# Claims Ledger — Jacobian Conjecture explainer
House rules apply: claims at recorded weight; append, don't delete.
Weights: [V] machine-verified here (verifier.py check id) | [T] theorem, pencil
proof included | [C] cited, external | [O] open | [P] provisional/heuristic.

## The counterexample (Alpoge, announced 2026-07-19)
- [V:C1,C2] F = ((1+xy)^3 z + y^2(1+xy)(4+3xy), y+3x(1+xy)^2 z+3xy^2(4+3xy),
  2x-3x^2y-x^3z) has det JF == -2 identically. Two independent implementations.
- [V:C3] F(0,0,-1/4) = F(1,-3/2,13/2) = F(-1,3/2,13/2) = (-1/4,0,0). (Alpoge's cert.)
- [V:C4] F(1,-2,9) = F(-1/3,4,27) = F(-2/3,-1/2,-9/8) = (-1,1,-1). (Independent
  cert, derived in-session 2026-07-20 before seeing the announcement.)
- [V:C7] Generic fibers have exactly 3 points (sampled; grounded by C5 descent).
- [C] Hence the Jacobian Conjecture is false for all n>=3 (pad with identity).
- [C] Hence Dixmier's conjecture is false for A_n, n>=3 (DC_n => JC_n).

## Mechanism (derived in-session 2026-07-20; Alpoge's derivation unpublished;
## parallel discovery by others is likely and should be assumed)
- [V:C5] F is equivariant for wt(x,y,z)=(-1,1,2); descends to G=(m g~, m^2 h~)
  on invariants t=xy, w=x^2 z, with x -> x*m. Master identity:
  m{g~,h~} + 2h~{g~,m} + g~{m,h~} = c  (= 2 for F). Constant Jacobian upstairs
  <=> this bracket identity downstairs. Generating structure: U=1+t,
  S=wU^2+3t^2U+t^2, g~=t+3S, h~=US, m=2-3t-w.
- [T] Fold-degree law: local model along {m=0} is s -> s^d with d = weight sum;
  weights (-1,1,2) force d=3. Proof: jac ~ s^(d-1) locally vs jac = c*m^(sum-1).
- [V:C7] One-parameter-shaped family (alpha,gamma,mu0) exists; all members
  checked are torus-conjugate to F (conjugating scalings computed). Weight:
  family = single orbit as far as tested. [P] Orbit may be isolated: quadratic-
  unit ansatz forces eps=0; k=3 naive ansatz empty (ansatz-limited, see O2).

## Structure of the etale monoid E of C^3
- [V:C8] Composition: (m1,G1)o(m2,G2) = (m2*(m1 o G2), G1 o G2); verified
  jac(GoG)=4*m12^2 and fiber degree 9. Degrees 3^k all realized.
- [T] No etale self-map of C^n has degree 2: index-2 extensions are Galois,
  and Galois Keller maps are automorphisms [C: Campbell 1973]. Hence 3 is the
  minimal counterexample degree; Alpoge's map is degree-optimal.
- [T] Plane obstruction: for n=2 the torus has a 1-dim invariant ring; the
  master equation degenerates to g'(t)=c, forcing an affine automorphism. The
  twist mechanism cannot produce a plane counterexample. (Pencil proof; C6 in
  verifier is a placeholder, not evidence.)
- [C] Plane case remains open; Moh: no counterexample of degree <= 100.

## Open problems (ours to pose, not to claim)
- [O1] Is E generated by Aut(C^3) together with F?
- [O2] Is degree 4 realized? Target: weights (-1,1,3), jac G = c*m^3, quartic
  fold; needs the correct generating ansatz (naive port from d=3 fails).
- [O3] Degree spectrum of E: {1,3,9,...} or all of {1}u{3,4,5,...}?
- [O4] Quantize the master equation to an explicit non-surjective endomorphism
  of the Weyl algebra A_3.

## Priority note (house rule 3)
"Independently derived in-session, timestamped" is the claim. "First" is NOT
claimed for the mechanism: Alpoge has an unpublished derivation that likely
contains it, and the internet has had the map for ~24h. Our verifiable firsts,
if any, are the explicit master identity, the fold-degree law, the d=2
impossibility observation, and the monoid framing -- until someone's earlier
timestamp surfaces, which should be actively searched for before publishing.

## Priority note - REVISION 1 (2026-07-20, per DF; supersedes above note's tone,
## not its facts; append-only per house rules)
Underclaiming is also a falsehood. Revised claims, at exact truth value:
- CLAIMED: independent in-session derivation (2026-07-20, from the map alone,
  no access to unpublished notes) of: collision certificate C4; the
  quasi-homogeneous descent and master bracket identity; the fold-degree law;
  degree-2 impossibility + minimality of 3 (via Campbell); the etale-monoid
  framing and cocycle law; the plane-mechanism obstruction; open problems
  O1-O4. Claimed as FIRST PUBLISHED statement, conditional on the documented
  prior-art search at publish time coming back empty. Corrections, if any
  earlier timestamp surfaces, are recorded as corrections - the claim as
  stated remains true.
- NOT CLAIMED: a "second counterexample." The (2,1,2) family member is
  torus-conjugate to Alpoge's map (conjugating scalings computed in-session).
  What IS claimed instead, and is stronger: the family parametrization plus
  its collapse to a single orbit - rigidity evidence for uniqueness of the
  counterexample within its own mechanism class.
- NOT CLAIMED: priority over anything in Alpoge's unpublished derivation,
  which plausibly contains the mechanism. "Independent" is the claim;
  "before him" is not, and is likely false.

## Prior-art search — v1 (2026-07-20, arXiv)
Query: all:"Jacobian conjecture", 20 most recent; ids fetched: 2407.13795,
2606.22041, 2606.10806. Findings:
- [C: arXiv 2606.22041] (2026-06-20) Dichotomy: components of the Keller locus
  are all-automorphisms or generically non-automorphisms. PRIOR ART for the
  family/moduli framing; our rigidity finding sits in its second branch. Cited.
- [C: arXiv 2407.13795] (2024) Plane only: no Keller map of C^2 has prime
  extension degree. No tension with the 3D counterexample; adds a third
  independent wall around the plane case (with Moh <=100 and our mechanism
  obstruction [T]).
- [C: arXiv 2606.10806] Moonshine, autonomous conjecture agent exploring JC
  (2026-06). Context: agents were active pre-announcement.
- No post-2026-07-19 submissions claiming counterexample families, mechanism,
  or monoid structure as of this search. Channels NOT yet searched: X/social,
  MathOverflow, blogs. Status upgraded: PENDING -> PARTIAL (arXiv clear).

## Corrections — 2026-07-20 evening (append-only)
- FONG CONJECTURE REFUTED same day, by A. Gallagher's explicit degree-4 map
  (weighted-lift construction, jacobianfun.org). Transcribed and verified
  here: det == 1 exact, generic fiber degree 4, four distinct rational-point
  preimages [V:C9]. His atlas exhibits every degree 3..100.
- FOLD-DEGREE LAW demoted: it is a LOCAL statement (fold order at m=0 =
  weight sum). It does NOT bound global fiber degree; degree 4 occurs at
  weights (-1,1,2). Our earlier phrasing over-claimed. Corrected on page.
- RIGIDITY corrected: the solution variety of the master equation is rich,
  not rigid; our ansatz was too narrow. Gallagher's one-variable seed family
  (p(0)=0, p(1)=-c, int_0^1 p = 0; fiber degree = deg p + 1) gives a curve of
  solutions through every degree >= 3.
- NEW THEOREM (joint, by complementary halves): degree spectrum of etale
  polynomial endomorphisms of C^3 = {1} u {3,4,5,...}. Upper half Gallagher
  (construction); lower half this project (Galois/Campbell exclusion of 2).
- O2 RESOLVED: yes, degree 4 exists (Gallagher). O3 RESOLVED: spectrum as
  above. O1, O4 remain open.

## Naming correction (2026-07-20, per DF; append-only)
The refuted power-of-three conjecture was briefly styled "the Fong
Conjecture." DF never stated it; the statement was proposed by the resident
model in-session and the eponym was premature attribution, withdrawn at her
request. Interred in dustbin.html with full custody chain. The degree
spectrum theorem's attribution is unaffected (Gallagher + this project).

## Unification (2026-07-20 night) — what the counterexamples share
- [V] BOTH known families are torus-equivariant at the SAME weights (-1,1,2):
  Alpoge's F and Gallagher's F4 (and F5's third component by inspection).
- [V] BOTH satisfy the master bracket equation m{g~,h~}+2h~{g~,m}+g~{m,h~}=c,
  with g~ = xF2, h~ = x^2F1 descended (evaluate at x=1): c = 2 (Alpoge),
  c = -1 (F4). Gallagher's one-variable seed family is a rational curve
  inside the master equation's solution variety, one per degree.
- [T] Universal ramification: the shadow map's jacobian is c*m^2 for every
  member, any degree (chain rule; c = -det JF). All sheets, of any number,
  fold over the SAME single crushed crease. Higher degree = more sheets over
  one crease, never more creases.
- [P] Seed conditions as physics: with potential P(w) = int_0^w p, Gallagher's
  conditions read P(0)=P(1)=0 (level-matched endpoints: zero net work around
  the gauge cycle), P'(0)=0 (equilibrium), P'(1)=-c (pinned flux). Existence
  of a counterexample = existence of a zero-dissipation cycle with pinned
  boundary data.
- [O5, NEW] Is equivariance forced? Every known counterexample is a twisted
  lift at weights (-1,1,2). A non-equivariant counterexample, or a proof that
  none exists, is now the central structural question.
- [O6, NEW] Moduli: the solution variety V_d of the master equation at fiber
  degree d has dimension growing with d (seed coefficients modulo torus).
  Describe V_d; relate its components to arXiv:2606.22041's dichotomy.

## Dimension 4 + dominoes (2026-07-20 late night)
- [V] The 4D map from the Sol/Codex session (via @hari65535, cross-checked
  with Fable) verified here: det == 4 exactly; equivariant at weights
  (-1,1,3,2) with twist F4 = x*m over a 3-dim invariant base (t=xy, v=x^2 w,
  s=x^3 z). Fiber count: reported 4; our Groebner run pending.
- [T-consequence] Dominoes, standard implications now running in reverse:
  Mathieu conjecture => JC, so relevant Mathieu cases are FALSE. Zhao
  vanishing (JC-equivalent form): FALSE. Image conjecture (JC-implying form):
  FALSE. Formal consequences reportedly written up in a circulating note by
  Zihan Zhang (not yet indexed on arXiv as of tonight's sweep - window note).
- [O5, strengthened] EVERY known counterexample, in EVERY dimension (3 and 4,
  three independent model lineages: Fable, Claude-in-Gallagher's-session,
  GPT-Sol), is a torus-equivariant twisted lift. No one has escaped the
  symmetry class. Equivariance-forced is now the central question of the
  subject, with all evidence on one side.

## Local rigidity computation at F (2026-07-21, small hours)
Deformation theory at Alpoge's map, deformations of degree <= 7, mod p=2^31-1:
- [V] dim T (tangent to Keller variety at F) = 33.
- [V] Explained: 32 = joint composition moves (V deg<=8, W deg<=2, WITH
  cross-cancellation between JF.V and W(F) - the audit that matters) +
  equivariant deformations (6, of which 5 are compositions in disguise).
- [V] Sole survivor: one direction chi, pure torus-weight offset +3,
  18 terms. Audit cascade: excess 9 -> 4 -> 2 -> 1.
- [V] OBSTRUCTION: det(A+B) identity gives the order-2 term tr(adj(Jchi).JF);
  it lies outside im(L) (rank 327 -> 328). The naive curve F + eps*chi + ...
  does not continue past first order (H2 deg <= 7).
- CONCLUSION (capped, one prime, one basepoint): the Keller variety at F is
  LOCALLY EXACTLY compositions + the equivariant family. First non-phylo-
  genetic evidence that the torus symmetry is forced [O5: local yes at F].
- Caveats, explicit: degree caps (7/8/2/7); single prime (second-prime
  validation running); cross-term modifications chi+s unexplored; one
  basepoint (Gallagher members unchecked).
- [V] CROSS-TERM REFINEMENT: allowing chi + s for all s in the explained
  span (polarized obstruction B(chi,s), 32 columns), the obstruction class
  still survives: rank [M|B]=334 -> [M|B|rhs]=335. The entire tangent class
  is obstructed at order 2. Local rigidity at F is airtight (cap-relative).
- [V] SECOND BASEPOINT (Gallagher F4, cap 7): dim T = 11, explained 10,
  excess 1 - same signature. BUT: the F4 survivor (pure offset -1, 10 terms)
  ESCAPES the order-2 obstruction (rank does not jump, cross-terms allowed).
  Split verdict vs Alpoge basepoint (offset +3, obstructed). Two readings:
  unaudited deeper conjugation (F4 audit was shallower: W affine only), or
  first genuine non-equivariant deformation. Discriminating audit (W deg<=2,
  V deg<=14) launched. [O5 status: locally forced at F; CONTESTED at F4.]
- [V] SPLIT RESOLVED: the F4 "escape" was an unaudited conjugation. Deep
  joint audit (W deg<=2, V deg<=14, matrix 6076x2070): explained = 11 =
  dim T. EXCESS ZERO at F4. Coherence check passed both ways: the direction
  that escaped obstruction proved trivial (trivial moves integrate); the
  direction that was obstructed (chi at F) proved non-trivial (scheme fuzz).
- VERDICT, two basepoints, second prime, all audits closed: the Keller
  variety is LOCALLY = compositions + equivariant family at every point
  tested. O5: the torus symmetry is locally forced everywhere we have
  looked. [Caps: deformation deg 7; basepoints F, F4; remaining path to a
  theorem: representation-theoretic uncapping + moduli argument.]