A global lift exists exactly when every cycle closes with product +1.
The signed-graph kernel is classical balance. The companion use is the weld: local one-bit choices can be joined into one global choice only when their admitted transitions are jointly coherent.
For a finite signed coherence datum G=(V,E,∂,σ), there exists a vertex-sign assignment s:V→{−1,+1} satisfying σ(e)=s(u)s(v) on every edge if and only if the transition product around every cycle is +1.
If a coherent datum has c connected components, it has exactly 2^c lifts. If coherent bridges reduce the component count to c′, they remove exactly c−c′ independent selector bits.
global lift ⇔ every cycle positive |L(G)| = 2^c
An exact geometric derivation returning sign 0 does not become a graph edge. It returns the typed state DERIVATION_DEGENERATE. The instrument therefore separates degeneracy, incoherence and invalid input rather than forcing all three through one binary verdict.
Classical provenance: Harary balance (1953–1954), with earlier equivalent ideas in Kőnig (1936). The theorem is reproved in the carried treatment; graph-theoretic priority is not claimed.
selector reduction
Many local choices can collapse to one global choice.
Each disconnected local system contributes one free whole-component sign. Coherent bridge relations join components without inventing a preferred absolute orientation.
Before bridging
After coherent bridging
obstruction / anchor
A negative cycle blocks the weld; an anchor acts only after coherence.
Flip one bridge sign. The cycle product changes exactly. If coherence survives, one supplied anchor fixes the whole connected component.
Put your own signed relation network through it.
Vertex and relation identifiers are strings. Supplied graph signs are literal integer +1 or −1; derived candidates may return −1, 0 or +1 before admission. Duplicate JSON fields, duplicate identifiers, Boolean aliases, decimal/exponent sign spellings, undeclared endpoints and self-loops are refused rather than coerced.
Exact trace / witness
Not evaluated.
Receipt
No receipt yet.
The finite theorem has no 256/4096 bound. Those limits are a browser resource envelope for this single-file implementation and are not promoted into the mathematics.
What this frame permits.
The instrument answers questions that factor through the exact signed datum it receives. Provenance records where a relation says it came from; provenance does not replace the algebra or authenticate an external physical interpretation.
Global signed-network coherenceEstablished
| target | existence of one global ±1 lift |
|---|---|
| frame | finite signed multigraph |
| result | coherent lift / exact negative-cycle witness |
| warrant | Theorem 4.1, classical signed-graph balance, proved in the carried treatment |
Selector reduction under coherent bridgingEstablished
| target | remaining independent global sign choices |
|---|---|
| frame | connected-component structure before and after admitted coherent edges |
| result | c → c′ removes exactly c−c′ selector bits |
| warrant | Corollary 7.4 / componentwise lift count |
Local common-lift verificationEstablished
| target | whether one declared local block admits a common lift |
|---|---|
| frame | the block's induced signed subgraph |
| result | same coherence predicate, applied locally |
| bridge | partition meaning and external provenance remain separate admission questions |
Exact geometric sign zeroTyped boundary
| target | whether a candidate defines a binary edge |
|---|---|
| frame | tri-valued exact derivation result −1,0,+1 |
| result | 0 → DERIVATION_DEGENERATE; no edge emitted |
| bridge | omitting that candidate requires a separately scoped resubmission |
External relation authenticity / physical identityOutside frame
| target | whether a supplied bridge is physically or semantically true |
|---|---|
| result | not inferred from algebra, provenance label or hash |
| bridge | requires an external warrant or separately verified derivation module |
Distance to coherence / uniquely culpable edgeSeparate problem
| target | minimum repair or unique fault localization |
|---|---|
| result | not returned by this instrument |
| warrant | frustration-index optimization is NP-hard in general; minimum repair sets need not be unique |
A successful weld establishes compatibility of the admitted relations. It does not prove completeness of an unsupplied larger carrier, manifold orientability, a physical observation mechanism, or a uniquely faulty relation when the weld fails.
and sources
Check it rather than trust it.
The carried conformance checks, frozen compiled treatment and closed design specification are embedded in this HTML. Nothing loads from the network.
Browser conformance
Runs 50 contract checks spanning empty/isolated cases, positive and negative cycles, parallel edges, anchors, exact-zero precedence, provenance boundaries, metamorphic invariance and selector counts.
Not run in this session.
Carried-document integrity
Recomputes byte counts and SHA-256 for the compiled treatment and closed specification embedded below.
Not checked in this session.
Documents
Carried artifacts
Astra Companion I — coherence and global orientation transport. Lu Semita, EmergenceByDesign.
Exact finite sign arithmetic throughout. The diagrams are illustrations of records decided algebraically by the embedded engine. This instrument is a standalone companion to Astra 15A; it does not alter Astra 15A's closed claim.