When do small views determine the whole?
Research preprint · priority unconfirmedA structure can look consistent in every small view and still fail to fit together globally. This paper computes exactly when that happens for binary-cut diagrams. It also proves which information must be relinquished to make finite reconstruction possible, and identifies the largest quotient that can be preserved.
31 October 2026 · Version 2 · Petri Mikael Autio · AI-assisted research and internal mathematical critique. External human review pending.
Seeing every small part can still leave something missing
Start with a finite collection of labels. A cut divides them into two nonempty groups, without choosing which group comes first. Restricting to a smaller collection gives a smaller cut, or nothing if one group disappears.
The paper forms linear combinations of these cuts over a field where adding a quantity to itself gives zero. It then discards terms beyond a specified algebraic degree. The resulting family has a precise restriction rule for every function between finite sets, including functions that identify labels.
Suppose we know this family only on sets with at most r labels. There are three possible failures of reconstruction: different global values can have the same local views; compatible local values can lack a global source; and higher compatibility can carry additional obstructions. The calculation identifies all three.
An exact threshold
Two parameters control the answer. The integer d specifies where the algebraic filtration is cut off. The integer h places the resulting vector space in homotopy degree h, forming an Eilenberg–Mac Lane space. For this family, knowing all sets with at most d + h + 1 labels is sufficient. One fewer is insufficient.
| Homotopy degree h | Smallest sufficient size | Failure one size earlier |
|---|---|---|
| 0 | 4 labels | Compatible values need not glue. |
| 1 | 5 labels | The reconstructed space has extra components. |
| 2 | 6 labels | The reconstructed space has extra components. |
The threshold holds for arbitrarily large ambient sets. It is proved by identifying the obstruction groups, rather than inferred from the finite checks below.
The mathematical statement
For a nonempty finite set S, let ES = 𝔽₂S/⟨1⟩, let JS be the augmentation ideal of k[ES], and set Fd(S) = JS/JSd, with char(k) = 2 and d ≥ 2. The least cardinality cutoff for the homotopy right-Kan reconstruction of K(Fd, h) is d + h + 1.
For 2 ≤ r < |S|, the positive derived groups are Λr(kS/k1) in exactly the degrees max(1, r − d) through r − 2. The ordinary unit has kernel JSr/JSd when r < d and zero otherwise; its cokernel is Λr(kS/k1) when r ≤ d and zero otherwise.
The dual complement-invariant Boolean module Wd−1 also has an exact FI-homology calculation: HiFI(Wd−1) on n labels is one trivial representation for i + 2 ≤ n ≤ i + d, and zero otherwise.
What must be given up?
Finding a quotient that works leaves an important question unanswered: could another quotient preserve more information and still reconstruct from the same small views? Version 2 answers that question for every natural linear quotient of the cut diagram.
Fix the largest size r we can observe and the homotopy degree h, with r ≥ h + 3. The theorem proves that every successful quotient must kill augmentation products of degree r − h − 1 and higher. Killing exactly those products is sufficient. Every other successful quotient discards at least as much.
| Homotopy degree h | Largest quotient | Information retained |
|---|---|---|
| 0 | J/J³ | Linear and quadratic augmentation terms. |
| 1 | J/J² | Linear augmentation terms. |
| 2 | 0 | No nonzero natural quotient qualifies. |
The proof first classifies every linear subfunctor compatible with all finite-set maps. They are precisely the augmentation powers and zero; an ideal condition does not need to be assumed. This classification is an elementary consequence of classical pointed cross-effect theory. Combining it with the exact reconstruction threshold gives the universal result.
The universal statement
For every natural objectwise surjection q: J → A and r ≥ h + 3, the diagram K(A,h) reconstructs from cardinality r if and only if Jr−h−1 is contained in ker(q). Equivalently, q factors uniquely as J → J/Jr−h−1 → A. For 1 ≤ r < h + 3, only A = 0 qualifies. This concerns quotients of the fixed cut diagram, rather than all possible mathematical structures.
What is already known?
The coefficient family has established predecessors. Richter studied its pointed version in Taylor towers for Γ-modules. The Boolean polynomial filtration is classical, and Powell describes it in work on functors between finite vector spaces. General FI presentation and homology methods also predate this calculation.
The subfunctor classification used in version 2 follows from Pirashvili’s classical pointed cross-effect equivalence. It is included with a direct proof and is not claimed as a new classification method.
The proposed contribution is the explicit calculation at every cardinality cutoff, its natural obstruction modules, the sharp reconstruction bound, and the resulting largest quotient theorem. The literature comparison has not established priority for these exact results. Equivalent formulas may exist in another language.
The prior-work report gives theorem-level comparisons and names the full texts that could not be inspected. This is a research preprint, with external review and priority still open.
How the question began
The project grew out of reading Hegel on induction, analogy and necessity. Collecting examples and establishing why a connection must hold are different tasks. That distinction suggested a mathematical question: what specified relations make bounded observations enough to determine the whole?
The first version selected an algebraic quotient and proved that it worked. The stronger question is whether the requirement itself determines that modification. Version 2 proves that it does within this family: once the initial diagram, its maps, the observation size and the homotopy degree are fixed, the theorem determines the least information loss that permits reconstruction.
This makes the connection more precise. We can prove what must be relinquished and what can be preserved under a stated demand. The theorem still presupposes its initial diagram and observational requirements; it does not derive those choices or establish a mathematical formalization of Hegel’s entire account of necessity.
My running commentary on the Science of Logic explores the philosophical question. The earlier preservation experiment gives more of the route into the mathematics.
Proofs and reproducible checks
The preprint contains the proofs. Four independent or complementary matrix calculations provide finite checks over 𝔽₂:
- 27 ordinary reconstruction cases, on up to seven labels.
- 55 derived reconstruction cases, on up to five labels, and 504 checks of maps between cochain complexes.
- 54 FI-homology cases, on up to eight labels, including 104,139 chain-equivariance checks and 385 checks of the induced permutation action on homology.
- Every linear subspace of the cut modules through four labels, without assuming ideality: 29,231 subspaces in total. At four labels, exactly the four predicted subspaces survive all 256 endomorphisms. The program also checks 310 injection-image agreements and 2,161 maps between the resulting smaller diagrams.
All recorded checks pass. The programs use exact arithmetic and the Python standard library. These finite computations support the proofs; they do not prove the universal statements or settle priority.
I directed the research. AI assistants helped develop the arguments, write code and text, compare literature, and conduct separate internal mathematical critiques. No external peer review or proof-assistant verification is claimed.
Version history: changes in version 2. The original version remains available as an 11-page PDF and source archive.