Method Lines of the Perfect Cuboid: FORC, GERES, SIO-X4, X4-DMSD, IGPC and TIFV

Methodologies and frameworks · 5 pages

FORC, GERES, SIO-X4 (the start and the temporary closure), X4-DMSD and IGPC: 5 reader pages and 6 complete bodies, all available in Chinese and English. The originals remain authoritative for their research claims; translation is not independent mathematical verification and does not upgrade the evidence level.

Results · 5 pages

A bounded face-height certificate, the G2 fibres, body-cuboid near-misses, X4 shared-edge coupling and TIFV: 5 reader pages and 10 complete bodies, all available in Chinese and English; all remain bounded results or frameworks. The originals remain authoritative for their research claims; translation is not independent mathematical verification and does not upgrade the evidence level. The other 45 documents remain original-only materials in the download table below.

All documents · Files

The original bytes of the documents with pages and of the other 45 (the main md of each zip and loose md files), unmodified; the code, CSV, JSON and SQLite inside the zips are not on this site. The others carry a one-line claim and a flag ('none' = no overclaim found; 'mild' = wording a reader could over-read, figures that could not be reproduced, or superseded by a later paper). The complete archive is kept in AMRAL's research archive and verified per file by SHA-256.

• FORC_分數本體論比例微積分_X4虛擬曲率與比例積分_方法論_v0.1.md — FORC · The methodology
• GERES_全域等價重建與縮放_方法論命題猜想_v0.1.md — GERES · The methodology
• SIO_X4_P0_Completion_Fibers_Latent_Body_Freedom_and_Generator_Quotient_v0.1.md — SIO-X4 · the start and the temporary closure
• SIO_X4_P20_Global_Generating_Kernel_Compositional_Reconstruction_and_Temporary_Closure_v0.1.md — SIO-X4 · the start and the temporary closure
• X4_DMSD_X四邊卯榫約束域_基礎方法論_v0.1.md — X4-DMSD · The methodology
• IGPC_資訊幾何完美長方體_命題猜想與世界分層_v0.1.md — IGPC · The information-geometric framework
• FORC_P7_Frontier_Only_Height_Bounded_Enumeration_and_Coverage_Bridge_v0.1.md — FORC-P7 and P8 · a bounded face-height certificate
• FORC_P8_Incremental_Height_Ladder_and_Local_Long_Run_Bounded_Proof_Package_v0.1.md — FORC-P7 and P8 · a bounded face-height certificate
• GERES_P8_G2_Fiber_Arithmetic_Geometry_and_Birational_Analysis_v0.1.md — GERES-P8 and P9 · the G2 fibres
• GERES_P9_Twin_Elliptic_Perfect_Gate_and_Fiber_Product_Analysis_v0.1.md — GERES-P8 and P9 · the G2 fibres
• GERES_P5_Perfect_Cuboid_World_Encoding_and_Candidate_Ingestion_v0.1.md — GERES-P5 and P7 · body-cuboid near-misses
• GERES_P7_Rational_Point_Fiber_Search_and_World_Deformation_Runtime_v0.1.md — GERES-P5 and P7 · body-cuboid near-misses
• X4_DMSD_P1_Exact_X4_Joint_Algebra_v0.1.md — X4-DMSD-P1 and P2 · X4 shared-edge coupling
• X4_DMSD_P2_Shared_Edge_Scale_Coupling_Engine_v0.1.md — X4-DMSD-P1 and P2 · X4 shared-edge coupling
• Perfect_Cuboid_Trajectory_Information_Freedom_Verification_Methodology_v0.2.md — TIFV · trajectory-information freedom verification
• Perfect_Cuboid_TIFV_Research_Notes_2026-10-06.md — TIFV · trajectory-information freedom verification
• FORC_P1_Exact_Rational_Difference_Algebra_and_Virtual_Curvature_Runtime_v0.1.md — FORC-P1 · Exact rational difference algebra and a virtual-curvature runtime: fixes the four canonical defects E1…E4 (a = 1 normalisation) so that E = 0 is exactly a rational perfect cuboid, implements exact-Fraction divided differences / curvature / integrals, and shows with a √2/√3 convergent branch that the closure energy shrinks while exact clicks stay 0. Flag: none.
• FORC_P2_Stern_Brocot_X4_Branch_Search_and_Curvature_Guided_Enrichment_v0.1.md — FORC-P2 · Stern–Brocot X4 branch search and curvature-guided candidate enrichment: shows the defects have degree ≤ 2 along affine rational paths so a 3-sample quadratic predictor is exact, and reports 100% oracle recall on 1000 synthetic affine branches, stating this is not global search superiority. Flag: mild — the 100% recall is a tautology of the exact-quadratic model (disclosed in the paper).
• FORC_P3_Coupled_Stern_Brocot_World_Branches_and_Real_X4_Candidate_Benchmark_v0.1.md — FORC-P3 · Coupled Stern–Brocot world branches and a real X4 candidate benchmark: builds X4 worlds from four Stern–Brocot Pythagorean joints, benchmarks Near / Linear / Curvature rankers on 2000 synthetic coupled branches (97 / 100 / 100% recall), and reconstructs the Euler brick (44, 117, 240) with E1 = E2 = E3 = 0 and 0 PerfectClicks over 28 body refinements. Flag: none.
• FORC_P4_Projective_Defect_Calculus_Height_Aware_Long_Jump_and_X4_DMSD_Gate_Fusion_v0.1.md — FORC-P4 · Projective defect calculus, height-aware long jump and X4-DMSD gate fusion: proves E3 = c²π_F(π_F + 2) and E4 = c²π_B(π_B + 2) for projective defects, compares coordinates on synthetic branches (about 99% recall), and uses a mod-7 gate and a CRT root-lattice gate to show that (44, 117, 240) and (1575, 1672, 9120) have no integer space diagonal. Flag: mild — 'IMPOSSIBLE' for the two known bricks is a pipeline test (a non-square S is excluded by isqrt anyway).
• FORC_P5_Adaptive_Projective_Trust_Regions_Ratio_Integrals_and_Persistent_Long_Range_Search_v0.1.md — FORC-P5 · Adaptive projective trust regions, ratio integrals and persistent long-range search: adds trust-region / coordinate switching and a SQLite ledger (251 branches, 1251 nodes) with a search manifest whose mode is 'discovery_nonexhaustive'; adaptive depth matches 99.712% on synthetic branches, and the paper says it is not a proof search. Flag: mild — the five-seed benchmark numbers differ from P4 for the same seeds (actionable 2152 vs 2075; canonical 98.25% vs 98.89%) without reconciliation.
• FORC_P6_Exhaustive_Product_Tree_Coverage_Resume_Cursor_and_Dual_Runtime_v0.1.md — FORC-P6 · Exhaustive product-tree coverage, a resume cursor and a proof/discovery dual runtime: partitions (0, 1)⁴ × 16 orientations by a synchronous Stern–Brocot product tree to depth 4 with sound interval pruning: 142,240 cells evaluated, 35,914 of 1,048,576 leaf-equivalents remain FRONTIER (96.57% pruned), resume hash equal; stated to be neither denominator-height exhaustive nor a nonexistence proof. Flag: mild — 'Exhaustive' names exhaustive accounting of regions at depth 4, not a search; 20,812 of the 35,914 frontier cells touch t = 0/1 and cannot be removed by deepening.
• GERES_P1_Formal_World_Semantics_v0.1.md — GERES P1 · Formal world semantics: formal semantics of world / representation / arithmetic state, gauge, certificate and state machine, and the cuboid encoding; no algorithm, code or result. Flag: none.
• GERES_P2_Global_Scale_Engine_v0.1.md — GERES P2 · The global scale engine: an algorithm specification of a world-synchronous scale engine with invariant and replay gates, stating that scaling cannot repair an irrational ratio; no code shipped. Flag: none.
• GERES_P3_Synthetic_Rational_Recovery_v0.1.md — GERES P3 · Synthetic rational recovery: a protocol for synthetic recovery experiments on known rational worlds (with negative and ambiguity controls); its own claim boundary says reconstruction is not implemented. Flag: none.
• GERES_P4_Irrational_and_False_Positive_Gate_v0.1.md — GERES P4 · The irrational and false-positive gate: rejection semantics and a false-positive firewall (the closure vector is a conjunction, never an average); no perfect-cuboid claim. Flag: none.
• GERES_P6_Perfect_Cuboid_Reconstruction_and_Neighbor_World_Search_v0.1.md — GERES P6 · Perfect-cuboid reconstruction and neighbour-world search: reparametrizes shape by P(u) = (1 − u²)/(2u): the perfect cuboid is a rational point of F(r, s, t) = 0 and G2 = (3/5, 4/7, 1073/840) is a rational point of the structural surface S(r, s, C) = 0 whose last open gate is whether C² − 1 is a rational square; novelty is disclaimed. Flag: none.
• GERES_P10_Effective_Covering_Reduction_Local_Sieve_and_Chabauty_Bridge_v0.1.md — GERES P10 · Effective covering reduction, a local squareclass sieve and the elliptic-Chabauty bridge: by factorization, gcd bounds and local congruence sieves any perfect point on the fibres lies in squareclasses {1, 34} (r = 3/5) or {1, 65, 165, 429} (s = 4/7) at every height; the Sage/Magma bridge scripts are stated as not executed. Flag: none.
• GERES_P11_Selmer_Cover_Completion_and_Elliptic_Chabauty_Execution_Package_v0.1.md — GERES P11 · Selmer cover completion and an elliptic-Chabauty execution package: height-independent 'modular cages' (for example 7·13·19·59 divides pq, so q ≥ 320) and Sage/Magma scripts for Selmer / Mordell–Weil / elliptic Chabauty that were 'generated but NOT EXECUTED in the current ChatGPT runtime'; no whole-fibre result is claimed. Flag: mild — the title says 'Execution Package' although nothing was executed (the README says so plainly).
• SIO_X4_P1_Completion_Fiber_Transport_and_First_Manifestation_Search_v0.1.md — SIO-X4-P1 · Completion-fibre transport and first-manifestation search: tracks the bounded structure across H = 5000 / 10000 / 20000 (36 and 56 dormant states activate; 156 and 264 new generators; third-face fibres are singletons). Flag: none.
• SIO_X4_P2_Activation_Kernel_Generator_Birth_Law_and_Manifestation_Pressure_v0.1.md — SIO-X4-P2 · The activation kernel, the generator-birth spectrum and a non-probabilistic manifestation pressure: defines an exact activation kernel and a non-probabilistic 'manifestation pressure'; at H = 20000 all 104 worlds are BLOCKED at the theorem gate and near-product proximity did not predict activation. Flag: none.
• SIO_X4_P3_Activation_Causality_Graph_Container_Memory_and_Quotient_Lineage_Dynamics_v0.1.md — SIO-X4-P3 · The activation-causality graph, container memory and quotient-lineage dynamics: builds a 'structural enabling' event graph of the bounded data (8134 nodes, 6838 edges; explicitly not physical causation). Flag: mild — the latent-memory counts are later superseded by P8 v0.2 (7692 nodes, 5512 edges; 53 worlds / 116 events instead of 99 / 558); this package was not regenerated.
• SIO_X4_P4_Structural_Causal_Motifs_Catalyst_Classes_and_Information_Flow_Compression_v0.1.md — SIO-X4-P4 · Structural causal motifs, catalyst classes and information-flow compression: shows the P3 graph is generated by four edge-production motif families (activation, generator, obstruction, latent memory) covering 6838/6838 edges. Flag: mild — the figures rest on the superseded P3 graph (P8 says the conclusion survives).
• SIO_X4_P5_Motif_Transition_Algebra_Grammar_Closure_and_Cross_Method_Coupling_v0.1.md — SIO-X4-P5 · Motif transition algebra, grammar closure and cross-method coupling: gives the observed motif transition vocabulary and a candidate operator basis (N, γ, π, α, χ, λ) with N scheduler-only and χ holding theorem authority. Flag: none.
• SIO_X4_P6_Operator_Laws_Rewrite_Confluence_and_Canonical_Information_Flow_Normal_Forms_v0.1.md — SIO-X4-P6 · Operator laws, rewrite confluence and canonical information-flow normal forms: bounded rewrite confluence and normal forms (for example 104/104 quotient confluence, 624/624 path associativity); explicitly not a global Church–Rosser theorem. Flag: none.
• SIO_X4_P7_Critical_Pair_Completion_Product_State_Semantics_and_Height_Extension_Stability_v0.1.md — SIO-X4-P7 · Critical-pair completion, product-state semantics and height-extension rewrite stability: critical-pair joinability and height-extension stability 5000 → 10000 → 20000 (no deactivation, retargeting or truth change); not an all-height theorem. Flag: none.
• SIO_X4_P8_Persistent_Semantic_Object_Direct_Limit_and_Information_Flow_Invariants_v0.2.md — SIO-X4-P8 v0.2 · Persistent semantic object, direct limit and information-flow invariants: corrects a latent-memory initialization error found in P3 (direct-min mismatches 0), defines finite semantic objects S_H with embeddings and a candidate direct limit; Λ_∞ = 0 would not imply a finite exact hit. Flag: none (the correction note is a plus).
• P3_to_P8_LATENT_MEMORY_CORRECTION_NOTE.md — The P3 → P8 latent-memory correction note: the correction note that ships with P8 v0.2. Flag: none.
• SIO_X4_P9_Colimit_Morphisms_Observable_Functors_and_Cross_Representation_Invariance_v0.1.md — SIO-X4-P9 · Colimit morphisms, observable functors and cross-representation invariance: cross-representation invariants — the reciprocal-invariant metric D_sym = t/r + r/t − 2, global scale invariance 416/416 — and the distinction that the body-square class is scale-invariant truth while a modular witness prime is only certificate metadata. Flag: none.
• SIO_X4_P10_Encoding_Equivalence_Canonical_Serialization_and_Radix_Free_Invariants_v0.1.md — SIO-X4-P10 · Encoding equivalence classes, canonical serialization and radix-free information invariants: five exact rational codecs form a groupoid with identical canonical semantic hashes for all 104 worlds; a float64 control is lossy for 727 of 936 exact fields. Flag: none.
• SIO_X4_P11_Information_Quotient_Tower_Minimal_Semantic_Kernel_and_Observable_Sufficiency_v0.1.md — SIO-X4-P11 · The information-quotient tower, the minimal semantic kernel and observable sufficiency: natural algebraic kernels — J = r + 1/r for local shapes (6372 → 3186) and (e₂, e₃) for S₃-symmetric world kernels; 'minimal' is stated to mean invariant-theoretic, not bit-count or Kolmogorov. Flag: none.
• SIO_X4_P12_Relation_Kernel_Incidence_Reconstruction_and_Semantic_Sufficiency_v0.1.md — SIO-X4-P12 · The relation kernel, incidence reconstruction and semantic sufficiency: an orientation-free incidence law R(J1, J2, J3) = J1² + J2² + J3² − J1·J2·J3 − 4 = 0 (it holds for J(a), J(b), J(ab)): 52 relation classes, each covering 2 worlds ('dual sheets'). Flag: mild — the 'dual sheet' is the classical brick duality (a, b, c) → (bc, ca, ab) (104 worlds = 52 dual pairs, none self-dual), which the text does not name.
• SIO_X4_P13_Dual_Sheet_Semantics_and_World_Reconstruction_Without_Stored_World_Tables_v0.1.md — SIO-X4-P13 · Dual-sheet semantics and world reconstruction without stored world tables: regenerates the 104 worlds and 624 labelled generators from the 52 relation classes without stored world tables (all post-generation audits pass). Flag: none.
• SIO_X4_P14_Relation_Class_Discovery_Without_Stored_Incidence_Tables_v0.1.md — SIO-X4-P14 · Relation-class discovery without stored incidence tables: rediscovers the 52 relation classes from the 3186 J-kernels through pair roots (5,076,891 pairs, 1060× fewer than naive triples; exact set match). Flag: none.
• SIO_X4_P15_Vieta_Mutation_Relation_Graph_Generative_Closure_and_Seed_Compression_v0.1.md — SIO-X4-P15 · The Vieta-mutation relation graph, generative closure and seed compression: a negative result — Vieta mutation preserves R but connects none of the 52 classes (52 distinct multiplicative lattices). Flag: none.
• SIO_X4_P16_Orbit_Space_Quotient_Multiplicative_Lattice_Kernel_and_Carrier_Changing_Operators_v0.1.md — SIO-X4-P16 · The orbit-space quotient, the multiplicative-lattice kernel and carrier-changing operators: CarrierExchange between relation classes sharing a carrier gives 29 edges and 30 components (bounded, filtration-dependent). Flag: none.
• SIO_X4_P17_Carrier_Exchange_Filtration_Component_Merge_Dynamics_and_Hub_Centrality_Laws_v0.1.md — SIO-X4-P17 · Carrier-exchange filtration, component merge dynamics and hub-centrality laws: a height filtration of that graph with laws such as |E_H| = Σ binom(d_H(J), 2); the component count is not monotone. Flag: none.
• SIO_X4_P18_Persistent_Orbit_Space_Object_Merge_Forest_and_Relational_Direct_Limit_v0.1.md — SIO-X4-P18 · A persistent orbit-space object, merge forest and relational direct limit: a persistent orbit-space object with a 22-edge merge forest preserving all 73 pair merge heights. Flag: none.
• SIO_X4_P19_Three_Level_Semantic_Tower_Inter_Level_Correspondences_and_Naturality_v0.1.md — SIO-X4-P19 · The three-level semantic tower, inter-level correspondences and naturality: a three-level tower (carrier, relation, world, component) with 5681/5681 naturality tests passed. Flag: none.
• X4_DMSD_P3_Ratio_Valuation_Modular_Signature_Engine_v0.1.md — X4-DMSD P3 · The ratio / valuation / modular signature engine: ratio keys, prime-valuation signatures and projective modular signatures avoid 99.9958% of exact-key computations at m ≤ 80. Flag: mild — the headline reduction figures are not reproducible from the shipped 55-line demo.
• X4_DMSD_P4_Multiscale_Block_Semantics_and_Sound_Hyperblock_Pruning_v0.1.md — X4-DMSD P4 · Multiscale block semantics and sound hyperblock pruning: decimal-block summaries prune 99.97% of 493,039 single-m hyperblocks while all 288 exact constructions survive; the summaries are derived from materialized children. Flag: mild — the figures are not reproducible from the shipped code.
• X4_DMSD_P5_Symbolic_Block_Operators_and_First_True_Scale_Jump_v0.1.md — X4-DMSD P5 · Symbolic block operators and the first true scale jump: a closed-form shape operator r(t) makes block summaries independent of decimal scale; includes an exact IMPOSSIBLE certificate for a mixed 10¹⁰⁰ / 10²⁵⁰ / 10¹⁰⁰⁰ hyperblock with 0 seeds materialized. Flag: mild — the certificate is a scale-independent shape-interval exclusion (any exponent works) and the exponents are nominal; the paper says it is not a search of 10¹⁰⁰⁰ seeds.
• X4_DMSD_P6_Symbolic_Residue_Valuation_Operators_v0.1.md — X4-DMSD P6 · Symbolic residue / valuation operators over huge Euclid seed blocks: a 2-adic orientation rule and finite residue supports make a shape-POSSIBLE huge hyperblock arithmetic-IMPOSSIBLE with zero materialization. Flag: mild — the same 'huge-scale' framing as P5.
• X4_DMSD_P7_pAdic_Depth_Primitive_Automata_and_Body_Square_Gate_v0.1.md — X4-DMSD P7 · p-adic depth, primitive-state automata and the body-square symbolic gate: truncated p-adic automata; a mod-7 body-square gate rejects 18.18% of face-coupled local-state triples and one specific nonempty huge hyperblock. Flag: mild — the same framing as P5.
• X4_DMSD_P8_CRT_World_Automata_and_Symbolic_PerfectClick_Hyperblocks_v0.1.md — X4-DMSD P8 · CRT world automata and symbolic PerfectClick hyperblocks: CRT world automata over primes (7, 17, 19) reject 99.8688% of symbolic arithmetic state-blocks (a state-mass measure, not seed density). Flag: mild — the percentages are block mass and the paper says they are not natural densities.
• X4_DMSD_P9_Nonempty_CRT_Shape_Cells_and_Eventual_Decade_Lifting_v0.1.md — X4-DMSD P9 · Nonempty CRT shape cells and eventual decimal-decade lifting: a nonempty-cell theorem — for a fixed modulus and shape interval every admissible CRT cell contains infinitely many coprime Euclid pairs, so P8's empty-block caveat is a finite-height issue; explicit witnesses at k = 10, 100, 1000. Flag: none.
• X4_DMSD_P10_Exact_Coupling_Lattices_and_Multi_Modulus_Convergence_v0.1.md — X4-DMSD P10 · Exact coupling lattices and multi-modulus convergence: the coupling defect D = A·C·B′ − B·C′·A′: |D| ≤ bound and modulus > bound force D = 0; the m ≤ 60 modular ladder 'converges' to 192 exact; modulus budgets for k = 10, 100, 1000; the body root lattice (32 roots, minimum 20332) excludes (1575, 1672, 9120). Flag: mild — the intermediate ladder values (for example 6198 at prime 19) are not reproduced by a plain recount of D = 0 mod Q (52,002; 192 is reached at 31, not 29); the filter is undefined in the text and there is no code.
• X4_DMSD_P11_Interval_CRT_Lattice_Refinement_and_Exact_World_Cells_v0.1.md — X4-DMSD P11 · Interval–CRT lattice refinement and exact world cells: an interval–CRT lattice classifier — the m ≤ 60 face cells resolve to 192 EXACT and 212 IMPOSSIBLE at shape resolution 2000. Flag: mild — 'EXACT' means an exact face coupling (an Euler brick), not a solution.
• X4_DMSD_P12_Adaptive_Exact_Cell_Scheduler_v0.1.md — X4-DMSD P12 · The adaptive exact-cell scheduler: an adaptive controller chooses shape / prime / root refinements autonomously; bounded replay closes all 6198 face cells and all 48 body worlds. Flag: mild — the same caution about the 'EXACT' label.
• X4_DMSD_P13_Distributed_Exact_Cell_Runtime_and_Persistent_Search_Ledger_v0.1.md — X4-DMSD P13 · Distributed exact-cell runtime and persistent search ledger: a content-addressed SQLite ledger with checkpoint/resume and proof replay (6198 cells; 192 EXACT / 6006 IMPOSSIBLE); the workers are simulated identities in one process. Flag: mild — 'Distributed' in the title (there is no network), the 'EXACT' label; the 10.8 MB SQLite is not on this site.
• Perfect_Cuboid_Trajectory_Information_Freedom_Verification_Methodology.md — TIFV v0.1 · The trajectory-information freedom verification methodology (v0.1, 2026-10-06): the loose md, byte-identical to the v0.1 inside the zip; expanded by v0.2 (see the TIFV page). Flag: none.