SOURCE AND PROOF LEDGER
Version 1.0 working release candidate, 9 November 2026

The numbered results below refer to addendum.tex, not the parent's numbering.
Mathematical validity, computational evidence, and novelty are separate claims.

C1. Relative two-term complex (Section 2).
Assumptions: finite poset; height counts strict inequalities and is <=2;
same-carrier coordinatewise two-bit thinning. All vertices survive.
Proof: order-complex dimension <=2 and C0(relative)=0. Orient by poset order.
Source relationship: inherited setup from Binary Order Thinning, Section 2.
Not a new claim of priority.

C2. Reduced signed-incidence form (Section 2).
Extra assumption: (C), no triangle loses all three edges.
Proof: every deleted triangle loses one or two edges; boundary coefficients
are +/-1. Two entries define sigma=-B_e B_f; singleton defines root attachment.
Same-sign Boolean words are exactly 10,01,10 and 01,10,01.
Classical algebra with model-specific local recognition. No inference that
an arbitrary signed graph is Boolean-realizable.

C3. Lemma 1: H1=Z^b plus (Z/2)^u; H2=Z^(m-r+b).
Assumptions: C1,C2. Components are internal two-entry graph components, each
with its own root if any singleton is attached; isolated rows count.
Proof: signed spanning-tree relations, root kills generator, negative cycle
imposes exactly 2g=0; free kernel and rank. Field ranks follow directly.
Primary antecedents: Zaslavsky 1982 Sections 8A-8B; Zaslavsky arXiv:1303.3083,
Sections IV.B-IV.C. Elementary integer proof included rather than imported
as a claimed new signed-graph theorem.

C4. Theorem 2: integer/F2/all-field acyclicity, rooted forest, collapse and
homotopy equivalence are equivalent.
Assumptions: C1,C2; actual inclusion L->K, not abstract endpoint equivalence.
Proof: C3 excludes rootless components; per-component edge count forces a
tree with one root attachment; nonroot leaf is an actual free deleted edge.
Retained triangles cannot contain deleted edges. Height excludes higher cofaces.
Empty relative pair handled explicitly. Known collapse mechanism; two-bit
restricted specialization is the addendum's extension of the parent result.
Closest collapse antecedents: Bernardi-Klivans and Contreras-Tawfeek Theorem4.2.

C5. Proposition 3: switching/TU and odd-field criterion.
Proof: switch along a spanning tree; unbalanced cycle supplies determinant
two minor. Singleton columns impose no row-switch constraint. Odd-field
acyclicity allows precisely rooted trees or rootless unbalanced unicycles.
Assumptions: C1,C2, and fixed field characteristic !=2 for last statement.
Classical signed algebra. A minor alone is not used to prove full torsion.

C6. Theorem 4: nine-state constrained realization, full SNF diag(1^8,2).
Input: exactly the 13 covers, 9 labels and all 10 triangles printed in Section4.
Proof: full 9x9 relative matrix, eight positive tree relations and one negative
cycle relation; mod2 quotient certifies nonzero generator. No missing singleton.
Computation: both exact_checks.py and independent_checks.py rebuild from input,
produce unimodular U,V, and verify UBV and determinants exactly. verify_package.py
checks both certificates and their agreement. No floating-point rank inference.
Contribution status: explicit realization within the Boolean rule and coefficient
consequence; not first discovery of signed incidence or Mobius-strip torsion.

C7. Absolute homology, surface, and inclusion degree (Section5).
Proof: printed free-pair sequences reduce both endpoints to connected unicyclic
graphs; pair long exact sequence forces multiplication +/-2. Independent
primitive cocycle evaluations give -2 in the recorded orientations.
Surface identification: all links are paths except vertex5 circle; unique
boundary cycle; connected compact surface chi0 with one boundary is Mobius.
Full matrix/collapse proof does not depend on surface recognition.
Classical antecedent: Contreras and Andrew R. Tawfeek, Example4.9 explicitly
uses a Mobius strip with its boundary and relative H1=Z/2.

C8. Higher-height warning and protocol examples (Section6).
Four-chain: displayed d3 kills ker d2 and printed collapse verifies it.
Four-state fan: explicit matrices and collapse pairs show block legality
does not imply legality of every execution order. verify_package.py reconstructs
all four matrices. No universal sequentialization theorem is claimed.
Nine-state individual bits: each 5x6 relative matrix has full-row minor +/-1,
so H1=0,H2=Z. Block and both first bits are illegal under the actual gate.

C9. Operational capacity interpretation (Section6; companion patches).
Assumptions: finite state set, noiseless signatures, fixed prior and target,
zero-one loss. Formula proved by independent fiberwise decoder optimization.
Anchor example: all quotient complexes are stars; projections contract new
leaf edges; uniform identity success is (number of fibers)/n. Tests isotone
on the anchor background and on every induced support.
Cost example: four equiprobable states, antichain background, named four-test
libraries and unit costs. Two bits attain depth2; singleton leaf depths
1,2,3,3 give expectation9/4 and worst3. General Bayes/decision-tree machinery
is not claimed new. Parent acquisition classification remains restricted to
its endpoint-antichain background, and its R subset E_* criterion is unchanged.

C10. Unresolved questions.
The addendum's existence and all-field conclusions rely on proof/full Smith
form, not enumeration. Nine is an upper bound, not a minimum. No exhaustive
minimum-size claim is made or needed. No connected minimum bound follows
from disjoint-union realization of (Z/2)^k.
Exclusive K2,2 Question19.1 belongs to a different interaction construction.
External priority and human peer review remain open. No empirical or
consciousness conclusion is made.

SOURCE VERSION CONTROL
source_ledger.json supplies URLs, precise roles, inspected version dates and
available hashes. Public files use relative names. Earlier source snapshots
and raw provenance remain in the local research record and are deliberately
not republished in this archive. Hashes identify bytes, not public priority.
