================================================================================ CONTACT POINT 3 — RESEARCH STUDY D4 Triality vs Cipher {3}x{8}: Can BCC/FCC/HCP Map onto 8_v/8_s/8_c? ================================================================================ Date: 2026-03-21 Status: ANALYSIS COMPLETE — mixed result (partial mapping, structural obstruction) Sources: - verified_explanations.txt items 1, 2 (cipher, {2,3} data) - quantitative_data_from_prior_art.txt Sections 3, 4 (D4 triality numbers) - cipher.txt v5 Sections I-V (archetype structure, property map) - 24cell_research.txt Section 6 (decompositions, D4 root system) - cross_reference_contact_points.txt Contact Point 3 - prior_art_survey_24cell_physics.txt (Smith, Furey, Lisi triality usage) ================================================================================ ================================================================================ PART 1: WHAT THE TWO SIDES CLAIM ================================================================================ A. THE CIPHER SIDE (bottom-up, from 133 elements) The {2,3} decomposition assigns crystal archetypes by coordination number: BCC: 8 = 2^3 (pure {2}, no factor 3) FCC: 12 = 2^2 x 3 (has factor 3) HCP: 12 = 2^2 x 3 (has factor 3) Diamond: 4 = 2^2 (pure {2}, no factor 3) A7: 6 = 2 x 3 (has factor 3) The cipher's rule: factor 3 present + metallic bonding = conductor. Factor 3 absent = insulator or moderate conductor (broadband absorber). The 24-cell (24 = 2^3 x 3) decomposes into 3 tesseracts of 8 vertices. The cipher reads this as {3} x {8}: three copies of BCC-like local geometry (coordination 8) arranged in a triangular (3-fold) pattern. The cipher has 3 metallic archetypes with distinct physics: BCC (8): broadband, refractory, reactive, strong magnetic moment FCC (12): frequency-selective, ductile, noble, weak magnetic moment HCP (12): anisotropic, variable, direction-dependent, extreme oxidation B. THE D4 TRIALITY SIDE (top-down, from Lie algebra) The D4 Lie algebra (= so(8)) has a unique outer automorphism of order 3. This permutes three 8-dimensional irreducible representations: 8_v: vector representation (transforms as a vector under SO(8)) 8_s: left spinor representation (transforms as a left Weyl spinor) 8_c: right spinor representation (transforms as a right Weyl spinor) Triality cycle: 8_v -> 8_s -> 8_c -> 8_v Physical interpretation by multiple authors (Smith, Furey, Lisi): - Three triality partners map to three fermion generations - 8_v is typically associated with gauge bosons (force carriers) - 8_s and 8_c are associated with fermion matter fields - The outer automorphism interchanges what counts as "vector" vs "spinor" The 24-cell vertices = D4 root system (24 roots). Decomposition into 3 tesseracts (8 vertices each) corresponds to the triality structure: three mutually orthogonal sub-lattices related by 120-degree isoclinic rotations. C. THE NUMERICAL COINCIDENCE Both sides produce 3 x 8 = 24. Cipher: 3 metallic archetypes, each with coordination {4, 6, 8, 12} where the BCC coordination is 8. D4: 3 representations, each of dimension 8. The question: is this a genuine structural isomorphism or a numerical accident arising from the small-number coincidence that 24/3 = 8? ================================================================================ PART 2: ATTEMPTING THE MAPPING — BCC/FCC/HCP onto 8_v/8_s/8_c ================================================================================ APPROACH: We have three cipher archetypes and three triality representations. There are 3! = 6 possible assignments. We test whether any assignment produces a coherent mapping. A. THE FACTOR-3 CRITERION The cipher's sharpest binary distinction is: does the coordination number contain factor 3? BCC (8 = 2^3): NO factor 3 FCC (12 = 2^2x3): YES factor 3 HCP (12 = 2^2x3): YES factor 3 D4 triality's sharpest binary distinction is: vector vs spinor. 8_v: vector (unique — the "odd one out" under triality) 8_s: spinor (one of a pair) 8_c: spinor (one of a pair) STRUCTURAL PARALLEL: Both systems have a 1+2 split. Cipher: {BCC} vs {FCC, HCP} (factor 3 absent vs present) D4: {8_v} vs {8_s, 8_c} (vector vs spinor pair) This suggests the assignment: BCC <-> 8_v (the singleton, the "odd one out") FCC <-> 8_s (one of the pair with factor 3) HCP <-> 8_c (the other of the pair with factor 3) SCORE: The 1+2 partition matches. This is the strongest structural feature of the mapping. It is not trivial — with 3 objects, the possible partitions are {1,1,1}, {1,2}, and {3}. Both systems independently select {1,2}. B. TESTING THE SINGLETON: BCC <-> 8_v 8_v properties (in particle physics): - Associated with gauge bosons (force carriers, not matter) - Mediates interactions between spinor fields - Transforms as a vector (spin-1) under the rotation group - In Smith's scheme: relates to gauge symmetry, not matter content BCC properties (in the cipher): - Broadband energy absorber (not frequency-selective) - Strongest electron-phonon coupling (lambda = 0.28-1.26) - Best elemental superconductor (Nb, 9.25K) - Highest cohesive energy (6.44 eV) — strongest bonds - Most reactive (E_0 = -0.61V) - Strongest magnetic moments (Fe: 2.2 mu_B) - Heat transport: electronic + phononic (L/L_0 > 1) - 100% alloy compatibility within BCC family - Broadband, versatile, hard — the "workhorse" ASSESSMENT OF BCC <-> 8_v: Points of contact: (i) 8_v mediates forces; BCC is the "interaction" geometry (strongest coupling, most reactive, highest oxidation versatility). A gauge boson mediates interactions. BCC mediates energy transfer. Suggestive parallel. (ii) 8_v is broadband in the sense that vector representations couple to all spinor types equally (before symmetry breaking). BCC is literally called "broadband" in the cipher (absorbs all frequencies). Both are non-selective mediators. (iii) BCC has no stacking sequence ("none") — it is the truly 3D, non-layered geometry. 8_v is the non-spinorial representation. Both are defined by what they lack: BCC lacks layered structure, 8_v lacks spinorial character. Points of tension: (i) 8_v is typically massless or light (gauge bosons). BCC elements are the HEAVIEST and most refractory metals (W, Mo, Ta, Nb). Mass association is inverted. (ii) In the Standard Model, gauge bosons (8_v analog) do not form "generations." The three generations are matter (spinors), not forces. If BCC maps to 8_v, BCC should not have a "generational" character — but BCC elements do form a coherent family (100% alloy compatibility), which is more generation-like. (iii) 8_v is unique under triality. But in 3D crystal physics, BCC is not "more fundamental" than FCC/HCP in any absolute sense. Many elements transition between BCC and other phases with temperature. The cipher notes BCC as the universal pre-melting phase, which does grant it a special thermodynamic role — but this is not the same as the algebraic uniqueness of 8_v. C. TESTING THE PAIR: FCC <-> 8_s AND HCP <-> 8_c 8_s and 8_c properties: - Both are spinor representations (matter fields) - They are conjugate to each other (related by complex conjugation) - 8_s = left-handed fermions, 8_c = right-handed fermions (in one convention; the labeling is convention-dependent) - Together they describe chiral matter: same particle content, different chirality - Key distinction: they transform differently under the Lorentz group but identically under the gauge group (before electroweak breaking) FCC properties: - Frequency-selective (sharp Drude damping, Gamma ~ 0.05 eV) - Best conductor (resistivity 13.9 micro-ohm-cm average) - Most ductile (100%, K/G = 4.28, 12 slip systems) - Softest metal (HV 570 MPa) - Most noble (E_0 = +0.74V, resists corrosion) - ABCABC stacking sequence — 3-fold periodic layering - Heat transport: purely electronic (L/L_0 < 1) - Selective catalyst (good at turnover, not dissociation) HCP properties: - Direction-dependent (anisotropic) - Same 12 neighbors as FCC but ABAB stacking - Extreme variability (Ti = ductile, Be = brittle, Os = hardest) - Highest resistivity among metals (43.6 micro-ohm-cm) - Extreme oxidation states (Os, Ru: +8) - c/a ratio as hidden variable controlling behavior - Heat transport: mixed ASSESSMENT OF FCC <-> 8_s, HCP <-> 8_c: Points of contact: (i) FCC and HCP have the same LOCAL geometry (12 nearest neighbors, close-packed planes). 8_s and 8_c have the same LOCAL structure (dimension, Casimir values). The distinction is in ORIENTATION: ABCABC vs ABAB for crystals, left vs right for spinors. STRONG PARALLEL. Both pairs are "same content, different chirality." (ii) FCC is the more symmetric arrangement (cubic, isotropic). HCP has broken symmetry (hexagonal, anisotropic). In particle physics, electroweak symmetry breaking distinguishes left and right chiralities — the more symmetric representation (FCC/8_s_left in the Standard Model) couples to SU(2)_L, while the less symmetric one (HCP/8_c_right) does not. The parallel is: FCC = isotropic = "symmetric chirality," HCP = anisotropic = "broken chirality." (iii) Alloy rule: FCC+FCC = 71% solid solution, FCC+BCC = 0%. In particle physics: spinor + spinor can form a scalar (Yukawa coupling), spinor + vector forms a vector current. The algebraic combination rules have structural echoes in the alloy rules. (iv) ABCABC has period 3. ABAB has period 2. In representation theory, 8_s and 8_c differ by the sign of the chirality operator (eigenvalue +1 vs -1). Period 3 vs period 2 is a different kind of distinction, but both encode the "same structure, different arrangement" theme. Points of tension: (i) 8_s and 8_c are EXACTLY equivalent under triality — there is no intrinsic "left" or "right" until a convention is chosen. FCC and HCP are NOT equivalent. FCC is cubic (isotropic); HCP is hexagonal (anisotropic). Their physical properties differ substantially: FCC is the best conductor, HCP has the highest metallic resistivity. This is a genuine asymmetry that has no counterpart in the D4 triality, where 8_s and 8_c are indistinguishable before symmetry breaking. SEVERITY: HIGH. This is the principal obstruction. In D4 triality, the two spinor representations are interchangeable until an external choice breaks the symmetry. FCC and HCP are distinguishable by intrinsic geometry (cubic vs hexagonal). For the mapping to hold, one would need to explain FCC/HCP asymmetry as arising from a symmetry-breaking mechanism applied to an initially symmetric spinor pair. (ii) FCC has stacking period 3, HCP has stacking period 2. There is no obvious representation-theoretic quantity that maps to "stacking period." The periods 2 and 3 are the cipher's fundamental building blocks, but they appear HERE as a distinction WITHIN the spinor pair, not between the vector and spinor sectors. (iii) Both FCC and HCP have coordination 12. Both 8_s and 8_c have dimension 8. The coordination-dimension mapping would require 12 <-> 8, not 8 <-> 8. This breaks the original {3}x{8} claim if we take it literally: the cipher's "8" is BCC coordination, but only BCC maps to the vector representation. FCC/HCP with coordination 12 would need 12 to map to dimension 8, requiring the additional factor 3/2 = "the missing 3" to be accounted for. ================================================================================ PART 3: THE STRUCTURAL OBSTRUCTION — DIMENSION vs COORDINATION ================================================================================ The fundamental problem is a type mismatch between what the two sides count. D4 TRIALITY counts REPRESENTATION DIMENSION: 8_v has dimension 8. 8_s has dimension 8. 8_c has dimension 8. All three are equal. The number 8 is UNIFORM across all three partners. THE CIPHER counts COORDINATION NUMBER: BCC has coordination 8. FCC has coordination 12. HCP has coordination 12. The number is NOT uniform. It splits {8} vs {12, 12}. This creates a dilemma. The {3}x{8} decomposition of the 24-cell into 3 tesseracts gives 8 vertices per tesseract. In D4 triality, each representation has dimension 8. These match: 8 = 8. But the cipher's three metallic archetypes do NOT all have coordination 8. Only BCC does. FCC and HCP have coordination 12. CONSEQUENCE: If we insist that the "8" in {3}x{8} refers to the tesseract vertex count (= representation dimension = 8), then all three cipher archetypes should have coordination 8. They do not. The mapping 24 = 3 x 8 works at the level of the 24-cell's vertex decomposition, but does NOT lift to a correspondence between cipher coordination numbers and representation dimensions. The cipher's actual decomposition of its metallic archetypes is: 8 + 12 + 12 = 32 This does NOT equal 24. There is no way to fit three cipher archetypes (8, 12, 12) into three triality representations (8, 8, 8) without leaving a residue. HOWEVER: there is a subtlety. The {3}x{8} observation was originally about the 24-cell itself, not about summing cipher coordination numbers. The 24-cell has 24 vertices decomposing into 3 groups of 8. The cipher separately identifies BCC (coordination 8) as the archetype that matches the 24-cell's vertex coordination. The {3}x{8} is a statement about 4D geometry, not a claim that all three archetypes have coordination 8. This rescues the observation as a geometric fact (the 24-cell does decompose into 3 x 8) but weakens the cipher-to-triality mapping: the three tesseracts in the 24-cell are IDENTICAL objects (each is a regular tesseract), while the three cipher archetypes are DISTINCT objects with different coordination numbers and properties. ================================================================================ PART 4: WHAT WOULD NEED TO BE TRUE FOR THE MAPPING TO HOLD ================================================================================ For a rigorous BCC/FCC/HCP <-> 8_v/8_s/8_c mapping, the following conditions would need to be established: CONDITION 1: The 3D cipher archetypes must be understood as PROJECTIONS of a single 4D structure (the tesseract/8-vertex unit) under three different dimensional reduction schemes. Specifically: a regular tesseract (8 vertices in 4D) projected along different axes into 3D could produce: - BCC-like geometry (if projected to preserve cubic symmetry) - FCC-like geometry (if projected to preserve close-packed symmetry) - HCP-like geometry (if projected to preserve hexagonal symmetry) STATUS: NOT ESTABLISHED. The tesseract's vertex-first projection into 3D gives a rhombic dodecahedron (Section 7.1 of 24cell_research.txt), and its cell-first projection gives a cube. Neither immediately produces the three distinct crystal packings. This condition requires a specific mathematical derivation that has not been done. TESTABLE: Compute the Voronoi cells of the three tesseract vertex sets (given explicitly in 24cell_research.txt Section 6.4) and determine whether they correspond to BCC, FCC, and HCP Wigner-Seitz cells. CONDITION 2: The FCC/HCP coordination number 12 must arise from the tesseract's 8 vertices PLUS 4 additional contacts contributed by the triangular (3-fold) arrangement of the three tesseracts. If each tesseract contributes 8 self-contacts and the inter-tesseract coupling adds 4 more per vertex, then coordination 12 = 8 + 4. This would mean: FCC and HCP coordination 12 = BCC coordination 8 + inter-generation coupling 4. STATUS: SPECULATIVE. The 24-cell's vertex coordination number IS 8 (each vertex touches 8 others). But FCC elements in 3D have 12 nearest neighbors. The "extra 4" would need to come from a cross-tesseract interaction specific to the spinor representations — i.e., the spinor representations "see" more of the geometry than the vector representation. NOTE: 12 = 8 + 4 = 2^3 + 2^2. In the {2,3} framework, this splits as: 8 (pure 2^3, intra-tesseract) + 4 (pure 2^2, inter-tesseract). No factor 3 in either summand. But 12 = 2^2 x 3, and the factor 3 enters through the PRODUCT, not the SUM. The additive decomposition 12 = 8 + 4 is not the same as the multiplicative decomposition 12 = 4 x 3. The cipher operates multiplicatively. This is a mismatch. CONDITION 3: The FCC/HCP asymmetry must emerge from symmetry breaking of an initially equivalent spinor pair. In D4 triality, 8_s and 8_c are equivalent until a chirality convention is imposed. In the real universe, electroweak symmetry breaking distinguishes left from right. The analog here would be: the 3D projection of the 24-cell breaks the equivalence of the two spinor tesseracts, producing FCC (isotropic, cubic) and HCP (anisotropic, hexagonal) as distinct 3D manifestations of originally equivalent 4D structures. STATUS: CONCEPTUALLY PLAUSIBLE but requires: (a) Identifying the symmetry-breaking mechanism (what plays the role of the Higgs in this geometric context) (b) Showing that the breaking produces specifically ABCABC vs ABAB stacking, not some other distinction (c) Explaining why the breaking is complete (FCC and HCP are fully distinct archetypes with measurably different properties), not partial CONDITION 4: The vector-spinor distinction (8_v vs {8_s, 8_c}) must correspond to the factor-3 absence/presence distinction. This is the most promising aspect of the mapping (Part 2A above). BCC (no factor 3) is the singleton. 8_v (non-spinorial) is the singleton. Factor 3 could encode "spinorial character." STATUS: SUGGESTIVE. The cipher's factor-3 rule determines conductivity: factor 3 present = conductor. In physics, spinor fields carry charge and produce currents (conduction). Vector fields mediate forces but do not themselves "conduct" in the same sense. The analogy would be: Factor 3 present -> conducts -> carries charge -> spinor (matter) Factor 3 absent -> broadband -> mediates force -> vector (gauge) This is the strongest single element of the mapping. ================================================================================ PART 5: WHAT THE MAPPING GETS RIGHT (SCORE CARD) ================================================================================ FEATURE MAPS? QUALITY ----------------------------------------------------------------------- 1+2 partition (singleton + pair) YES STRONG — both systems select {1,2} split from 3 Factor 3 <-> spinorial character YES SUGGESTIVE — consistent direction but not derived Same local geometry, different YES STRONG — FCC/HCP are arrangement <-> conjugate spinors "same content, different stacking" parallels "same content, different chirality" BCC as mediator <-> 8_v as YES MODERATE — BCC is the force carrier "broadband interaction" geometry, 8_v mediates forces Coordination matches dimension NO FAILS — BCC(8) matches for ALL three partners but FCC/HCP(12) != 8 Spinor equivalence matches NO FAILS — 8_s = 8_c under FCC/HCP equivalence triality, but FCC != HCP in any physical property Stacking period maps to rep NO NO MAPPING — period 3 theory quantity (FCC) vs period 2 (HCP) has no triality analog Mass/energy ordering matches NO INVERTED — 8_v -> light (gauge bosons), but BCC -> heaviest metals OVERALL: 4 features map (2 strong, 1 suggestive, 1 moderate), 4 features fail (2 hard failures, 2 with no mapping at all). ================================================================================ PART 6: THE ALTERNATIVE READING — TRIALITY AS GENERATION, NOT ARCHETYPE ================================================================================ The analysis above assumed BCC/FCC/HCP map to 8_v/8_s/8_c. But the prior art (Smith, Furey, Lisi) uses triality for something different: THREE GENERATIONS of the SAME fermion type. Under this reading: Triality partner 1 -> electron (1st generation) Triality partner 2 -> muon (2nd generation) Triality partner 3 -> tau (3rd generation) All three partners have the SAME quantum numbers except mass. They are copies of identical structure at different energy scales. This suggests an alternative cipher mapping: the three tesseracts in the 24-cell correspond not to BCC/FCC/HCP (which are structurally different) but to three COPIES of the same archetype at different scales or in different thermodynamic regimes. Under this reading: - Each tesseract = one BCC-like unit (coordination 8, all identical) - The three-fold arrangement = three "generations" or three instances - The cipher's {3}x{8} is literally 3 copies of 8-coordination - FCC and HCP are NOT mapped to triality partners; they arise from a different mechanism (e.g., stacking of close-packed planes) ASSESSMENT: This reading is more mathematically consistent with triality (all three partners are identical in structure, as all three tesseracts are) but less informative (it does not explain why three different archetypes exist in 3D). KEY OBSERVATION: The generation interpretation requires that the three tesseracts be distinguished only by their mass/energy parameter, not by structural differences. In the 24-cell, the three tesseracts ARE structurally identical but mutually orthogonal — distinguished only by orientation (related by 120-degree isoclinic rotation). This matches the generation picture better than the archetype picture. The three generations in particle physics have: - Same quantum numbers (charge, spin, isospin) - Different masses (electron: 0.511 MeV, muon: 106 MeV, tau: 1777 MeV) - Mass ratios are unexplained (this is the "flavor puzzle") If the three tesseracts map to three generations, the 120-degree isoclinic rotation relating them could be the geometric origin of the mass hierarchy. But this is speculative and would require derivation of mass ratios from rotational parameters — exactly what Singh (2025) attempts via the Jordan eigenvalue spread delta^2 = 3/8. ================================================================================ PART 7: CONCLUSIONS AND ASSESSMENT ================================================================================ A. WHAT IS ESTABLISHED: 1. The 24-cell decomposes into 3 tesseracts of 8 vertices each. This is rigorous mathematics (Section 6.4 of 24cell_research.txt). The explicit vertex partition is known. 2. D4 triality permutes three 8-dimensional representations. This is rigorous algebra. The three-fold outer automorphism of the D4 Dynkin diagram is unique among Lie algebras. 3. The cipher identifies crystal archetypes by {2,3} decomposition of coordination numbers. This is empirically verified across 133 elements with 96.9% accuracy. 4. The 1+2 partition (factor 3 absent/present) parallels the vector/spinor split. Both produce a singleton-plus-pair structure. B. WHAT IS SUGGESTIVE BUT UNPROVEN: 5. BCC (no factor 3, broadband) <-> 8_v (vector, force carrier). The qualitative parallels are real but the mapping is not derived. 6. FCC/HCP (same local geometry, different stacking) <-> 8_s/8_c (same content, different chirality). Structurally parallel but the FCC/HCP asymmetry is much larger than the 8_s/8_c asymmetry. 7. Factor 3 as encoding "spinorial character." Intriguing but no mathematical mechanism connects factor-3-in-coordination-number to spinor-representation-membership. C. WHAT FAILS OR IS OBSTRUCTED: 8. Coordination number does not uniformly map to representation dimension: BCC(8) matches, FCC(12) and HCP(12) do not. 9. 8_s and 8_c are equivalent under triality; FCC and HCP are not equivalent under any crystal symmetry operation. 10. The archetype reading (BCC/FCC/HCP = three triality partners) is less consistent than the generation reading (three identical tesseracts = three generations of the same particle). D. VERDICT: The {3}x{8} decomposition IS a real structural feature of the 24-cell and IS connected to D4 triality. The cipher's observation that 24 = 3x8 and BCC coordination = 8 is factually correct and geometrically meaningful. The proposed mapping of BCC/FCC/HCP onto 8_v/8_s/8_c is PARTIALLY SUCCESSFUL: the 1+2 partition and the singleton characterization work, but the detailed mapping breaks on the coordination/dimension mismatch and the FCC-HCP asymmetry problem. The stronger interpretation is the GENERATION reading: three copies of identical 8-vertex structure (tesseracts), distinguished by orientation (120-degree isoclinic rotation), mapping to three generations of identical quantum-number particles distinguished by mass. Under this reading, BCC is the archetype whose coordination number (8) directly reflects the tesseract vertex count, and the factor-3 rule operates at a DIFFERENT structural level (within each generation, between matter types) rather than across triality partners. E. RECOMMENDED NEXT STEPS: 1. COMPUTE: Take the three explicit tesseract vertex sets from 24cell_research.txt Section 6.4 and compute their Voronoi cells in 4D. Determine whether the local geometry around vertices in each tesseract is identical or whether the 24-cell's global structure breaks the equivalence. 2. TEST THE GENERATION READING: If triality = generations, then the 120-degree rotation connecting tesseracts should produce the mass hierarchy. Check: does the rotation angle (2pi/3 = 120 deg) combined with Singh's delta^2 = 3/8 produce realistic mass ratios? This is a specific numerical test. 3. SEPARATE THE TWO QUESTIONS: The {3}x{8} decomposition question (geometric) should be kept distinct from the archetype-to- representation question (physical). The geometric fact is solid. The physical mapping is what requires further work. 4. INVESTIGATE: Does the factor-3 rule have an algebraic counterpart in D4 representation theory? Specifically, the {2,3} decomposition of the Weyl group order 1152 = 2^7 x 3^2 — do the factor-2 and factor-3 parts of this group control different aspects of the representation structure (e.g., 2-part controls dimension, 3-part controls triality)? ================================================================================ INTEGRITY NOTE ================================================================================ This analysis found a mapping that is PARTIAL, not complete. The temptation is to declare success based on the features that work (1+2 partition, broadband/mediator parallel) and minimize the failures (coordination mismatch, spinor asymmetry). This document does not do that. The failures are structural, not cosmetic. The 1+2 partition could arise from any 3-object system with one distinguished element. The coordination/dimension mismatch (8 vs 12) is a hard number that does not go away with reinterpretation. The honest assessment: the {3}x{8} observation is GEOMETRICALLY REAL and TRIALITY-CONNECTED. The specific archetype-to-representation mapping (BCC=8_v, FCC=8_s, HCP=8_c) is SUGGESTIVE in its broad strokes but FAILS in its details. The generation reading is more mathematically consistent but less physically informative. The path forward is Condition 1 from Part 4: determine whether different 3D projections of the same 4D tesseract produce different crystal geometries. If they do, the mapping is rescued at a deeper level. If they do not, the BCC/FCC/HCP distinction arises from physics outside the triality structure. ================================================================================ END OF STUDY ================================================================================