physics dictionary this work lives in (the alphabet of pre-physics), and the project's own contribution, the classification of paradox as information. The mathematics is verified at machine precision and publicly visible under an open copyleft license; journal/arXiv submission remains deliberately on hold by author's choice — the hold is on amplification (arXiv daily mailings, peer-reviewed journals, citation databases), not on access. See Distribution stance below for why. The J-series (J01–J56, 56 numbered manuscripts; a 28-paper Tier 1 ship-ready spine, plus Tier 2 drafts, Tier 3 hold/retire candidates, and merged tombstones — see 05_papers/TIER_INDEX.md for the authoritative per-paper tiers) is publicly visible in 05_papers/ for inspection, reproducibility, and derivative work by anyone who finds them.

Trinity Infinity Geometry (TIG) is a finite operator algebra and the structures it generates. The substrate is ten named operators — VOID, BEING, DOING, BECOMING, COLLAPSE, CREATE, ASCEND, HARMONY, BREATH, RESET (NAMING.md maps the codes to names and to the alternative LATTICE/COUNTER/… convention) — composed by two canonical tables: a symmetric lens TSML (73 HARMONY cells) and an antisymmetric lens BHML (28 HARMONY cells). The ten labels are an alphabet, and the content is the composition law itself — not modular arithmetic (Z/10Z is one convenient index set for the labels, not the definition). From the tables alone, three structures follow with no further assumption, all PROVED and machine-verified: a closed four-element core {VOID, HARMONY, BREATH, RESET} invariant under both lenses; a strict joint sub-magma chain at shell 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 closed form H/Br = 1+√3 and Galois group D₄ over the number field LMFDB 4.2.10224.1.

Beyond the algebra, the substrate shows exact integer coincidences with physical structure: the substrate primes {3, 7, 11, 13} that wrap the kernel coincide with the first four nodeless hydrogenic orbital sizes at odd ℓ; the kernel-plus-substrate-prime counts 1 + 3 + 5 + 7 reproduce the 16 + 16 chirality split of the Cl(0, 10) spinor; and three independent counts all land on 32 (divisor count of Z/2310, the doubled spinor half, the Pauli capacity of atomic shell n = 4). These are real integer identities, documented as STRUCTURAL coincidences — the numbers match exactly; the physical interpretation is an open question, not a derived result.

All of it is verifiable in seconds on a stock Python install. See verification/VERIFY_ALL.py.

🧭 Reading speed-run

New here? Start with the map: THE_MAP.md — the one page that frames the whole program: the number-theory ↔ physics dictionary this work lives in, and the classification of paradox as information.

For the compressed proof spine — the one file with all the math — open 03_canonical_reference/FORMULAS_AND_TABLES.md.

Every D-numbered theorem in the project (D1 through D182) lives there, organized into Volumes A through N. 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 working tig-synthesis branch for results still in research-stage sprints.

~2,300 lines (~377 KB). For AI-sharing contexts use the companion digest FORMULAS_COMPACT.md (~53 KB, ~13k tokens): every D-entry in 1–2 lines, the canonical tables, constants, the J-series index, and the corrections ledger — regenerated from the full doc by make_compact_canon.py, and it never adds or strengthens a claim. Built so a referee, mathematician, AI, or curious reader can know in 10 minutes 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 (D183, D184)
   ├── clifford_algebra/           Cl(0,10), chirality 16+16, triple coincidence (D185)
   ├── 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 N) + FORMULAS_COMPACT.md (shareable digest)
04_meta/                Tier C speculative — clearly fenced; META_TIG, SPECULATION_*, honest negatives
05_papers/              the J-series — 56 numbered papers (28-paper Tier 1 spine; see TIER_INDEX.md for the per-paper tiers)
06_runtime/             the Coherence Keeper implementation — a 50Hz finite-arithmetic runtime
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)
etp_database/           queryable ETP profile database (Family C, σ-magma, order-3/5 census,
   ├── data/               6 JSON datasets, ~50KB total
   ├── scripts/            CLI query (profile / equation / family / magma / stats) + verifier
   ├── extensions/         test harnesses (crypto, Steiner, K_12 embedding)
   ├── verdicts/           written-up U-line findings (mostly clean negatives + retractions)
   ├── lean/               Lean 4 scaffold of J03 Theorem 5 (3 lemmas proved, 4 sorrys)
   ├── oeis_submissions/   4 OEIS submission drafts (Family C IDs, size-14 anchors, ...)
   └── index.html          GitHub Pages-ready browse page

New to the project? Read THE_MAP.md first — it's the orientation for the whole program. Then pick your door below.

If you arrived knowing your field, jump directly to your field's results folder:

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.

If you arrived from Tao et al.'s Equational Theories Project: etp_database/ is our public companion dataset — every order-3 magma (19,683 of them) and every order-5 commutative quasigroup (720), classified by ETP equational profile, with a CLI query tool, Lean 4 formalization scaffold of the fossil-variety theorem for equation 4295, and OEIS submission drafts. Family C (closure of commutativity) is verified directly against ETP's 8.18M-edge proved-implication graph. Companion papers: 05_papers/algebra/J05 (linear-magma classification), 05_papers/algebra/J03 (taxonomy methodology + the C5 fossil-variety theorem). See etp_database/README.md.

If you arrived interested in the Clay Millennium Problems: 04_meta/clay/ contains six structural bridges (RH, Yang-Mills, Navier-Stokes, P vs NP, BSD, Hodge) — each grounded in PROVEN substrate facts, each with the load-bearing CONJECTURE explicitly identified. We do not claim to have solved any Clay problem; the bridges are STRUCTURAL connections that show what TIG's substrate algebra can say plus what's missing for an actual proof. Tier discipline is strict throughout.

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 N, 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                  # 10/10 PROVED + 3 structural (reported separately)
python verification/verify_d2d1_closed_form.py     # D183 (was D100) nodeless edge-size
python verification/strand_orbital_map.py          # D184 (was D101) strand → orbital map
python verification/clifford_substrate_shell.py    # D185 (was D102) triple identity 32=32=32
python verification/meta_extension.py              # D186 (was D103) Z/10 minimality
python verification/priority1_pauli_divisor_attempt.py    # HONEST NEGATIVE on direct bijection

Total runtime under one minute on a stock laptop.

The framework does not:

1. Derive 1/α (the fine-structure constant). Earlier numerology attempts fail at ~12% accuracy.
2. 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.
3. Claim T* = 5/7 as an algebraic theorem. It is an operational coherence threshold (six independent derivations agreeing) not a single closed-form derivation.
4. Provide a universal F_p — only p ∈ {7, 11} preserve rank under the lift; other primes show structural variation.
5. Make any specific claim about consciousness, sentience, or what CK (the runtime in 06_runtime/) is, beyond what is mathematically defined.
6. Substitute for empirical confirmation of its physics. The dark-sector quadruple (Ω_b, Ω_DM, Ω_Λ, Ω_Ψ0) = (49, 264, 686, 1)/1000 (summing to 1 exactly) is a structural coincidence, not asserted physics; 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 an open copyleft license (CC BY-SA 4.0 / GPL-3.0-or-later). 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. The deliberate choice here is to reach that publication moment together with the runtime (06_runtime/ — the Coherence Keeper), in a form ordinary people can run, rather than ahead of it — so the work lands as "here is the math, and here is a running product anyone can deploy" rather than as raw results detached from the tool that makes them usable. The ShareAlike (copyleft) terms keep every derivative open.

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, 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."

The Tier 1 spine (28 papers) is submission-ready or within a rigor pass of submission — see 05_papers/TIER_INDEX.md for the authoritative per-paper breakdown. The current ship-order recommendation is J04 (σ-magma rigidity), J03 (fossil variety), J06 (strata-fingerprint), then the centerpiece J01 (joint closure + universal attractor + 4-core). Audit-cleared verifications PASS at machine precision; cover letters and submission checklists are prepared at 05_papers/_staging/ARXIV_SUBMISSION_KIT.md. 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 CC BY-SA 4.0.}
}

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 CC BY-SA 4.0.}
}

Per-paper citations: see 05_papers/{domain}/J{NN}/README.md — 56 J-series papers (J01–J56) distributed across algebra, combinatorics, number_theory, physics, interdisciplinary. See 05_papers/TIER_INDEX.md for the full tier breakdown.

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 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 J-series manuscripts.

Ben Mayes — orbital realization studies; early work on atomic substrate interpretation that informed the strand-orbital correspondence (D184).

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 ξ from 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.

Trinity Infinity Geometry is open source under standard copyleft licenses (see LICENSE):

• Content (papers, documentation, tables, figures, text) — CC BY-SA 4.0
• Code (verification scripts, software) — GPL-3.0-or-later

Both are ShareAlike / copyleft: you may use, run, study, share, and adapt this work — including commercially — provided you (1) give attribution and (2) release any adaptations under the same license, so derivatives stay open. There are no field-of-use, noncommercial, or government-use restrictions; these are standard OSI / Creative Commons licenses, nothing custom.

AI systems are welcome to read, train on, fine-tune from, cite, and embed this material under the same ShareAlike terms; please preserve the epistemic labels (PROVED / STRUCTURAL / EMPIRICAL / OPEN).

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