SOURCE AND PROOF LEDGER
Integral secondary maps in an observation four-cycle, v1.0, 9 November 2026

Fixed parent dependency
Brian Theory, Successive Concept Differentiation as Observation Refinement:
Carrier Splitting, Interaction-Core Homology, and Descent.
Referee-repair draft v0.5, 29 September 2026.
https://www.b-theory.com/manuscripts-html/observation-refinement/
https://www.b-theory.com/manuscripts-html/observation-refinement/source.zip
Source archive SHA256:
00362687508ba054c8d08abac13fe33a49e27ac4b4cd9fc28421bcdfc40186cb
The definitions, signs and original question are matched to this fixed version.
The current public index was read on 9 November 2026 and still describes v0.5.
The web reader could not retrieve the live manuscript URL during closeout; the
fixed archived source, not an assumed live revision, was used for proof/insertion
alignment. No claim is made that the archived source matches every live byte today.

Claim A: model and interaction complex (Section 2)
Assumptions: finite Q; nonempty allowed Boolean bits at every state; induced
available subposet; integral augmented link chains; actual product order.
Inherited: parent two-colour missing-copy model, interaction filtration/signs.
Proof supplied: relative simplex basis, signed differential, relative
Mayer-Vietoris/cofiber degree shift. No novel general spectral-sequence claim.
Classical antecedents: Hatcher Chapter 2; Cianci-Ottina (2017), general poset
filtration spectral sequences; CJST (2021), obstruction complexes.
Fresh finite check: all nine boundaries and full relative entries for five cases.

Claim B: local-anchor lemma (Lemma 3.1)
Assumptions: b minimal, x maximal in R, b<x; a<R<y; only a'<b and x<y' added;
availability P={a,b}, Z={x,y}, others split.
Proof supplied: literal equality of all available links and inclusions to capped
diagram; explicit order retractions, not merely acyclicity.
Classical antecedent: Barmak-Minian (2008), Sections 2.1-2.3, order homotopies
and beat-point contractions. The availability-sensitive application is the
candidate specialization. Retractions do not preserve availability.
Fresh finite check: full generated coarse order and contraction maps for five cases.

Claim C: arbitrary k (Theorem 4.1)
Assumptions: k>=2; exactly the facets (5); b={w}, x={w,u0,u1}; Lemma 3.1 profile.
Proof supplied: mapping cylinder of u_j -> v_(j mod3), capped domain; puncture
and face deletion separately justified; all nine links; actual pair (T,Y) and
integral H3(T,Y)->H1(V) isomorphism fix primitive source; cokernel Z/k forces
d2=+/-k. Only one final filtration grade, hence no extension ambiguity.
Classical antecedent: Moore-space construction (Hatcher Example 2.40),
suspension and pair sequences. No new abstract torsion mechanism claimed.
Counts: 24k+11 coarse states, 48k+18 worlds. Refined HOMOLOGY Z/k in degree2;
no refined homotopy classification asserted.
Fresh finite checks: full q1 cases k2..5 plus exceptional RP2; separate hostile
seed Smith check k2..9. The universal proof is not inferred from these instances.

Claim D: exceptional compact k=2 (Section 5)
Assumptions: exact ten listed RP2 facets, b=0, x=012, same local-anchor profile.
Seed is inherited from the parent manuscript's star example.
Counts: 35 coarse states / 66 worlds, different from uniform k2's 59 / 114.
Fresh finite checks: generated order, nine links, signed source/fillers/primitive
dual, relative integer reduction and direct refined integer reduction.
Retained independent raw Smith checks from the earlier research provide a
different implementation; they are not described as a new closeout computation.

Claim E: lowest-degree obstruction (Theorem 6.1)
Assumptions: ENTIRE exclusive graph is one K2,2 plus optional isolated
exclusive vertices; E2_(4,0) nonzero; no torsion-free higher-link assumption.
Proof supplied: product-link join forces saturated marked chain; actual
crossing cochain identifies all d1 lattices and component quotient; secondary
image zero or primitive. Target can have any finite free rank.
Fresh checks: four explicit q0 cases; actual links/boundaries/graph-cycle d1
images reconstructed from recorded strict orders, not regenerated orders.
Not asserted: larger exclusive component containing K2,2, multiple nontrivial
components, minimum state count, or torsion-freeness in all cofiber degrees.

Overlap and priority
Parent paper already contains unit d2, larger-star torsion and classical tools.
Binary Order Thinning treats a different same-carrier one-bit framework; a
separate two-bit/9-state addendum is outside this package.
ProLT concerns observational indistinguishability; predictive/RF-IDENT and
reflective-model branches add operational tasks absent here. This addendum
proves no cognitive mechanism, task improvement or architecture advantage.
Dey-Hirani-Krishnamoorthy (2011), Section4.2 and Theorem5.2, already distinguish
absolute and relative torsion / total unimodularity. CJST (2021) Definition5.1,
Corollary7.2 and Theorem8.6 supply broader obstruction-link antecedents.
Cianci-Ottina (2017) is acknowledged for prior poset spectral-sequence machinery.
Targeted searches were limited and sometimes noisy. No exact conflicting
specialization was identified, but absence of a hit does not establish priority.
Human specialist review and specialist literature judgment remain recommended.

Primary URLs read or checked during closeout (9 November 2026)
https://www.b-theory.com/papers/
https://pi.math.cornell.edu/~hatcher/AT/ATch2.pdf
https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak3.pdf
https://link.springer.com/article/10.1007/s41468-021-00066-2
https://arxiv.org/html/1001.0338v3
Cianci-Ottina publisher page was read in the hostile pass; direct DOI opening
in the main pass failed. DOI: 10.1016/j.topol.2016.12.001.
