Successive Concept Differentiation as Observation
Refinement
Carrier Splitting, Interaction-Core Homology, and Descent
Brian Theory
Referee-repair draft v0.5 — 29 September 2026
Abstract
A 2015 proposal for concept unfolding combined Boolean distinction, simplicial boundary, and
local-to-global reconstruction. We separate these operations and study one precise refinement
model: a new Boolean observation splits the states of a finite observational posetQ, producing
an induced subposetQψ⊆Q×{0< 1}with projectionπ.
Carrier splitting factors through a homotopy-neutral split completion followed by order
thinning. In weighted height at most two its cone has a complete integral rooted-incidence
classification. At arbitrary height, a two-colour missing-copy model identifies the obstruction. If
the exclusive inversion graph is a matching, cone homology is the direct sum of coarse edge-link
reduced homology shifted by two, with no torsion-freeness assumption.
Beyond matching, a filtration by mixed-core size has first page given by available-link
homology. When cores have at most three vertices, an explicit chain mapping cone computes the
defect and the spectral sequence collapses atE2, with integral extensions retained. Maximum-
degree-two exclusive graphs stabilize atE3; an eight-stateK2,2 example proves this bound
sharp. A seven-state star rules out unconditionalE2 collapse for forests, Ferrers graphs, and
chordal-bipartite graphs. An all-exclusive star family further shows that interaction can create
torsion even when all local links have torsion-free homology.
The Boolean historical reconstruction and cover/descent framework are retained as separate
interpretations. The general cofiber, obstruction-complex, spectral-sequence, nerve, and sheaf
machinery is classical. The contribution claimed here is the explicit observation-product special-
ization and its positive classifications and integral failure boundaries; publication priority for the
exact specialization remains subject to review.
Contents
1 Purpose and scope 3
1.1 Conventions and result hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Historical operator: what survives and what does not 4
2.1 Fixed-factor unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Representation dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Boolean derivative interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 What the attached “Logical Complexes” notes recover 5
3.1 XOR is a difference vector, not a scalar distance . . . . . . . . . . . . . . . . . . . . 5
3.2 The three-variable tetrahedron is genuine . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Why the signs disappear overF2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
14 Exact unfolding layers and hypersimplices 6
5 A canonical cumulative unfolding complex 7
6 Observation systems and three kinds of refinement 8
7 Carrier splitting under a new Boolean observation 8
7.1 The exact inversion-chain complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.2 Immediate homotopy-neutrality criteria . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.3 Localization by lower fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.4 Exact low-height classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.5 Sharpness beyond weighted height two . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8 All-height carrier splitting under noninteracting exclusive inversions 13
8.1 The full-doubling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8.2 Sharpness: interaction begins at the first shared endpoint . . . . . . . . . . . . . . . 15
9 Interaction cores beyond matching 16
9.1 Higher differentials are necessary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
9.2 An unbounded interacting class and new torsion . . . . . . . . . . . . . . . . . . . . 18
9.3 Whole-relation Horn closure is neutral . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10 Cover refinement and canonical homology maps 19
11 Observation–Refinement Descent and Cone Theorem 20
11.1 What the theorem means for unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . 21
12 Descent defects for data that are not already a sheaf 21
13 Exact simplicial gluing and the no-filler warning 22
14 Finite prototypes 22
14.1 Three attributes: vertices, cycle, filler . . . . . . . . . . . . . . . . . . . . . . . . . . 22
14.2 Four attributes: a tetrahedral history complex . . . . . . . . . . . . . . . . . . . . . 23
15 Symmetry, indistinguishability, and homology are different invariants 23
16 Relation to observation topologies and order thinning 23
17 Persistence and zigzags 24
18 Priority and novelty position 24
19 Next theorem target 24
20 Conclusion 25
A Reproducibility 25
A.1 Carrier-splitting and interaction verification . . . . . . . . . . . . . . . . . . . . . . . 25
21 Purpose and scope
The motivating question is
Can successive differentiation of a concept be represented rigorously as a
transformation between complexes?
The answer developed here is yes, provided that four levels are not conflated:
(i) a semantic state space and its attributes;
(ii) an observation or distinction process;
(iii) a combinatorial/topological representation of the observed structure;
(iv) a genuine chain differential on that representation.
The historical thesis is used only as a source of ideas. The mathematical claims below stand on
newly stated definitions.
The resulting architecture has two primary operators:
d :Cn(Kk;R)−→Cn−1(Kk;R), d 2 = 0,
and
Uk :C•(Kk;R)−→C•(Kk+1;R),
where d is the ordinary chain differential andUk is induced by an unfolding/refinement map. When
Uk is a chain map, it induces homology maps; when the represented support changes,H∗(Cone(Uk))
measures the change relative to the chosen representation.
1.1 Conventions and result hierarchy
Finite observational posets are assumed nonempty. An order complex includes the empty face. Its
augmented chain complex is denoted˜C∗; in particular ˜H−1(∆ ∅;R) = R. Joins and suspensions
of empty links use this simplicial convention; the formal(−1)-sphere has no vertices. Ordinary
homology is zero in negative degrees. A coefficient ring is commutative with identity.
For a chain mapf : A∗→B∗, use Cone(f)n = Bn⊕An−1 and d(b,a) = (dBb +f(a),−dAa).
For a map of spaces or simplicial complexes,Cone(f) denotes its homotopy cofiber and its homology
is explicitly reduced. For nonempty source and target these conventions agree under the usual chain
comparison. A simplicial map that repeats an image vertex sends that oriented simplex to zero on
normalized chains. A shiftC[k]n =Cn−k has differential(−1)kdC.
The main mathematical results are organized as follows.
(1) Theorem 7.2 converts a changing observational carrier into a neutral split completion followed
by order thinning.
(2) Theorem 7.9 gives the exact integral low-height result; Theorem 8.3 gives the all-height
coarse-edge-link formula.
(3) Theorems 9.2 and 9.3 describe controlled overlap, including the sharp distinction betweenE2
and E3 collapse.
3(4) Theorem 11.2 treats cover refinement and descent separately. It reconstructs data on the
support actually covered; restriction fromX to a proper subspaceY need not preserve or
uniquely extend global sections.
The historical Boolean and simplex-skeleton constructions motivate the model; they are not a claim
to new general Boolean, polyhedral, or sheaf theory.
2 Historical operator: what survives and what does not
2.1 Fixed-factor unfolding
Let A ={a1,...,am}be a finite attribute set and letB(A) be the Boolean functions on{0,1}A.
Definition 2.1(Factorized concept). A factorized concept is an ordered tupleF = (f1,...,fm)
with fi∈B(A). Its conjunction semantics is
Π(F) =
m⋀
i=1
fi.
Definition 2.2(Historical one-step unfolding). For a chosen factorizationF = (f1,...,fm), define
Uth(F) =
m⋁
j=1

¬fj∧
⋀
i̸=j
fi

.
Proposition 2.3(Exact-one-false characterization). At a valuationx, Uth(F)(x) = 1 if and only if
exactly one factor valuefi(x) is false.
Proof. The jth disjunct requiresfj(x) = 0 and every other factor to be1. Distinct disjuncts are
mutually exclusive. Hence the disjunction is true exactly on the one-false layer.
For the positive conjunctionX1∧···∧Xm, the one-step output is therefore the Hamming layer
with exactly one zero. This is a precise mathematical version of “one further distinction.”
2.2 Representation dependence
Theorem 2.4(Non-congruence of factorized unfolding). Fix the aritym =|A|≥2. There is no
Boolean-function operatorU : B(A)→B(A) satisfying
U(Π(F)) =Uth(F)
for everym-factor tupleF∈B(A)m.
Proof. The constant Boolean functions belong toB(A). At the fixed aritym, take
F = (0,1,1,...,1), F ′= (0,0,1,...,1).
Both conjunctions are the zero function. Exactly one factor is false inF, soUth(F ) = 1; exactly
two are false inF′, soUth(F′) = 0. One operator on the conjunction cannot give both.
Form = 1, by contrast,Π((f)) =f and Uth((f)) =¬f; ordinary negation is such an operator.
Thus the failure of semantic descent begins at arity two. Form≥2, either internal factoriza-
tion/provenance is part of the concept, a canonical normal form must be chosen, or the historical
rule must be replaced by a semantic operator.
42.3 Boolean derivative interpretation
For the monomialf(x) = ∏
ixi, the Boolean partial derivative
∂B
j f =f(xj = 0)⇕f(xj = 1)
is ∏
i̸=jxi. Therefore
Uth(f) =
⋁
j
(
(1−xj)∧∂B
j f
)
.
The historical rule is thus a negation-gated Boolean sensitivity operator, not the simplicial boundary.
3 What the attached “Logical Complexes” notes recover
The attached notes contain a useful idea, but two algebraic levels must be separated.
3.1 XOR is a difference vector, not a scalar distance
For Boolean state vectorsx,y∈Fm
2 ,
x⇕y =x +y
is the vector difference in characteristic two. The scalar Hamming distance is
dH(x,y) = wt(x⇕y).
Thus expressions such as “XOR is the distance” should be replaced by: XOR is the coordinatewise
disagreement vector, whose Hamming weight is the distance.
3.2 The three-variable tetrahedron is genuine
Set
G = (1,1,1), D= (1,1,0), E= (1,0,1), F= (0,1,1).
Then
D⇕G = (0,0,1), E⇕G = (0,1,0), F⇕G = (1,0,0).
These three differences form a basis ofF3
2; viewed as points inR3, the corresponding real differences
are also linearly independent. Hence{G,D,E,F}is an affine3-simplex, a tetrahedron. The four
points are not redundant.
A low-arity coincidence in the notes is also correct:
D⇕E =F,
and cyclically. Here{0,D,E,F}is the even-parity plane inF3
2. This closure does not persist for
general m: XOR of two one-flip children of1m has Hamming weight2, notm−1.
53.3 Why the signs disappear over F2
The notes also anticipate a valid characteristic-two phenomenon. If[v0,...,vn] is a formal simplex,
then over F2
d[v0,...,vn] =
∑
i
[v0,...,ˆvi,...,vn]
contains no visible alternating signs because−1 = +1. Every codimension-two face appears twice
in d2, sod2 = 0.
The important correction is that the addition above occurs in thechain group on formal simplices.
It is not XOR of the bit-vector labels themselves. Label-space XOR and chain addition happen to
use the same field in this model, but they are distinct operations on distinct vector spaces.
4 Exact unfolding layers and hypersimplices
Let the fully undifferentiated positive state be1 = (1,...,1)∈{0,1}m. ForS⊆[m], define
xS = 1⇕χS,
so precisely the coordinates inS are switched from1 to 0.
Definition 4.1(Exact distinction layer). The kth exact unfolding layer is
L(m)
k ={xS :|S|=k}.
Throughout this section, the layer index is an integer satisfying0≤k≤m.
Theorem 4.2(Boolean unfolding simplex and hypersimplex theorem). Form≥2:
(i) |L(m)
k |=
(m
k
)
and everyx∈L(m)
k has Hamming weightm−k;
(ii){1}∪L(m)
1 is an affinem-simplex;
(iii) the convex hull of the exactkth layer is the hypersimplex
conv(L(m)
k ) =
{
x∈[0,1]m :
∑
i
xi =m−k
}
;
(iv) for 0<k <m , this polytope has dimensionm−1 and is a simplex exactly in the extremal
casesk = 1 or k =m−1.
Proof. Part (i) follows by choosing thek zero coordinates. For (iii), the displayed intersection of
the cube with the integer-sum hyperplane is a bounded polytope. If two coordinates of a point
lie strictly between0 and 1, perturb one upward and the other downward by a sufficiently small
amount; the point is not a vertex. Exactly one fractional coordinate is impossible because the
coordinate sum is an integer. Hence every vertex is a0–1 vector with exactlyk zero coordinates,
proving the hull equality. For (ii), ifui = 1⇕ei, thenui−1 =−ei in Rm, so them difference
vectors are linearly independent. Part (iv) is the standard dimension and vertex description of
the hypersimplex [16]; in the two extremal nontrivial weights its vertices are the standard simplex
vertices up to complementation.
Form = 4 the exact layers have the particularly transparent geometry
point −→tetrahedron −→octahedron −→tetrahedron −→point,
since ∆(4,2) is an octahedron. Thus repeated unfolding does not naturally stay inside the category
of simplices if one geometrizes each exact state layer by convex hull.
65 A canonical cumulative unfolding complex
There is, however, a canonical simplicial model ofdistinction histories. A subsetS⊆[m] records
the attributes that have been distinguished so far.
Definition 5.1(Cumulative distinction complex). For 0≤r≤m−1, define
K(m)
r ={S⊆[m] : 1≤|S|≤r + 1}.
Thus K(m)
r = skr ∆ m−1, ther-skeleton of the full simplex on the attribute set. We also retain the
empty face implicitly.
The inclusions
K(m)
0 ↪→K(m)
1 ↪→···↪→K(m)
m−1 = ∆ m−1
form a genuine filtration. Here increasingr means allowing larger jointly distinguished attribute
sets.
Theorem 5.2(Exact cone profile of cumulative unfolding). LetR be any coefficient ring and let
ir :K(m)
r−1↪→K(m)
r , 1≤r≤m−1.
Then
Hq
(
Cone(C∗(ir;R))
)∼=Hq(K(m)
r ,K(m)
r−1;R)∼=



R( m
r+1), q=r,
0, q ̸=r.
Proof. The relative chain complexC∗(Kr,Kr−1;R) has no new cells below degreer and has one
degree-r basis element for every(r + 1)-subset of [m]. There are no relative cells above degreer.
Hence every relative differential is zero and the displayed relative homology follows. The mapping
cone of an inclusion computes the relative homology of the pair [7].
Corollary 5.3(Absolute homology of intermediate distinction stages). For 0≤r<m −1 and a
field k,
˜Hq(K(m)
r ; k) = 0 ( q̸=r), dim ˜Hr(K(m)
r ; k) =
(
m−1
r + 1
)
.
The final stageK(m)
m−1 = ∆ m−1 is contractible.
Proof. The augmented chain complex of the full simplex is exact. In each degreeq <r, the skeleton
has the same chain groups and adjacent boundary maps, so its reduced homology vanishes there.
There are no chains above degreer. Its reduced Euler characteristic is
−1 +
r∑
j=0
(−1)j
(
m
j + 1
)
= (−1)r
(
m−1
r + 1
)
.
Thus its only possible reduced Betti number has the asserted rank. The final stage is the full simplex
and is contractible.
This gives a useful correction to the older suggestion that cycles should represent symmetry.
In this exact model the transient cycles are forced combinatorially by missing higher-order fillers.
They measure incomplete compatibility, not group-theoretic symmetry.
Example 5.4(Four attributes). Form = 4 the cumulative stages have Betti profiles
K0 : (β0) = (4), K 1 : (1,3), K 2 : (1,0,1), K 3 : (1,0,0,0).
The relative/cone ranks at successive steps are6 in degree1, 4 in degree2, and 1 in degree3, exactly
the numbers of newly admitted edges, triangles, and tetrahedra.
76 Observation systems and three kinds of refinement
The word “refinement” is overloaded. Three operations should be distinguished.
Definition 6.1(Signature refinement). An observation map isqk :X→Ok. We sayqk+1 refines
qk when there isrk :Ok+1→Ok with qk =rk◦qk+1. Equivalently, everyqk+1-fiber lies in aqk-fiber.
For finite propositional observation families, adding observables refines the generated topology,
can split observational equivalence classes, thins the specialization preorder, and can increase
signature distance. This is an observation-theoretic notion of differentiation.
A second regime occurs when a new observation is constant on every currentT0 equivalence class.
Then the carrier does not split; only order relations can be removed. This becomes a same-carrier
order-thinning problem, which is treated separately in companion work. If the new observation
separates previously identified worlds, the refined carrier gains points and a different construction is
required.
A third notion, used in the main theorem below, is acover refinement. It concerns local pieces
representing a support, not merely enlargement of an observation vocabulary.
Remark 6.2 (Presentation-dependent nerve warning). For an indexed family of truth regions
{ς(φi)}, the observation-side nerve records joint satisfiability, but it is presentation-dependent.
Adding a tautological observation can cone the nerve without changing the generated observation
topology. Therefore arbitrary extension of an observation list should not be identified with a
topologically meaningful nerve filtration. The main theorem uses actual refinement of covers with
explicit containment data.
7 Carrier splitting under a new Boolean observation
We now treat the refinement regime left open in the previous draft: a new observation is not constant
on every current observational-equivalence class, so one oldT0 state can split into two refined states.
This operation is not a cover refinement and is not, on its face, a same-carrier order thinning. It
does, however, admit a canonical factorization through one.
Let W be the finite world set, letΦ be the current observation family, and write
Q =oΦ (W )⊆{0,1}I
for the finiteT0 quotient ordered coordinatewise. Add one Boolean observationψ. The refined
quotient is
Qψ=oΦ∪{ψ}(W )⊆Q×{0,1},
with the product order. Let
π:Qψ−→Q, π(s,a) =s,
be the coordinate-forgetting map. Fora∈{0,1}put
Aa :={s∈Q : (s,a)∈Qψ}, C :=A0∩A1.
ThusA0∪A1 =Q, andC is precisely the set of old observational states that split.
Definition 7.1(Split completion). On the same carrier asQψ, define the split-completion order
ˆQψby
(s,a)≤ˆQ (t,b) ⇐⇒
(
s<t in Q
)
or
(
s =t and a≤b
)
.
8Equivalently, ˆQψis the lexicographic sum overQ of the nonempty fiber chains
Fs :={a∈{0,1}: (s,a)∈Qψ}.
Letb(s,a) =a be the new-bit label on this expanded carrier.
The distinction between Qψand ˆQψis exact. The split completion remembers every old
comparison between coarse states, independently of the new bit. The actual refined order keeps
only those comparisons along which the new bit is nondecreasing.
Theorem 7.2(Carrier-splitting factorization). With the notation above:
(i) the projectionp : ˆQψ→Q, p(s,a) =s, is a homotopy equivalence of finite posets; in factˆQψ
beat-reduces to a subposet isomorphic toQ;
(ii) the actual refined quotient is exactly the one-bit thinning of the split completion,
Qψ= ( ˆQψ)b,
meaning that a comparisonx< ˆQy is retained precisely whenb(x)≤b(y);
(iii) if i : ∆( Qψ) ↪→∆( ˆQψ) is the inclusion, then ∆(π) = ∆( p)◦i, and there is a homotopy
equivalence of homotopy cofibers
Cof(∆ π)≃Cof(i).
Consequently, for every coefficient ringR,
˜Hn(Cone(∆π);R)∼=Hn
(
∆( ˆQψ),∆(Qψ);R
)
.
Proof. For every split states∈C, the lower copy(s,0) is an up-beat point ofˆQψ: its strict upper
set has the minimum(s,1), because (s,1) lies below every copy over everyt > s. Remove these
lower split copies one at a time. What remains has one representative over eachs∈Q and is
order-isomorphic toQ. Beat-point removal is a strong deformation retraction for finiteT0 spaces [19],
giving (i).
For distinct coarse statess < t, the split completion always contains(s,a)< (t,b) whenever
those copies exist. The product order onQψretains exactly the casesa≤b. The same statement
holds within a split fiber, where(s,0)< (s,1) is retained. This proves (ii). Part (iii) follows from
π=p◦i and homotopy invariance of the homotopy cofiber under postcomposition by the homotopy
equivalence ∆(p). Since i is a simplicial inclusion, its homotopy cofiber computes the relative
homology of its simplicial pair.
Remark 7.3(What is standard and what is specific). Beat-point reduction is classical finite-space
topology [19], and homotopy cofiber invariance is standard; Quillen-type poset equivalences even
admit simple-homotopy refinements in the standard setting [20]. The project-specific content is the
observation-theoretic factorization: an actual split ofT0 observational states is converted canonically
into a homotopy-neutral lexicographic expansion followed by the precise one-bit monotonicity thinning
already studied on a fixed carrier. Thus carrier splitting is not a fourth unrelated operation; it is a
neutral blow-up plus an order defect.
97.1 The exact inversion-chain complex
Theorem 7.2 gives a chain-level classification without any nerve hypothesis.
Theorem 7.4(Inversion-chain model). For any coefficient ringR, the relative group
Ck
(
∆( ˆQψ),∆(Qψ);R
)
is free on the strict chains
x0< ˆQx1< ˆQ···< ˆQxk
whose bit word
b(x0)b(x1)···b(xk)
is not weakly increasing. Equivalently, a relative generator is exactly a split-completion chain
containing an inversion1···0. Its relative boundary is the ordinary alternating simplicial boundary
with every monotone-bit face set to zero.
Proof. The two order complexes have the same vertex set. A split-completion chain is a simplex of
∆(Qψ) exactly when every comparison in the chain survives the one-bit thinning, which is equivalent
to the bit word being weakly increasing. Quotienting the simplicial chain groups therefore leaves
exactly the nonmonotone chains, and the relative differential is the induced simplicial differential.
This formulation is useful computationally: the defect depends only on the old signature poset
and the three-valued fiber profile
Fs∈
{
{0},{1},{0,1}
}
,
not on the number of worlds inside each old equivalence class.
7.2 Immediate homotopy-neutrality criteria
The factorization also reveals when splitting is automatically conservative.
Proposition 7.5 (Monotone-section criterion). If A1 is an upward-closed subset ofQ, thenπ:
Qψ→Q is a homotopy equivalence. Dually, ifA0 is downward closed, thenπis a homotopy
equivalence.
Proof. Assume A1 is upward closed. Define
r+(s) =
{
(s,1), s∈A1,
(s,0), s /∈A1.
The upset condition makesr+ order preserving, andπr+ = idQ. Moreover every (s,a)∈Qψ
satisfies (s,a)≤r+(s), so idQψ≤r+π. The order-homotopy lemma givesr+π≃id and henceπis a
homotopy equivalence. TheA0 statement is dual, using the minimum available bit in each fiber.
Corollary 7.6(Common-outcome anchor). If every old observational state has at least one refined
world withψ= 1 (A1 =Q), or every old state has at least one withψ= 0 (A0 =Q), then carrier
splitting is homotopy-neutral. In particular, if every old state genuinely splits, soA0 =A1 =Q,
then Qψ=Q×{0< 1}and the projection is a homotopy equivalence.
Thus the act of duplicating a state is not itself the source of topological defect. The defect is
caused by incompatible outcome availability across comparable coarse states: a lower state that can
realize 1 together with an upper state that can realize0 creates a potentially deleted comparison.
107.3 Localization by lower fibers
Fors∈Q define the lower fiber
Ls :=π−1(Q≤s).
If s∈A1, then (s,1) is a maximum ofLs, hence Ls is contractible. Therefore all lower-fiber
obstructions to Quillen-type equivalence localize to the0-only statesA0\A1. Dually, all upper-fiber
obstructions localize to the1-only statesA1\A0.
This observation does not replace the exact relative complex above. It does, however, connect the
carrier-splitting problem to the classical poset-fiber framework. Under the connectivity hypotheses
of the Björner–Wachs–Welker fiber theorem [21], the source∆(Qψ) admits a wedge decomposition
in terms of the noncontractible lower fibers and upper links. The important point here is that the
split fibers themselves disappear from that list whenever their top copy is present.
7.4 Exact low-height classification
The general inversion complex can have arbitrarily high dimension. In the first nontrivial dimensional
regime it collapses to graph incidence and admits a closed integral classification.
Definition 7.7(Weighted split height). LetC =A0∩A1 be the split set. Define
hψ(Q) := max
s0<···<sr in Q
(r + #{j :sj∈C}).
Lemma 7.8.hψ(Q) = dim ∆( ˆQψ).
Proof. A strict coarse chain contributes one vertex from every fiber, and a split fiber can contribute
both copies consecutively. Taking both copies at every split state on the chain attains the displayed
number, and no split-completion chain can contain more.
Suppose hψ(Q)≤2. An inversion edgeis a deleted order-complex edge of∆( ˆQψ), equivalently
a pair
(s,1)< ˆQ (t,0), s<t,
with s∈A1 and t∈A0. A deleted triangle has one of the four nonmonotone bit patterns
010, 100, 101, 110.
It therefore contains either one or two inversion edges. Introduce one formal root for each connected
component ofQ (equivalently of ˆQψ). The carrier-split defect graphDψ(Q) has one vertex for
every inversion edge; a deleted triangle with two inversion edges joins their vertices, while a deleted
triangle with one inversion edge joins that vertex to the root of its component. Parallel graph
edges are allowed. Here a rooted forest means a forest with exactly one formal root in each graph
component. This incidence graphDψis different from the exclusive inversion graph introduced
later.
Theorem 7.9(Carrier-splitting cone classification in weighted height two). Assumehψ(Q)≤2.
Letc0(Dψ) be the number of connected components ofDψ(Q) containing no formal root. Then
˜Hn(Cone(∆π); Z)∼=



Zβ1(Dψ), n= 2,
Zc0(Dψ), n= 1,
0, otherwise.
The relative boundary matrix is totally unimodular. Hence the same Betti ranks hold over every
field. Moreover the following are equivalent:
11(i) the carrier-splitting cone is acyclic overZ;
(ii) it is acyclic over every field;
(iii)Dψ(Q) is a rooted forest;
(iv) ∆( ˆQψ) collapses onto ∆(Qψ) through deleted edge–triangle pairs;
(v) ∆π: ∆(Qψ)→∆(Q) is a homotopy equivalence.
Proof. The relative complex has onlyC2→C1, since all vertices are retained and the split completion
has dimension at most two. A deleted triangle of bit type100 or 110 has two inversion edges
with opposite boundary signs; a triangle of type010 or 101 has one. Consequently its matrix
is the oriented incidence matrix ofDψwith formal-root rows deleted. Each graph component
contains at most one root, because triangles cannot connect different coarse components. Relative
graph homology therefore givesH2 = Zβ1(Dψ) and H1 = Zc0(Dψ). Incidence matrices and their row
submatrices are totally unimodular.
Vanishing of both groups is exactly the rooted-forest condition. In a rooted tree with an edge,
choose a nonroot leaf. Its inversion edge occurs in just one remaining deleted triangle and in no
retained triangle, so the pair is an elementary simplicial collapse. Iterating leaf removal deletes every
relative edge and triangle. Conversely a relative collapse gives a homotopy equivalence and hence
vanishing relative homology. Finally the split-completion projection is a homotopy equivalence, so
two-out-of-three gives the statement about∆π. The free groups displayed above also show the
equivalence with acyclicity over every field.
Remark 7.10(Status of the low-height theorem). The graph-incidence algebra, total unimodularity,
rooted forests, and beat-point technology are standard. The same height-two one-bit result is developed
in the companion order-thinning manuscript [25]; the proof above makes the transfer used here self-
contained. The new point here is the carrier-splitting reduction that makes that theorem applicable
to a process which changes theT0 carrier. Accordingly, this theorem should be positioned as an exact
transfer theorem, not as a new general theory of rooted forests.
7.5 Sharpness beyond weighted height two
No coefficient-independent forest criterion can hold in unrestricted dimension. The carrier-splitting
family contains the no-split caseC = ∅, which is exactly arbitrary same-carrier one-bit thinning.
Companion results [25] show that one-bit thinning in height three can already have relative integral
torsion, and in unrestricted height can encode arbitrary finite-poset homotopy types inside a
contractible coarse cone. If one insists that at least one old state genuinely split, a disjoint neutral
split component may be adjoined without changing positive-degree cone homology. Thus the
weighted-height-two hypothesis in Theorem 7.9 marks a real boundary rather than a presentational
convenience.
The appropriate all-height object is therefore the inversion-chain complex of Theorem 7.4,
supplemented where useful by poset-fiber or homotopy-colimit methods. The next two sections give
all-height results under matching and interaction-core hypotheses, together with explicit obstructions
to stronger graph-class conclusions.
128 All-height carrier splitting under noninteracting exclusive inver-
sions
The low-height rooted-incidence theorem gives a complete answer when the weighted height is
at most two. In higher dimensions a universal rooted-forest criterion is impossible: the same-
carrier boundary case already supports integral torsion and arbitrary finite-poset homotopy types.
Nevertheless, an all-height exact classification survives when the genuinely incompatible carrier
outcomes do not interact.
Let
Aa ={s∈Q : (s,a)∈Qψ}, a ∈{0,1},
and put
P =A1\A0, Z =A0\A1, C =A0∩A1.
ThusP consists of the1-only coarse states,Z of the 0-only coarse states, andC of the states that
genuinely split.
Definition 8.1(Exclusive inversion graph). The exclusive inversion graphis the bipartite graph
Gψ= (P⊔Z,Eψ), E ψ={(p,z)∈P×Z :p<z in Q}.
An edge records a coarse comparison whose earlier endpoint admits only the new outcome1 while its
later endpoint admits only0.
8.1 The full-doubling model
Let
D =Q×{0< 1}, K = ∆D, L = ∆Qψ.
The actual refined carrier is the induced subposet ofD obtained by deleting exactly
M0 ={(p,0) :p∈P}, M 1 ={(z,1) :z∈Z}.
Let KP be obtained by deletingM0, letKZ be obtained by deletingM1, and put
U =KP∪KZ, L =KP∩KZ.
The complexKP closes upward to the full top row, whileKZ closes downward to the full bottom row.
Hence bothKP ↪→K and KZ ↪→K are homotopy equivalences. SinceH∗(K,KP ) =H∗(K,KZ) =
0, the triple (K,KP,L), simplicial excision forU = KP∪KZ, and the triple (K,U,KZ) give
Hn(K,L)∼=Hn(KP,L)∼=Hn(U,KZ)∼=Hn+1(K,U). Thus
Hn(K,L;R)∼=Hn+1(K,U;R) (1)
for every coefficient ringR. SinceD→Q is a homotopy equivalence,Hn(K,L;R) is the reduced
homology of the homotopy cofiber of the refined projection∆π: ∆Qψ→∆Q.
A simplex ofK\U contains at least one missing lower copy(p,0) and at least one missing
upper copy (z,1). Because it is a product-order chain, every such pair satisfiesp<z . Therefore
the missing vertices of a relative simplex project to a complete bipartite subgraph ofGψ. This
observation isolates the source of higher interaction: bicliques with more than one edge couple
otherwise local defects.
13Proposition 8.2(Empty-exclusive-graph neutrality). If Eψ= ∅, then ∆πis a homotopy equiva-
lence.
Proof. Set b(s) = 1 exactly when somep∈P satisfiesp≤s. This isotone choice takes an available
value at every state: it is1 on P and cannot be1 on Z without an exclusive edge. The section
r(s) = (s,b(s)) obeys πr= id. OnQψ, the mapc(s,a) = (s,max{a,b(s)}) is available and isotone,
with id≤c≥rπ. Comparable poset maps are homotopic, sor is a homotopy inverse.
Theorem 8.3(Exclusive-Inversion Matching Theorem). Suppose that the exclusive inversion graph
Gψis a matching. Then, for every coefficient ringR and every integern,
˜Hn(Cone(∆π);R)∼=
⨁
(p,z)∈Eψ
˜Hn−2
(
lk∆Q[p,z];R
)
. (2)
Consequently ∆πis anR-homology equivalence if and only if every coarse edge linklk∆Q[p,z] indexed
by an exclusive inversion isR-acyclic.
Proof. A relative simplex of(K,U) contains at least one vertex ofM0 and one vertex ofM1. If
it contained two missing lower vertices and one missing upper vertex, the corresponding upper
endpoint would have degree at least two inGψ; the dual statement holds for two missing upper
vertices. The matching hypothesis therefore forces every relative simplex to contain a unique mixed
pair
e = ((p,0)< (z,1)), (p,z)∈Eψ.
Deleting either endpoint ofe sends the simplex intoU, while deleting any other vertex preserves
the same pair. Reorienting each simplex with the two core vertices first gives an even chain shift.
Hence the relative chain complex splits as
Cm(K,U;R) =
⨁
e∈Eψ
C(e)
m , C (e)
m ∼= ˜Cm−2(lkKe;R).
Thus
Hm(K,U;R)∼=
⨁
e∈Eψ
˜Hm−2(lkKe;R).
It remains to identify the product-order link. Fore = ((p,0)< (z,1)), its link factors into the
lower interval, the open middle interval, and the upper interval. The open middle interval is covered
by two contractible subposets: delete(z,0) and close upward to the bit-1 row, or delete(p,1) and
close downward to the bit-0 row. Their intersection is(p,z)Q×{0< 1}, homotopy equivalent to
(p,z)Q. Since (p,1) and (z,0) are incomparable, every middle-chain lies in one of these two pieces;
therefore
∆((p,0),(z,1))≃Σ∆( p,z)Q.
Joining with the lower and upper factors gives
lk∆(Q×2)e≃Σ lk∆Q[p,z].
Combining this suspension shift with (1) yields (2).
Corollary 8.4(Bounded-poset localization). Assume thatQ has a global minimumˆ0 and maximum
ˆ1, and thatGψis a matching. Then every exclusive edge except possibly(ˆ0,ˆ1) has contractible coarse
edge link. Hence only the global endpoint inversion can contribute to the cone homology.
14Proof. If p̸= ˆ0, the lower factor∆Q<p has the minimumˆ0; ifz̸= ˆ1, the upper factor∆Q>z has
the maximumˆ1. In either case the edge link is a join with a nonempty contractible complex.
Corollary 8.5 (Distributive endpoint profile). Let Q = J(R) be the lattice of order ideals of a
nonempty finite posetR. Make the bottom state1-only, the top state0-only, and every other state
split. IfR is anr-element antichain, so thatQ =Br, then
˜Hr(Cone(∆π); Z)∼= Z
and all other reduced cone homology vanishes. IfR is not an antichain, the cone is integrally acyclic.
Proof. Corollary 8.4 reduces the answer to the reduced homology of the proper part ofJ(R), shifted
by two. By the classical crosscut theorem [26], the atom crosscut is a full simplex unless the atoms
join to the top element. InJ(R) the atoms correspond to the minimal elements ofR, and they join
to the top exactly when every element ofR is minimal, i.e. exactly whenR is an antichain. In that
case J(R) =Br and its proper part is homotopy equivalent toSr−2.
8.2 Sharpness: interaction begins at the first shared endpoint
The matching condition cannot be weakened merely to “Gψis a forest.” For the three-element chain
0< 1< 2 with availability profile
0 :{1}, 1 :{0}, 2 :{0},
the exclusive graph is the two-edge starK1,2. Direct integral Smith normal form gives
˜H1(Cone(∆π); Z)∼= Z,
while the sum of the two individual coarse-edge-link contributions is zero. The shared endpoint
creates a three-vertex interaction core and a differential between the two edge defects.
Nor does matching imply total unimodularity or torsion-freeness. Given any nonempty finite
poset R, take
Q = ˆ0⊕R⊕ˆ1
and use the endpoint profile above. There is exactly one exclusive inversion, and Theorem 8.3 gives
˜Hn(Cone(∆π); Z)∼= ˜Hn−2(∆R; Z).
Thus arbitrary finite-poset homology, including torsion, can occur even inside the matching class.
In particular, total unimodularity must be controlled by stronger conditions on the relevant links;
global torsion-freeness alone is not a TU criterion [24].
The matching hypothesis is sufficient, not necessary: the same proof applies whenever there is
no mixed three-vertex core. Section 9 makes this criterion and the nonmatching interaction maps
explicit.
Remark 8.6 (Priority boundary). The deletion/cofiber technology used here is classical. In
particular, vertex-cover obstruction complexes already provide general homotopical tools for comparing
a complex with unions of induced subcomplexes [23], while poset-fiber decompositions give broad
results under fiber-connectivity hypotheses [21]. The candidate contribution of Theorem 8.3 is the
observation-theoretic two-colour recognition and the resulting exact coarse edge-link formula. No
claim is made that simplicial vertex deletion or homotopy cofibers themselves are new.
159 Interaction cores beyond matching
Retain K = ∆( Q×2), L = ∆ Qψ, the missing setsM0 = P×{0}and M1 = Z×{1}, and
U =KP∪KZ. Cone homology in this section is reduced homology of the topological homotopy
cofiber, equivalently homology of the algebraic mapping cone. The two-colour shift is
˜Hn(Cone(∆π);R)∼=Hn+1(K,U;R).
For a mixed missing coreS, putΛS = lkK(S)[V (L)]. All link chain complexes below are augmented,
including degree−1.
Proposition 9.1 (Interaction spectral sequence). The finite-filtration construction [8] gives a
convergent spectral sequence
E1
p,q=
⨁
|S|=p
˜Hq(ΛS;R) = ⇒Hp+q(K,U;R),
with dr of bidegree(−r,r−1). The first differential is the signed sum of the inclusion maps obtained
by deleting one missing vertex, omitting nonmixed faces. If all mixed cores have size at mostM≥2,
then EM−1 =E∞.
Proof. Orient a relative simplex by listing itsp missing vertices first and its available vertices second,
with the corresponding shuffle sign. Its boundary is
d(S∧T) =
∑
i
(−1)i(S\{si})∧T + (−1)pS∧∂T.
Terms with nonmixed core vanish in the relative complex. Filtering byp gives the displayed page
and differential. The filtration is finite and exhaustive. The abutment is the associated graded of
homology, so integral extension data may remain. Only columns2,...,Mcan occur, so everydr
with r≥M−1 has zero source or target.
Theorem 9.2(Exact two-column interaction theorem). Suppose no mixed core has four or more
vertices. Define
A•=
⨁
|S|=2
˜C•(ΛS;R), B •=
⨁
|S|=3
˜C•(ΛS;R),
and leth :B•→A•be the signed sum of core-deletion inclusions. Then
C•(K,U;R)∼= Cone(h)[2], ˜Hn(Cone(∆π);R)∼=Hn−1(Cone(h)).
The interaction spectral sequence collapses atE2, and there is a natural exact sequence
0−→cokerHn−1(h)−→˜Hn(Cone(∆π);R)−→kerHn−2(h)−→0.
In particular the cone isR-acyclic if and only ifh is a quasi-isomorphism.
Proof. The degree-m relative group isAm−2⊕Bm−3 and its boundary is(a,b)↦→(∂a+hb,−∂b).
This is the asserted shifted chain cone. Apply the two-colour shift, the mapping-cone long exact
sequence, and Proposition 9.1 withM = 3.
The exact sequence splits noncanonically over a field. OverZ it splits when the right-hand
kernel is free abelian; no splitting is asserted in general. Thus collapse alone does not replace the
integral answer by a direct sum of link-homology kernels and cokernels. Signed incidence coefficients
on link chains likewise do not imply total unimodularity of the induced maps on homology.
16Theorem 9.3(Degree-two classification and its sharp page bound). Suppose the exclusive graph
has maximum degree at most two. Every mixed core has size at most four, soE3 =E∞. If noC4
component has both its same-colour pairs comparable inQ, Theorem 9.2 applies andE2 =E∞. In
every cycle component of length at least six the cone contribution is the coarse-edge-link direct sum
of the matching theorem.
Proof. Transitivity givesN(p′)⊆N(p) when p<p ′in P, andN(z)⊆N(z′) when z <z′in Z. A
mixed core is a biclique with each colour class a chain. Degree two bounds both class sizes by two.
A four-core requires a wholeK2,2 component with both colour pairs comparable. This proves the
bounds. On a longer cycle, two same-colour degree-two vertices have distinct neighbourhoods of the
same cardinality and cannot be comparable if they share a neighbour. Hence there are no triple
cores. No missing vertex can then lie in a product-edge link, so the original matching proof applies
componentwise without change.
More explicitly, a path with at least three edges supports triples only on its first two or last
two edges: a comparable common-neighbour pair must include a leaf. For a path of at least four
edges these end triples have disjoint edge supports, giving independent star blocks and interior
edge blocks. A three-edge path has up to two triples sharing its middle edge and is treated by one
instance of Cone(h). Every surviving relative face remains in the same exclusive graph component,
so the component decomposition is at chain level. Links must nevertheless be calculated in the
whole coarse poset.
For a two-edge star with ordered leaves,h is the signed pair of inclusions from its triple link to
its two edge links. On the frozen three-chain profile(1,0,0), all three available links have reduced
H0 = Z and h0 is the primitive signed diagonalZ→Z2. This gives coneH1 = Z, although both
coarse edge links are contractible.
9.1 Higher differentials are necessary
For aC4 component whose two pairs are comparable, restrict to its relative chain summand and
write the core ordera < b < x < y. Let A,B,Vbe the sums of edge, triple, and four-core link
chains respectively. Then
Cm(K,U) =Am−2⊕Bm−3⊕Vm−4, d (a,b,c) = (∂a+hb,−∂b+kc,∂c),
where h,kare the signed core-deletion inclusions andhk = 0. In triple order(abx,aby,axy,bxy),
the signs ofk are (−,+,−,+), while h(abx) = bx−ax, h(aby) = by−ay, h(axy) = ax−ay,
and h(bxy) = bx−by, each term meaning its corresponding inclusion. The only possible higher
differential is
d2 :E2
4,q−→E2
2,q+1.
If ∂c= 0 and ∂b=kc, it is represented byhb. Thus a differential lowering core size by at most one
can still produce a nonzero secondary differential.
An explicit eight-state example makes this bound sharp. TakeQ =A2⊕R⊕A2, with bottom
antichain{a,a′}, top antichain{y,y′}, andR ={v,b,w,x}with relationsv < x, b < w, b < x.
Makea,bone-only andx,yzero-only; split the other states. The only nonzero first-page groups are
E1
2,1 = Z and E1
4,0 = Z. The four-core link consists ofb1,x0. The triple linksabx and bxy are cones
with apexy′
1 and a′
0, respectively; the linksaby and axy are the two paths displayed below. The
edge linksax, by, andbx are cones, with apexy′
1, a′
0, and either of those vertices. The remaining
edge linkay has maximal simplices
[a1,b1,w1], [v0,x0,y0], [a1,v1], [v0,v1], [w0,w1], [w0,y0].
17Collapsing its two triangle ears leaves a cycle. Thus the stated first-page groups follow directly from
all nine available links. The paths
x0−v0−v1−a1−b1, x 0−y0−w0−w1−b1
in the two relevant triple links fill the four-core difference. Give each path orientation fromx0 to
b1. In theay component of the signed zigzag, the first path minus the second traverses the loop
once; the other edge components are cycles in contractible links. Therefored2 is a unitZ→Z, and
E3 = 0. The supplementary signed-chain certificate verifies these fillings and the integral homology
directly.
Nor do Ferrers or chordal-bipartite graphs forceE2 collapse. The seven-state poset with covers
0< 1, 0< 2, 1< 3, 2< 3, 2< 5, 3< 4, 3< 6, 5< 6, and profile(1,2,0,0,2,2,0) has exclusive
graphK1,3. Its integralE1 =E2 has Z2 in bidegree (2,1) and Z in (4,0); d2 is a primitive injection.
Its only nonzero cone group isH2 = Z. Here profile value 2 means split. The four-core link is
{01,20}≃S0. The three triple links, indexed by their upper missing states023, 026, and 036,
contain respective paths
A : 20−41−01, B : 20−51−01, C : 20−30−10−11−01.
All three triple links are contractible. The02 edge link is the square on bipartition{01,20}⊔{41,51};
the 03 edge link is a cone with apex41. The 06 edge link has maximal faces
[01,11],[01,51],[10,11],[10,30,60],[20,30,60],[20,50,51],[20,50,60].
Collapsing its triangle ears leaves the cycle01−11−10−30−20−51−01. OrientA,B,Cfrom 20
to 01. With the missing-core deletion signs, their horizontal boundary has componentsB−A in the
02 link,A−C in the03 link, andC−B in the06 link. The first traverses the02 square once; hence
d2 : Z→Z2 has a unit coordinate and is primitive. This gives a signed witness independent of the
certificate’s rank calculation. For the abutment, writeT =Q\{0}and C ={1,4,5}. The complex
∆T is contractible: its two maximal-element lower ideals are cones whose intersection is the lower
ideal of 3. The complex∆C is an edge and an isolated point. In a star profileQ ={p}⊕T, withp
one-only,C split, andT\C zero-only, the refined complex is the union of∆((T×{0})∪(C×{1}))
and the cone∆(({p}×{1})∪(C×{1})). The first piece retracts to∆T, and the intersection is
∆C. The homotopy-pushout description consequently gives˜Hn(Cone(∆π))∼=Hn−1(∆T,∆C). Here
this is Z only atn = 2; the sole possibled2 must be a primitive injection. The supplementary
certificate records all link matrices and signed chains. These are explicit counterexamples, not global
minimality claims.
9.2 An unbounded interacting class and new torsion
Proposition 9.4(All-exclusive star). LetR be a nonempty finite poset. TakeQ ={p}⊕R, make
p one-only and every state ofR zero-only. ThenE2 =E∞integrally and
˜Hn(Cone(∆π); Z)∼=Hn−1(∆R; Z),
where the right side is ordinary homology.
Proof. Every core is{p0}∪σ1 for a nonempty chainσin R. Its available link is the isolated vertex
p1 disjoint from the cone∆(R≤minσ×{0}). The difference-of-components generators identify all
nonzero link homology withZ in degree zero and identify their inclusion maps with the identity.
Consequentlyd1 is, up to a uniform sign, the ordinary simplicial boundary of∆R. The single row
collapses onE2 and has only one filtration grade in each total degree.
18For the nonempty-face poset of a triangulated projective plane, all these available links have
torsion-free homology, and every coarse edge link is a vertex link in its barycentric subdivision,
hence a circle. Nevertheless the interaction differential createsZ/2 in cone degree two. The explicit
six-vertex triangulation in the supplement gives a 32-state example, with every link and the reduced
interaction complex checked by integral Smith normal form.
9.3 Whole-relation Horn closure is neutral
Proposition 9.5. If the nonempty refined Boolean relation is closed under coordinatewise meet,
then ∆πis a homotopy equivalence. The dual statement holds for coordinatewise join closure.
Proof. Finiteness and meet closure give a minimum in the refined relation. Its coordinate projection
is again meet closed and nonempty, so also has a minimum. Both order complexes are contractible.
Join closure gives maxima.
The hypothesis concerns the whole realized refined relation; it is not merely a closure assumption
on a selected subset of observations. This is an elementary application of closure-system topology,
not a new Horn theorem.
Remark 9.6(Scope and priority). Filtered-complex spectral sequences, link/cofiber recursions, and
their extension issues are classical. The complexesΛS are precisely vertex-cover obstruction complexes
in the sense of Chachólski–Jin–Scolamiero–Tombari [23]. The results here explicitly identify which
such complexes interact in the observation product model, with positive class restrictions and integral
obstructions. They do not introduce a new general obstruction theory or supersede poset-fiber
theorems.
10 Cover refinement and canonical homology maps
Definition 10.1(Nerve). After discarding empty members of a finite coverU ={Ui}i∈I of X, its
nerve Nrv(U) has vertex setI and simplexσ⊆I exactly when
Uσ:=
⋂
i∈σ
Ui̸= ∅.
Definition 10.2(Refinement map). LetV ={Vj}j∈J andU ={Ui}i∈I cover the same space, or let
the union ofV lie in the union ofU. A refinement choice is a mapρ:J→I such that
Vj⊆Uρ(j)
for everyj.
Lemma 10.3(Contiguity of refinement choices). Every refinement choice induces a simplicial map
N(ρ) : Nrv(V)→Nrv(U).
Any two refinement choicesρ,ρ′are contiguous and therefore induce the same homomorphism on
homology.
Proof. If σ∈Nrv(V), choosex∈⋂
j∈σVj. Thenx belongs to everyUρ(j) and everyUρ′(j). Hence
the union of the two image vertex sets spans a simplex ofNrv(U), which is precisely contiguity.
This lemma is standard. In particular, Leitão’s 2026 framework [14] for persistent homology
of cover refinements makes this contiguity principle explicit at the cover level. It should not be
presented here as a new theorem.
1911 Observation–Refinement Descent and Cone Theorem
We now state the central structural result in a form adapted to successive concept differentiation.
Definition 11.1(Pair-good subcomplex cover). LetY ⊆X be a finite simplicial pair. A finite
familyU ={Ui}i∈I of subcomplexes coveringX is good relative toY if every nonempty intersection
Uσ= ⋂
i∈σUi is contractible and every nonemptyUσ∩Y is contractible. The restricted cover
U|Y ={Ui∩Y}i∈I is indexed after deleting its empty members. A finite subcomplex cover ofY is
good if all its nonempty intersections are contractible. The subcomplex condition is part of these
definitions, not a consequence of intersection contractibility alone.
Theorem 11.2 (Observation–Refinement Descent and Cone Theorem). Let Y ⊆X be a finite
simplicial pair. LetU be a finite pair-good subcomplex cover ofX and letV be a finite good subcomplex
cover ofY refiningU|Y. Delete empty indexed members before taking nerves. Thus all three covers
U,U|Y, andV lie in the functorial subcomplex-cover nerve theorem. Choose any refinement map
ρ:V−→U|Y
and let
u : Nrv(V)
N(ρ)
−−−→Nrv(U|Y )↪→Nrv(U)
be the induced simplicial map. Then:
(a) the homotopy class up to contiguity, and hence the induced homology mapu∗, is independent
of the refinement choice;
(b) N(ρ) : Nrv(V)→Nrv(U|Y ) is a homotopy equivalence;
(c) for any coefficient ringR,
Hn(Cone(C∗(u;R)))∼=Hn(X,Y;R)
naturally up to the standard nerve identifications;
(d) in the conservative caseY =X, the cone is acyclic, equivalentlyu is a quasi-isomorphism;
(e) assume additionally thatX, Y, the cover members, and their intersections are admissible
objects of a site with inclusion morphisms for the displayed subcomplexes, their pullbacks are
represented by these intersections, and the displayed families are covers in that site. Take the
sheaf and cosheaf values in a category admitting the finite limits and colimits used below, for
example abelian groups. For a sheafF, compatible data onU reconstructF(X); compatible
data onU|Y orV reconstructF(Y ). Refinement between the latter two covers preserves that
section onY. No unique extension fromY to X is asserted;
(f) under the same site hypothesis, a cosheafG is assembled as the colimit of the intersection
diagram on each covered support: the colimits areG(X) andG(Y ), respectively. Refinement
on Y preserves its colimit; inclusion induces the comparisonG(Y )→G(X), which need not be
an isomorphism.
Proof. Part (a) is Lemma 10.3. For (b), the finite subcomplex-cover nerve theorem [27, 15] supplies
compatible equivalences forV andU|Y with Y. Functoriality for the refinement map identifiesN(ρ)
with the identity ofY through these equivalences. HenceN(ρ) is a homotopy equivalence.
20Let j : Nrv(U|Y ) ↪→Nrv(U). Since N(ρ) is a homotopy equivalence, the mapping cones of
j◦N(ρ) and j have isomorphic homology; this follows, for example, by comparing the long exact
sequences of the two cones and applying the five lemma. The mapping cone of the chain inclusion
C∗(j) computes
Hn(Nrv(U),Nrv(U|Y );R).
The relative nerve theorem identifies this withHn(X,Y;R), proving (c). IfY =X, the relative
groups vanish, giving (d).
For (e), apply the sheaf axiom [9] separately on each support: compatible families on a cover of
S are the equalizer describingF(S), forS =X or S =Y. The coversU|Y andV cover the sameY,
so their gluing maps agree with refinement. RestrictionF(X)→F(Y ) is a separate map and need
not be bijective. Part (f) is the dual cosheaf colimit statement, again on each support separately.
Example 11.3(Restriction is not unique extension). Take the discrete spaceX ={a,b}, Y ={a},
and the sheaf of integer-valued functions. Singleton covers satisfy the stated good-cover conditions.
Data onY determine an element ofZ, whereasF(X) = Z2; restriction is the projection onto the
first coordinate. A section onY therefore has many extensions toX. This distinguishes descent on
a fixed support from an extension problem.
11.1 What the theorem means for unfolding
The theorem separates two questions that were previously mixed:
(i) Does the chosen map fail to be a homology equivalence?Cone/relative homology detects this
failure; its vanishing alone does not detect every change of support or homotopy type.
(ii) Can compatible local descriptions be reconstructed globally?This is answered by descent or
colimit conditions.
A finer description can therefore be topologically conservative while still providing more local
information. Conversely, one may have perfect local gluing on a support whose topology has
genuinely changed.
12 Descent defects for data that are not already a sheaf
If one assumes a sheaf from the beginning, reconstruction is built into the axioms. For modeling it
is useful to measure how a candidate local-data system fails to be a sheaf.
LetF be a presheaf of abelian groups andU ={Ui}a finite cover. Define
C0(U;F) =
∏
i
F(Ui), C 1(U;F) =
∏
i<j
F(Ui∩Uj),
and
(δ0s)ij =sj|Ui∩Uj−si|Ui∩Uj.
Let Z0 = kerδ0 be the compatible local families and let
res :F(X)→Z0(U;F)
be global restriction.
21Definition 12.1(Descent defects). Define
Duniq(U;F) = ker(res), D exist(U;F) = coker(res).
Proposition 12.2 (Exact reconstruction criterion). Compatible local data reconstruct a unique
global element if and only if
Duniq = 0 =Dexist.
In particular both defects vanish for a sheaf.
This motivates a two-axis defect signature for an unfolding stepu:
D(u;F) = (H∗(ConeC∗(u)), Duniq, Dexist).
The first entry is structural; the latter two are semantic/local-to-global. They should not be collapsed
into one number.
13 Exact simplicial gluing and the no-filler warning
Proposition 13.1(Exact gluing as a colimit). If a simplicial complexK is covered by subcomplexes
{Ki}i∈I, thenK is the colimit of the full diagram consisting of theKi and all their finite intersections,
with the evident inclusion maps.
This is the rigorous combinatorial version of rebuilding an object from pieces with overlap
provenance.
Proposition 13.2 (Bare proper faces do not determine a filler). The proper-face data of an
n-simplex do not by themselves determine whether then-simplex is present.
Proof. The boundary complex∂∆ n and the filled simplex∆ n have exactly the same proper faces,
but only the latter contains the top-dimensional simplex.
Therefore the historical idea of an “integrator” requires a filler rule, provenance, a flag/clique
convention, a colimit diagram containing the intended top cell, or semantic descent data. Conjunction
of face-formulas is not a general simplicial gluing operation.
14 Finite prototypes
14.1 Three attributes: vertices, cycle, filler
The cumulative distinction filtration form = 3 is
K0 ={1,2,3}↪→K1 =∂∆ 2↪→K2 = ∆ 2.
Over F2:
H0(K0)∼= F3
2, H 1(K1)∼= F2, H >0(K2) = 0.
The first step adds three edges and has
H1(K1,K0)∼= F3
2.
The second adds one triangle and has
H2(K2,K1)∼= F2.
The one-dimensional cycle at the middle stage is therefore not a symmetry invariant; it is the
obstruction created by having all pairwise compatibilities without the triple filler.
2214.2 Four attributes: a tetrahedral history complex
Form = 4:
K0⊂K1⊂K2⊂K3 = ∆ 3,
where K1 is the complete graphK4 and K2 is the tetrahedral boundary. The nonzero reduced
homology is
˜H0(K0)∼=R3, H 1(K1)∼=R3, H 2(K2)∼=R.
The next stage fills the tetrahedron and kills the2-cycle.
At the exact-state level, however, thek = 2 Hamming layer has six points and convex hull an
octahedron. This illustrates why the state-layer geometry and the distinction-history complex are
related but not identical constructions.
15 Symmetry, indistinguishability, and homology are different
invariants
Let Aut(K) be the simplicial automorphism group. Let an observation mapq : X→O define
indistinguishability by equal fibers. Neither is determined by homology.
A filled triangle has automorphism groupS3 but trivial positive-dimensional homology. Con-
versely, an asymmetric unicyclic graph can haveH1∼= Z and trivial automorphism group. Thus
cycles cannot generally be identified with symmetry.
There is nevertheless a rigorous symmetry-breaking statement at the observational level. If a
group G acts onX, define
Gq ={g∈G :q(gx) =q(x)∀x∈X}.
If qk+1 refinesqk, then
Gqk+1≤Gqk.
Finer observation can therefore break observational symmetry by shrinking the subgroup that
preserves all observations. This is separate from the homology of the chosen complex.
16 Relation to observation topologies and order thinning
The observation-topology program provides a natural semantic base for the present construction.
An indexed familyΦ = (φi) defines signatures and an observation-generated topology. ExtendingΦ
by new coordinates makes the topology finer, refines observational equivalence, and thins the special-
ization preorder. These facts supply a mathematically clean interpretation of “more differentiated
observation.”
However, the passage from observation topology to complex must be chosen explicitly. The
nerve of the indexed truth regions records joint satisfiability but is not an invariant of the generated
topology. This is why the present paper uses a cover-refinement contract when applying the nerve
theorem.
There is also an important same-carrier subcase. If a new Boolean observation factors through the
currentT0 quotient, it does not split observational states; it only removes comparisons incompatible
with the new bit. Companion work shows that, for a height-two current signature poset and one
new bit, homotopy neutrality can be recognized exactly by a rooted-forest criterion and certified by
simplicial collapse. That result complements the present theorem: the present paper treats changing
carriers, matching and interacting cone defects, and cover-level descent, while the companion
order-thinning paper develops the same-carrier incidence theorem and its separate sharpness results.
2317 Persistence and zigzags
A monotone sequence of inclusions of concept complexes
K0↪→K1↪→···
produces an ordinary persistence module on homology. If differentiation alternates between additions,
deletions, support restrictions, or reinterpretations, the natural object is instead a zigzag
K0→K1←K2→K3←···.
This is preferable to forcing every cognitive or observational change into a monotone filtration.
Cover-level persistence is already an active theory. In particular, Leitão develops persistent
homology directly from cover refinements and proves stability statements that propagate through
nerve and co-nerve functors. Therefore any novelty claim here must be made at the level of the
specific observation semantics, relative defect interpretation, or descent coupling, not at the level of
cover-refinement persistence itself.
18 Priority and novelty position
The standard ingredients include Boolean/Hamming and hypersimplex geometry, chain cones, beat-
point reductions, relative nerve theorems, sheaf descent, poset-fiber methods, and filtered-complex
spectral sequences. For the coverX0 =V (K)\M0, X1 =V (K)\M1, withA =X0∩X1 =V (L),
each ΛS is precisely the available-vertex obstruction complexSt(S,A) of Chachólski–Jin–Scolamiero–
Tombari [23], with our augmented empty-face convention. Their published Corollary 7.2 treats
several outside vertices and Theorem 8.6 controls the homotopy fibers of the vertex-cover inclusion
through these obstruction complexes.
The Björner–Wachs–Welker fiber theorem [21] gives decompositions under explicit fiber-
connectivity or null-attachment hypotheses. Here the attachment maps are retained and may
produce torsion or higher differentials. This is a narrower exact computation, not a replacement
for general fiber theory. Likewise, Ferrers graph resolutions, graph independence complexes, and
unweighted biclique complexes do not by themselves identify the available-link diagram used here.
The candidate contribution is the explicit observation-product recognition, the coarse-edge-link
matching formula, the transitivity restrictions on interacting cores, and the sharp integral positive
and negative examples. The matching deletion step is a direct specialization of the obstruction
decomposition above; identifying its product-edge link with a suspension of the coarse link is the
model-specific calculation. The interaction first page, filtration-width collapse, and two-column
kernel/cokernel sequence use standard filtered-chain and mapping-cone algebra after the available-
link diagram has been identified. These tools are not claimed as independent new constructions.
The historical audit and descent framework provide interpretation rather than priority for classical
machinery. No claim of first discovery of the exact specialization is made solely from its absence in
a targeted search. The separate convergence report records the primary-source comparison and the
remaining publication-level judgment.
19 Next theorem target
The matching case, the two-column interaction case, and the degree-twoE3 bound are now
established. UnrestrictedE2 collapse in the proposed graph classes is false, so it is no longer an
open target.
24Question 19.1(Integral secondary maps in a four-cycle). For each integerk≥2, can an exclusive
K2,2 profile with torsion-free available-link homology realized2 : Z→Z as multiplication byk?
Alternatively, what restrictions does the product-poset geometry impose on the possible integral
secondary maps?
This isolates interaction-created torsion at the smallest exclusive graph that supports a four-
core. The unit example above does not resolve it. The smallest possible higher-differential or
interaction-torsion realization also remains open globally; induced-restriction minimality is weaker.
20 Conclusion
Observation refinement is a transformation between representations, not a chain boundary. Carrier
splitting has a neutral-completion factorization, an exact low-height incidence description, and
an all-height matching edge-link formula. Beyond matching, available-link inclusions form the
interaction complex. Two core columns give an exact chain cone andE2 collapse; degree two permits
a genuined2 and only the sharpE3 bound in general. Interaction can create integral torsion even
from torsion-free local links.
Cover refinement and descent answer a separate question on the support actually covered. The
historical Boolean reconstruction, local-to-global data, and structural homology therefore remain
distinct. The mathematical claims of this draft are the stated specialized formulas and failure
boundaries; general topological machinery and publication priority are not silently strengthened.
A Reproducibility
The accompanying scriptverify_unfolding_complexes.py checks the following finite claims over
F2 for 2≤m≤8:
• affine independence of the parent plus all one-flip children;
• exact-layer cardinalities and constant Hamming weights;
• Betti numbers of the cumulative simplex skeletons;
• the relative/mapping-cone rank formula of Theorem 5.2;
• the three-variableG,D,E,Fcalculation extracted from the attached logical-complex notes.
The earlier audit scripts separately verify the non-congruence counterexample, the gated Boolean
derivative identity, the characteristic-two incidence differential, and the failure of the thesis’s printed
gluing identity.
A.1 Carrier-splitting and interaction verification
The transferred matching gate records 3,242 integral matching cases through four coarse states,
1,707 product-link checks overF2, and 639 additional nonempty matching five-state checks overF2.
The interaction gate enumerates 90,201 naturally labelled poset/profile cases through five states
over F2, plus 5,000 seeded six-state trials. Coarse-poset isomorphism postprocessing yields 87 poset
types and 16,761 profiles; exclusive graphs give 70 colour-preserving types with isolated exclusive
vertices retained. These are different counts of the same bounded exhaustive evidence, not additional
independent cases.
25The main interaction run makes 435 selected integral comparisons, supplemented by nine exact
local models, signed integer certificates for bothd2 witnesses, and a separate 32-state torsion
construction. The latter checks 181 available links, 31 coarse edge links, and the proved reduced
interaction matrix by Smith normal form, not the whole doubled pair. The projective-plane
triangulation has facets
012,013,024,035,045,125,134,145,234,235.
An additional 20,000 targetedK2,2 trials locate the sharp secondary differential; these trials are not
an exhaustive or uniform isomorphism sample. All coefficients, seeds, scripts, and integer certificates
are recorded in the accompanying interaction-gate reports. Computations support the proofs and
do not establish global minimality or publication priority.
References
[1] I. Matte-Blanco,The Unconscious as Infinite Sets: An Essay in Bi-Logic, Duckworth, 1975.
[2] B. Droncheff,On the Ultrametric Unconscious, Quantum Consciousness, and Mental Spaces, Master’s
thesis, 2015.
[3] A. Yu. Khrennikov, “Modelling of psychological behavior on the basis of ultrametric mental space:
Encoding of categories by balls,”BioSystems 90 (2007), 656–675.
[4] F. Murtagh, “Matte Blanco’s bi-logic, metric and ultrametric modelling,”Language and Psychoanalysis
3 (2014).
[5] C. H. Dowker, “Homology groups of relations,”Annals of Mathematics56 (1952), 84–95.
[6] A. Björner, “Nerves, fibers and homotopy groups,”Journal of Combinatorial Theory, Series A102(1)
(2003), 88–93. DOI: 10.1016/S0097-3165(03)00015-3.
[7] A. Hatcher,Algebraic Topology, Cambridge University Press, 2002.
[8] C. A. Weibel,An Introduction to Homological Algebra, Cambridge University Press, 1994.
[9] J. Curry, Sheaves, Cosheaves and Applications, Ph.D. thesis, University of Pennsylvania, 2014;
arXiv:1303.3255.
[10] J. M. Curry, “Dualities between cellular sheaves and cosheaves,”Journal of Pure and Applied Algebra
222(4) (2018), 966–993. DOI: 10.1016/j.jpaa.2017.06.001.
[11] G. Carlsson and V. de Silva, “Zigzag persistence,”Foundations of Computational Mathematics10 (2010),
367–405.
[12] H. Edelsbrunner, D. Letscher, A. Zomorodian, “Topological persistence and simplification,”Discrete &
Computational Geometry28 (2002), 511–533.
[13] N. J. Cavanna and D. R. Sheehy, “The generalized persistent nerve theorem,” arXiv:1807.07920, 2018.
[14] A. Leitão, “It’s All About Covers: Persistent Homology of Cover Refinements,” arXiv:2602.22784, 2026.
[15] D. A. Ramras, “Variations on the Nerve Theorem,”Discrete & Computational Geometry75 (2026),
871–903.
[16] I. M. Gelfand, M. Goresky, R. MacPherson, V. Serganova, “Combinatorial geometries, convex polyhedra,
and Schubert cells,”Advances in Mathematics63 (1987), 301–316.
26[17] F. Grande, A. Padrol, R. Sanyal, “Extension complexity and realization spaces of hypersimplices,”
Discrete & Computational Geometry59 (2018), 621–642.
[18] P. Gärdenfors,Conceptual Spaces: The Geometry of Thought, MIT Press, 2000.
[19] R. E. Stong, “Finite topological spaces,”Transactions of the American Mathematical Society123 (1966),
325–340. DOI: 10.1090/S0002-9947-1966-0195042-2.
[20] J. A. Barmak, “On Quillen’s Theorem A for posets,”Journal of Combinatorial Theory, Series A118
(2011), 2445–2453. DOI: 10.1016/j.jcta.2011.06.008.
[21] A. Björner, M. L. Wachs, and V. Welker, “Poset fiber theorems,”Transactions of the American
Mathematical Society357 (2005), 1877–1899. DOI: 10.1090/S0002-9947-04-03496-8.
[22] X. Fernández and E. G. Minian, “The cylinder of a relation and generalized versions of the Nerve
Theorem,”Discrete & Computational Geometry63 (2020), 549–559. DOI: 10.1007/s00454-018-0028-7.
[23] W. Chachólski, A. Jin, M. Scolamiero, and F. Tombari, “Homotopical decompositions of simplicial and
Vietoris–Rips complexes,”Journal of Applied and Computational Topology5 (2021), 215–248. DOI:
10.1007/s41468-021-00066-2.
[24] T. K. Dey, A. N. Hirani, and B. Krishnamoorthy, “Optimal homologous cycles, total unimodularity, and
linear programming,”SIAM Journal on Computing40(4) (2011), 1026–1044. DOI: 10.1137/100800245.
[25] B. Theory,Binary Order Thinning of Finite Posets: Homology, Collapses, and Sharp Limits, unpublished
companion research manuscript, 27 September 2026, reviewed v3. The archived title page uses the name
Brian Theory.
[26] A. Björner, “Topological methods,” inHandbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel,
and L. Lovász (eds.), Elsevier, 1995, pp. 1819–1872.
[27] U. Bauer, M. Kerber, F. Roll, and A. Rolle, “A unified view on the functorial nerve theorem and its
variations,”Expositiones Mathematicae41(4) (2023), 125503. doi:10.1016/j.exmath.2023.04.005.
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