teerthsharma/monodromy: monodromy v0.1.0 — injectivity without the Jacobian

Deciding injectivity, symmetry, dimension and sensitivity from multi-point geometry rather than from the Jacobian determinant.

Motivation

The working invertibility certificate across applied machine learning is local: sample the domain, evaluate det DF, and conclude invertibility if it never vanishes or never changes sign. This inference underwrites the change-of-variables density in normalizing flows, the folding check in diffeomorphic image registration, and element validity in mesh generation.

The forward implication is the inverse function theorem and is sound. The converse — local invertibility everywhere implying global injectivity — is what fails. The Jacobian conjecture (Keller, 1939) asked whether a polynomial map of C^n with det DF a nonzero constant must be a polynomial automorphism. It was refuted in July 2026; arXiv:2608.00222 (Gao, 31 July 2026) generalises the refutation above dimension two and describes the counterexamples as "everywhere unramified, and fail to be injective only through points escaping to infinity" — étale coverings that are not proper. JC_2 remains open.

Under a hypothesis that is now known to be insufficient, a certificate reading one point at a time cannot decide the question. This release provides certificates that read more than one point.

Principal results

Jacobian-free injectivity. Define

free_ratio(F; X) = [ min_{p≠q} |F(p)−F(q)| / |p−q| ]
                 ÷ [ median_{p≠q} |F(p)−F(q)| / |p−q| ]

Numerator and denominator are distances between sampled points; no derivative is evaluated. On eight maps with closed-form ground truth drawn from the settings above, the determinant test scores 6/8 and this statistic 8/8, separating the classes by a factor of 3.02 (injective 1.53e-02 … 7.06e-01; non-injective 1.04e-03 … 5.07e-03). The two determinant failures are exactly those the refutation predicts: a planar flow layer with w·u = −2, whose det DF = 1 − 2 sech²(z) is strictly positive for |z| > 0.8814 while g(z) = z − 2 tanh z gives g(1.915) = g(−1.915) = 0; and a log-polar registration map with det = e^{2x} > 0 satisfying F(x,y) = F(x, y+2π).

A normalisation branching on the hypothesis. ρ = λ/s with s = min σ_min(DF) uses the determinant as reference rather than answer. It is valid only where F is étale, since it divides by a quantity vanishing exactly under ramification; the decision branches on min|det| / median|det|.

Properness as a growth exponent, and the covering chain. With m(R) = min_{|x|=R} |F(x)|, F is proper iff m(R) → ∞. A proper local homeomorphism is a covering map; its image is open and closed, hence the whole connected target; C^n is simply connected, so the covering is trivial and the degree is 1. Hence étale ∧ proper ⟹ injective. Both estimator traps are documented: a sampled sphere minimum cannot locate a zero on a measure-zero subvariety (unrefined sampling reports a clean exponent of 1.000 for a map whose m(R) is identically 0), and the exponent alone remains insufficient after refinement because a numerical floor times R has the same slope.

A certified proof on convex regions. For x, y ∈ B convex, F(y) − F(x) = M(y − x) with M in the entrywise interval hull of DF over B; 0 ∉ [det] proves injectivity on B. Measured: 3 proofs, 0 false certificates across 7 adversarial cases, at 0.0017 s against 3.5 s for the sampled certificate. This is classical interval analysis — Lagrange, Delanoue & Jaulin, Reliable Computing 13 (2007), Theorems 5 and 7 — and is included with that attribution.

Symmetry: periodicity is not symmetry. harmonic_order establishes that a defect profile repeats with period 360/n; it does not establish that the defect at 360/n is small. make_swiss_roll(400, random_state=0)[:, :2] was reported C2 at power 0.8897 in four of six configurations. The discriminator is the sample's own mean nearest-neighbour spacing (true symmetries 0.0000–1.6223, false positives 2.3737–2.8243). Depth below the profile median was measured and rejected at a separation of 1.013.

Thresholds as minimax equilibria. V(t) = max(FN(t), FP(t)), t* = argmin V. Reported with the negative result that the game value is 0.000 for every separable statistic and therefore cannot rank a gap of factor 3.01 against one of 1.01; the margin can, and maximising the minimum margin is the same minimax problem.

Verification

• Mutation score 15/15. Fifteen deliberate reversions of real fixes, applied one at a time from an isolated git worktree, each required to turn the suite red. Three survived an earlier pass and were bound, including one that the trace identity Σλ = ln|det| provably could not detect because the mutation conserves it.
• 116 passed, 5 xfailed, 0 failed.
• Clean-room install, independently audited: wheel into a fresh virtualenv, imported from a working directory with the source tree absent from sys.path, under numpy 2.4.6; all entry points executed on 25,934 real exchange trades; map_spectrum called, Σλ = ln(0.3) to 15 digits; torch absent from pip list, from sys.modules, and import torch raises.
• Hénon reference check: λ₁ = +0.42084 against Sprott's +0.41922, D_KY = 1.25901 against 1.25827.
• Sierpinski gasket: 1.5998 against exact log 3 / log 2 = 1.5850.

Limitations

Stated in docs/corrections.md and in each module rather than implied by omission. Only the box certificate proves injectivity; every other instrument is one-sided and reports collision_found, never injective. certify_injective returns True or None and never False, because a missing hypothesis is not evidence against a conclusion. Symmetry recovery is sound and not complete: 6/6 clean, 2/6 at 10% jitter, 0/24 false positives, and every miss is order 1 or a proper divisor of the truth. The persistent-homology dimension estimator is trustworthy to roughly intrinsic dimension 3; a uniform 10-D cube reads 8.660 and degrades with additional samples. so3_scan costs 73.5 s at n=100 and exceeds 300 s at n=500. Every calibrated constant rests on fewer than thirty cases; FREE_LEVEL was measured not to transfer below n=400 or across dimension and now refuses to run outside its calibrated range. Most of the benchmark is two-dimensional while the refutation is for n ≥ 3.

Novelty

None of the mathematics is new: Fekete–Kingman subadditivity, covering-space theory, mean-value interval hulls, Steele's theorem, Schweinhart's estimator. Whether any statement is new as a machine-learning instrument went to a literature sweep whose verdicts, search queries and caveats are in docs/novelty.md, including a deliberate control (the interval-hull certificate) that correctly returned already-published. Every "not found" is a not-found over stated queries, not a proof of absence; two paywalled venues returned HTTP 403 and went unchecked.

Dependencies

numpy, scipy, ripser, persim, mpmath. Python ≥ 3.11. No torch — it was present at module scope and again inside a function body, carrying eight lines of linear algebra that numpy provides natively.

References

• Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222 (2026)
• Lagrange, Delanoue & Jaulin, Reliable Computing 13 (2007), DOI 10.1007/s11155-007-9042-9
• Schweinhart, arXiv:1808.02196 · Jaquette & Schweinhart, arXiv:1907.11182
• Steele, Growth rates of Euclidean minimal spanning trees, Ann. Probab. (1988)
• Sprott, Chaos and Time-Series Analysis, OUP (2003)