Currency Completeness: Sources of Relational Coherence

provisional

Overview

This derivation answers an accounting question left open by the interaction classification: where does relational coherence come from?

Three Interaction Types (Step 6) established the two-level reading of interaction-mediated correlation: Type I quanta are the currency that crosses between observers; the Type III relational invariant is the ledger entry left at the endpoints once the transfer clears. That remark asserted the reading for “most physically realized Type III correlations.” This derivation closes the gap between “most” and “all” by proving a completeness statement in source/redistribution form:

• Creation is currency. Every event that increases the relational coherence between observers is a Type I phase-transfer coupling — at zero range when the participants’ coherence domains overlap (contact), or decomposed as contact–transit–contact when a propagating quantum carries the phase between separated domains (mediation).
• Everything else is redistribution. Decay correlations, decoherence, entanglement swapping, and the physical transport of correlated observers move relational coherence around the network without net creation.
• Redistribution has finite ancestry. Every chain of inheritance terminates, in finitely many steps, at a currency event or at the primordial $t_0$ condensation — the pre-geometric regime where the interaction classification does not yet apply.

The engine of the proof is condition C5 (strong subadditivity) of Coherence Conservation. Under the entropy correspondence of Coherence as Physical Primitive, relational coherence is quantum mutual information, and C5 is the Lieb–Ruskai theorem; monotonicity of mutual information under local processing — the data-processing inequality — is equivalent to it. The axiom condition that forces the observer network to exist (Multiplicity Theorem 7.2) is the same condition that forbids the network’s correlations from arising without connection.

Why this matters. The completeness statement converts the bootstrap’s qualitative dependence on mediation into an exact source law: currency events are the only faucets of relational structure; all other processes are plumbing. Downstream, it generalizes Claim A of Spectral Complementarity from the spatially-separated case to all creation events, and it gives the ER=EPR channel picture a dynamical counterpart: every persistent channel is the geometric record of a currency ancestry.

An honest caveat. The core lemma (Step 1) is rigorous given the entropy correspondence. The locality and causal-propagation steps (Steps 2–3) rest on Coherence Lagrangian Theorem 6.0 (second-order locality from Axiom 3) and Speed of Light Proposition 4.2 (maximal signaling speed), the latter of which carries the speed-of-light S1 structural postulate. The page inherits that provisional status.

Statement

Theorem (Currency Completeness). In the post-geometric regime:

1. (No sources without coupling) Transformations that factor as product channels across the cut $\{\mathcal{O}_1\} \,|\, \{\mathcal{O}_2\}$ — each observer evolving with third parties that share no interaction with the other side — cannot increase the relational coherence $\mathcal{C}(\Sigma_1 : \Sigma_2)$.
2. (Creation vertices are phase transfer) Every elementary event that increases relational coherence between its direct participants is a local coupling at shared support whose transferable dynamical quantity is the conjugate phase — a Type I currency event, of which propagating mediation is the contact–transit–contact composition.
3. (Redistribution is source-free) Decay-product correlation, decoherence, entanglement swapping, and transport of correlated observers redistribute existing relational and binding coherence with exact accounting and no net creation.
4. (Well-founded ancestry) Every inheritance chain of relational coherence terminates in finitely many steps at a currency event or at the primordial $t_0$ condensation.

Corollary (Source law). Currency events are the only sources of relational coherence. Every Type III relational invariant in the post-geometric network is currency-built, up to redistribution of primordial content.

Derivation

Step 1: No Sources Without Coupling

Lemma 1.1 (Separable invariance and monotonicity). Let $\mathcal{O}_1, \mathcal{O}_2$ be observers on a Cauchy slice and let $R_1, R_2$ be disjoint sets of third-party observers with no interaction between $R_1 \cup \{\mathcal{O}_1\}$ and $R_2 \cup \{\mathcal{O}_2\}$ during the process considered. Then:

(a) A transformation factoring as $T_1 \times T_2$ with each $T_i$ an admissible automorphism of its own factor preserves $\mathcal{C}(\Sigma_1 : \Sigma_2)$ exactly.

(b) A transformation factoring as a product channel — $\mathcal{O}_1$ evolving jointly with $R_1$ only, $\mathcal{O}_2$ with $R_2$ only — cannot increase $\mathcal{C}(\Sigma_1 : \Sigma_2)$.

Proof. Under the correspondence of Coherence as Physical Primitive (Theorem 1.1 and the identification table), subsystem coherence is von Neumann entropy of the reduced state and relational coherence is quantum mutual information: $\mathcal{C}(\Sigma_1 : \Sigma_2) = S(\rho_1) + S(\rho_2) - S(\rho_{12}) = I(1:2)$.

(a) A factorized automorphism acts on the joint state as a product of automorphisms; each of $S(\rho_1)$, $S(\rho_2)$, $S(\rho_{12})$ is invariant, hence so is $I(1:2)$.

(b) Tracing out the third parties, the pair’s reduced evolution is a product channel $\Phi_1 \otimes \Phi_2$. Mutual information is monotone under local channels:

$I\big((\Phi_1 \otimes \Phi_2)\,\rho_{12}\big) \leq I(\rho_{12})$

This is the data-processing inequality, which is equivalent to strong subadditivity of entropy [Lieb & Ruskai, 1973] — condition (C5) of Coherence Conservation under the correspondence (Coherence as Physical Primitive, condition (C5) of Theorem 1.1). $\square$

Remark 1.2 (C5 does double duty). The condition that forces the observer network to exist — C5 non-triviality requires at least three observers and ultimately a full network (Multiplicity Theorem 7.2, Step 7) — is the same condition that forbids relational coherence from arising between unconnected parts of that network. One axiom condition supplies both the demand for correlation and the constraint on how correlation can be supplied.

Remark 1.3 (No LOCC loophole). In quantum information, “local operations and classical communication” is the standard class of correlation-non-creating maps, with classical communication granted as a free resource. The framework grants nothing for free: a classical signal between the wings is itself carried along a directed path of the interaction graph by physical carriers (Speed of Light Proposition 4.2). Classical communication is currency traffic, so the framework statement is cleaner than LOCC: any channel between the wings — quantum or classical — falls on the coupling side of Lemma 1.1, and the source-free class is exactly the product-channel class.

Step 2: Creation Vertices Are Phase-Transfer Events

Proposition 2.1 (Elementary creation is local phase coupling). Any elementary interaction event that increases the relational coherence of its direct participants is (i) non-separable, (ii) local — the participants’ coherence domains share support at the event — and (iii) a transfer of conjugate phase.

Proof. (i) is Lemma 1.1(a): separable dynamics preserves relational coherence, so a creation event is non-separable (condition I1 of Three Interaction Types Definition 1.1 is necessary, not merely definitional). (ii) The dynamics is generated by the coherence action, whose field equations are second-order and local (Coherence Lagrangian Theorem 6.0); a coupling term exists only where the participating degrees of freedom share support. (iii) Through a Type III event both participants’ invariants persist (Three Interaction Types Definition 4.4). While the invariants persist, the only transferable dynamical quantity is the phase conjugate to each Noether charge — the argument of Three Interaction Types Proposition 4.2 applied at the kinematic level of the coupling. The relational invariant registered by the event is a functional of the joint phase history. $\square$

Remark 2.2 (The ledger entry is built from currency). The framework’s own consistency model realizes Proposition 2.1(iii) literally: the Type III model invariant on $S^1 \times S^1$ is $I_{12} = \cos(\theta_1 - \theta_2)$ (Three Interaction Types Theorem 9.1) — a function of nothing but the phases whose transfer is the Type I mechanism. Currency is not merely the trigger of the ledger entry; it is the raw material the entry is written in.

Remark 2.3 (Contact and mediation are one mechanism at two transit lengths). Propagating mediation decomposes as contact at the emitter, Type I transit, contact at the absorber — the three-layer pattern of the atom-emits-photon worked example in Three Interaction Types Step 6. Zero-range creation (overlapping domains) is the same phase coupling with the transit segment absent. The standard-physics counterpart of this identification: apparent contact interactions in effective theories resolve into mediation at finer resolution (the Fermi four-fermion vertex resolves into $W$ exchange), never the reverse.

Step 3: Long-Range Creation Requires Connected Chains

Theorem 3.1 (Chain requirement). Let $\mathcal{O}_1, \mathcal{O}_2$ have disjoint coherence domains. Any process that increases $\mathcal{C}(\Sigma_1 : \Sigma_2)$ contains a causally connected chain of currency events linking the two domains: a sequence of creation vertices (Proposition 2.1) joined by carrier segments, each segment either a Type I quantum in transit or a ledgered observer in transport.

Proof. By Lemma 1.1(b), the increase is impossible if the third-party structure splits into disjoint halves $R_1, R_2$ with no interaction across the split. Therefore every partition of the participating observers separating $\mathcal{O}_1$ from $\mathcal{O}_2$ is crossed by at least one interaction — the participation graph contains a path from $\mathcal{O}_1$ to $\mathcal{O}_2$. Each edge of that path is an interaction event, which by Proposition 2.1 is a local phase coupling at shared support; each traversal between consecutive event sites is carried by something physical moving within the causal cones, since influences propagate only along directed paths of the interaction graph at speeds $\leq c$ (Speed of Light Proposition 4.2). The carriers are of exactly two kinds: off-ledger Type I quanta (Loop Closure, Region 1: null-boundary phase-transfer carriers; massive short-range channels per Observer Pattern Signal) or ledgered observers physically transported between vertices. $\square$

Remark 3.2 (Transport is not creation). A messenger observer that interacts with $\mathcal{O}_1$ at contact, travels, and interacts with $\mathcal{O}_2$ at contact does not create $\mathcal{O}_1$–$\mathcal{O}_2$ relational coherence during transit: free propagation is separable relative to the wings (Lemma 1.1), so the transit segment only transports the correlation established at the first vertex, and the final $\mathcal{O}_1$–$\mathcal{O}_2$ invariant is assembled by redistribution (Step 4) from the vertex-created pairs $\mathcal{O}_1$–messenger and messenger–$\mathcal{O}_2$. This is the structure of engineered long-range entanglement: quantum repeaters and heralded atom–atom entanglement build remote correlations from chains of local creation vertices, photon transits, and joint-detection redistribution — never from action at a distance. Creation happens only at vertices; distance is covered by carriers.

Step 4: Redistribution Is Source-Free

Proposition 4.1 (The redistribution class creates nothing). The following processes change the location of relational coherence with exact accounting and no net creation:

(i) Decay correlation. The mutual correlation of decay products is inherited: the binding coherence of the composite is released as product relational coherence plus emitted radiation, with $\mathcal{C}(\Sigma_{12}) = \mathcal{C}(\Sigma_1 \cup \Sigma_2) + \mathcal{C}(\text{emitted})$ (Three Interaction Types Proposition 7.3). The products’ correlation is the composite’s internal coherence relabeled by the split, not new relational structure.

(ii) Decoherence. Pairwise relational coherence delocalizes into many-body correlation with third parties, total conserved (Three Interaction Types Proposition 7.5).

(iii) Swapping. A joint operation at a middle node converts $\mathcal{O}_1$–$M$ and $M$–$\mathcal{O}_2$ correlations into $\mathcal{O}_1$–$\mathcal{O}_2$ correlation. By Lemma 1.1(b) applied to each wing, the created pairwise coherence is bounded by what the consumed pairs supplied; the swap is a change of bookkeeping basis, not a source.

Proof. (i) and (ii) are the cited coherence-accounting propositions of Three Interaction Types. (iii): the swap operation is local at the middle node — a product channel relative to the cut $\{\mathcal{O}_1\} \,|\, \{\mathcal{O}_2\}$ — so it cannot increase $\mathcal{C}(\Sigma_1 : \Sigma_2)$ beyond what the pre-existing chain correlations already carried through the node (Lemma 1.1(b)). $\square$

Proposition 4.2 (Composite formation is currency-coupled). Every Type II fusion event with nonzero binding excess is itself a currency event: the excess must be disposed of by Type I transfer.

Proof. Coherence conservation runs the decay accounting of Three Interaction Types Proposition 7.3 in reverse (Step 7 of that derivation: every forward process has a well-defined reverse with the same accounting): at fusion, $\mathcal{C}(\Sigma_1 \cup \Sigma_2) = \mathcal{C}(\Sigma_{12}) + \mathcal{C}(\text{emitted})$. A positive binding excess must leave the fusing pair — as emitted radiation (radiative capture: hydrogen formation emits the 13.6 eV binding quantum) or as phase transfer to a third body (three-body recombination). A negative excess (endothermic fusion) must be supplied by the same channels in reverse. In every case the fusion event includes Type I transfer with the rest of the slice. $\square$

Corollary 4.3 (Inheritance bottoms out in currency). The binding coherence that funds decay-product correlations (Proposition 4.1(i)) was itself balanced through currency at the composite’s formation (Proposition 4.2). Inherited correlation is deferred currency accounting, not an independent source.

Step 5: Ancestry Is Well-Founded

Proposition 5.1 (Finite inheritance chains). Every chain of inheritance — each step deriving a correlation from a prior redistribution event — terminates in finitely many steps.

Proof. Each inheritance step references a strictly earlier event: the redistribution event lies in the causal past of the correlation it explains. The network’s causal order is realized as a locally finite partial order — for any $x \preceq z$, the interval $\{y : x \preceq y \preceq z\}$ is finite (Causal Set Statistics Definition 1.1, with local finiteness derived from the holographic bound in Proposition 1.2(ii)). All events of a chain explaining a correlation at event $e$ lie in the interval between the condensation slice — every observer’s $t_0$ (Multiplicity Corollary 7.4) — and $e$. A strictly descending chain in a finite interval is finite. Independently, each redistribution step moves at least one $\hbar\omega_0$ quantum of the integer ledger (Bootstrap Corollary 2.3), and the ledger capacity of any bounded region is finite (Area Scaling). $\square$

Proposition 5.2 (Terminal characterization). A maximal inheritance chain terminates at exactly one of: (i) a currency event (Proposition 2.1 vertex, possibly within a chain per Theorem 3.1, or a fusion disposal per Proposition 4.2); or (ii) the primordial $t_0$ condensation.

Proof. By Propositions 4.1–4.3, every redistribution step defers to prior relational or binding coherence; by Proposition 5.1 the deferral cannot regress indefinitely. A terminal event either creates relational coherence — and is then a currency event by Proposition 2.1 — or is not an interaction at all. The only correlation-bearing structure not produced by any interaction is the primordial pairwise structure of the simultaneous condensation, which is a topological transition, not a classified interaction (Multiplicity, remark following Corollary 7.4: ”$t_0$ is not a Type III interaction”). That regime is pre-geometric — observers have winding numbers but no metric properties (Bootstrap, simultaneous-condensation remark) — so the interaction classification, and with it the currency/anchor distinction, does not apply there. $\square$

Step 6: The Completeness Theorem

Theorem 6.1 (Currency Completeness). Every unit of relational coherence in the post-geometric network was created at a currency event — a Type I phase-transfer coupling, zero-range or carried — and reached its present location by transport and redistribution; every inheritance chain terminates at a currency event or at the primordial condensation.

Proof. Assemble Steps 1–5. Creation requires non-separable coupling (Lemma 1.1); every creation vertex is local phase transfer (Proposition 2.1); long-range creation is a connected chain of such vertices (Theorem 3.1); all non-creating processes are the redistribution class with exact accounting (Propositions 4.1–4.3); ancestry is finite with the stated terminals (Propositions 5.1–5.2). $\square$

Corollary 6.2 (The currency sector never idles). A bootstrap network requires permanently ongoing currency events.

Proof. Sustaining mutual definition requires ongoing Type III structure (Bootstrap Theorem 3.1), and the generated structure irreversibly expands (Relational Invariants Proposition 6.2) while decoherence continually redistributes existing pairwise coherence into the background network (Three Interaction Types Proposition 7.5). Maintaining any pair’s mutual definition against that redistribution requires ongoing creation, and by Theorem 6.1 all creation is currency. A network epoch with no active currency events is an epoch in which relational coherence only delocalizes — mutual definition degrades and the multiplicity requirement fails. $\square$

Consistency Models

Each part of Theorem 6.1 has an exact counterpart in established physics:

Remark (Elementary vertex structure). The Standard Model contains no fundamental vertex with only fermion legs: every elementary interaction vertex includes at least one integer-winding field. Under Theorem 6.1 this is the expected shape of a currency-complete interaction structure — every creation vertex has a currency leg — but the framework does not yet derive the exclusion of fundamental fermion-only vertices from its own Lagrangian machinery (Open Gap 1).

Remark (Exchange symmetrization is not relational coherence). The (anti)symmetrization of far-separated identical observers is winding-class topology (Spin and Statistics Proposition 4.1) — kinematic structure of the configuration space, carrying zero mutual information in the proper mode accounting. It is outside the scope of Theorem 6.1 and requires no currency ancestry.

Remark (Gravitational channel interface). Whether the emergent-geometry channel can itself create Type III invariants is an open interface question between this theorem and the Einstein-equations chain. If gravitationally mediated entanglement exists (Bose–Marletto–Vedral-type experiments), Theorem 6.1 requires the geometric channel to be a genuine currency channel — a quantum phase carrier. If the framework’s bookkeeping reading of gravity (Einstein Equations) restricts the channel to Type I accounting with no invariant generation, the prediction is a null result. Either experimental outcome sharpens the framework’s commitment; the framework does not yet fix which.

Consequences

C1. The three-types currency remark is upgraded to completeness. “Most physically realized Type III correlations are produced through Type I-mediated traffic” (Three Interaction Types Step 6) becomes: currency events are the only sources; everything else redistributes, with finite ancestry.

C2. Spectral complementarity’s Claim A is general. The currency-necessity direction of Spectral Complementarity no longer carries a spatial-separation scope restriction: contact creation is the zero-transit case of the same mechanism, and inherited correlation is deferred currency accounting.

C3. C5 unifies existence and connectivity. The condition that forces the network into existence (multiplicity) also forces the network’s correlation structure to be built along connected chains of interaction (Remark 1.2). The completeness of the $K_3$ relational structure required at the smallest scale (Minimum Bootstrap Closure) is achieved, post-geometrically, only through currency traffic.

C4. ER=EPR consonance. The persistent channel $\gamma_{12}$ that carries a relational invariant (ER=EPR Proposition 1.2c) is the geometric record of the invariant’s currency ancestry: the chain of vertices and carriers of Theorem 3.1 is the dynamical history of which the channel is the standing geometry.

C5. No currency-free epoch. Corollary 6.2 forbids any epoch of a bootstrap network in which the currency sector is inactive. Combined with Spectral Complementarity Claim B (currency is integer-winding), no epoch of any bootstrap network is purely fermionic.

Rigor Assessment

Rigorous given the entropy correspondence:

• Lemma 1.1 (separable invariance and monotonicity): standard invariance of mutual information under local automorphisms; data-processing inequality equivalent to strong subadditivity, which is C5 under Coherence as Physical Primitive Theorem 1.1
• Proposition 4.1 (redistribution class): coherence accounting of Three Interaction Types Propositions 7.3 and 7.5, plus Lemma 1.1(b) for swapping
• Proposition 5.1 (finite chains): causal-set local finiteness (Causal Set Statistics Proposition 1.2) plus the integer ledger

Semi-formal:

• Proposition 2.1 (creation vertices are phase transfer): combines Lemma 1.1 with Coherence Lagrangian Theorem 6.0 locality and the Proposition 4.2 phase-uniqueness argument of Three Interaction Types; the application of the phase-uniqueness argument at the kinematic level of a Type III coupling is structural rather than fully formal
• Theorem 3.1 (chain requirement): the partition-crossing argument is rigorous given Lemma 1.1; the carrier classification (Type I quanta or transported observers, exhaustively) rests on the framework’s carrier taxonomy
• Proposition 4.2 (fusion disposal): time-reversal of the decay accounting; the genericity of nonzero binding excess is physical rather than proven

Provisional inheritance:

• Speed of Light Proposition 4.2 (maximal signaling speed) carries structural postulate speed-of-light-S1 (smooth ambient geometry). Steps 3 and 5 inherit that status; promotion of S1 (via Continuous-Discrete Duality Conjecture 4.1) would upgrade them.

Assessment: The source/redistribution decomposition is the load-bearing structure, and its core (Lemma 1.1) reduces to C5 via the entropy correspondence — the same reduction the framework uses to ground the coherence axioms themselves. The locality and causal-propagation inputs are the framework’s own derived results, inherited with their existing statuses. The completeness claim is therefore as strong as the framework’s spacetime chain, no stronger.

Open Gaps

1. Fermion-only vertex exclusion. Derive from the Coherence Lagrangian’s exclusion machinery (Theorem 6.0 and its extensions) that no fundamental creation vertex couples half-integer-winding fields alone — making the integer-winding leg at every currency vertex a theorem rather than a consistency observation, and grounding Spectral Complementarity Claim B at the vertex level. Difficulty: MODERATE.
2. Interaction-graph formalization. Recast Theorem 6.1 graph-theoretically: decorate the causal set with currency events and prove that relational-coherence increase is supported exactly on the connected components of the currency-event graph. Difficulty: MODERATE.
3. Quantitative source law. Promote the completeness statement to a continuity equation: relational coherence density with currency flux as its source term, with rates per vertex fixed by the Born Rule and the coupling constants. Difficulty: HARD.
4. Gravitational channel commitment. Determine whether the framework’s emergent-geometry channel generates Type III invariants (predicting positive Bose–Marletto–Vedral-type results) or is restricted to Type I bookkeeping (predicting null results). Formulating the commitment either way is a prerequisite for treating BMV-class experiments as framework tests. Difficulty: MODERATE.