PROSE AID ONLY — use PDF/page images and exact source for mathematics.

PDF PAGE 1
ProLT Observation Refinement Classification note v0.2
Observation Refinement in Propositional Logical Topologies
Fiber profiles, adjoint criteria, Quillen fibers, refinement defects, and the limits of complete
classification
Working research note v0.2 — 23 September 2026
Follow-up to Homological Foundations for Propositional Logical Topologies, built on the ProLT core manuscript v0.8.
Status convention. [S] denotes standard finite-poset/algebraic-topological mathematics. [A] denotes
a direct ProLT specialization or elementary theorem derived from the v0.8 signature framework. [C]
denotes a potentially publishable ProLT-specific formulation, invariant, or algorithmic consequence
whose novelty should be subjected to a dedicated literature audit before any priority claim.
1 Question and answer in one page
LetΦ= (𝜙𝑖)𝑖∈𝐼 be an observation family and letΨ extendΦ by new observations. The v0.8 ProLT theory
gives the realized-signature posets
𝑃Φ=𝑜Φ(Ω𝑃), 𝑃 Ψ=𝑜Ψ(Ω𝑃),
ordered by inclusion, and a canonical forgetful refinement map
𝑟ΨΦ :𝑃Ψ−→𝑃Φ
obtained by deleting the new coordinates. The corresponding order complexes are
OΦ=Δ(𝑃Φ), OΨ=Δ(𝑃Ψ).
The central problem is:
When does adding observations change only the resolution of the presentation, and when does it change the
homotopy or homology of the observable-state space?
The main results of this note give a hierarchy of answers.
R1. Every refinement admits an exact fiber-profile normal form: over each coarse signature 𝑝, record
exactly which new-coordinate signatures occur. This completely reconstructs𝑃Ψ and the map𝑟ΨΦ.
R2. The refinement is topologically redundant exactly when every coarse state has one new profile and
those profiles are monotone. Equivalently,𝑟ΨΦ is an order isomorphism.
R3. If the refinement does not split coarse states, thenOΨ is literally a subcomplex ofOΦ. A coarse chain
survives exactly when its new-coordinate profiles increase along the chain. For one new Boolean
observation, the deleted chains are precisely those containing a 1→ 0 descent.
R4. There are simple signature-level sufficient criteria for homotopy invariance. If every lower refinement
fiber has a greatest element, or every upper refinement fiber has a least element, then𝑟ΨΦ has an order
adjoint and is a homotopy equivalence. For one observation 𝜓, this says: possibility of𝜓 or forced
truth of𝜓 must be monotone in the coarse information order.
1

PDF PAGE 2
ProLT Observation Refinement Classification note v0.2
R5. More generally, Quillen–McCord/Barmak fiber criteria apply. Contractible principal fibers imply a
simple homotopy equivalence; acyclic principal fibers imply homology equivalence.
R6. Homology equivalence has an exact finite test: the refinement mapping cone is acyclic. In the
non-splitting case it reduces to ordinary relative homology𝐻∗(OΦ,OΨ).
R7. There is no unrestricted algorithmic classification of homotopy-neutral refinements: arbitrary finite
posets can be realized as refined signature posets above a one-point coarse space, so the problem
contains finite-complex contractibility, which is undecidable. Homology neutrality, by contrast,
remains decidable by finite boundary-matrix computation.
2 Base ProLT data and refinement map
Let 𝑃 ={𝑝1,...,𝑝 𝑛} be a finite propositional variable set and Ω𝑃 ={0, 1}𝑃 its valuation space. For a
formula𝐿, write
𝜍(𝐿)={𝑣∈Ω𝑃 :𝑣|=𝐿}.
For an indexed observation familyΦ= (𝜙𝑖)𝑖∈𝐼, define
𝑜Φ(𝑣)={𝑖∈ 𝐼 :𝑣|=𝜙𝑖}⊆ 𝐼.
The v0.8 ProLT results identify topological indistinguishability with equality of signatures and the specialization
preorder with inclusion:
𝑣∼Φ 𝑤 ⇐⇒ 𝑜Φ(𝑣)=𝑜Φ(𝑤), 𝑣 ⪯Φ 𝑤 ⇐⇒ 𝑜Φ(𝑣)⊆ 𝑜Φ(𝑤).
Hence the Kolmogorov quotient is the finite poset
𝑃Φ=𝑜Φ(Ω𝑃)⊆ 2𝐼
with inclusion order.
Suppose now that
Ψ= (Φ,Θ)
extendsΦ by a disjointly indexed familyΘ= (𝜃𝑎)𝑎∈𝐴. Then
𝑜Ψ(𝑣)=  𝑜Φ(𝑣),𝑜Θ(𝑣)∈ 2𝐼× 2𝐴,
and the coordinate-forgetting map
𝑟 =𝑟ΨΦ :𝑃Ψ→𝑃Φ, 𝑟 (𝑠)=𝑠∩𝐼,
is surjective and monotone.
3 The fiber-profile normal form
Definition 3.1 (Coarse state and refinement profile). For𝑝∈𝑃Φ, let
𝐶𝑝 :=𝑜−1
Φ(𝑝)
be the coarse observational state represented by𝑝. Define its new-coordinate fiber profile
B𝑝 :=𝑜Θ(𝐶𝑝)={𝑜Θ(𝑣) :𝑣∈𝐶𝑝}⊆ 2𝐴.
EachB𝑝 is nonempty.
2

PDF PAGE 3
ProLT Observation Refinement Classification note v0.2
Theorem 3.2 (Fiber-profile normal form). [A]/[C] Let
𝑄B :={(𝑝,𝑎) : 𝑝∈𝑃Φ, 𝑎∈B 𝑝}⊆ 𝑃Φ× 2𝐴
with product order
(𝑝,𝑎)≤( 𝑞,𝑏) ⇐⇒𝑝≤𝑞 in𝑃Φ and𝑎⊆𝑏.
Then
𝑃Ψ 𝑄B
as posets, under separation of old and new coordinates. Under this isomorphism the refinement map is exactly
the first-coordinate projection
𝑟(𝑝,𝑎)= 𝑝.
Consequently the family(B𝑝)𝑝∈𝑃Φ determines the refined signature poset and the refinement projection
completely.
Proof. Every realizedΨ-signature comes from a valuation𝑣 and splits uniquely as
(𝑜Φ(𝑣),𝑜Θ(𝑣)) =(𝑝,𝑎)
with𝑎∈B 𝑝. Conversely, every pair in𝑄B is realized by the definition ofB𝑝. Inclusion of the combined
signatures is coordinatewise inclusion because the old and new index sets are disjoint. The projection
statement is immediate. □
Remark 3.3 (Point fibers are not Quillen fibers). The direct point fiber
𝑟−1(𝑝)={𝑝}×B 𝑝
records only how the coarse state𝑝 splits. Quillen-type theorems use instead the preimages of principal ideals
or filters,
𝑟−1(↓𝑝), 𝑟 −1(↑𝑝),
which also remember how the split pieces compare with states below or above𝑝. This distinction is essential:
every point fiber can be contractible while the refinement still changes homotopy type.
4 Exact classification of topological redundancy
The v0.8 manuscript calls a new observation topologically redundant when its truth region is already open in
the old ProLT. The fiber normal form turns this into an exact signature-level theorem for an arbitrary finite
block of added observations.
Theorem 4.1 (Redundancy/order-isomorphism theorem). [A]/[C] For a refinementΨ= (Φ,Θ), the following
are equivalent.
(i) 𝜏Ψ=𝜏Φ on the fixed valuation carrierΩ𝑃.
(ii) Every new truth region 𝜍(𝜃𝑎),𝑎∈ 𝐴, is already𝜏Φ-open.
3

PDF PAGE 4
ProLT Observation Refinement Classification note v0.2
(iii) Every fiber profile is a singleton,
B𝑝 ={𝛼(𝑝)},
and the resulting map
𝛼 :𝑃Φ→ 2𝐴
is isotone:
𝑝≤𝑞=⇒𝛼(𝑝)⊆ 𝛼(𝑞).
(iv) The refinement projection 𝑟 :𝑃Ψ→𝑃Φ is an order isomorphism.
Proof. The equivalence of (i) and (ii) is the definition of equality of the generated topologies: 𝜏Φ⊆ 𝜏Ψ
automatically, and equality holds exactly when each new subbasic truth region is old-open.
Assume (i). If two valuations lie in the same old signature class𝐶𝑝, they are topologically indistinguishable
in𝜏Φ; equality of topologies makes them indistinguishable in𝜏Ψ as well. Thus the new signature is constant
on𝐶𝑝, givingB𝑝 ={𝛼(𝑝)}. If 𝑝≤𝑞, choose𝑣∈𝐶𝑝,𝑤∈𝐶𝑞. The old specialization relation gives 𝑣⪯Φ 𝑤.
Equality of topologies gives𝑣⪯Ψ 𝑤, hence𝛼(𝑝)⊆ 𝛼(𝑞). Therefore (iii) holds.
Assume (iii). For each new coordinate𝑎∈ 𝐴, the set
𝑈𝑎 :={𝑝∈𝑃Φ :𝑎∈𝛼(𝑝)}
is upward closed. By the signature representation of the ProLT topology,
𝜍(𝜃𝑎)=𝑜−1
Φ(𝑈𝑎)
is𝜏Φ-open. Hence (ii).
Under (iii), Theorem 3.2 identifies𝑃Ψ with the graph{(𝑝,𝛼(𝑝))} . Monotonicity of 𝛼 says
𝑝≤𝑞 ⇐⇒ (𝑝,𝛼(𝑝))≤( 𝑞,𝛼(𝑞)),
so projection is an order isomorphism. Conversely, if projection is an order isomorphism then every fiber has
one point and its inverse is necessarily isotone, giving (iii). □
Remark 4.2. This theorem separates three notions which should not be conflated:
(a) no splitting: eachB𝑝 is a singleton;
(b) no order deletion: the singleton profile map is isotone;
(c) full redundancy: both hold, equivalently the topology is unchanged.
A refinement can leave every coarse equivalence class intact while still deleting one-way specialization
comparisons.
4

PDF PAGE 5
ProLT Observation Refinement Classification note v0.2
5 Non-splitting refinements and chain deletion
Definition 5.1 (Non-splitting refinement). A refinement is non-splitting if everyB𝑝 is a singleton. Write
B𝑝 ={𝛼(𝑝)}.
No monotonicity of𝛼 is assumed.
Theorem 5.2 (Order deletion theorem). [A]/[C] For a non-splitting refinement, identify the fine and coarse
vertex sets through the bijection𝑟. Then
𝑝≤Ψ 𝑞 ⇐⇒ 𝑝≤Φ 𝑞 and𝛼(𝑝)⊆ 𝛼(𝑞).
Consequently
≤Ψ⊆≤ Φ
and
OΨ⊆O Φ
as a simplicial subcomplex on the same vertex set. A coarse chain
𝑝0< 𝑝1<··· < 𝑝𝑘
survives the refinement exactly when
𝛼(𝑝0)⊆ 𝛼(𝑝1)⊆···⊆ 𝛼(𝑝𝑘).
Proof. In the fiber-profile normal form the unique fine point over 𝑝 is(𝑝,𝛼(𝑝)). Product order gives the
displayed criterion. Every fine chain is therefore a coarse chain; conversely, a coarse chain is fine exactly
when its new profiles are coordinatewise nondecreasing. □
Corollary 5.3 (One-bit descent criterion). Suppose one new Boolean observation 𝜓 is constant on each
coarse class, and write
𝑏(𝑝)∈{ 0, 1}
for its value on𝐶𝑝. Then a coarse chain 𝑝0<··· < 𝑝𝑘 survives exactly when
𝑏(𝑝0)≤ 𝑏(𝑝1)≤···≤ 𝑏(𝑝𝑘).
Equivalently, it is deleted exactly when it contains a 1→ 0 inversion.
Corollary 5.4 (Post-𝑇0 refinement filtration). [A]/[C] If𝑋Φ is already𝑇0 on the valuation carrier, then𝑜Φ is
injective, so every later observation refinement is automatically non-splitting. Thus a sequence
Φ0⊆Φ1⊆···⊆ Φ𝑇
with𝑋Φ0𝑇0 yields, after identifying vertices with valuations,
OΦ𝑇⊆···⊆O Φ1⊆O Φ0.
Hence ordinary persistence can be applied after reversing the refinement index; zigzag persistence is not
required while observations are only added.
Proof. A coarse signature class contains a single valuation when𝑜Φ is injective, so it cannot split further as a
set. Theorem 5.2 then applies at every later stage. □
5

PDF PAGE 6
ProLT Observation Refinement Classification note v0.2
6 Relative defect chains in the non-splitting case
The preceding inclusion lets the mapping-cone defect from the foundation note be sharpened.
Theorem 6.1 (Refinement defect becomes relative homology). [S]/[A] For a non-splitting refinement, under
the vertex identification of Theorem 5.2, the induced simplicial map is the inclusion
OΨ ↩→O Φ.
Therefore the mapping-cone refinement defect satisfies
𝐷𝑘(Ψ/Φ;𝑅)  𝐻𝑘(OΦ,OΨ;𝑅).
Definition 6.2 (Refinement-descent chain complex). For a non-splitting refinement define𝐶desc
𝑘 (Ψ/Φ;𝑅) to
be the free𝑅-module generated by the coarse𝑘-chains that fail the survival condition
𝛼(𝑝0)⊆···⊆ 𝛼(𝑝𝑘).
Its boundary is the ordinary simplicial boundary followed by deletion of every face that does survive the
refinement.
Proposition 6.3 (Exact descent-chain computation). [A]/[C] There is a natural chain isomorphism
𝐶desc
∗ (Ψ/Φ;𝑅) 𝐶∗(OΦ,OΨ;𝑅).
Hence
𝐷𝑘(Ψ/Φ;𝑅)  𝐻𝑘(𝐶desc
∗ (Ψ/Φ;𝑅)).
For one added Boolean observation, the generators are exactly coarse chains containing a1→ 0 descent.
Proof. The simplicial chain group of OΦ is free on its simplices, and the subcomplex OΨ is spanned by
exactly the surviving simplices. The quotient chain group therefore has the nonsurviving simplices as a basis.
Its differential is the coarse simplicial differential modulo the surviving submodule, which is precisely the
stated rule. □
7 Adjoint criteria: an elementary homotopy-neutral class
For𝑝∈𝑃Φ, define the principal refinement fibers
𝐿𝑝 :=𝑟−1(↓𝑝), 𝑈 𝑝 :=𝑟−1(↑𝑝).
The following test is much easier than computing arbitrary homotopy types.
Theorem 7.1 (Greatest-lower-fiber / least-upper-fiber criterion). [S]/[A]/[C] Let
U(𝑝) :=
Ø
𝑞≤𝑝
𝑎∈B 𝑞
𝑎, L(𝑝) :=
Ù
𝑞≥𝑝
𝑎∈B 𝑞
𝑎.
Then:
6

PDF PAGE 7
ProLT Observation Refinement Classification note v0.2
(i) 𝐿𝑝 has a greatest element if and only if
U(𝑝)∈B 𝑝.
When this holds, the greatest element is(𝑝, U(𝑝)).
(ii) 𝑈𝑝 has a least element if and only if
L(𝑝)∈B 𝑝.
When this holds, the least element is(𝑝, L(𝑝)).
If (i) holds for every𝑝, then𝑟 has a right adjoint. If (ii) holds for every 𝑝, then𝑟 has a left adjoint. Either
global condition implies that
Δ(𝑟) :OΨ→O Φ
is a homotopy equivalence.
Proof. Suppose𝐿𝑝 has greatest element(𝑞,𝑎∗). Since 𝑟 is surjective, choose some(𝑝,𝑏)∈ 𝑟−1(𝑝)⊆ 𝐿𝑝.
Greatness gives(𝑝,𝑏)≤( 𝑞,𝑎∗), so𝑝≤𝑞≤ 𝑝 and therefore𝑞= 𝑝. The second coordinate 𝑎∗ contains every
new signature occurring over every𝑞≤ 𝑝, hence contains U(𝑝). Since 𝑎∗ itself occurs in the union, equality
follows. Conversely, if U(𝑝)∈B 𝑝, then every(𝑞,𝑎)∈ 𝐿𝑝 satisfies
𝑞≤ 𝑝, 𝑎 ⊆ U(𝑝),
so(𝑝, U(𝑝)) is greatest. This proves (i). The proof of (ii) is dual.
If every lower fiber has a greatest element, define
𝑠𝑅(𝑝) :=(𝑝, U(𝑝)).
Then
𝑟(𝑥)≤ 𝑝 ⇐⇒ 𝑥≤𝑠𝑅(𝑝),
so𝑟⊣𝑠𝑅. In particular 𝑟𝑠𝑅= id and
id≤𝑠𝑅𝑟.
Order-preserving maps that are pointwise comparable induce homotopic maps on order complexes, soΔ(𝑠𝑅)
is a homotopy inverse toΔ(𝑟). The left-adjoint case is dual. □
7.1 A sharp one-observation interpretation
Let𝜓 be one newly added observation. For each coarse state𝐶𝑝, define
𝑀𝜓(𝑝)=
(
1, some𝑣∈𝐶𝑝 satisfies𝜓,
0, otherwise,
and
𝑚𝜓(𝑝)=
(
1, every𝑣∈𝐶𝑝 satisfies𝜓,
0, otherwise.
Thus𝑀𝜓 records possibility of the new observation inside a coarse state, while𝑚𝜓 records forced truth. The
possible fiber profiles are
{0}, {1}, {0, 1},
with
𝑚𝜓(𝑝)= minB𝑝, 𝑀 𝜓(𝑝)= maxB𝑝.
7

PDF PAGE 8
ProLT Observation Refinement Classification note v0.2
Corollary 7.2 (Possible/forced monotonicity criterion). [A]/[C] For a one-observation refinement:
(i) 𝑟 has a right adjoint if and only if𝑀𝜓 :𝑃Φ→{ 0, 1} is isotone.
(ii) 𝑟 has a left adjoint if and only if𝑚𝜓 :𝑃Φ→{ 0, 1} is isotone.
Equivalently, either of the following is sufficient for homotopy invariance:
• the coarse states in which𝜓 is possible form an upset;
• the coarse states in which𝜓 is forced form an upset.
Topological redundancy is the stronger special case
𝑚𝜓 =𝑀𝜓 =𝑏
with𝑏 isotone.
Proof. For one bit,
U(𝑝)= max
𝑞≤𝑝
𝑀𝜓(𝑞).
The membership conditionU(𝑝)∈B 𝑝 is equivalent toU(𝑝)=𝑀𝜓(𝑝), which for every𝑝 is exactly isotonicity
of𝑀𝜓. Dually,
L(𝑝)= min
𝑞≥𝑝
𝑚𝜓(𝑞)
and L(𝑝)∈B 𝑝 for every𝑝 is exactly isotonicity of𝑚𝜓. □
8 Quillen–McCord/Barmak fiber criteria
The adjoint criterion forces a cone point in every principal fiber. Quillen’s fiber theorem permits much more
general contractible fibers.
Theorem 8.1 (Contractible-fiber criterion). [S]/[A] If for every𝑝∈𝑃Φ, the order complex
Δ 𝑟−1(↓𝑝)
is contractible, then
Δ(𝑟) :OΨ−→OΦ
is a simple homotopy equivalence, hence a homotopy equivalence. Dually, the same conclusion follows if
every
Δ 𝑟−1(↑𝑝)
is contractible.
Proof. The lower-fiber statement is the finite-poset McCord–Quillen theorem together with Barmak’s
strengthening to simple homotopy equivalence. The upper-fiber statement follows by applying the theorem to
opposite posets. □
8

PDF PAGE 9
ProLT Observation Refinement Classification note v0.2
Theorem 8.2 (Acyclic-fiber homology criterion). [S]/[A] Fix coefficients𝑅. If
e𝐻𝑖

Δ(𝑟−1(↓𝑝));𝑅

= 0
for every𝑝∈𝑃Φ and every𝑖≥ 0, then
𝑟∗ :𝐻𝑖(OΨ;𝑅)−→ 𝐻𝑖(OΦ;𝑅)
is an isomorphism for every𝑖. The dual upper-fiber statement also holds.
Remark 8.3 (Sufficient, not necessary) . The principal-fiber conditions are powerful but not necessary.
Section 11 gives a concrete ProLT refinement for which𝑟 is a homotopy equivalence even though one lower
principal fiber and one upper principal fiber are each two disconnected points. Thus neither orientation of the
Quillen test succeeds, despite homotopy neutrality.
9 Exact homology classification by the refinement cone
The homotopy problem is difficult in full generality, but the homology problem has an exact algebraic answer.
Definition 9.1 (Refinement mapping cone). Let
𝑟# :𝐶∗(OΨ;𝑅)−→ 𝐶∗(OΦ;𝑅)
be the induced chain map. Define
D∗(Ψ/Φ;𝑅)= Cone(𝑟#)
and
𝐷𝑘(Ψ/Φ;𝑅)=𝐻𝑘(D∗(Ψ/Φ;𝑅)).
Theorem 9.2 (Exact homology-equivalence criterion). [S]/[A]/[C] The following are equivalent:
(i) 𝑟∗ :𝐻𝑘(OΨ;𝑅)→ 𝐻𝑘(OΦ;𝑅) is an isomorphism for every𝑘;
(ii)
𝐷𝑘(Ψ/Φ;𝑅)= 0 for every𝑘.
Thus the refinement cone gives an exact, finite homological classification of whether the refinement is
homology-neutral.
Proof. The standard mapping-cone long exact sequence is
···→ 𝐻𝑘(OΨ)
𝑟∗
−→𝐻𝑘(OΦ)→ 𝐷𝑘(Ψ/Φ)→ 𝐻𝑘−1(OΨ)→··· .
Exactness gives the equivalence. □
Corollary 9.3 (Quick obstructions). If any of the following differs betweenOΨ andOΦ, the refinement is not
homology-neutral:
(a) a Betti number over the chosen field;
(b) the number of connected components;
(c) the Euler characteristic.
The converses are false in general.
9

PDF PAGE 10
ProLT Observation Refinement Classification note v0.2
9.1 An exact 𝐻0 criterion
Proposition 9.4 (Connected-component criterion). [A] Let 𝜋0(𝑃) denote connected components of the
undirected comparability graph of a finite poset𝑃. Since 𝑟 is surjective, the induced map
𝑟∗ :𝐻0(OΨ;𝑅)→ 𝐻0(OΦ;𝑅)
is always surjective. It is an isomorphism if and only if, for every connected component𝐶⊆ 𝑃Φ, the inverse
image
𝑟−1(𝐶)
is connected.
Proof. A connected component of𝑃Ψ maps into one component of𝑃Φ because comparability paths map to
comparability-or-equality paths. Surjectivity of𝑟 ensures every coarse component receives at least one fine
component. On 𝐻0, each fine component generator maps to the generator of its coarse component. The map
is injective exactly when no coarse component receives more than one fine component. □
10 A strict hierarchy of refinement-neutrality tests
It is useful to distinguish five levels.
Class Condition
R 𝑟 is an order isomorphism (topological redundancy)
A 𝑟 has a left or right order adjoint
Q all lower or all upper principal refinement fibers are contractible
H Δ(𝑟) is a homotopy equivalence
M 𝑟∗ is a homology isomorphism
Standard implications and the preceding theorems give
R⊆ A⊆ Q⊆ H⊆ M.
The examples below show that the first three inclusions are already strict in very small Boolean ProLTs. The
universality result of Section 13 shows thatH ⊊ M also occurs in unrestricted ProLT refinements by importing
any finite acyclic noncontractible complex.
11 Worked Boolean examples
Throughout, signatures are ordered by inclusion and coefficients for displayed homology groups are F2.
11.1 Redundant: adding 𝑋∧𝑌 to(𝑋,𝑌)
Let
Φ= (𝑋,𝑌), 𝜓 =𝑋∧𝑌.
10

PDF PAGE 11
ProLT Observation Refinement Classification note v0.2
The coarse signatures are
𝐷=∅, 𝐴 ={𝑋}, 𝐶 ={𝑌}, 𝐵 ={𝑋,𝑌},
forming the Boolean diamond. The new bit is
𝑏(𝐷)=𝑏(𝐴)=𝑏(𝐶)= 0, 𝑏 (𝐵)= 1.
It is isotone, so Theorem 4.1 applies. The refinement is inR: the ProLT topology and canonical order complex
are unchanged, although a weighted signature metric may change if the redundant coordinate is assigned
positive weight.
11.2 Adjoint-neutral but nonredundant: adding 𝑋 to no observations
Take one propositional variable𝑋, letΦ be the empty observation family, and refine byΨ= (𝑋). The coarse
poset is a point; the fine poset is the two-element chain
0< 1.
The projection to a point has both a left adjoint (choose the minimum) and a right adjoint (choose the
maximum), so it is a homotopy equivalence. The topology nevertheless changes from indiscrete to Sierpi´nski.
Thus this refinement lies in A\ R.
11.3 Quillen-neutral but no adjoint: adding XOR to (𝑋,𝑌)
Let
Φ= (𝑋,𝑌), 𝜓 =𝑋⇕𝑌.
The new-bit profile on(𝐷,𝐴,𝐶,𝐵 ) is
(0, 1, 1, 0).
No coarse class splits, but the old comparisons 𝐴 < 𝐵and𝐶 < 𝐵are deleted. The fine poset has 𝐷 as a
minimum and three elements𝐴,𝐶,𝐵 above it. Hence its order complex is a cone and is contractible.
The bit is not isotone, so neither the possible nor forced map is isotone; no adjoint criterion applies. However
every lower Quillen fiber is contractible: over 𝐷,𝐴,𝐶 it is a point or edge, and over𝐵 it is the entire fine
poset, again a cone. Therefore the refinement lies in
Q\ A.
11.4 Homotopy-neutral while both Quillen orientations fail
Take
Φ= (𝑋∧𝑌, 𝑋, 𝑋∨𝑌).
On the four valuations 00, 01, 10, 11, the realized signatures form a four-element chain
𝑝0< 𝑝1< 𝑝2< 𝑝3
11

PDF PAGE 12
ProLT Observation Refinement Classification note v0.2
with
00 ∅
01 {𝑋∨𝑌}
10 {𝑋,𝑋 ∨𝑌}
11 {𝑋∧𝑌,𝑋,𝑋 ∨𝑌}.
Add
𝜓=¬𝑌.
Its profile along the chain is
1, 0, 1, 0.
No coarse class splits. The surviving strict comparisons are
𝑝0< 𝑝2, 𝑝 1< 𝑝2, 𝑝 1< 𝑝3.
ThusOΨ is a three-edge tree, whileOΦ is a 3-simplex. Both are nonempty and contractible, so the refinement
projection is a homotopy equivalence.
Nevertheless,
𝑟−1(↓𝑝1)={𝑝0,𝑝 1}
is a two-point antichain, and
𝑟−1(↑𝑝2)={𝑝2,𝑝 3}
is also a two-point antichain. Hence the lower-fiber Quillen criterion fails and the dual upper-fiber criterion
fails. This gives a concrete refinement in
H\ Q.
11.5 Homology-active: adding negation to (𝑋)
Let
Φ= (𝑋), Ψ= (𝑋,¬𝑋).
The coarse signature poset is a two-element chain, soOΦ is an edge. The fine signatures{𝑋} and{¬𝑋} are
incomparable, soOΨ is two isolated points. Hence
𝐻0(OΦ)  F2, 𝐻 0(OΨ)  F2
2.
The refinement is not homology-neutral. The mapping cone gives
𝐷1(Ψ/Φ; F2)  F2, 𝐷 0(Ψ/Φ; F2)= 0.
This is the smallest example in which adding an observation splits a connected coarse information structure
into disconnected fine states.
11.6 Splitting can still be homotopy-neutral
LetΦ= (𝑋) on two variables𝑋,𝑌 , and add𝑌. Each coarse 𝑋-state splits into the two possible𝑌-values, so
B∅=B{𝑋} ={0, 1}.
The coarse poset is an edge and the fine poset is the Boolean diamond. Both are contractible. In fact both the
possible map𝑀𝑌≡ 1 and the forced map𝑚𝑌≡ 0 are isotone, so the projection has both adjoints. This shows
that state splitting by itself does not imply a topological defect.
12

PDF PAGE 13
ProLT Observation Refinement Classification note v0.2
12 Small exhaustive verification
To test the logical hierarchy independently of the proofs, the accompanying script exhaustively enumerates
every one-bit refinement profile over every nonempty subposet of the Boolean square𝐵2. A coarse point may
have new-bit fiber{0},{1}, or{0, 1}. There are 255 such refinement instances in total.
The verifier checks redundancy, left/right section-adjoint criteria, lower-fiber contractibility by finite beat-point
reduction, lower-fiber acyclicity overF2, the Betti numbers of the coarse and fine order complexes, and the
mapping-cone Betti numbers. It finds exactly five logical pattern types under those tests:
(R, A, Q↓, Acyc↓, M) count
(0, 0, 0, 0, 0) 24
(0, 0, 0, 0, 1) 16
(0, 0, 1, 1, 1) 20
(0, 1, 1, 1, 1) 144
(1, 1, 1, 1, 1) 51
These finite checks are not substitutes for the theorems, but they verify the example calculations and
demonstrate that the sufficient conditions are genuinely non-equivalent even at very small size.
13 Universality of refinement and the limit of complete homotopy classification
The preceding criteria invite the question whether one could push further to a complete effective homotopy
classification. In unrestricted dimension the answer is negative.
Theorem 13.1 (Universality of refined signature posets). [A]/[C] Every finite poset𝑄 is isomorphic to the
realized-signature poset of some finite propositional observation family. More strongly,𝑄 can occur as𝑃Ψ in
a refinement
Φ⊆Ψ
for which𝑃Φ is a single point.
Proof. Take enough propositional variables that|Ω𝑃|≥| 𝑄|, and choose a surjection
ℎ :Ω𝑃 ↠𝑄.
Let the coarse observation familyΦ be empty, so𝑃Φ is one point. Index new observations by the elements
𝑞∈𝑄. For each 𝑞, prescribe the truth set
𝑇𝑞 :={𝑣∈Ω𝑃 :𝑞≤ℎ(𝑣)}.
Every subset of a finite valuation space is definable by a propositional formula, for example by the disjunction
of the complete minterms of its members. Choose 𝜃𝑞 with𝜍(𝜃𝑞)=𝑇𝑞.
Ifℎ(𝑣)=𝑥, then the new observation signature is
𝑜Ψ(𝑣)={𝑞∈𝑄 :𝑞≤𝑥}=↓𝑥.
The map𝑥↦→↓𝑥 is an order embedding of𝑄 into 2𝑄, because
𝑥≤𝑦 ⇐⇒ ↓𝑥⊆↓𝑦.
Surjectivity ofℎ realizes every such signature and no others. Hence 𝑃Ψ 𝑄. □
13

PDF PAGE 14
ProLT Observation Refinement Classification note v0.2
Corollary 13.2 (Every finite simplicial homotopy type occurs). For every finite simplicial complex𝐾, choose
its face poset𝑄=X(𝐾). Then
Δ(𝑄)= sd(𝐾),
the barycentric subdivision of𝐾. Therefore a refinement above a one-point coarse ProLT can realize the
homotopy type of any finite simplicial complex.
Theorem 13.3 (Undecidability of unrestricted homotopy-neutral refinement) . [S]/[A]/[C] There is no
algorithm that, for arbitrary finite ProLT observation refinements in unrestricted dimension, always decides
whether
Δ(𝑟ΨΦ)
is a homotopy equivalence.
Proof. Given a finite simplicial complex𝐾, apply Corollary 13.2 to construct a refinement whose coarse
signature poset is a point and whose fine order complex has the homotopy type of 𝐾. The refinement
projection to the point is a homotopy equivalence exactly when𝐾 is contractible. Algorithmic recognition of
contractibility for arbitrary finite simplicial complexes is undecidable. Thus a universal decision algorithm
for homotopy-neutral ProLT refinements would decide an undecidable problem. □
Corollary 13.4 (Homology remains effectively computable) . For any fixed computable coefficient field,
homology-neutrality of a given finite refinement is decidable: construct the finite mapping-cone boundary
matrices and compute their ranks. Thus the homotopy and homology classification problems diverge sharply
in unrestricted generality.
Remark 13.5 (Strictness of H ⊊ M). The universality theorem also imports standard acyclic noncontractible
finite complexes. For example, a finite triangulation of a punctured Poincar´e homology sphere is acyclic but
has nontrivial fundamental group. Refining a point to its face poset gives a homology equivalence to the point
which is not a homotopy equivalence.
14 A practical classification algorithm
The theory suggests the following release-quality workflow for a concrete observation refinement.
Step 1. Build the coarse poset. Compute the realized old signatures𝑃Φ and their inclusion order.
Step 2. Build fiber profiles. For each𝑝∈𝑃Φ, compute
B𝑝 =𝑜Θ(𝐶𝑝).
No fine-poset construction is conceptually necessary beyond these profiles.
Step 3. Check exact redundancy. If everyB𝑝 is singleton and the profile map is isotone, the refinement is
an order isomorphism and the classification stops.
Step 4. Check adjoints. Compute U(𝑝) and L(𝑝). If all U(𝑝)∈B 𝑝, or all L(𝑝)∈B 𝑝, the refinement is
homotopy-neutral.
Step 5. Check principal fibers. Apply beat-point/collapse tests or another contractibility certificate to
𝑟−1(↓𝑝) and𝑟−1(↑𝑝). Uniform contractibility certifies simple homotopy equivalence. Uniform
acyclicity certifies homology equivalence.
14

PDF PAGE 15
ProLT Observation Refinement Classification note v0.2
Step 6. Compute the exact homology defect. Build Cone(𝑟#). Nonzero 𝐷𝑘 proves that the refinement
changes homology. Vanishing of all𝐷𝑘 proves homology equivalence.
Step 7. For non-splitting refinements, use descent chains. Replace the general mapping cone by the
relative/descent chain complex of Proposition 6.3.
Step 8. Do not treat an inconclusive homotopy test as failure. The Quillen and adjoint criteria are
sufficient, not necessary. In unrestricted dimension no complete decision algorithm exists. For a
research theorem, restrict to a structurally controlled subclass.
15 Promising theorem targets after this classification
The classification above narrows the next research questions substantially.
T1. Logical criteria for contractible Quillen fibers.Find syntactic or semantic conditions on newly added
formulas that force every𝑟−1(↓𝑝) to have a cone point, beat-point reduction, or nerve decomposition.
T2. Minimal homology-changing observations. GivenΦ, characterize or optimize observations𝜓 for
which𝐷𝑘((Φ,𝜓)/Φ)≠ 0, possibly with a cost on formulas or observations.
T3. Post-𝑇0 persistence. Study the reverse filtration of Corollary 5.4. Determine whether logical properties
of the added observations constrain barcode births/deaths in the relative descent complexes.
T4. Refinement/update interaction. Combine the contravariant-looking refinement deletionOΨ⊆O Φ
after𝑇0 with premise restriction subcomplexes. This should yield a two-parameter family indexed by
observational resolution and admissible-state restriction.
T5. Cross-layer comparison. Relate refinement defects in canonical order-complex homology to changes
in the presentation-sensitive Dowker compatibility complex and to changes in metric/Rips persistence.
A nontrivial comparison theorem would be substantially stronger than defining the layers independently.
T6. Restricted decidable classes. Because unrestricted homotopy neutrality is undecidable, identify
meaningful ProLT classes where the problem becomes decidable or efficiently certifiable: bounded
poset height, bounded number of observations, distributive-lattice images, monotone formula fragments,
or bounded treewidth of the comparability/Hasse graph.
16 Preliminary novelty positioning
The finite-poset fiber theorems, adjunction-homotopy principle, mapping cones, and undecidability of arbitrary
finite-complex contractibility are standard mathematics. They should not be claimed as new. The current
contribution candidates are instead the ProLT-specific synthesis and sharpenings: the refinement fiber-profile
normal form; the exact redundancy theorem in signature terms; the possible/forced monotonicity criterion; the
descent-chain description after𝑇0; the refinement-defect hierarchy; and the universality reduction formulated
for propositional observation refinement.
A preliminary literature check found adjacent but distinct areas. Formal concept analysis and information-
system theory study binary object–attribute data and attribute reduction. Separately, Gaucher used the phrase
“refinement of observation” in directed homotopy theory for flows and finite bounded posets, with preservation
15

PDF PAGE 16
ProLT Observation Refinement Classification note v0.2
results for branching/merging homology and underlying homotopy type. That is not the same object as the
ProLT signature-projection refinement studied here, but the terminology overlap should be acknowledged in
any publication. A dedicated adversarial novelty audit remains necessary before promoting any [C] item to a
novelty claim.
17 Conclusion
Observation refinement in a ProLT is not a single phenomenon. It may:
• add a coordinate without changing the topology at all;
• split observational states while preserving homotopy type;
• leave all states unsplit but delete specialization comparisons;
• change homotopy while preserving homology;
• or change homology itself.
The fiber-profile normal form makes these possibilities explicit. It reduces every refinement to a structured
surjective poset map
𝑄B−→𝑃Φ.
Topological redundancy is exactly order isomorphism. Adjoint and Quillen-fiber tests give progressively
broader homotopy-neutral classes. Mapping-cone homology gives an exact finite test for homology neutrality.
Once the old ProLT is𝑇0, refinement becomes especially transparent: it can no longer split vertices, so it
acts only by deleting chains from the canonical order complex, and the defect is ordinary relative homology
generated by the deleted descent chains.
The universality theorem also tells us where to stop looking for an all-purpose homotopy decision rule:
unrestricted refinement already contains the full contractibility problem for finite complexes. The productive
research direction is therefore not a mythical universal criterion, but a theory of logically natural subclasses
in which the fiber geometry can be controlled and related back to observations, premises, and eventually
time-evolving logical state spaces.
A Computational verification summary
The accompanying verifier constructs order complexes and normalized simplicial chain maps explicitly over
F2. It verifies:
(i) the redundant 𝑋∧𝑌 example has identical coarse/fine Betti vector(1, 0, 0) and zero cone homology;
(ii) the XOR refinement has coarse Betti(1, 0, 0), fine Betti(1, 0), and zero cone homology;
(iii) the negation split has coarse Betti (1, 0), fine Betti(2), and cone Betti(0, 1);
(iv) the splitting-but-neutral 𝑌 refinement of(𝑋) has zero cone homology;
(v) the four-chain alternating profile(1, 0, 1, 0) yields contractible coarse and fine complexes while failing
both lower and upper Quillen-fiber tests;
16

PDF PAGE 17
ProLT Observation Refinement Classification note v0.2
(vi) all 255 one-bit profiles over nonempty subposets of 𝐵2 agree with the logical hierarchy reported in
Section 10.
References
[1] B. Theory, Propositional Logical Topology: Observation, Distinguishability, and Inference , core
manuscript v0.8, September 2026, unpublished.
[2] B. Theory, Homological Foundations for Propositional Logical Topologies, working research note v0.1,
September 2026, unpublished.
[3] M. C. McCord, “Singular homology groups and homotopy groups of finite topological spaces,” Duke
Mathematical Journal 33 (1966), 465–474.
[4] D. Quillen, “Homotopy properties of the poset of nontrivial 𝑝-subgroups of a group,” Advances in
Mathematics 28 (1978), 101–128.
[5] J. A. Barmak, “On Quillen’s Theorem A for posets,”Journal of Combinatorial Theory, Series A 118
(2011), 2445–2453.
[6] J. A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in
Mathematics 2032, Springer, 2011.
[7] M. Tancer, “Recognition of collapsible complexes is NP-complete,”Discrete & Computational Geometry
55 (2016), 21–38. The appendix discusses undecidability of contractibility for finite simplicial complexes.
[8] P. Gaucher, “T-homotopy and refinement of observation. III. Invariance of the branching and merging
homologies,” New York Journal of Mathematics 12 (2006), 319–348.
[9] P. Gaucher, “T-homotopy and refinement of observation. IV. Invariance of the underlying homotopy
type,” arXiv:math/0505331 (2005).
17