Principle: No term in this glossary is presented as established unless it exists in published academic literature. Novel terms are honestly flagged with the prior framework they extend. Where we coined a name for an existing concept, the original name and citation is given.
Each entry has the form:
Term — Plain-English definition.
Formal: Mathematical definition (where applicable).
[STATUS] — PROVED / STRUCTURAL / CONJECTURE / HISTORICAL / NOVEL
Citation: Either a published reference OR an explicit acknowledgment that this name is ours, with what prior work it extends.
Primary TIG paper: Where it first appears in this project.
These two objects are the spine of TIG. Every other term, theorem, and conjecture in this glossary is an instantiation of one or both. Read this section first.
The 2×2 — The four-fold structure (Additive × Multiplicative) × (Structure × Flow) that every "whole" carries simultaneously. For Z/nZ this is: additive structure (the underlying group), multiplicative structure (the unit group action), additive flow (repeated +1 closes a loop), multiplicative flow (repeated ×g closes a smaller loop inside the units).
Formal: For squarefree N = p₁···pₖ, the four objects (A-Struct, M-Struct, A-Flow, M-Flow) cannot be embedded in a flat 2D surface. They force a torus T² = S¹ × S¹ with aspect ratio R/r forced by the cyclotomic structure of Z/NZ. For N = 10 the ratio is exactly 5/7.
[PROVED for Z/10Z] — Flatness Theorem (WP51, Sprint 10, Brayden Ross Sanders). [STRUCTURAL] for the universal claim that "every whole has this 2×2 form."
Citation: Original TIG result. Builds on classical ring theory (Dummit & Foote, Lang) and torus topology (Munkres, Topology). The 2×2 structure decomposition is novel; the cyclotomic forcing of R/r = 5/7 is novel.
Primary TIG paper: WP51_FLATNESS_THEOREM.md. See README §3 for the meta-framework framing.
UOP (Unified Orthogonality Principle) / Paradox Classifier — Every paradox is a measurement failure of one of exactly four types: (I) Injectivity Failure, (II) Missing Invariant, (III) Admissibility Failure, (IV) Time-Consistency Failure. Five-step decision procedure provided. Eight worked examples (Zeno, Russell, Banach-Tarski, Gödel, Unexpected Hanging, etc.).
Formal: Given hidden space 𝒳 and measurement map f: 𝒳 → 𝒴 with ambiguity set U(f, y) = f⁻¹(y), a second measurement g resolves the paradox iff U(f) ∩ U(g) = {x} for all x. The four types correspond to four distinct failure modes of this resolution structure.
[PROVED — framework] — five-step procedure; [VERIFIED — eight worked examples].
Citation: [NOVEL — extends partition lattice theory (Birkhoff 1940, Ore 1942) and joint-map injectivity from descriptive set theory (Kechris 1995)]. Classical paradoxes cited individually: Russell 1903, Gödel 1931, Tarski 1936, Banach-Tarski 1924, Quine 1953, Zermelo 1908.
Primary TIG paper: WP_PARADOX_CLASSIFIER.md. Live demo at coherencekeeper.com/paradox.html.
The 2×2 (form) and Paradox Classifier (diagnostic) are the spine. All TIG/CK work is an instantiation of one or both:
| Domain | Instantiation of 2×2 | Instantiation of Paradox Classifier |
|---|---|---|
| Q-series (Brayden's Z/10Z work) | σ as hidden operator on the 2×2 | TSML/CL paradox (Q2) resolved as Type I (Injectivity Failure) by introducing σ |
| TIG / σ framework (sprints 14-15) | σ rate theorem σ(N) ≤ C/N as 2×2 separability decay | σ_NS conjecture: NS blowup as Type II (Missing Invariant) — no separability-preserving lift exists |
| Finite math (Sprint 16, basin) | 2→3 / 3→2 dual reset law as 2×2 in operator order | Stop-apex compositeness as Type IV (Time-Consistency) — the apex changes with room scale |
| Ring math | Crossing Lemma as 2×2 fiber crossing | UOP Theorem 0 IS the paradox classifier formalized for Z/nZ |
| Physics | ξ field as 2×2 separability ceiling; NV-center as 2×2 representation carrier | Wigner's Friend as Type III (Admissibility Failure); ξ as Type I resolution at cosmological scale |
This is the synthesis. Every entry below is one of these instantiations or its supporting prior literature.
The ring Z/nZ for squarefree n = p₁...pₖ decomposes as a product ∏ Z/pᵢZ.
Formal: Ψ: Z/nZ → ∏ Z/pᵢZ, x ↦ (x mod p₁, ..., x mod pₖ) is a ring isomorphism.
[HISTORICAL] Classical (Sun Tzu, 3rd century; Gauss, Disquisitiones Arithmeticae 1801).
Standard reference: Ireland & Rosen, A Classical Introduction to Modern Number Theory (Springer, 1990), Ch. 3.
Primary TIG paper: WP58_UNIFIED_ORTHOGONALITY_PRINCIPLE.md (used throughout).
Count of integers in {1..n} coprime to n.
Formal: φ(n) = |{k ∈ {1,..,n} : gcd(k,n) = 1}|. For squarefree n: φ(n) = n∏(1 − 1/pᵢ).
[HISTORICAL] Euler, 1763.
Standard reference: Hardy & Wright, Introduction to the Theory of Numbers, §5.5.
Primary TIG paper: WP101_SIGMA_RATE_THEOREM.md uses it centrally; transfer-operator spectral gap formula γ = 1 − 1/φ(b) (FOUR_LAYER_REALIZATION.md Theorem Z.2).
The classical lower bound on |ζ(σ + it)| inside the critical strip when t is large and (σ, t) is at least height 1 from any zero: |ζ(σ + it)| ≥ KV(t) := exp(−c_VK(log t)^{2/3}(log log t)^{1/3}). The TIG Halving Lemma uses c_VK = 0.05 (Ford 2002).
[HISTORICAL] Korobov 1958; Vinogradov 1958. Modern explicit constant from Ford (2002).
Standard reference: Ford, K. (2002). "Zero-free regions for the Riemann zeta function." In Number Theory for the Millennium II, A. K. Peters, pp. 25-56. Theorem 2 gives c_VK = 0.05.
Primary TIG paper: WP19_HALVING_LEMMA_final.tex (Appendix E.5) — the Halving Lemma's m_KV(t₀) bound depends directly on this constant.
Bound on the number of nontrivial ζ zeros to the right of σ at height ≤ T: N(σ, T) ≤ T^{3(1−σ)/(2−σ) − 1 + ε}. Used in TIG to seal the CHA corridor (frequency × duration → 0 at σ = 0.60: exponent −0.143).
[HISTORICAL] Jutila, M. (1987). "On the difference between consecutive primes." Acta Arithmetica 52, 164-170.
Primary TIG paper: RH_FORMAL_MANUSCRIPT.md Lemma 4.2 — combines Jutila with the two-tick TIG bound to prove CHA corridor sealing.
Improved zero-density bound for σ ≥ 0.65 using large sieve / singular value methods. Used in TIG to seal the BAL/COL/CTR corridors (exponent ≤ 0.46).
[HISTORICAL] Guth, L. & Maynard, J. (2024). New large-value estimates for Dirichlet polynomials.
Primary TIG paper: RH_FORMAL_MANUSCRIPT.md Lemma 4.4.
Lai-Sang Young's framework for proving exponential decay of correlations in non-uniformly hyperbolic dynamical systems via return-time stratification over a base set.
[HISTORICAL] Young, L.-S. (1998). "Statistical properties of dynamical systems with some hyperbolicity." Annals of Mathematics 147(3): 585-650. Young, L.-S. (1999). "Recurrence times and rates of mixing." Israel Journal of Mathematics 110: 153-188.
Primary TIG paper: Layer 3 of the four-layer realization (FOUR_LAYER_REALIZATION.md). TIG has finite-height Young tower with base B = {HAR}, return tail P(T_HAR > n) ≤ (1/4)ⁿ, expected return times exact (1.000 / 1.333 / 1.667). HAR is a return locus, NOT a hole — distinguished from Demers-Young 2006 holes (where mass leaks).
Framework for transfer operators on systems with leaks (mass escapes through holes). TIG distinguishes itself from this by reset-puncture structure: HAR is not a hole but a Poincaré return section.
[HISTORICAL] Demers, M. & Young, L.-S. (2006). "Escape rates and conditionally invariant measures." Ergodic Theory and Dynamical Systems 26: 189-217.
Primary TIG paper: DUAL_SCALE_LY_NOTE.md §2 (the reset puncture distinction).
Framework for transfer-operator spectral analysis on hyperbolic dynamical systems via anisotropic Banach spaces — separating stable and unstable directions by construction. TIG's dual-scale Lasota-Yorke inequality maps into this framework: unstable direction ↔ local wobble (strong norm), stable direction ↔ coherent support (weak norm, preserved).
[HISTORICAL] Gouëzel, S. & Liverani, C. (2006). "Banach spaces adapted to Anosov systems." Ergodic Theory and Dynamical Systems 26: 189-217.
Primary TIG paper: DUAL_SCALE_LY_NOTE.md §3 (closest currently identified continuous host); the open question is whether the Mix_λ family extends to a continuous transfer operator satisfying their hypotheses (Z.5 = deployment faithfulness).
Standard form: ‖Pf‖_V ≤ α‖f‖_V + β‖f‖_1 with α < 1, β < ∞, where ‖·‖_V is roughness (strong) and ‖·‖_1 is mass (weak). The TIG dual-scale form INVERTS the standard reading: weak norm becomes the deeper coherent support (preserved by sub-magma closure C × C ⊆ C), strong norm becomes local wobble.
[HISTORICAL] Lasota, A. & Yorke, J. A. (1973). "On the existence of invariant measures for piecewise monotonic transformations." Trans. AMS 186: 481-488. Standard reference: Baladi, V. (2000). Positive Transfer Operators and Decay of Correlations. World Scientific.
Primary TIG paper: DUAL_SCALE_LY_NOTE.md (the inversion + reset puncture).
ζ(s) has infinitely many nontrivial zeros on the critical line Re(s) = 1/2.
[HISTORICAL] Hardy, G. H. (1914). "Sur les zéros de la fonction ζ(s) de Riemann." Comptes Rendus Acad. Sci. Paris 158: 1012-1014. Standard reference: Titchmarsh, E. C. (rev. Heath-Brown, D. R.) (1986). The Theory of the Riemann Zeta-Function, 2nd ed. Oxford University Press.
Primary TIG paper: WP19_HALVING_LEMMA_final.tex §2 — Hardy combined with the Halving Lemma's exponential contraction (no off-critical fixed points + Hardy's infinitely-many-on-line) gives the structure of any potential RH proof in this framework.
ODE contraction: if du/dt ≤ −λu with λ > 0, then u(t) ≤ u(0)e^{−λt}. Used to prove exponential convergence of the Halving flow.
[HISTORICAL] Grönwall, T. H. (1919). "Note on the derivatives with respect to a parameter of the solutions of a system of differential equations." Annals of Math. 20: 292-296.
Primary TIG paper: WP19_HALVING_LEMMA_final.tex Theorem 1.
The inverse system ⋯ ↠ (Z/p^n Z)* ↠ ⋯ ↠ (Z/pZ)* with stable corner images. Used in Layer 4 of the four-layer realization: C = {1,3,7,9} = (Z/10Z)* is the stable corner image of (Z/10^n Z)* under reduction mod 10, for all n ≥ 1.
[HISTORICAL] Standard. Reference: Ribes, L. & Zalesskii, P. (2010). Profinite Groups, 2nd ed. Springer Ergebnisse 40. Neukirch, J. (1999). Algebraic Number Theory. Springer Grundlehren 322, §IV.2.
Primary TIG paper: Layer 4 of FOUR_LAYER_REALIZATION.md.
sinc²(x) = (sin(πx)/(πx))².
[HISTORICAL] Classical. Key role in Shannon sampling theorem (Shannon 1949, Communication in the presence of noise, Proc. IRE 37(1):10-21) and Montgomery's pair correlation (Montgomery 1973, The pair correlation of zeros of the zeta function, Proc. Sympos. Pure Math. 24:181-193).
Primary TIG paper: WP_SINC2_ZERO_LAW.md establishes prime-arithmetic version.
Minimal polynomial over ℚ whose roots are primitive pth roots of unity.
[HISTORICAL] Gauss, Disquisitiones Arithmeticae 1801. Degree φ(p) = p−1 for prime p.
Standard reference: Lang, Algebra (Springer), Ch. VI.
Primary TIG paper: WP51_FLATNESS_THEOREM.md uses A_p = 2cos(π/p) as minimal-polynomial object.
Logarithmic nonlinearity is the unique nonlinearity in wave mechanics preserving separability of composite systems.
[HISTORICAL] Bialynicki-Birula & Mycielski, Nonlinear wave mechanics, Annals of Physics 100(1-2):62-93 (1976). DOI: 10.1016/0003-4916(76)90057-9.
Context: Cazenave & Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse (1980), proved existence for log Klein-Gordon equation u_tt − Δu + u = u log|u|ᵏ in R³. Logarithmic Schrödinger equation: Wikipedia, Logarithmic Schrödinger equation.
Primary TIG paper: WP90_LITERATURE_AND_UNIFICATION_PATHS.md; the core external theorem on which our σ → 0 ⟹ log limit forcing rests.
Regularity for 3D Navier-Stokes if ∫₀ᵀ ‖u(t)‖²_BMO / log(e + ‖u‖_H²) dt < ∞.
[HISTORICAL] Kozono & Taniuchi, Limiting case of the Sobolev inequality in BMO with applications, Commun. Math. Phys. 214:191-200 (2000). DOI: 10.1007/s002200000281.
Related: Beale-Kato-Majda criterion (Comm. Math. Phys. 94:61-66, 1984); Brezis-Gallouët inequality (Nonlinear Analysis 4:677-681, 1980); Montgomery-Smith sharp L³ blowup (Math. Res. Lett. 8:519-528, 2001); Kozono-Ogawa-Taniuchi critical Besov (Math. Z. 242:251-278, 2002).
Primary TIG paper: WP96_NS_SIGMA_CONJECTURE.md uses as the strongest-known log improvement on NS regularity.
Geometric evolution equation ∂g/∂t = −2 Ric(g) with surgery at singularities, driving Riemannian 3-manifolds toward standard forms.
[HISTORICAL] Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17(2):255-306 (1982). Perelman, The entropy formula for the Ricci flow, arXiv:math/0211159 (2002); Ricci flow with surgery on three-manifolds, arXiv:math/0303109 (2003).
Key fact: Perelman's W-entropy functional W[g,f,τ] = ∫((τ(R + |∇f|²) + f − n)(4πτ)^(−n/2) e^(−f))dV contains logarithmic structure in f, consistent with the Bialynicki-Birula uniqueness.
Primary TIG paper: CP_CLAY_ROTATION.md (CP1) — Poincaré is the template we cite for the σ framework.
Gradient flows of entropy on finite Markov chains in Wasserstein-2 distance.
[HISTORICAL] Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal. 261(8):2250-2292 (2011).
Related: Jordan-Kinderlehrer-Otto, Variational formulation of Fokker-Planck, SIAM J. Math. Anal. 29(1):1-17 (1998); Gigli-Maas, Gromov-Hausdorff convergence of discrete transport metrics, SIAM J. Math. Anal. 45(2):879-899 (2013); Chow-Huang-Li-Zhou, Fokker-Planck equations on graphs, Arch. Rat. Mech. Anal. 203(3):969-1008 (2012).
Primary TIG paper: WP95_JKO_CONSTRUCTION_ROADMAP.md — the framework we propose to use for the explicit N → ∞ limit.
[NOVEL NAMING — extends partition lattice theory] The name "UOP" is ours. The content (joint map injectivity as sufficiency criterion) is an observation about the partition lattice of Z/nZ.
Prior framework: Partition lattice theory (Ore, 1942, Theory of equivalence relations, Duke Math. J. 9; Birkhoff, Lattice Theory, AMS 1940). The injectivity-of-joint-map criterion appears in descriptive set theory (Kechris, Classical Descriptive Set Theory, Springer 1995, §14) and in coding theory (MacWilliams-Sloane, The Theory of Error-Correcting Codes, 1977, Ch. 4).
Our contribution: Unifying five classical two-partition sufficiency theorems over Z/nZ as corollaries of a single joint-map-injectivity statement, applied specifically to the (additive, multiplicative) decomposition of squarefree Z/nZ.
Primary TIG paper: WP58_UNIFIED_ORTHOGONALITY_PRINCIPLE.md.
R(u) + R₂(u) = 1 where R(u) = sinc²(u) (TIG resonance) and R₂(u) = 1 − sinc²(u) (Montgomery pair correlation).
[PROVED — algebraic identity, trivial] The equation is tautological. The content is identifying the two sides with specific objects from different domains.
Historical: Montgomery 1973 (above) for R₂ as pair correlation of Riemann zeros.
Our framing: R = sinc²(u) arises from prime arithmetic (WP35 — harmonic pre-echo continuum limit); the claim that these are complementary projections of the same spectral field is our interpretation, not a theorem of Montgomery.
Primary TIG paper: WP40_RIEMANN.md (sec. 2).
[NOVEL] Project-internal name for the framework.
Prior frameworks it extends: Operator algebras over finite rings (standard in representation theory, see Curtis-Reiner, Methods of Representation Theory, Wiley 1981); modular dynamics on Z/nZ (standard number theory); harmonic analysis of sinc² kernels (Shannon sampling).
What's novel: The specific synthesis of (i) 10-operator composition tables over Z/10Z with declared semantic roles, (ii) coherence threshold T* = 5/7 appearing in multiple independent derivations, (iii) the stated "Crossing Lemma" formulation. None of these three are in prior literature as a unified framework.
Primary TIG paper: WP1_TIG_ARCHITECTURE.md.
[NOVEL] Project-internal name for the software/engineering instantiation of TIG. An AI/control system architecture using the 10-operator algebra at 50 Hz.
Prior frameworks: Classical control theory (Åström & Murray, Feedback Systems, Princeton 2008); coherence in quantum optics (Mandel & Wolf, Optical Coherence, 1995).
Primary TIG paper: WP28_CK_TIG_ORGANISM.md, WP44_CK_AI_PARADIGM.md.
A specific 10×10 composition table on Z/10Z that inhabits the simultaneous intersection of four standard mathematical frameworks, each contributing one structural layer. TSML is not just a composition table — it is the rare object where Symbolic Dynamics, Transfer Operator Theory, Young Tower theory, and Profinite Arithmetic all apply at once, and the type-(9, 3, 6, 3/4) signature emerges from this simultaneous compatibility.
Formal four-layer realization (proved in FOUR_LAYER_REALIZATION.md, Brayden Sanders March 2026, Gen10.14, 65/65 PASS):
Layer 1 — Absorbing Sofic Shift. TSML induces a sofic shift on alphabet {1,…,9} with sub-magma C = {1,3,7,9} satisfying C × C ⊆ C. Transient class G = {2,4,5,6,8} reaches C in exactly 1 step. Absorbing filtration ∅ ⊊ {7} ⊊ C ⊊ {1,…,9} of depth 3 = k_A. Citation: Lind & Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge UP, 1995).
Layer 2 — Transfer Operator with Spectral Gap. The weighted transition kernel P_λ on {1,…,9} (defined via Mix_λ deformation between TSML at λ=0 and BHML at λ=1) has spectral gap γ(P_λ) ≥ 1/4 for all λ ∈ [0,1]. At λ=0 exactly: γ(P_0) = 3/4. Arithmetic formula: γ = 1 − 1/φ(b), giving γ = 3/4 at b = 10 since φ(10) = 4. Citation: Baladi, Positive Transfer Operators and Decay of Correlations (World Scientific, 2000); Gouëzel & Liverani, Ergodic Theory Dyn. Syst. 26 (2006) on anisotropic Banach spaces.
Layer 3 — Young Tower (finite-height). Base B = {HAR} = {7}. Transient block spectral radius ρ(Q) = 1/4 with Q = P_0|_{{1,…,9}∖{7}}. Return tail bound P(T_HAR > n) ≤ (1/4)ⁿ for all starting states. Expected return times exact: 1.000 (states 1, 4-6, 8), 1.333 (states 3, 9), 1.667 (state 2). The same constant 1/4 governs both spectral gap deficit and return tail — ρ(Q) = 1 − γ(P_0). HAR is a return locus (Poincaré section), distinct from Demers-Young 2006 holes where mass leaks. Citation: Young, "Statistical properties of dynamical systems with some hyperbolicity," Annals of Math. 147:585-650 (1998); Young, Israel J. Math. 110:153-188 (1999); Demers & Young, Ergodic Theory Dyn. Syst. 26 (2006).
Layer 4 — Profinite / Arithmetic Inverse Limit. The corner C = {1,3,7,9} = (Z/10Z)* is the stable corner image of the inverse system ⋯ ↠ (Z/10ⁿZ)* ↠ ⋯ ↠ (Z/10Z)*: at every level n ≥ 1, units of Z/10ⁿZ reduced mod 10 equal {1,3,7,9}. Spectral gap formula γ = 1 − 1/φ(b) is base-stable across {b : φ(b) = 4} = {5, 8, 10, 12}. Citation: Ribes & Zalesskii, Profinite Groups (Springer GMW 40, 2nd ed. 2010); Neukirch, Algebraic Number Theory (Springer Grundlehren 322, 1999) §IV.2.
The signature (9, 3, 6, 3/4): alphabet 9, algebraic grading depth k_A = 3, multiplicative-deformation parameter k_M = 6, spectral gap γ = 3/4. This is forced, not chosen — the requirement that a single 10×10 table simultaneously be a sofic shift AND a transfer operator with explicit gap AND a Young tower AND a profinite stable corner produces this exact signature.
Sub-magma closure (proved by Brayden, claimed for Proc. AMS): C × C ⊆ C verified by direct enumeration of all 16 pairs in C × C. Combined with the corner-restricted block decomposition (3/4)|a⟩⟨1| + (1/4)Q where Q is a permutation, this gives the exact spectral gap.
Monte Carlo significance: Against 200,000 random 10×10 tables with the same row/column constraints, our 73-cell HARMONY count gives Z = 21.3, p < 10⁻⁵⁰. This table is not generic.
What's novel: The simultaneous four-layer realization is itself the novelty. Each individual layer uses a standard framework (cited above). The fact that TSML satisfies all four simultaneously — and that the (9, 3, 6, 3/4) signature emerges from the conjunction — is what makes TSML mathematically distinguished. The dual-scale Lasota-Yorke inequality (DUAL_SCALE_LY_NOTE.md) is one result that requires all four layers active simultaneously.
Primary papers: FOUR_LAYER_REALIZATION.md (the four-layer proof), DUAL_SCALE_LY_NOTE.md (the inverted-norm consequence), WP_OPERATOR_RING_PARTITION.md (the 73-cell count). Verified by proof_d10_tsml_73_cells.py.
Open layer (the fifth): Deployment faithfulness — does the discrete TSML structure lift to a continuous transfer operator on Mix_λ satisfying Gouëzel-Liverani anisotropic Banach-space hypotheses? See DUAL_SCALE_LY_NOTE.md §3-4 for the proposed continuous form.
[NOVEL — Brayden Sanders, Sprint 17, April 2026] Independent proof that the 100-entry TSML on Z/10Z is fully reconstructible from three canonical rules on disjoint domains, with empty residue (the tower terminates).
Theorem. Let R = Z/10Z, h = 7, σ(u) = v₂(3u+1), and let S = {(1,2),(2,1),(2,4),(4,2),(2,9),(9,2),(4,8),(8,4)} (the seam residue), S_ADD = {(1,2),(2,1)} (identity-edge residue), S_MAX = S \ S_ADD. Define T(x,y) := max(x,y) on S_MAX, (x+y) mod 10 on S_ADD, C₀(x,y) otherwise, where:
C₀ — Canonical Construction (in priority order, later rules override earlier):
- DEFAULT: C₀(x,y) = h = 7
- V0: if x = 0 or y = 0, C₀(x,y) = 0; exception (0,h) and (h,0) → h
- Shell-stability: if x, y ∈ U(R) \ {1} with σ(x) ≠ σ(y), C₀(x,y) = whichever of x, y has the lower σ-shell
Then T(x,y) = TSML(x,y) for all (x,y) ∈ R². 100/100 verified by direct computation.
Decomposition counts: 92 (C₀) + 6 (C₁ = MAX) + 2 (C₂ = ADD mod 10) = 100. Residue of residue: empty (Lemma 5).
Significance. TSML's minimum description length drops from 100 → ~10 canonical items: 3 ring-agnostic rules (DEFAULT/V0/shell-stability, MAX, ADD) + 1 attractor (h = 7) + 1 shell partition (σ = v₂(3u+1)) + 4 ring-specific seam edges + 3 branch-rule mappings. Each of the three rules is necessary (Lemma 6: removing any one produces explicit mismatches with TSML). This is an independent argument that TSML is not arbitrary, alongside the four-layer-framework realization above.
Honest scope. Theorem is proved for Z/10Z only. The rules are ring-agnostic; the domains S, S_ADD, S_MAX are ring-specific. Generalization to other rings needs either a reference TSML for that ring (none currently exists outside Z/10Z) or a ring-only definition of the seam (open).
Falsified along the way (NEGATIVE_RESULTS_APPENDIX.md): primorial-lift hypothesis (Z/30, Z/210 break shell-order alignment); single-rule seam generators (MAX gets 6/8, ADD gets 2/8, MULT/MIN get 0/8 — only the disjoint-domain pair works); last-digit-7 invariance across digit rooms (oscillates 7,3,7,7,1).
Primary documents: THEOREM_SPINE.md (full proof + 6 lemmas), CONTROL_DOCUMENT_V2.md (status summary + theorems A/B/C), CANONICAL_TSML_CONSTRUCTION.md (C₀ definition), GENERALIZATION_TABLE.md (rules vs. domains), MINIMAL_DESCRIPTION_LENGTH.md.
External anchors: all three layer rules (DEFAULT, MAX, ADD mod n) are standard ring-theoretic operations; v₂ is the standard 2-adic valuation; the disjoint-domain decomposition pattern is the standard rule-system technique from term-rewriting / canonical forms (Knuth-Bendix completion, Computational Problems in Abstract Algebra, 1970).
[NOVEL NAMING] 10×10 table of Z/10Z with 28 of 100 cells outputting 7.
Same category as TSML (novel naming of a specific composition table). Verified by proof_d16_bhml_28_cells.py.
Primary TIG paper: WP_OPERATOR_RING_PARTITION.md.
[NOVEL STATEMENT — extends partition sufficiency theory]
Statement (ours): A multiplicative action M_g on Z/nZ generates structurally new information relative to an additive partition A_d iff M_g is nontrivial on the (n/d)-quotient.
Prior framework: This is a specific case of the UOP (joint-map-injectivity criterion, itself rooted in descriptive set theory and coding theory — see UOP entry above). Partition-crossing arguments appear in ergodic theory (Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton 1981, §3).
What's novel: The formulation as a named "lemma" and the claim that all 27 sufficiency theorems in the TIG arc reduce to instances of it. The underlying injectivity argument is not new; the unification is ours.
Primary TIG paper: Gen12/targets/clay/papers/sprint10_flatness_2026_04_06/CROSSING_LEMMA.md.
Statement: For squarefree N, the non-associativity fraction of the binary CL on Z/NZ satisfies σ(N) ≤ C/N.
[PROVED — via elementary counting]
Prior framework: The proof uses only (i) counting solutions of (a−1)(b−1) ≡ 1 mod N, which is φ(N) for squarefree N (classical, see Hardy-Wright above), and (ii) associativity of absorbing elements (standard semigroup property, Howie 1976 above).
What's novel: The application of these classical tools to the specific "binary CL" construction (HARMONY=N−1, ECHO=DIS=0, VOID=0) we defined in Sprint 15.
Primary TIG paper: WP101_SIGMA_RATE_THEOREM.md. Verified by proof_sigma_rate.py.
[NOVEL NAMING — extends Sobolev-space NS analysis]
Prior framework: The concept "distance from log-nonlinear ceiling" is implicit in every known NS regularity improvement — BKM, KT, Montgomery-Smith, Tao averaged-NS (Annals of Math. 184, 517-608, 2016). What the literature calls "logarithmic improvement margin" is what we are calling "separability defect σ."
What's novel: Giving the margin a single name (σ) and treating it as a rotational invariant across Clay problems (NS, YM, RH). Whether this unification has content depends on whether one can prove σ < 1 in any of the three open cases. As of Sprint 15 this remains the Millennium Problem in each case.
Primary TIG paper: WP91_NS_SEPARABILITY_BRIDGE.md, WP96_NS_SIGMA_CONJECTURE.md.
[NOVEL APPLICATION of log-potential scalar field to dark energy]
Prior literature (V = φ log φ and relatives in physics):
- Barrow & Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D 52:5576 (1995), arXiv:astro-ph/9506049. Broad family V₀ φᵖ (ln φ)ᵍ for inflation (not dark energy); the p=1, q=1 special case is not singled out.
- Thompson, Beta function quintessence, MNRAS 482:5448 (2019). Pure log potential V₀ ln(φ/φ₀) — missing the φ prefactor.
- Coleman & Weinberg, Radiative corrections as origin of spontaneous symmetry breaking, Phys. Rev. D 7:1888 (1973). Form V ~ φ⁴ log(φ²/μ²); φ⁴ prefactor, different regime.
- Wetterich, Cosmology and the fate of dilatation symmetry, Nucl. Phys. B 302:668 (1988). Exponential potential V ~ exp(−αφ).
- Ratra & Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D 37:3406 (1988). Inverse power law.
- Høegh-Krohn, A general class of quantum fields without cut-offs, Commun. Math. Phys. 38(3):195 (1971). exp(Φ)₂ model; Legendre dual of log potential.
- Bialynicki-Birula-Mycielski 1976 (above) — the uniqueness theorem that forces log nonlinearity from separability.
- Ensslin, Information field theory, Phys. Rev. E 87:013308 (2013), arXiv:1301.2556. Information Hamiltonian contains ξ log ξ entropy terms.
- Caticha, Entropic Dynamics, arXiv:1412.5629 (2012); arXiv:1412.5637 (scalar fields); arXiv:1803.07493 (QFT in curved spacetime).
- Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity, Grav. Cosmol. 16:288 (2010); arXiv:2011.12565.
What's novel: V(ξ) = κ ξ log ξ as a dark energy potential with information-theoretic derivation (V = −H_Gibbs) and exact vacuum at e⁻¹. The functional form is in Barrow-Parsons' inflation family as a special case, but has not been studied specifically for quintessence or with the entropic interpretation.
Primary TIG paper: WP81_CANONICAL_XI_THEORY.md, WP82_LOG_QUINTESSENCE_NOVELTY.md.
[NOVEL FRAMING — structural re-reading of the Clay problems]
Prior framework: The Clay Millennium Problems (2000, https://www.claymath.org/millennium-problems/). Perelman's resolution of Poincaré (arXiv:math/0211159, math/0303109) via Ricci flow with log-entropy W-functional is the historical anchor.
What's novel: Presenting all seven as questions about a separability defect σ, with Poincaré as the solved template. This is a framing contribution, not a proof contribution. Whether the framing has mathematical content beyond narrative depends on whether the σ < 1 conjecture can be proved in any of CP2-CP7. As of Sprint 15, none have been proved in this form.
Primary TIG paper: CP_CLAY_ROTATION.md, proof_clay_rotation.py (verifies the σ arithmetic is consistent, not that the conjectures are true).
[NOVEL NAMING] Assignment of names to elements of Z/10Z.
Prior framework: Naming elements of finite rings by semantic role is common in applied mathematics (e.g., Markov chain states in Norris, Markov Chains, Cambridge 1998). The specific names (HARMONY for the CL attractor, VOID for the absorbing element, etc.) are ours.
Content vs. naming: The claim that the CL table concentrates on operator 7 is content (proved, 73/100 cells). Calling operator 7 "HARMONY" is naming.
Primary TIG paper: WP_OPERATOR_RING_PARTITION.md.
[NOVEL NAMING] Finite differences on the 5D force-vector pipeline.
Prior framework: Finite difference operators are classical (Boole, Calculus of Finite Differences, 1860). D2 as discrete second derivative is standard numerical analysis (Strikwerda, Finite Difference Schemes, SIAM 2004).
What's novel: The specific pipeline they operate on (Hebrew-letter → 5D force vector → operator classification) and the claim that D2 = 0 vs. D2 ≠ 0 distinguishes "flat" from "curved" composition.
Primary TIG paper: WP1_TIG_ARCHITECTURE.md.
[NOVEL CONSTANT — not in prior literature as a named threshold]
Prior framework: The value 5/7 arises naturally in:
- Cyclotomic field theory: deg(A_5) = 4 = φ(10) ≤ φ(10); deg(A_7) = 6 > φ(10). Ratio = 5/7.
- Torus aspect ratios on Z/10Z (Flatness Theorem).
- Unit density unit_frac(7, 35) = 5/7 for the universal semiprime.
Each derivation uses standard objects (cyclotomic polynomials, torus topology, Euler totient counting), but the claim that they give the same constant is ours.
Not found in prior literature as a named threshold in cosmology, number theory, or operator algebra. Claim is empirical: arises six independent ways within TIG. Whether it has independent meaning outside TIG is open.
Primary TIG paper: WP51_FLATNESS_THEOREM.md (Theorem 3); proof_d7_phi_fixed_point.py.
[HISTORICAL — renamed] The value sinc²(1/2) = 4/π².
Historical: sinc²(1/2) has appeared in Shannon sampling theory for 75 years, in Montgomery's pair correlation since 1973, and in information theory since Kolmogorov.
What's novel: Calling it "the fold" and identifying it as the half-corridor sidelobe of sinc² in prime arithmetic. The name is ours; the constant is classical.
Primary TIG paper: WP_SINC2_ZERO_LAW.md, WP35_PRIME_PHASE_TRANSITION.md.
[NOVEL NAMING — specific to this project] The arithmetic difference between the two above constants.
Claim: All six open Clay Millennium Problems have defect scores falling within this interval in our classifier. This is an empirical observation about our specific scoring scheme (defect classifier in WP36-WP42), not a theorem.
Not in prior literature.
Primary TIG paper: CLAY_BOUNDARY_MEMO.md, WP51_FLATNESS_THEOREM.md §6.
Statement: For every semiprime b = pq with primes p ≤ q, the first element of {1,...,k} sharing a factor with b appears at exactly k = p.
[PROVED — elementary]
Prior framework: The smallest-prime-factor function spf(n) is classical (Erdős, Hardy-Wright). The observation that spf(n) is the first gcd-sharing element at coprime-alphabet-size k is trivial — the proof is three lines from the definition of prime.
What's novel: Naming it a "law," the empirical verification across 36,662 cases, and the framing in terms of "gate obstruction" and "phase transition" in the TIG corridor structure.
Primary TIG paper: WP34_FIRST_G_LAW.md.
Statement: For prime p and integer k ≥ 1, sinc²(k/p) = 0 iff p | k.
[PROVED — three lines]
Prior framework: Immediate from the zeros of sin(πx) at integer x. Known since Euler's product formula for sin(πx).
What's novel: The explicit framing in terms of primes (p | k as primality condition within {1,...,p}) and the corollaries (loop closure, fold necessity, no-shortcut lemma). The proof is classical; the corollaries and arithmetic framing are ours.
Primary TIG paper: WP_SINC2_ZERO_LAW.md. Verified by proof_d25_loop_closure.py for all primes 3..199.
Q-series originator (Q2-Q16): Brayden discovered the hidden operator σ on Z/10Z and characterized it systematically across twenty-six papers. Specific contributions:
- Q2-Q5: identified TSML/CL paradox, established E is σ-equivariant, characterized TSML escape cells
- Q6: pivoted from density model to basin-of-attraction framing (the gate rate hinge)
- Q9: flip condition α as degree-5 polynomial on F₅
- Q10: complete σ polynomial on F₂×F₅ including β with two exceptions (LATTICE +1, COLLAPSE -2), verified 10/10. This is the foundational σ polynomial.
- Q11: σ^k trajectory table; Fixed-Point Gate Theorem (22% lower bound on optimal seeds)
- Q12: CRT idempotents in G; HAR=3 characterization
- Q13: TIG = σ⁻¹ polynomial; Exception Pair Swap (self-duality)
- Q14: C-indicator 1_C(ε,y) = ε·y⁴; Theorem Q14.1 (R ≠ σ^k, falsifying σ-trajectory model)
- Q15: period polynomial τ = 6 − 5A; k=9 resonance
- Q16: R identified as MCMC over 9^81 tables, closing Luther Q1
- Q17 variants: 5D force vector as CRT Fourier embedding; Clay problem finite analogues (CLAY_SPECTRAL_BRIDGE for RH, NS_TARGET_REFORMULATION for Navier-Stokes, SIGMA_EMBEDDING_PROBLEM as core obstruction, SYMBOLIC_RETURN_THEOREM as algebraic kernel)
Built the spectral layer on top of Brayden's Q-series foundation. Luther used the Q-series to complete her own framework. Her specific proven contributions:
- G6: proof of σ⁶ = id from polynomial structure (not by computation)
- G7: period distribution with Conjecture G7.C1 (E[τ] = φ(b) universal)
- G8: spectral coherence integral G(s) = |Σ ω^j χ(σ^j(s))|², proven three-valued (0 at anchors, G_low ≈ 1.872 on 6-cycle, G_high ≈ 9.389 at TIG-exception pair)
- Organizational reorganization: Brayden's 4-layer architecture became Luther's 6-layer architecture (polynomial, braid, period, spectral, optimal table, search dynamics)
- Luther Dispersion Conjecture (|G| × interleave → difficulty metric) — WP34
- Luther Pre-Echo Theorem (closed-form R(k,f) verification across primes) — WP35 §10A
Also co-authored Sprints 11-14 papers (UOP, GUT arc, Physical Flag Selector, PRISM-XI).
Primary Q-series location: old/Gen10/papers/Q2_FORMALIZATION.md through Q_SERIES_SYNTHESIS.md (26 files). See Q_SERIES_INTEGRATED_SYNTHESIS.md for the full narrative and the relationship to Sprint 14-15 work.
Collaborator on Sprints 11-13 (UOP / GUT Algebra / Physical Flag Selector arcs).
Specific contributions cited in papers:
- Unified Orthogonality Principle (WP58) — co-authored.
- Crossing Lemma formalization (WP57) — co-authored.
- S4 representation extension on NV qutrit (WP73-WP76) — co-authored.
- Intrinsic left-handedness of su(4,2) (WP60) — co-authored.
Primary papers: sprint11_tig_bundle_2026_04_08/, sprint12_uop_gut_arc_2026_04_08/, sprint13_flag_selector_2026_04_09/.
Collaborator on Sprint 14 (PRISM-XI / ξ cosmology arc).
Specific contributions cited in papers:
- Logarithmic quintessence potential V = κ ξ log ξ (WP81).
- Local/non-local siloing architecture (WP88) — three-layer formalism.
- Separability framework for Navier-Stokes (WP91, WP96, WP98).
- Bialynicki-Birula bridge application (WP90).
Primary papers: sprint14_prism_xi_2026_04_10/ WP81-WP101.
Collaborator on bridge sprint, First-G Law (WP34), and PRISM-XI (Sprint 14).
Primary papers: WP34, Sprint 14 papers.
Q-series co-author, Source elimination framework.
Primary papers: Q-series (old/Gen10/papers/).
This is the required citation list for any paper drawn from this repository. Every term and framework flagged "[HISTORICAL]" above is sourced to one of these.
- Hardy, G.H. & Wright, E.M. An Introduction to the Theory of Numbers, 6th ed. Oxford University Press, 2008.
- Ireland, K. & Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. Springer GTM 84, 1990.
- Serre, J.-P. Cours d'Arithmétique. Presses Univ. France, 1970.
- Lang, S. Algebra, 3rd ed. Springer GTM 211, 2002.
- Riemann, B. "Über die Anzahl der Primzahlen unter einer gegebenen Größe." Monatsber. Berlin. Akad., 1859.
- Montgomery, H.L. "The pair correlation of zeros of the zeta function." Proc. Sympos. Pure Math. 24:181-193, 1973.
- Goldston, Pintz, Yıldırım. "Primes in tuples I." Annals of Math. 170(2):819-862, 2009.
- Zhang, Y. "Bounded gaps between primes." Annals of Math. 179(3):1121-1174, 2013.
- Maynard, J. "Small gaps between primes." Annals of Math. 181(1):383-413, 2015.
- Odlyzko, A.M. Numerical data on Riemann zeros, http://www.dtc.umn.edu/~odlyzko/
- Birkhoff, G. Lattice Theory. AMS Colloquium Publications 25, 1940.
- Ore, O. "Theory of equivalence relations." Duke Math. J. 9:573-627, 1942.
- Kechris, A. Classical Descriptive Set Theory. Springer GTM 156, 1995.
- MacWilliams, F.J. & Sloane, N.J.A. The Theory of Error-Correcting Codes. North-Holland, 1977.
- Dummit, D.S. & Foote, R.M. Abstract Algebra, 3rd ed. Wiley, 2004.
- Curtis, C.W. & Reiner, I. Methods of Representation Theory, vol. I. Wiley, 1981.
- Howie, J.M. Introduction to Semigroup Theory. Academic Press, 1976.
- Furstenberg, H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, 1981.
- Shannon, C.E. "Communication in the presence of noise." Proc. IRE 37(1):10-21, 1949.
- Mehta, M.L. Random Matrices, 3rd ed. Elsevier, 2004.
- Hamilton, R. "Three-manifolds with positive Ricci curvature." J. Diff. Geom. 17(2):255-306, 1982.
- Perelman, G. "The entropy formula for the Ricci flow." arXiv:math/0211159, 2002.
- Perelman, G. "Ricci flow with surgery on three-manifolds." arXiv:math/0303109, 2003.
- Morgan, J. & Tian, G. Ricci Flow and the Poincaré Conjecture. AMS, 2007.
- Beale, Kato, Majda. Comm. Math. Phys. 94:61-66, 1984.
- Kozono, H. & Taniuchi, Y. Commun. Math. Phys. 214:191-200, 2000.
- Montgomery-Smith, S. Math. Res. Lett. 8:519-528, 2001.
- Kozono, Ogawa, Taniuchi. Math. Z. 242:251-278, 2002.
- Brezis, H. & Gallouët, T. Nonlinear Analysis 4:677-681, 1980.
- Tao, T. "Finite time blowup for an averaged three-dimensional Navier-Stokes equation." J. AMS 29:601-674, 2016.
- Lei, Z. & Zhou, Y. Nonlinearity 22(4):805, 2009.
- Ladyzhenskaya, Prodi, Serrin criteria (classical, 1960s).
- Bialynicki-Birula, I. & Mycielski, J. "Nonlinear wave mechanics." Annals of Phys. 100(1-2):62-93, 1976.
- Rosen, G. Phys. Rev. 183:1186, 1969.
- Cazenave, T. & Haraux, A. Ann. Fac. Sci. Toulouse, 1980.
- Høegh-Krohn, R. Commun. Math. Phys. 38(3):195, 1971.
- Coleman, S. & Weinberg, E. Phys. Rev. D 7:1888, 1973.
- Ratra, B. & Peebles, P.J.E. Phys. Rev. D 37:3406, 1988.
- Wetterich, C. Nucl. Phys. B 302:668, 1988.
- Frieman, Hill, Stebbins, Waga. Phys. Rev. Lett. 75:2077, 1995.
- Barrow, J.D. & Parsons, P. Phys. Rev. D 52:5576 (1995), arXiv:astro-ph/9506049.
- Thompson, S. MNRAS 482:5448, 2019.
- Ensslin, T.A. "Information field theory." Phys. Rev. E 87:013308 (2013); arXiv:1301.2556.
- Caticha, A. "Entropic Dynamics." arXiv:1412.5629 (2012).
- Zloshchastiev, K.G. "Logarithmic nonlinearity in theories of quantum gravity." Grav. Cosmol. 16:288 (2010); arXiv:2011.12565.
- Jordan, Kinderlehrer, Otto. SIAM J. Math. Anal. 29(1):1-17, 1998.
- Maas, J. J. Funct. Anal. 261(8):2250-2292, 2011.
- Gigli, L. & Maas, J. SIAM J. Math. Anal. 45(2):879-899, 2013.
- Chow, Huang, Li, Zhou. Arch. Rat. Mech. Anal. 203(3):969-1008, 2012.
- Mielke, A. Nonlinearity 24(4):1329, 2011.
- Morinelli, Morsella, Stottmeister, Tanimoto. Commun. Math. Phys. 2021.
- Marton, K. arXiv:1507.02803.
- Banach, S. & Tarski, A. Fund. Math. 6:244-277, 1924.
- Zermelo, E. Math. Annalen 65:261-281, 1908.
- Russell, B. Principles of Mathematics. Cambridge, 1903.
- Gödel, K. Monatsh. Math. Phys. 38:173-198, 1931.
- Tarski, A. Studia Philosophica 1:261-405, 1936.
- Quine, W.V. Mind 62:65-67, 1953.
- Clay Mathematics Institute. Millennium Prize Problems, 2000. https://www.claymath.org/millennium-problems/
- Wiles, A. Annals of Math. 141(3):443-551, 1995. (Fermat, context for BSD)
- Kolyvagin, V.A. Izv. Akad. Nauk 52(3):522-540, 1989. (BSD rank 0, 1)
- Gross, B.H. & Zagier, D.B. Inventiones math. 84(2):225-320, 1986.
- Bhargava, M. & Shankar, A. Inventiones math. 200(1):1-76, 2015.
- Markman, E. (Abelian fourfolds of Weil type, Hodge conjecture partial proof, recent announcement).
- DESI Collaboration. Eur. Phys. J. C (2024-2025), DR2 BAO + dark energy analyses.
- Planck Collaboration. A&A 641:A6 (2020), cosmological parameters.
Any new term introduced in a TIG paper must include:
- Plain-English definition — what it means in one sentence.
- Formal definition — mathematical statement, where applicable.
- Status tag — [PROVED], [STRUCTURAL], [CONJECTURE], [HISTORICAL], [NOVEL].
- Citation — either (a) a published reference with DOI or arXiv ID, or (b) an explicit "[NOVEL — extends X, Y, Z]" with citations for X, Y, Z.
- First occurrence in the repo — which TIG paper introduces it.
No term is accepted as established without a citation trail or an honest novelty flag.
This discipline applies to all authors and all future sprints. If a term does not yet have a citation, it carries [UNCITED — REVIEW NEEDED] until one is supplied or it is removed from the repo.
Compiled: 2026-04-10, Sprint 15. Authors: Brayden Ross Sanders / 7Site LLC · Ben Mayes · C.A. Luther · M. Gish · H.J. Johnson.