The revision records the exact marked-origin cancellation strategy and corrects the three-use certificate to distinguish one three-event lineage from four total marked-origin events.
Evidence posture · Reported resultToric geometry · smooth rational fans · rewriting systems
Oda’s Strong Factorization Conjecture
Collaboration betaGiven two smooth rational fans in one lattice with the same support, the conjecture asks whether both can be refined to one smooth fan using only ordinary smooth star subdivisions.
Known results and sources
Research problem
Exact mathematical statement
Here and are smooth rational fans in one lattice with the same support, is required to be smooth, and each displayed relation means a finite sequence of ordinary smooth star subdivisions.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Oda’s Strong Factorization Conjecture stands
Selected highlights from the Revision 19 source material: Oda's ordinary smooth strong factorization conjecture remains open; two conditional bridges remain incomplete; the corrected direct program uses a 64-element S3-equivariant frontier profile, symbolic corridor formulas, and exact route counterexamples; the current frontier is the reachable FSAFE_1 grammar together with the CMPSTACK_2 comparator-stack descent. Packaged computations are retained only as reported evidence and do not establish an all-length theorem or the conjecture.
Build a closed production grammar on profile-labelled residual genealogy trees and prove FSAFE_1 at every first productive-use cut.
Route status · Active routeThe fixed-frame quotient is not equivariant under one-sided suffix transport and must not be used as the reachable grammar state.
Route status · Refuted routeThe current work begins from the direct R6WF_1 bridge and the separate geometric PNORM_3/PCTX_3/MIG_4/UNR_3 bridge, while explicitly retaining open status.
Evidence posture · Reported reductionThe current work gives exact piecewise height formulas for A^2V^nX^2v and its y-axis partner, closing the stated p=2 two-leading-source families.
Evidence posture · Source-reported route statement · dependencies incompleteGive a finite production list for complete one-token computations on 64-profile-labelled binary forests and prove every reachable first productive-use cut is accepting.
Task status · Ready to work onRevision 19 reports a deterministic PASS checker for the profile algebra, finite frontier comparisons, corridor regressions, and exact adversarial certificates.
Research stage 10Work mapped so far
Oda’s Strong Factorization Conjecture in numbers
6kretained lines of mathematical investigation5,959 in the current working snapshot
- Argument development
- 4,737 · 79%
- Explored or eliminated routes
- 190 · 3%
- Computational analysis
- 333 · 6%
- Open obligations
- 425 · 7%
- Definitions and setup
- 274 · 5%
21selected mapped statements8routes investigated10reported milestones6open questions5contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Prove RBL_1 and FSAFE_1
Give a finite production list for complete one-token computations on 64-profile-labelled binary forests and prove every reachable first productive-use cut is accepting.
Suggested move: Write the finite productions and verify closure under split, cancellation, coordinate-permutation relabelling, and productive-cut evaluation.
What would count as progress
- Cover every recursively reachable boundary.
- Accept every first productive-use cut.
- Admit reachable terminal recaptures while excluding or closing arbitrary gate-theft classes.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Build a closed production grammar on profile-labelled residual genealogy trees and prove FSAFE_1 at every first productive-use cut.
Route status · Active routeReplace the refuted fixed-overhead rank by a well-founded stack of weighted capacities, row count, phase, and cyclic frame.
Route status · Active routeContinue PCTX_3, PNORM_3, MIG_4, and UNR_3; Revision 19 adds no new geometric theorem.
Route status · Active routeExpand the cancellation-free, blocked off-corner, and comparator classifications into complete transition tables without using finite regressions as proof.
Route status · Active routeExplored alternatives
Other routes
Extend through a third leading origin, arbitrary A^mV^nX^pe, and arbitrary source-block stacks after CMPSTACK_2.
Route status · Route held in reserveThe fixed-frame quotient is not equivariant under one-sided suffix transport and must not be used as the reachable grammar state.
Route status · Refuted routeBounded cancellation work and individual scar counts are too fine-grained; only macro-compressed return phases remain viable.
Route status · Eliminated routeBrowse 1 more explored route
The ceil(i/q)-based scalar remains useful evidence and a possible stack component, but is not justified as an all-length theorem.
Route status · Not yet justifiedRoute statements and reductions
Statements the next route can inspect and build on
PNORM_3, PCTX_3, MIG_4, and UNR_3 together would imply finite one-center developments and hence Oda's conjecture.
Source-reported route statement · dependencies incompleteAt every first productive use of a current origin during a one-token computation from a certified reachable boundary, the normalized complete surviving same-origin left frontier is accepted by the six-state automaton.
Source-reported route statement · dependencies incompleteRBL_1 together with FSAFE_1 would imply ROC_1, then one-token termination, R6WF_1, and Oda's conjecture.
Source-reported route statement · dependencies incompleteFor D_(q,i,s)=A^2V^qaxBUX^iA(yA)^s with s≥2, a finite stack of weighted comparator capacities, remaining rows, phase, and cyclic frame should decrease on every Rule-6b return edge.
Source-reported route statement · dependencies incompleteMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Prove RBL_1 and FSAFE_1
Give a finite production list for complete one-token computations on 64-profile-labelled binary forests and prove every reachable first productive-use cut is accepting.
Suggested move: Write the finite productions and verify closure under split, cancellation, coordinate-permutation relabelling, and productive-cut evaluation.Ready to work on
Prove CMPSTACK_2
Classify every first-generation Rule-6b child of D_(q,i,s), define a well-founded comparator-stack order, and prove decrease on every return edge and sibling.
Suggested move: Classify the child exposed after a completed top comparator reaches the next yA row and prove its lower profile is no larger in the same cyclic frame.Ready to work on
Expand load-bearing production tables
Replace retained classification sentences by complete transition, child, and terminal tables for the cancellation-free grammar, blocked off-corner proof, and alternating comparators.
Suggested move: Write the exact six-input cancellation-free transitions, the four-template off-corner map, and the C/R/Q/Q^x recurrences.Ready to work on
Complete the geometric interfaces
Prove PCTX_3, PNORM_3, MIG_4, and UNR_3 with the exact context and noetherian assembly requirements inherited from the governing appendix.
Suggested move: Continue the named geometric interfaces without repeating bare five/six-pivot searches.Ready to work on
Prove or reject blocked noncorner dominance
Determine whether every noncorner child of E_(c;q,i,s) has height at most R_q(i,s)-1 for the conjectural sharp blocked rank.
Suggested move: Expand the off-corner classification into exact child-to-template transitions and test whether the candidate rank controls each class.Ready to work on
Extend through arbitrary source stacks
After CMPSTACK_2, introduce a third leading origin, prove arbitrary homogeneous A^mV^nX^pe termination, then extend to arbitrary source-block stacks with frame rotation and transparent returns.
Suggested move: Complete CMPSTACK_2 before introducing a third leading origin.Blocked by the current route
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
The ordinary smooth, or unweighted, conjecture remains open. Adiprasito and Pak prove the weighted version, whose stellar subdivisions may use rationally smooth rather than smooth toric centers; that theorem does not settle the ordinary smooth statement.
[2]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedAdiprasito and Pak prove that any two triangulations of a geometric complex have a common stellar subdivision, yielding the weighted Oda theorem and common toric blowups at rationally smooth points.[4] PreprintStrong factorization was confirmed for the special class arising from the braid arrangement fan.[1] Peer reviewedDa Silva and Karu give an algorithm conjectured to construct a smooth strong factorization, prove reductions and special cases, and identify termination as the unresolved issue.[5] PreprintThe weak factorization problem, in which subdivisions and inverse operations may be interleaved, was proved in full generality; it is distinct from strong factorization.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusOda's Strong Factorization Conjecture
The open ordinary statement requires smooth star subdivisions; the weighted theorem permits the broader rationally smooth setting.
[2]Strong factorization orders all blowups before all blowdowns, whereas weak factorization permits them in any order.
[6]The toric conjecture is equivalent to asking for a nonsingular fan reachable from two nonsingular fans with the same support by smooth star subdivisions.
[6]Formal and computational footholds
Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.
- computation · source linked; not reproduced by ProofAtlasReported factorization experiments and tables
The paper reports computer experiments and tabulated factorization counts for finite parameter families, but the scoped source does not provide a public code repository.
[6]
How the route was assembled
Argument structure
These stages follow the mathematical order of the supplied argument.
10 mapped milestonesretained argument map
Browse all 10 mapped stages
- stage 1Conjecture and incomplete bridge map
- stage 2First-use automaton retained
- stage 3Unsafe 78-label compression withdrawn
- stage 4Equivariant 64-profile algebra installed
- stage 5Symbolic corridor calculus retained
- stage 6Exact route certificates restored and corrected
- stage 7Fixed second-A overhead eliminated
- stage 8Sharp blocked rank retained only as a candidate
- stage 9Direct frontier split into two concrete open bridges
- stage 10Exact audit checker packaged
Conjecture and incomplete bridge mapThe current work retains the unresolved ordinary smooth conjecture and two distinct incomplete routes to it.
Research stage 1Mapped research milestoneInitial research sequence
First-use automaton retainedThe six-state productive-recapture automaton survives the Revision 19 audit unchanged.
Research stage 2Mapped research milestoneInitial research sequence
Unsafe 78-label compression withdrawnA concrete suffix-relabel witness shows that the fixed-frame 13-element quotient cannot be paired with a frame tag as a sufficient 78-label state.
Research stage 3Mapped research milestoneInitial research sequence
Equivariant 64-profile algebra installedThe corrected frontier state stores all six relabelled transformations, with componentwise composition and coordinate-permutation relabelling, and the current work reports 64 realized profiles.
Research stage 4Mapped research milestoneInitial research sequence
Symbolic corridor calculus retainedExact project formulas cover the R_q, rectangle, equality-comparator, blocked, one-leading-source, alternating-comparator, and p=2 axis families.
Research stage 5Mapped research milestoneInitial research sequence
Exact route certificates restored and correctedRevision 19 restores the R_q cancellation strategy and clarifies that the three-use branch has three events on one lineage but four total marked-origin events.
Research stage 6Mapped research milestoneInitial research sequence
Fixed second-A overhead eliminatedThe complete reported W_15 return tree has height 15 where 2s+4 gives 14, eliminating the fixed-overhead rank.
Research stage 7Mapped research milestoneInitial research sequence
Sharp blocked rank retained only as a candidateThe ceil(i/q)-based rank passes stated finite ranges but still needs an all-length proof of blocked noncorner dominance.
Research stage 8Mapped research milestoneInitial research sequence
Direct frontier split into two concrete open bridgesThe live direct program is now organized around CMPSTACK_2 and the 64-profile RBL_1 plus FSAFE_1 production grammar, with proof hardening and source-stack extension behind them.
Research stage 9Mapped research milestoneInitial research sequence
Exact audit checker packagedRevision 19 reports a deterministic PASS checker for the profile algebra, finite frontier comparisons, corridor regressions, and exact adversarial certificates.
Research stage 10Mapped research milestoneInitial research sequence
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
17 standing statements4 proposed statements10 mathematical milestones6 open questions7 conditional results2 completed special cases
- theorem candidate
4 of 21 4 - reduction
4 of 21 4 - lemma
10 of 21 10 - computational claim
2 of 21 2 - counterexample
1 of 21 1
Selected mathematical clusters7 mathematical clusters
Conjecture and conditional bridgesThe unresolved ordinary smooth statement and its direct and geometric bridge architecture.7 displayed rows · 1 route included
- retained route statementOda ordinary smooth strong factorization conjecture
- retained route statementDirect Algorithm-A bridgeconditional
- retained route statementGeometric finite-development bridgeconditional
- Recorded relationshipThe retained direct bridge would settle the conjecture once its open termination premise is supplied.supports · reported by source
- Recorded relationshipThe retained geometric route is a separate conditional route to the same unresolved conjecture.supports · reported by source
- Research targetComplete the geometric interfacesopen
- Active routeGeometric normalization and assemblyContinue PCTX_3, PNORM_3, MIG_4, and UNR_3; Revision 19 adds no new geometric theorem.
First-use frontier algebraThe six-state automaton, fixed-frame monoid, failed 78-label compression, and corrected 64-profile algebra.9 displayed rows · 1 route included
- retained route statementSix-state first-use frontier automatonintermediate
- retained route statementThirteen fixed-frame transformationscomputational
- retained route statementWithdrawn 78-label compressionintermediate
- retained route statementS3-equivariant profile calculusintermediate
- retained route statementSixty-four realized equivariant profilescomputational
- ChallengeThe fixed-frame equality τ(V)=τ(VV) is destroyed by the +2 suffix shift because τ(X) differs from τ(XX), so the proposed quotient action and 78-label state are invalid.counterexample · reported resolved
- Useful failureCompress each frontier to one of 13 fixed-frame automaton transformations together with a six-valued current frame.reported failure
- ComputationPackaged exact closure of the fixed-frame, cyclic-equivariant, and full S3-equivariant frontier transformations.The current work reports 13 fixed-frame transformations, 43 cyclic-equivariant profiles, and 64 full S3-equivariant profiles, with complete multiplication and relabelling tables. · reported unreproduced
- Refuted routeFixed-frame 78-label compressionThe fixed-frame quotient is not equivariant under one-sided suffix transport and must not be used as the reachable grammar state.
Reachable frontier safetyRBL_1, FSAFE_1, and the conditional chain from first-use acceptance to Oda.6 displayed rows · 1 route included
- retained route statementReachable frontier safety FSAFE_1conditional
- retained route statementFrontier-safety termination chainconditional
- DerivationOnce the complete frontier is accepting at every first use, the retained automaton criterion rules out productive recapture from the left; ordered separation and the finite residual-tree argument then give the stated termination chain.active reported
- Research targetProve RBL_1 and FSAFE_1open
- ComputationReported comparison of the six-state frontier automaton against the direct local transducer for every frontier through length four in both first-use modes.The current work reports 3,110 comparisons with zero failures. · reported unreproduced
- Active routeReachable genealogy with equivariant profilesBuild a closed production grammar on profile-labelled residual genealogy trees and prove FSAFE_1 at every first productive-use cut.
Symbolic corridor calculusExact retained height formulas for R_q, rectangles, equality comparators, blocked rectangles, one-leading sources, comparator phases, and p=2 axes.11 displayed rows · 1 route included
- retained route statementFixed-height R_q familyintermediate
- retained route statementRectangle height formulaintermediate
- retained route statementEquality comparatorintermediate
- retained route statementUniform blocked-rectangle upper boundintermediate
- retained route statementComplete one-leading-source height formulaintermediate
- retained route statementAlternating comparator tablesintermediate
- retained route statementExact p=2 two-leading-source axesspecial case
- DerivationThe first-generation children reduce either to blocked rectangles covered by the uniform bound or to the terminal exact rectangle; the final child maximizes the height.active reported
- DerivationThe current work classifies the D and D^x child phases using their comparator lengths, then adds the root edge to obtain the two exact p=2 axes.active reported
- ComputationReported finite regressions for R_q, the marked-origin strategy, rectangles, equality comparators, blocked states, one-leading-source formulas, comparator tables, and p=2 axes.All stated packaged finite ranges report zero failures, but the current work explicitly separates them from all-length symbolic proofs. · reported unreproduced
- Active routePublication-grade proof hardeningExpand the cancellation-free, blocked off-corner, and comparator classifications into complete transition tables without using finite regressions as proof.
Route counterexamples and scope firewallsConcrete witnesses exclude the 78-label summary, fixed overhead, cancellation-scale ranks, and uniform two-use claims without challenging Oda itself.9 displayed rows · 1 route included
- retained route statementThree productive uses on one lineagespecial case
- retained route statementRefuted fixed additive second-A overheadconditional
- Useful failureCompress each frontier to one of 13 fixed-frame automaton transformations together with a six-valued current frame.reported failure
- Useful failureBound the general second-leading-source return height by a fixed additive overhead 2s+4.reported failure
- Useful failureUse individual cancellation scars or a bounded amount of productive work per cancellation as a termination rank.reported failure
- Useful failureProve a uniform two-use bound for one origin on all homogeneous three-source flat roots.reported failure
- ComputationComplete reported return-tree computation for W_15=A^2VaxBUX^11A(yA)^5 with a retained maximizing path.The current work reports height 15, 4,852 nodes, and an exact 15-edge maximizing path, refuting the proposed 2s+4 bound at s=5. · reported unreproduced
- ComputationReported exact 61-bit branch certificate for the marked third A in A^5V^10X^8v.The branch ends after 184 rewrite steps and records three final productive events on one lineage plus one earlier event on a disjoint lineage. · reported unreproduced
- Eliminated routeIndividual cancellation-scar rankingBounded cancellation work and individual scar counts are too fine-grained; only macro-compressed return phases remain viable.
Second-leading-source frontierThe conjectural sharp blocked scalar, refuted fixed overhead, open CMPSTACK_2 rank, and deferred arbitrary-stack extension.11 displayed rows · 3 routes included
- retained route statementSharp blocked-rank candidatecomputational
- retained route statementRefuted fixed additive second-A overheadconditional
- retained route statementCMPSTACK_2 comparator-stack descentconditional
- ChallengeFinite passing ranges do not prove blocked noncorner dominance for all parameters, so the sharper rank must remain conjectural.unsupported step · open
- ChallengeThe exact reported W_15 return tree has height 15 at s=5, exceeding the proposed 2s+4 bound of 14.counterexample · reported resolved
- Research targetProve CMPSTACK_2open
- Research targetProve or reject blocked noncorner dominanceopen
- Research targetExtend through arbitrary source stacksblocked
- Active routeSecond-leading-source comparator stackReplace the refuted fixed-overhead rank by a well-founded stack of weighted capacities, row count, phase, and cyclic frame.
- Route held in reserveArbitrary homogeneous source stacksExtend through a third leading origin, arbitrary A^mV^nX^pe, and arbitrary source-block stacks after CMPSTACK_2.
- Not yet justifiedSharp blocked-rank candidateThe ceil(i/q)-based scalar remains useful evidence and a possible stack component, but is not justified as an all-length theorem.
Computation and proof-strength auditReported checker outputs, finite-range scope limits, and open publication-grade table expansions.8 displayed rows · 1 route included
- ComputationPackaged exact closure of the fixed-frame, cyclic-equivariant, and full S3-equivariant frontier transformations.The current work reports 13 fixed-frame transformations, 43 cyclic-equivariant profiles, and 64 full S3-equivariant profiles, with complete multiplication and relabelling tables. · reported unreproduced
- ComputationReported comparison of the six-state frontier automaton against the direct local transducer for every frontier through length four in both first-use modes.The current work reports 3,110 comparisons with zero failures. · reported unreproduced
- ComputationReported finite regressions for R_q, the marked-origin strategy, rectangles, equality comparators, blocked states, one-leading-source formulas, comparator tables, and p=2 axes.All stated packaged finite ranges report zero failures, but the current work explicitly separates them from all-length symbolic proofs. · reported unreproduced
- ComputationComplete reported return-tree computation for W_15=A^2VaxBUX^11A(yA)^5 with a retained maximizing path.The current work reports height 15, 4,852 nodes, and an exact 15-edge maximizing path, refuting the proposed 2s+4 bound at s=5. · reported unreproduced
- ComputationReported exact 61-bit branch certificate for the marked third A in A^5V^10X^8v.The branch ends after 184 rewrite steps and records three final productive events on one lineage plus one earlier event on a disjoint lineage. · reported unreproduced
- ChallengeThe project theorem remains in the current research map, but publication-grade use requires a full alternating-gate induction showing that no omitted noncorner child exceeds the recorded cores.unsupported step · open
- Research targetExpand load-bearing production tablesopen
- Active routePublication-grade proof hardeningExpand the cancellation-free, blocked off-corner, and comparator classifications into complete transition tables without using finite regressions as proof.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
The current research map records this as an open mathematical step.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Cover every recursively reachable boundary.
- Accept every first productive-use cut.
- Admit reachable terminal recaptures while excluding or closing arbitrary gate-theft classes.
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Prepared starting pointProve RBL_1 and FSAFE_1
Oda’s Strong Factorization Conjecture · ready to start
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Research contextPrepared context for any AI agent
Given two smooth rational fans in one lattice with the same support, the conjecture asks whether both can be refined to one smooth fan using only ordinary smooth star subdivisions.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references6 cited works · next context review by Nov 2, 2026
The mathematical context was checked on Aug 2, 2026. Status can be refreshed sooner after a material result or claim.
- 1All triangulations have a common stellar subdivisionpreprint · accessed Aug 2, 2026
- 2All triangulations have a common stellar subdivisionpreprint · accessed Aug 2, 2026
- 3Lectures on Torus Embeddings and Applicationsoriginal source · accessed Aug 2, 2026
- 4All triangulations have a common stellar subdivisionpeer reviewed result · accessed Aug 2, 2026
- 5On Oda's Strong Factorization Conjecturepeer reviewed result · accessed Aug 2, 2026
- 6On Oda's Strong Factorization Conjectureoriginal source · accessed Aug 2, 2026
Important qualifications
- Do not collapse the 2026 weighted theorem into the ordinary smooth conjecture; the permitted blowup centers differ.
- Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.
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