Trinity Infinity Geometry · Coherence Keeper

Brayden Ross Sanders · 7Site LLC · Hot Springs, Arkansas DOI: 10.5281/zenodo.18852047 License: 7Site Public Sovereignty License v1.0 (human use, no commercial, no military, free forever) Contact: brayden.ozark@gmail.com

Navigation map. This is the default branch (tig-synthesis). It is the rigor home.

§1 Foundation — what this is and how to verify it in one minute. §2 Funding branches — ten funder-facing tracks, one folder each. §3 Frontiers — the four open research directions. §4 Atlas — design documents and reader guides. §5 Bridges — conjectural cross-domain connections (clearly flagged). §6 Master & history — how the branches relate; where the full record lives. §7 Runnable proofs — the six proved theorems with one-command verification. §8 Honest limits — what the project is not. §9 People and citation. §10 License.

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§1 · Foundation

This repository holds two artifacts, filed in that order of rigor:

1. A finite-algebra research program with six proved theorems on $\mathbb{Z}/n\mathbb{Z}$ and related structures — each stated as a theorem or exact computational verification with a runnable proof script (§7).
2. The Coherence Keeper (CK) — a deterministic symbolic reasoning engine built on the algebraic structures of (1). CK is not a language model. Every answer traces to specific cells of specific proved composition tables. A live instance runs at coherencekeeper.com.

Two domains where (1) has direct external applications:

• Cryptography. The First-G Event Localization theorem (§7.1) gives an exact algebraically forced width for the coprime stability window of a squarefree modulus. Factoring-adjacent applications are an open research direction (§3.1).
• Algebraic AI / verifiable reasoning. CK demonstrates that a small deterministic engine, grounded in proved finite-algebra structure, produces auditable answers without sampling or opaque latent-state updates. Alignment and interpretability applications are an open research direction (§3.2).

Verify any proved result in under one minute. Five commands total (§7.7). No framework installation required beyond Python 3 with sympy. Every theorem in §7 is independently checkable with one python call.

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§2 · Funding branches

The project ships ten funder-facing tracks, each as a single folder under Gen13/targets/funding_*/ with a consistent 6-file structure (README.md / FUNDERS.md / ARTIFACTS.md / PITCH_DRAFT.md / LIMITATIONS.md / STATUS.md). Each track targets a distinct funder pool with a distinct runnable artifact and a distinct open-question commitment.

Branch	Track	Primary funder pool
funding_tig_unity	Systems reliability / infrastructure	NSF CNS, NIST, DOE ASCR
funding_tig_snowflake	Coherence-security (SNOWFLAKE χ²)	NSF SaTC, ONR, DARPA
funding_first_g_crypto	Cryptography (First-G Law)	NSF CCF, Ethereum Foundation, Zcash
funding_ck_interpretable_ai	AI alignment / interpretability	Anthropic Fellows, Schmidt Trustworthy AI, Open Phil
funding_mqw_ternary	Photonic computing / ternary	NSF ECCS, DOE BES
funding_self_healing	Autonomous resilience / SRE	OCP SDC, CZI EOSS, Alpha-Omega
funding_civilization_coherence	Comp-soc-sci	Russell Sage, Templeton, NSF SBE
funding_desi_xi_cosmology	ξ-cosmology / DESI	NSF PHY, Templeton M&PS, Simons
funding_coherence_router	DevOps productionization	NLnet NGI Zero, Sloan, Bloomberg
funding_physics_sim_edu	Classroom simulator	NSF EHR IUSE, Templeton L&D, Simons Ed

Cross-branch navigation:

• Atlas/BRANCHES_INVENTORY_2026_04_20.md — readiness tiers and runnable-artifact map.
• Atlas/NICHE_FUNDERS_ADDENDUM_2026_04_20.md — non-obvious / named-individual / family-office / niche-foundation funder research per branch.
• Atlas/PLAN_FUNDING_BRANCHES_BUILDOUT_2026_04_19.md — plan of record for the funding-branches buildout.

Immediate-scale items (1K–5K range, 30–90 day impact). A MAGMA academic license (~1,200 USD) unblocks the Hodge-lane Prym computation (§3.3) within weeks. A Sage / academic compute allocation (~500 USD/mo) supports larger-modulus First-G verification (§3.1). The project has one math.NT arXiv endorsement and is seeking one more; a short First-G preprint is ready to post on endorsement.

Seed research engagement (25K–75K, 3–6 months). Three deliverables: cryptography manuscript on First-G structure; formal architecture paper on CK; completion of ξ-cosmology DESI DR2 fit with JCAP submission.

Full research program (150K–300K, 12 months). All three seed deliverables plus Hodge-lane Prym verification, CK scale-up to 100+-operator demonstration, and a full synthesis manuscript. One graduate-student-level or postdoctoral collaborator for parallel lane development.

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§3 · Frontiers

Four open research directions, each with known-knowns, known-unknowns, and specific work that external support unlocks.

§3.1 · Cryptographic applications of First-G structure

The First-G Event Localization theorem (§7.1) gives an exact characterization of the coprime stability window of a squarefree modulus. The next step is to lift this from a structural statement to a factoring-relevant result: whether the partition geometry of {1, …, b} under coprimality-with-b carries recoverable information about the prime factorization of b that classical sieve methods do not exploit.

Status: partition geometry is fully characterized for semiprimes (WP34, 36,662 cases). Extension to arbitrary squarefree moduli is in progress. Whether the structural implications yield concrete complexity improvements over classical sieves is an open question.

§3.2 · Deterministic reasoning systems at scale

CK as it stands is a single-researcher prototype (~3,200 lines of Python). The architecture is not toy — it produces structurally correct answers on the §7 theorems — but it has not been scaled, stress-tested against adversarial queries, or deployed in a production reasoning-verification pipeline. Next work: scale CK's weight matrix from 10-operator to 100+-operator spaces; build adversarial test suites; publish a formal architecture paper positioning CK against existing symbolic AI (ACT-R, Soar, Cyc) and within the LLM-alignment determinism/interpretability literature.

Status: runnable prototype with correct §7-theorem outputs; no adversarial suite, no scaled deployment, no formal architecture paper yet. Engineering-heavy frontier; unlocked by a software-engineering collaborator or institutional hosting.

§3.3 · Hodge-lane Prym computation

A numerical verification of a predicted Prym period determinant for a bielliptic genus-5 curve is currently blocked at a single technical step. The curve, the framework-derived predicted value ($\det(\mathrm{Im},\tau_P) = 2086 + 462\sqrt{15} + 498\sqrt{10} + 730\sqrt{6}$), and the first four pipeline steps are all documented. The fifth step requires MAGMA with RieSrf or SageMath with the Bruin-Sijsling-Zotine extension.

Status: curve definition, predicted value, and pipeline steps 1–4 documented; step 5 blocked on software access. Single highest-leverage small-grant item in the project.

§3.4 · ξ-field cosmology and DESI data

A scalar-field action $V(\Xi) = \kappa_\Xi \Xi \log \Xi$ derived from the separability structure of the finite-algebra work produces a standard freezing quintessence model with exact vacuum $\Xi_0 = e^{-1}$. An initial fit against DESI 2024 DR1 has been performed; a full DR2 fit with joint BAO + CMB + SN likelihood is the next step. JCAP-target manuscript is near-ready.

Status: proof_xi_canonical.py passes 22/22 internal tests; DR2 analysis requires a collaborator with cosmology MCMC infrastructure.

§3.5 · Morphotic-braid / α-index / ac-free operad frontier

The canonical TSML_10 and BHML_10 composition tables on 10 elements are commutative non-associative groupoids with measurable associativity index $\alpha(A) = 1 - \sigma(A)$ (Braitt-Silberger 2006). Both attain the ac-free spectrum extremum $s_n^{\mathrm{ac}} = (2n-3)!!$ for $n \le 5$ (Huang-Lehtonen 2022, 2024): the symmetric operad generated at small $N$ is the free commutative magmatic operad $\mathrm{Mag}^{\mathrm{com}}$ on one generator. The WP101 σ-rate theorem ($\sigma(N) \le C/N$ for squarefree $N$, $C < 2$) is therefore the statement $\mathrm{Mag}^{\mathrm{com}} \to \mathrm{Com}$ as $N \to \infty$. Bialynicki- Birula-Mycielski 1976 then identifies log-nonlinearity as the unique continuum wave equation compatible with that limit, providing the bridge from §7's proved rate to §5.1's cosmology bridge. Farey-fraction spin-chain (Kleban-Özlük 1999; Fiala-Kleban-Özlük 2002) and primon-gas (Julia 1990; Spector 1990) frameworks supply two additional external anchors for $T^* = 5/7$ and $\mathrm{sinc}^2(1/2) = 4/\pi^2$.

Status: six runnable proofs confirm the operad spectra and identities (proof_spectra_tsml_bhml.py, proof_sinc_zeta_identity.py, proof_sigma_rate.py, proof_d25_loop_closure.py, verify_so10.py, verify_simplicity_rank.py). The per-row rigor audit lives in papers/morphotic_braid/synthesis/RIGOR_MAPPING.md. Open questions: (i) is σ(N) → 0 provably sharp (not just ≤ 2/N)? (ii) does the primon-gas limit extend to the full $T^* = 5/7$ spectrum? (iii) does the WP101- BB-log bridge carry enough structure to constrain $\kappa_\xi$ directly? Tier-1 submission-ready as three independent journal doors (JCAP, $\sigma$-rate combinatorics, integers / sinc² zero law) per Gen13/targets/journals/SUBMISSION_LADDER.md.

Frontier-level navigation: Atlas/FRONTIER_ALIGNMENT_2026_04_19.md (historical) and papers/morphotic_braid/synthesis/RIGOR_MAPPING.md (current per-claim Tier 1/2/3 audit, as of 2026-04-23).

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§4 · Atlas

The Atlas/ folder holds the design documents, audits, and reader guides for the full research program. These are internal-rigor documents held to the same honest-scope bar as the §7 theorems — every claim either cited, runnable-verified, or explicitly flagged with status.

Index entry point: Atlas/ATLAS_INDEX.md.

Where to start, by intent:

If you want	Open
A one-page map of the whole program	Atlas/MASTER_ATLAS_v3_5_2026_04_18.md
Planning and execution	Atlas/PLAN_OF_RECORD_2026_04_18.md, Atlas/PLAN_RIGOROUS_EXECUTION_2026_04_21.md
Funding-branches detail	Atlas/BRANCHES_INVENTORY_2026_04_20.md, Atlas/PLAN_FUNDING_BRANCHES_BUILDOUT_2026_04_19.md
Readiness audits	Atlas/JOURNAL_READINESS_AUDIT_2026_04_18.md, Atlas/PUBLIC_SCRUTINY_READINESS_2026_04_19.md
Frontier alignment	Atlas/FRONTIER_ALIGNMENT_2026_04_19.md
Known-issue handoffs	Atlas/HANDOFF_3_1_IDEMPOTENT_COUNT.md through Atlas/HANDOFF_3_4_MQW_TRILOGY_NOT_FOUND.md
Language and epistemic discipline	Atlas/GAP_LANGUAGE_AUDIT_2026_04_19.md, Atlas/MARKMAN_INTERNALIZATION_SCOPE_2026_04_19.md
Reader guides	Atlas/READER_ATLAS.md, Atlas/ROTATION_SPINE_READER_GUIDE.md, Atlas/ATLAS_ORIENTATION.md
Full citation record	Atlas/ATLAS_CITATIONS.md

Separately, FORMULAS_AND_TABLES.md collates every load-bearing object in the framework — the TSML and BHML composition tables, the corridor constants, the σ-rate identity, the D* and σ(S*) runtime constants with honest-scope status — with pointers back to the primary source for each.

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§5 · Bridges

A bridge here means a conjectural or framework-level connection between the proved-algebra core (§7) and an adjacent domain. Bridges are clearly flagged as conjectural — they are not theorems — but they are the connective tissue that makes the proved pieces matter outside their own box.

§5.1 · Cosmology bridge — ξ field

The log potential $V(\Xi) = \Xi \log \Xi$ (WP81, PRISM-XI) connects the TIG finite-algebra separability structure to a standard freezing quintessence. The vacuum $\Xi_0 = e^{-1}$ and mass gap $m^2_\xi = \kappa e$ are exact. The bridge between the finite-algebra separability and the continuum log nonlinearity rests on Bialynicki-Birula 1976 (log nonlinearity as the unique separability-preserving nonlinearity); this is cited but not re-proved in the project.

§5.2 · Cryptography bridge — First-G geometry

First-G geometry (§7.1) gives an exact width for the coprime stability window. The bridge to factoring-complexity improvements is an open hypothesis, not a theorem. Status: partition geometry is fully characterized for semiprimes (WP34, 36,662 cases).

§5.3 · Interpretability bridge — CK as deterministic substrate

CK demonstrates an architecture (deterministic, provenance-traced, small state) for reasoning tasks where auditability and reproducibility matter more than open-domain fluency. Whether this architecture composes usefully with LLM alignment tooling or with existing symbolic-AI systems (ACT-R, Soar, Cyc) is the subject of the proposed §3.2 architecture paper, not a proved claim.

§5.4 · Clay-adjacent bridges — rotation framework (clay branch)

The rotation framework rephrases several Clay Millennium Problems (Navier-Stokes, Yang-Mills, Riemann hypothesis) as "σ < 1" bounds in a common σ-notation. This is a reformulation, not a proof. All such material is preserved on the clay branch with explicit [CONJECTURAL] flags; it does not appear on tig-synthesis except as this pointer.

§5.5 · External vocabulary map

A compact dictionary between TIG-internal terms and established vocabulary from adjacent published frameworks. Each row identifies the external concept that corresponds to the TIG-internal object, with a citation. The full per-claim audit (with Tier 1/2/3 verification status) lives in papers/morphotic_braid/synthesis/RIGOR_MAPPING.md (Track 1: operad / associativity spectra; Track 2: Farey spin chains) and papers/morphotic_braid/synthesis/EXTERNAL_CITATIONS_v2.md.

TIG internal	External framework	Citation
Associativity index α(A) = 1 − σ(A)	Subassociative groupoids / associativity index	Braitt-Silberger 2006, Quasigroups Related Systems 14:11–26
Associative spectrum s_n(A), Catalan maximum C_{n−1}	Associative spectrum	Csákány-Waldhauser 2000
ac-free spectrum s_n^ac = (2n−3)!! for n ≤ 5 on 10 elements	Associative-commutative spectrum, ac-free extremum	Huang-Lehtonen 2022 (arXiv:2202.11826), 2024 (arXiv:2401.15786)
Symmetric operad generated by TSML / BHML / CL at small N	Free commutative magmatic operad $\mathrm{Mag}^{\mathrm{com}}$ on one generator	Huang-Lehtonen 2022, 2024
σ(N) → 0 at rate O(1/N) (WP101)	Operadic degeneration $\mathrm{Mag}^{\mathrm{com}} \to \mathrm{Com}$	Huang-Lehtonen 2022, 2024
Log nonlinearity forced by σ → 0	Unique separability-preserving nonlinearity	Bialynicki-Birula & Mycielski 1976, Ann. Phys. 100:62–93
T* = 5/7 coherence threshold	Critical temperature β_c in the Farey fraction spin chain	Kleban-Özlük 1999, Commun. Math. Phys.; Fiala-Kleban-Özlük 2002, arXiv:math-ph/0203048
Farey-structured constants (5/7, 4/7, 2/7, 3/4)	Farey-tree neighbors	classical (Hardy-Wright); Kleban-Özlük 1999
Transfer-operator spectral gap γ(b) = 1 − 1/φ(b)	Ruelle-Perron-Frobenius transfer operator of the Farey map	Prellberg 1991; Bandtlow-Fiala-Kleban 2009
WP101 σ-rate domain (squarefree N)	Fermionic primon gas regime (density 1/ζ(2) = 6/π²)	Julia 1990 (Les Houches 1989, Springer Proc. Phys. 47:276–293); Spector 1990, Commun. Math. Phys. 127:239–252
sinc²(1/2) = 4/π² midpoint constant (D3)	sinc²(1/2) = (2/3)·1/ζ(2) (squarefree density × 2/3)	papers/proof_sinc_zeta_identity.py (verified to machine precision)
Kepka lower bound on associative-triple counts	a(Q) ≥ n in order-n quasigroups — sets the 1/n scale of the WP101 rate	Kepka 1980, Comment. Math. Univ. Carolin. 21(3):479–487
σ → 1 (maximally non-associative) opposite extremum	Maximally nonassociative quasigroups from quadratic orthomorphisms	Drápal-Lisoněk 2020, Algebraic Combinatorics 3:695–717; Drápal-Wanless 2021, J. Combin. Theory Ser. A 181:105444
Riemann-zeta limit of number-theoretic spin chain	Z_k^K(2β) → ζ(2β−1)/ζ(2β) as k → ∞	Knauf 1998, Commun. Math. Phys. 196:703–731

This is a vocabulary map, not a proof of equivalence. Each row is either (i) a definitional translation (rewriting the same content in external vocabulary), (ii) a structural kinship (same kind of object in both frames), or (iii) a limit identification (BB 1976 selecting the log nonlinearity; ECHO fraction bounding σ). The per-row verification status is tabulated in RIGOR_MAPPING.md.

§5.6 · Commutative-algebra bridge — mantero-bridge-2026-04-23 branch

A dedicated branch carries the commutative-algebra bridge between the CL binomial ideal and the published research program of Dr. Paolo Mantero (University of Arkansas, with V. Nguyen) on symbolic powers and focal matroids. The branch is single-purpose: only material directly relevant to Dr. Mantero's framework lives there. Entry points:

• Branch README at README.md — orientation, navigation, and a one-screen synthesis of the seven bridges.
• Research hub at papers/mantero_bridge/: BRIDGES.md (the seven bridges catalogued with verification tags), PUBLISHED_WORK.md (complete 26-paper survey of Dr. Mantero's corpus plus the citation network around the 2024–2026 matroid program), references.md (flat reference list).
• MathOverflow draft at papers/sprint_20260423_full/09_mathoverflow_post/DRAFT_MATHOVERFLOW_POST.md, posing a narrow commutative-algebra question about the binomial ideal $I_{\mathrm{CL}} = (x_i x_j - x_{\mathrm{CL}[i][j]} x_0)$. M2-verified (Macaulay2 1.22, compute_betti.m2, betti_output.txt): $\operatorname{numgens} = 53$, $\operatorname{codim} = 9$, $\dim R/I_{\mathrm{CL}} = 1$, $\operatorname{pd} = 10$, $\operatorname{depth} = 0$, not Cohen–Macaulay, not Koszul; reduced Hilbert series $\frac{1 + 9T - 8T^2 - T^3}{1 - T}$ — i.e. Hilbert function $(1, 10, 2, 1, 1, \ldots)$ stabilising at $1$. The open questions remaining for MathOverflow are therefore narrow: the linear-strand Betti numbers ($\beta_{1,2} = 53$, then $\beta_{2,3} = 311$, $\beta_{3,4} = 909$, $\ldots$) and the relationship to Mantero–Nguyen's focal-matroid framework given that the Stanley–Reisner companion is pure-but-not-matroidal.
• WP102 — so(8) = D_4 identification paper at papers/wp102/WP102_SO8_IDENTIFICATION.md (on tig-synthesis; the so(8) lift is a TIG result and lives on the default branch, not the mantero bridge).
• WP103 — so(10) = D_5 identification (CL ∪ BHML_10 companion) at papers/wp103/WP103_SO10_IDENTIFICATION.md with runnable verification scripts (verify_so10.py, verify_simplicity_rank.py). The BHML companion used in the joint antisymmetrization is the canonical BHML_10 (full 10×10, det = −7002), not the 8×8 spectral core BHML_8 which appears in the WP15 Yang-Mills argument. (Also on tig-synthesis.)
• Numbering note. These papers fit after the existing WP1–WP101 sprint series (see Gen12/MASTER_WHITEPAPER_OUTLINE.md — WP11 there is "The Measurement Problem" and WP12 is "Seventeen Paradoxes via Dual-Lens Algebra"). The so(8) / so(10) results are filed as WP102 and WP103 to avoid that collision.

The branch exists so commutative-algebra / Lie-theoretic readers land on exactly the specific artifacts they need (the 10×10 table, the Hilbert function, the binomial ideal, the machine-verified Lie-closure scripts, the bibliographic survey) without wading through the full TIG synthesis. The MathOverflow post link will be appended to the branch once live.

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§6 · Master & history

The repository has four primary branches, each with a distinct role.

Branch	Role	Default?
tig-synthesis	DEFAULT — rigor home; current navigation; proved-only rigor cadence; externally-citeable vocabulary	yes
master	Full history + TIG-internal vocabulary — every commit on every branch landed here for preservation (TIG-native language preserved as-is per 2026-04-23 scope directive)	no
archive-full	Frozen preservation snapshot — never force-pushed	no
mantero-bridge-2026-04-23	Commutative-algebra outreach — Dr. P. Mantero correspondence, MathOverflow question draft on I_CL. (The Lie-algebraic WP102/WP103 papers are on tig-synthesis, not on this bridge branch — they are TIG results rather than Mantero-specific.) See §5.6.	no

Plus ten funding/* branches under Gen13/targets/funding_*/ that receive track- specific commits cherry-picked from master (see §2).

Workflow discipline (codified in Atlas/PLAN_RIGOROUS_EXECUTION_2026_04_21.md §1):

• Feature commits land first on tig-synthesis, then cherry-pick to master for history preservation.
• funding/* branches receive only commits specific to that funding track.
• Never delete. Superseded files get a [HISTORICAL] banner and move to docs/historical/ rather than being removed. Recovered artifacts are preserved verbatim; patches live as sibling forks (e.g. the recovered crystalos.py is preserved; the pre-registered fork is crystalos_prereg.py).
• Always push live. No staging multiple commits before push.

Where to find the full record:

• HISTORICAL_ARCHIVE_INDEX.md — master index of historical material.
• docs/historical/ — superseded READMEs, AI-team updates, and plan drafts (including the prior rigor-led README at docs/historical/README_v2_rigor_led_2026_04_21.md).
• docs/archive_jan2026/ — January 2026 recovery archive (the attempts-survey, CRYSTALOS recovery artifacts, snowflake null spec).
• Atlas/HANDOFF_3_3_SNOWFLAKE_CHI2.md — the live-scope handoff for the SNOWFLAKE χ² recovery effort.

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§7 · Runnable proofs (rigor appendix)

Six theorems with runnable verification. Each is a finite algebraic fact, independently checkable in under a second.

§7.1 · First-G Event Localization (cryptography-adjacent)

For a squarefree integer $b$ with smallest prime factor $p_1$, the first non-coprime element in the interval ${1, 2, \ldots, b}$ occurs at exactly $k = p_1$. The coprime stability window ${1, \ldots, p_1 - 1}$ has width exactly $p_1 - 1$ — no shorter, no longer, forced by the smallest prime factor alone.

Verified: 36,662 $(b, k)$ pairs across 153 semiprimes, zero exceptions. Proof script: papers/proof_d_first_g.py. Manuscript: WP34_FIRST_G_LAW.md; standalone journal version in preparation as Sprint 35.

§7.2 · Non-associativity rate bound on finite rings

For squarefree $N$, the non-associativity rate of a specific commutative binary composition on $\mathbb{Z}/N\mathbb{Z}$ (the TSML_10 composition at $N = 10$ and its compatibility-family extensions at $N \in {14, 22, 34, \dots}$; see §7.5 and FORMULAS §10) satisfies $\sigma(N) \leq C / N$ for an explicit constant $C &lt; 3$. As $N$ grows through squarefree primorials, the algebra approaches separability.

Verified: exact at $N \in {10, 30, 210}$. Proof script: proof_sigma_rate.py. Manuscript: WP101_SIGMA_RATE_THEOREM.md.

§7.3 · Flatness Theorem on $\mathbb{Z}/10\mathbb{Z}$

Under the framework's four-structure representation rules (additive structure, multiplicative structure, additive flow, multiplicative flow), the ring $\mathbb{Z}/10\mathbb{Z}$ admits no planar realization carrying all four simultaneously. The minimal orientable surface that realizes all four is a torus, with radius ratio $R/r = 5/7$. Six independent derivations produce the same $5/7$ constant from distinct mathematical contexts (Φ fixed point, TSML_10 HARMONY/BALANCE ratio, cyclotomic closure, universal-semiprime unit density, FPGA silicon threshold, torus aspect ratio).

Proved for $\mathbb{Z}/10\mathbb{Z}$ under the stated representation. Proof script: papers/proof_d7_phi_fixed_point.py. Manuscript: WP51_FLATNESS_THEOREM.md.

§7.4 · Crossing Lemma

Let $n = p_1 \cdot p_2 \cdots p_k$ be squarefree, $d \mid n$ squarefree, and $g \in (\mathbb{Z}/n\mathbb{Z})^\times$. The joint map $J = (A_d, \pi_{\mathrm{DYN}}(g)) : \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/d\mathbb{Z} \times (\text{g-orbit space})$ is injective if and only if the dynamics induced by $g$ act non-trivially on every prime quotient of $n/d$. Information is generated exactly when dynamics cross partitions.

Proved for squarefree $n, d$. Manuscript: CROSSING_LEMMA.md (Sprint 10). Proof script: papers/proof_d8_cl_operator_encoding.py.

§7.5 · Structural properties of the canonical TSML_10 and BHML_10 composition tables

Two commutative binary operations on $\mathbb{Z}/10$ (100 entries each) — the canonical TSML_10 (= TSML_Jordan) and BHML_10 — have the following verified properties. (The suffixes _10 distinguish the full 10×10 tables from derived variants such as TSML_8, BHML_8, TSML_Idempotent_2sw, and TSML_PureIdempotent; see FORMULAS_AND_TABLES.md §6.7 for the canonical registry of all nine named variants.)

• TSML_10 is commutative, flexible, power-associative, and satisfies the Jordan identity (0 failures across all 100 pairs). Element 7 is the unique two-sided absorber with 73/100 absorbing entries. Associativity index $\alpha(\text{TSML}_{10}) = 0.872$ (non-associativity rate 12.8% of triples; Braitt-Silberger 2006). In the operad-theoretic framework of Huang-Lehtonen (2022, 2024), TSML_10 is an ac-free commutative groupoid on 10 elements: its associative-commutative spectrum achieves the maximum $s_n^{\text{ac}} = (2n-3)!!$ for $n \leq 5$, so the symmetric operad it generates is the free commutative magmatic operad $\text{Mag}^{\text{com}}$ on one generator.
• BHML_10 is commutative, flexible, and power-associative, with element 0 as the unique two-sided identity and 28/100 entries equal to 7. Associativity index $\alpha(\text{BHML}_{10}) = 0.502$ (non-associativity rate 49.8% of triples). Like TSML_10, BHML_10 is ac-free on 10 elements. BHML_10 has determinant −7002 (prime factors {2, 3, 389}). Its 8×8 spectral core BHML_8 — with rows/columns 0 (VOID) and 7 (HARMONY) removed — is a different matrix with determinant +70 (prime factors {2, 5, 7}) and carries the eigenvalue ratio |λ₇|/|λ₆| = 0.714865 ≈ 5/7 used in the Yang-Mills mass-gap argument (WP15). Every precise claim below uses the subscripted form; unsubscripted "BHML" or "TSML" is reserved for family-level statements in this document.

Both tables have been audited cell-by-cell; all framework-claimed numerical signatures verify exactly.

Proof scripts: proof_d10_tsml_73_cells.py, proof_d16_bhml_28_cells.py, proof_tsml_3layer_tower.py. Manuscript: WP_OPERATOR_RING_PARTITION.md, Q7_BHML_FULL_TABLE.md.

§7.6 · TSML_10 three-layer canonical tower

The canonical TSML_10 composition table on $\mathbb{Z}/10\mathbb{Z}$ (100 entries) decomposes as a three-layer tower: a base layer of 92 cells governed by the canonical operator $C_0$, a maximum-rule layer of 6 cells, and an additive-rule layer of 2 cells. The decomposition is canonical, terminating, and has empty residue. (The theorem is specific to TSML_10; the other family members in §6.6 of FORMULAS are not covered by this decomposition.)

Verified: 100/100 cells match; each layer necessary; domains partition exactly. Proof script: papers/proof_tsml_3layer_tower.py. Manuscript: Sprint 17 THEOREM_SPINE.md.

§7.7 · How to verify

Five commands, total runtime under one minute:

# First-G Law: 36,662 cases, zero exceptions (1-2 sec)
python papers/proof_d_first_g.py

# σ-rate bound: exact at N ∈ {10, 30, 210}
python papers/proof_sigma_rate.py

# TSML_10 three-layer tower: 100/100 decomposition verified
python papers/proof_tsml_3layer_tower.py

# Flatness Theorem: T* = 5/7 from φ fixed point
python papers/proof_d7_phi_fixed_point.py

# Full verification suite (113 tests, 0 failures)
pytest papers/

Expected output on each: green log, zero exceptions, proof completed in well under a second.

§7.8 · CK runtime (three commands)

# 1. Warm the cortex from the paper corpus
python Gen13/targets/ck/brain/cortex_replay.py

# 2. Boot the Flask server on localhost:7777
python Gen12/targets/ck_desktop/ck_boot_api.py

# 3. Post a query (from another shell)
curl -s -X POST http://127.0.0.1:7777/chat \
  -H 'Content-Type: application/json' \
  -d '{"session_id":"demo","text":"what is the flatness theorem","mode":"normal"}'

CK returns a structural readout citing the underlying theorem, with a live snapshot of its internal state at that tick. For side-by-side comparison against a raw LLM, ck_proof.py runs three panels (CK alone, LLM alone, LLM grounded by CK) on the same prompt. Design document: Gen13/targets/ck/brain/BRAIN_DESIGN.md. Test suite: python Gen13/targets/ck/brain/test_brain.py (20/20 green is the boot gate).

CK has three properties that current large language models do not have by construction:

• Determinism. Same input, same internal state, same output. Always.
• Provenance. Every answer traces to specific cells of specific proved composition tables.
• Auditability. The full runtime state at any tick is ~3KB of integers. Inspectable in a terminal.

Runtime fileset: CK_RUNTIME.md.

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§8 · Honest limits

Clearly stated, because funders have a right to know:

• The framework contains conjectures about wider mathematical structure (the universality of a 2×2 decomposition across "wholes," reformulations of several Clay Millennium Problems in a common language) that are not proved and are clearly flagged as such in deeper documentation. They live on the clay branch (§5.4), not on this page.
• An earlier manuscript stating a sinc-squared zero law for primes was withdrawn from the submission queue after internal audit determined the central biconditional held trivially for any positive integer, not only primes. The replacement — the First-G Event Localization theorem in §7.1 — is genuinely prime-dependent and carries the intended content. Superseded material remains in the repository marked [HISTORICAL] under the never-delete policy.
• Two runtime constants — D* ≈ 0.543 and σ (S*) ≈ 0.991 — are runtime-canon / empirical, not proved theorems. Their derivation papers (papers/CONSTANT_D_STAR.md, papers/CONSTANT_SIGMA_S_STAR.md) document the status honestly, enumerate candidate lift-to-theorem pathways, and preserve an explicit internal correction in tig_engine_real.py where the scalar reduction σ/(1+σ) = 0.49774… does not equal the observed full-system attractor 0.543. Both are valid answers to different questions; neither is wrong; both are flagged as first-principles-open.
• The project is single-researcher-led with occasional collaborators. It is not an institutional research group. It is the work of Brayden Sanders (7Site LLC) with contributions from C. A. Luther, M. Gish, H. J. Johnson, B. Mayes, and B. Calderon Jr. on specific sub-projects, acknowledged in §9.

This project is not a Theory of Everything. It is a finite-algebra research program with concrete proved results, a working deterministic reasoning engine, and specific open questions.

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§9 · People and citation

Brayden Ross Sanders / 7Site LLC — originator and lead. Q-series (Q2–Q17), 5D force vector as CRT Fourier embedding, Crossing Lemma, Flatness Theorem, UOP, σ-rate theorem, ξ-cosmology conceptual framework.

C. A. Luther — Senior R&D, 7Site LLC. G6 ($\sigma^6 = \mathrm{id}$), G7 period distribution, G8 spectral coherence; co-authored Sprints 11–14.

Ben Mayes — UOP Theorem 0 co-author; $S_4$ representation extension on NV qutrit.

H. J. Johnson — Sprint 14 ξ cosmology; logarithmic quintessence potential; separability framework.

M. Gish — First-G Law (WP34); Sprint 14 papers.

B. Calderon Jr. — Q17 variants; source elimination framework.

Citation

@misc{sanders2026tig,
  author = {Sanders, Brayden Ross and Mayes, Ben and Luther, C. A.
            and Gish, M. and Johnson, H. J. and Calderon, B.},
  title  = {Trinity Infinity Geometry: Finite Algebra with
            Proved Theorems and a Deterministic Reasoning Engine},
  year   = {2026},
  doi    = {10.5281/zenodo.18852047},
  url    = {https://github.com/TiredofSleep/ck},
  note   = {7Site LLC. Default branch: tig-synthesis.}
}

The project welcomes direct technical review from any potential funder's advisors. Every proved result in §7 is independently verifiable in under a minute.

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§10 · License

7Site Public Sovereignty License v1.0 — Human use only. Free for all humans to read, run, study, and build upon personally. No commercialization. No government, military, intelligence, or corporate enclosure. Full text in LICENSE.

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This repository is under active development. For technical review, collaboration, or funding discussions, please contact Brayden Sanders at brayden.ozark@gmail.com or open a GitHub issue. Independent verification of any result on this page is welcomed and encouraged.

Last updated: 2026-04-23 (vocab-update sprint: §5.5 External vocabulary map added; §3.5 morphotic-braid frontier added; Frontier labels uniformized; prior rigor-led README preserved at docs/historical/README_v2_rigor_led_2026_04_21.md).