PRIORITY FILING - CONSOLIDATED EDITION
A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
July 2026
Part of the formal foundations of Origin Chain of Thought (OCOT).
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
The books had advanced the canonical object to a six-facet Beier Tensor carrying a temporal, disposition, and composition metadata layer, while the formal corpus specified a five-field record with the extended price written T = (p, D, w). The seven papers in this volume bring the formal object level with the one the books use, defining the six-facet Tensor and building the metadata layer around it, ordered so that each result rests on the object the one before it established.
The keystone separates the record from the price it extends, writes the extended price as E = (p, T, w), carries the basis from five facets to six with Transmit established as independent, and recovers the published five-field treatment as the metadata-suppressed limit, so nothing already on Zenodo is overturned by what follows. The six papers after it build the metadata layer: strike-time against transmit-time, where a single timestamp cannot hold honest scoring and author delay at once and a tamper-evident strike-time is necessary and sufficient; the Transmit facet's scope test and the anti-enclosure result that removing the public option makes the commons enclosable; typed nulls, under which structured absence carries more than silence; the substrate marker that the self-monitoring gap makes necessary; the engagement level, scored effort-blind and read effort-aware; and the composition algebra whose independence requirements are the floor under emergence detection and the closure of laundering routes.
One outward connection deserves stating at the head of the set. The journal-facing companion to this work argues that every quantitative series carries an unreported property, the share of its variance that is coordination-dependent, and that producers of data owe their downstream users a report of it. The Tensor specified here is the record built to carry that report, and the Beier Protocol is the instrument that generates it. The obligation argued in the abstract is discharged by the object defined in these pages.
The published formal definition of the Beier Tensor (Beier, 2026z) treats the record as a triple T = (p, D, w) with the disclosure D living in a five-dimensional space spanned by Input, Method, Output, Identity, and Scaling. The canonical object has since moved past that definition. As specified in Good Judgment (Beier, 2026) and The Future Transparent (Beier, 2026), a Beier Tensor is a Nucleus wrapped in six facets - Identity, Input, Method, Output, Scaling, and Transmit - carrying two timestamps (strike-time and transmit-time), an engagement level, a substrate-type marker, a composition history, and an interface marker. The mathematics of 2026z describes a record the architecture no longer uses. This paper supplies the corrected formal object. The six facets span an information space V = R^6, with the Transmit coordinate carrying the disposition that governs who may read the record. The metadata sits as a fiber over each point of V: the two timestamps form a temporal fiber whose lag coordinate is itself informative, and the composition history indexes a directed acyclic graph of citations. The track record weight is a positive semidefinite bilinear form on V, now field-specific by construction. Four results follow. Proposition BT6-1 establishes that the Transmit coordinate is linearly independent of the other five, so the correct channel count is six. Proposition BT6-2 establishes that proper scoring is well-defined only on the strike-time section and that the strike-to-transmit lag is an admissible and strictly informative conditioning variable for a reader. Proposition BT6-3 establishes that the set of valid Tensors is closed under composition and that the induced lineage structure is acyclic, giving the algebra on which provenance tracking rests. Proposition BT6-4 generalizes the Blackwell dominance of 2026z to the six-facet extended price E = (p, T, w) and strengthens it for readers whose decisions depend on timing or disposition. The five-field tensor of 2026z is recovered as the metadata-suppressed, Transmit-constant limit, consistent with the inclusion principle that every prior coordination object survives as a special case of the next.
Keywords: Beier Tensor, extended price, disposition, strike-time, transmit-time, composition algebra, track record metric, information geometry, OCOT
A scalar price p in R carries one number, and every reader receives the same number. The Beier Tensor was introduced to carry what the number cannot. Beier (2026z) gave the first geometric account: a five-field disclosure as a vector, a track record weight as a metric, and a transformation law under change of analytical framework. That account was correct for the object as it stood in March 2026.
The object grew. Two structural additions arrived after 2026z was filed, and both are now load-bearing in the books that specify the architecture for practitioners. The first is the Transmit facet, which fixes the disposition of the record: private, shared with a named group, public, or public after a stated delay. The second is the temporal pair, a strike-time marking when the cognitive act occurred and a transmit-time marking when the record became readable, with the strike-time carrying the scored claim and the transmit-time never preceding it. Around these, four metadata coordinates were added: an engagement level recording the depth of cognitive work, a substrate-type marker recording whether the reasoning originated from a person, a model, or a combination, a composition history recording which earlier records this one builds on, and an interface marker recording which tool produced it.
None of this appears in 2026z. A reader who takes the published formal paper as the definition of the object will reason about a five-coordinate vector with a single timestamp implied and no disposition, and downstream papers that cite 2026z's metric, transformation law, and welfare dominance are citing results stated for the wrong coordinates. The present paper defines the six-facet Tensor, locates the metadata as fiber structure over the facet space, and re-derives the three results of 2026z on the corrected object, with one additional result for each new piece of structure.
Write a Beier Tensor as the tuple
T = (N, h, tau_s, tau_t, e, s, chi, iota).
The components are as follows. N is the Nucleus: the position or decision point under judgment, carrying the scalar price p in R when the Tensor is attached to a market quantity, and otherwise carrying the decision the judgment addresses. The hexagon h = (h_I, h_In, h_M, h_O, h_S, h_T) in R^6 holds the six facet coordinates, in canonical order Identity, Input, Method, Output, Scaling, Transmit. The scalars tau_s and tau_t in R are the strike-time and the transmit-time, constrained by tau_t >= tau_s. The engagement level e lies in an ordered set E (a graded scale from a single-glance assessment to a multi-month study). The substrate type s lies in a finite set Sigma = {human, model, hybrid}. The composition history chi is a finite set of identifiers of parent Tensors. The interface marker iota lies in a finite set of tool labels.
A facet coordinate may be filled or may carry a typed null. A typed null is not a zero. It records a specific kind of absence: not applicable, unknown to the author, withheld under the record's disposition, or carried by the author's default view rather than stated here (Beier, 2026, Good Judgment, Ch 11). The information content of these null types is the subject of a companion paper; for the present formalization a typed null is a labeled element of the facet's coordinate alphabet, distinct from any filled value and distinct from the other null types.
The extended price is the triple
E = (p, T, w),
where p in R is the scalar drawn from the Nucleus, T is the Tensor, and w is the originator's track record weight. The published paper wrote the tensor itself as (p, D, w), which folded the price, the record, and the weight into one symbol. Separating them is the first correction: T is the record, p is the number the record extends, and E is the pair bound together with the weight that tells a reader how much to lean on it. The Tensor extends the price. It does not replace it.
Definition 1 (Facet space). The six facets span the vector space V = R^6 with basis vectors e_1, ..., e_6 corresponding to Identity, Input, Method, Output, Scaling, Transmit. A hexagon h is the coordinate vector of a disclosure in this basis. The dimension is the number of independent information channels the record carries, which Section 4 fixes at six.
This corrects Definition 1 of 2026z, which set K = 5. The added basis vector e_6 is the Transmit coordinate. It does not measure the content of the assessment; it measures the disposition the author selected for that content, and Section 4 shows the disposition is not recoverable from the other five.
Definition 2 (Analytical framework). An analytical framework F is a linear map M_F from V to a decision-relevant representation space. A fund manager's framework extracts the components that bear on investment risk; a regulator's framework extracts the components that bear on compliance exposure; a hydrologist's framework extracts the components that bear on constraint trajectory. Each reader applies a different M_F to the same hexagon and recovers a different projection.
The metadata coordinates do not transform under M_F. A strike-time is the same instant whoever reads it; a substrate type is the same label; a lag is the same duration. The facets are the part of the record that different frameworks read differently. The metadata is the part every framework reads identically. This split is cleaner than the treatment in 2026z, which had no metadata to place.
Definition 3 (Track record metric). Let the originator carry a diagonal track record weight W = diag(w_I, w_In, w_M, w_O, w_S, w_T), with each w_j in [0, 1] determined by the originator's cumulative proper-scoring performance on facet j (Beier, 2026r, Propositions 5 through 7). The metric on the facet space is the bilinear form
g_W(h, h') = h^T W h'.
The form is positive semidefinite and framework-invariant: under a change of framework the hexagon and the metric transform so that g_W(h, h') is unchanged, which means two readers compute the same informational agreement between two records even though each reads the records through a different projection.
The diagonal weight is no longer an optional refinement. The published paper carried a scalar w and noted in a remark that a field-specific diagonal would be more faithful. With six facets, one of which is a disposition and not an assessment, the isotropic assumption is untenable: an author can be well calibrated on Output and poorly calibrated on Scaling, and the Transmit coordinate is not scored against an outcome at all, so its diagonal entry is fixed by convention rather than by performance. The field-specific form is the natural object here, and the scalar w of 2026z is the special case W = w I with the Transmit entry suppressed.
The first question a reader of the corrected object should ask is whether the sixth facet earns its place. If the disposition were a deterministic function of the other five, then Transmit would add no channel and the five-field count of 2026z would have been right after all.
Proposition BT6-1 (Transmit independence). Consider the cross-section of Tensors produced by a population of authors. If the disposition an author selects is ever sensitive to private constraints not encoded in the Identity, Input, Method, Output, and Scaling facets, then the conditional mutual information between the Transmit coordinate and the author's full information set, given the other five facets, is strictly positive. The Transmit basis vector e_6 is therefore not in the span of e_1 through e_5, and the channel count is six.
Proof sketch. Construct two Tensors that agree on Identity, Input, Method, Output, and Scaling but differ in Transmit: the same assessment of the same constraint by the same author, published in one case and held private under disposition in the other. The two records exist whenever an author holding a position with competitive sensitivity assesses that position - the assessment is identical to the one the author would publish, and the choice to seal it is driven by the sensitivity, which is private and is carried in none of the first five facets. The disposition thus separates two records that the first five facets identify. A coordinate that separates records the others do not is linearly independent of them as an information channel. The strict positivity of the conditional mutual information follows because the disposition is a nondegenerate function of the private constraint conditional on the assessment, so observing the disposition updates a reader's belief about that constraint. The full necessity argument, including the proof that a mandatory public option is required to keep the channel from being captured, is developed in the companion paper on disposition economics. The result needed here is only that the dimension is six. End sketch.
The disposition carries behavioral information the content does not. A record sealed for thirty days and then released tells a reader something about the author's situation that the released assessment alone does not, in the same way that the decision to stay silent carries information in a market where silence is observed (Beier, 2026aj). Transmit is the structural home for that information, and it is a sixth axis, not a relabeling of the fifth.
The two timestamps are not facets. They do not measure the assessment and they do not transform under a reader's framework. They sit over the facet space as a fiber.
Definition 4 (Temporal fiber). Over each disclosure point h in V, attach the fiber of admissible timestamp pairs
Phi(h) = { (tau_s, tau_t) : tau_t >= tau_s },
with fiber coordinate given by the lag delta = tau_t - tau_s >= 0. A Tensor is a point in V together with a choice of point in the fiber over it. The strike-time section is the set of records read at their strike-time, delta = 0.
Proposition BT6-2 (Strike-time scoring and lag information). Proper scoring of a Tensor is well-defined and incentive-compatible only when the score is computed against the outcome realized after the strike-time and the resulting track record update is attached to the strike-time. Tamper-evidence of the strike-time is necessary for incentive compatibility. The lag delta is an admissible conditioning variable for a reader and is strictly informative whenever the lag distribution differs across author types.
Proof sketch. A strictly proper scoring rule rewards a forecaster for reporting the belief held given the information available when the claim was made (Beier, 2026r, Proposition 5). The information set that makes the rule strictly proper is therefore the set held at the moment of the claim, which is the strike-time. Scoring against the strike-time means the forecaster cannot improve a score by waiting to see how events unfold, because the scored claim is fixed to the earlier instant. If the strike-time could be moved after the fact, a forecaster could relocate a poor claim to a moment when the relevant uncertainty had partly resolved, which would let the scoring set drift forward and destroy strict propriety. Tamper-evidence forbids that relocation, so it is necessary. The lag follows from Proposition BT6-1: authors whose dispositions reflect sensitive holdings exhibit systematically longer lags, so the lag distribution is type-dependent, and a reader who conditions a posterior on the observed lag updates strictly. The transmit-time governs when the signal reaches the market under the disposition; the strike-time governs what the record is scored on. The two coexist because they answer different questions. End sketch.
The two-timestamp structure is what lets accountability and competitive secrecy hold at once, a tension the architecture is repeatedly asked to resolve. A fund commits a judgment at the strike-time, the judgment is scored against the outcome on that strike-time, and the assessment becomes public after a bounded interval set in the disposition. The market receives the signal when the disposition allows; the record preserves what was actually believed and when. As baseline transparency rises across a population, a long lag becomes a costly signal rather than a free option, and the equilibrium lag compresses toward zero without any mandate forcing it (Beier, 2026, Good Judgment, Ch on the temporal architecture). The fiber coordinate, in other words, has its own dynamics, which a later paper formalizes.
A Tensor can be built from earlier Tensors. An analyst who reads three disclosures and synthesizes a fourth records the three as parents. The composition history is the structure that carries this, and it determines whether the corpus of records forms a coherent object or an unordered pile.
Definition 5 (Composition). For Tensors T_1, ..., T_k, a Tensor T is a composition of them when its composition history is chi(T) = { id(T_1), ..., id(T_k) } and its strike-time satisfies tau_s(T) >= max_i tau_s(T_i). Write T = comp(T_1, ..., T_k; h, ...) when T cites those parents and carries its own facets and metadata.
The strike-time condition is the content of the definition. A record cannot build on a parent that did not yet exist at the moment the record was struck. The condition is not a convention; it is the requirement that lineage respect time.
Proposition BT6-3 (Closure and acyclicity). The set of valid Tensors is closed under composition, and the directed graph whose edges point from each Tensor to its parents is acyclic. Composition is associative on this graph in the sense that the lineage reachable from a record is independent of the order in which intermediate compositions were formed.
Proof sketch. Validity of a Tensor is the conjunction of three conditions: every facet is filled or carries a typed null, the timestamps satisfy tau_t >= tau_s, and the strike-time is at least the maximum strike-time among parents. A composition is constructed to satisfy the first two by filling its own facets and timestamps, and satisfies the third by Definition 5, so the conjunction holds and the composition is valid. For acyclicity, every citation edge points from a record to a parent with a strictly earlier or equal strike-time, and ties at equal strike-time can be broken by the transmit-time, which is the publication order; a cycle would require a record to be its own ancestor, hence to have a strike-time strictly less than itself, which is impossible. Associativity of the reachable lineage is the associativity of set union over the parent sets along paths, which holds because reachability in a directed acyclic graph is the transitive closure of the edge relation and the transitive closure is order-independent. End sketch.
The algebra this defines is the foundation for provenance tracking and for the emergence detection that the cosci.app engine is built to perform (Beier, 2026, cosci.app specification). Convergence among independent authors shows up as distinct lineages arriving at agreeing Output facets; divergence shows up as a single lineage branching into conflicting assessments; a gap shows up as a region of the constraint space with no records reaching it. Each of these is a statement about the lineage graph, and each is well-posed only because the graph is acyclic and the composition is closed. Without Proposition BT6-3, the detection problems are not defined.
The welfare result of 2026z survives the move to six facets and gains strength from the metadata.
Proposition BT6-4 (Six-facet Blackwell dominance). For any reader whose decision problem is sensitive to at least one constraint dimension, the six-facet extended price E = (p, T, w) is strictly more Blackwell-informative than the scalar price p. Adding the Transmit coordinate and the temporal fiber cannot reduce informativeness, and is strictly informative for readers whose optimal action depends on the disposition or on the timing of the signal.
Proof sketch. The scalar price p is recoverable from the Nucleus of T, so E refines p, and a refinement is weakly Blackwell-superior: any decision rule available to the reader of p is available to the reader of E. Strictness on the facets is the excess-distortion result: a scalar channel compressing a constraint environment of more than one dimension leaves strictly positive distortion on the constraint dimensions (Beier, 2026h, Proposition C), so there is a decision problem in which a facet changes the optimal action, which is the definition of strict Blackwell improvement (Beier, 2026u, BE-3). The metadata coordinates are available to be ignored, so they cannot lower informativeness; they are strictly informative for a reader whose action depends on whether the signal is actionable now, which the disposition and the transmit-time determine, or on how much weight the signal deserves given its lag, which Proposition BT6-2 shows is type-dependent. The dominance is therefore at least as strong as in 2026z and strictly stronger on the set of timing-sensitive and disposition-sensitive decisions. End sketch.
The result has the same shape as BT-3: welfare dominance is the informational content of the extended price expressed in decision-theoretic language. The strengthening is that two new classes of reader - the counterparty deciding whether a signal can be acted on today, and the reader deciding how far to trust a signal that sat private for a year - now have a formal reason to prefer the extended price, and neither class was visible in the five-field object.
The five-field tensor of 2026z is the special case of the present object obtained by three suppressions. Set the Transmit coordinate to a single constant value across all records, so e_6 contributes no variation and the facet space collapses to the five-dimensional span of e_1 through e_5. Collapse the temporal fiber to a point by reading every record at its strike-time, so delta = 0 everywhere and the two timestamps coincide. Drop the composition history, the engagement level, the substrate type, and the interface marker by treating every record as uncomposed and unmarked. Under these three suppressions the tuple T = (N, h, tau_s, tau_t, e, s, chi, iota) reduces to a five-coordinate disclosure with a price and a weight, which is exactly T = (p, D, w) with D in R^5, and the metric, transformation law, and Blackwell result reduce to BT-1, BT-2, and BT-3.
The published results are thus not overturned. They are the metadata-suppressed, Transmit-constant limit of the corrected object, in the same way that a scalar price is the one-dimensional limit of the Tensor and eigenspace is the limiting case of Beierspace (Beier, 2026e). The corpus's inclusion principle holds at this level too: each coordination object the architecture introduces contains its predecessor as a degenerate case, and the six-facet Tensor contains the five-field tensor by setting the new structure to its trivial value.
The corrected object hands several questions to companion papers, each of which now has a defined foundation to cite. The disposition economics of the Transmit facet - why a mandatory public option is required, how dishonest scope is defeated by redundancy rather than enforcement, and what the disposition choice reveals - is the direct continuation of Proposition BT6-1. The information theory of typed nulls sharpens the silence result of 2026aj by replacing a binary present-or-absent signal with the labeled alphabet of Section 2. The economics of the substrate-type marker, the engagement level, and the lineage algebra each take one of the metadata coordinates and ask what it is worth and how it is gamed. The temporal fiber's own dynamics, gestured at in Section 5, are the subject of a paper on the lag distribution and its competitive compression.
A reader who wants the geometry of the deep price now has the object the books describe rather than the object that preceded them. The next paper in this sequence takes the Transmit coordinate established here as independent and asks what its disposition is worth.
A Beier Tensor carries two timestamps. The strike-time records when the cognitive act occurred and fixes the claim that proper scoring evaluates. The transmit-time records when the record became readable under its disposition and never precedes the strike-time. The published formal corpus, written around a single implied timestamp, does not explain why two are required or what the interval between them carries. This paper supplies both. The central impossibility result is that no single-timestamp record can preserve strict propriety of scoring while permitting an author-chosen publication delay: one timestamp forces a choice between honesty and secrecy, and the pair dissolves the choice. Tamper-evidence of the strike-time is then shown to be necessary and sufficient for honest scoring. The lag between the two timestamps is the paper's main object. Because the lag is chosen by the author and its benefit rises with the competitive sensitivity of the underlying position, the lag separates author types in equilibrium and is a strictly informative conditioning variable for a reader, though its sign as good or adverse news is not fixed in advance. The reputational cost of a long lag is endogenous to the population: opacity is cheap when most authors are opaque and expensive when most transmit at short lag, which produces a contraction dynamic in the lag distribution with a transparent rest point and an opaque rest point, and a coordination transition between them that the contractual adoption mechanism is built to trigger. The structure resolves the standing objection that public disclosure and competitive secrecy cannot coexist: a bounded, disclosed delay preserves the disclosure obligation and the honesty of the score while giving the author orderly time to manage exposure, and an unbounded or undisclosed delay collapses to non-disclosure.
Keywords: strike-time, transmit-time, proper scoring, costly signaling, lag, disclosure delay, coordination transition, OCOT
Every account of a timestamped judgment has to answer a question it usually leaves implicit: which instant does the timestamp mark? Two answers are available to a record carrying one timestamp, and each fails a requirement the architecture cannot give up.
Mark the instant the thought occurred. The record then dates the cognition honestly, and a proper scoring rule applied to it rewards the belief the author actually held at that moment. The cost is that the record must become public at the same instant, because there is no second mark to separate the moment of thought from the moment of publication. An author who holds a competitively sensitive position cannot use such a record. Publishing an assessment of a large position in real time moves the market against the position before the author can act on the assessment. Intelligence work, legal strategy, and pre-announcement corporate analysis carry the same exposure. A record that demands immediate publication excludes every domain where the value of the judgment depends on acting before others see it.
Mark the instant the record was published instead. The author can now delay, holding the assessment private and releasing it later. The cost this time falls on the score. A scoring rule applied to a claim dated at publication evaluates a claim that could have been conditioned on everything that happened between the cognition and the release. An author who waits until the uncertainty has partly resolved, then files a claim dated at the moment of filing, is scored as if the claim were a forecast when it was partly a memory. Strict propriety is gone, and with it the meaning of the track record.
The two costs are not symmetric nuisances to be traded off. They are the two halves of a single structural fact: a record carrying one timestamp cannot hold the moment that should be scored apart from the moment that should be published, and those moments are different whenever an author needs to think privately before speaking. The architecture's answer is to carry both. The strike-time marks the cognition and fixes the scored claim. The transmit-time marks the publication and governs when the signal reaches a reader. Neither does the other's work.
Let an author form a judgment at strike-time tau_s and release it at transmit-time tau_t, with tau_t >= tau_s. Write the lag as
delta = tau_t - tau_s >= 0.
The judgment is a claim about an outcome that resolves at some later time T_res > tau_s. A strictly proper scoring rule R evaluates the claim against the realized outcome, and the resulting update to the author's track record weight attaches to the strike-time (Beier, 2026z, BT-2; Beier, 2026, "The Six-Facet Beier Tensor," BT6-2). The strike-time is anchored on an immutable substrate so that it cannot be altered after the record is struck; the reference architecture supplies the anchor through a plain-text record placed in archive.org with an ORCID binding, optionally reinforced by a cryptographic timestamp (Beier, 2026, Good Judgment, Ch on publication infrastructure).
The disposition fixes the transmit-time relative to the strike-time. A real-time disposition sets delta = 0. A delayed disposition sets a positive delta, stated in the record. The temporal pair sits as a fiber over the facet space of the Tensor, with delta as the fiber coordinate (Beier, 2026, "The Six-Facet Beier Tensor," Definition 4). The remainder of this paper treats delta as the decision variable it is, chosen by the author under a cost structure, and asks what the choice reveals and how the population's choices move over time.
The argument of Section 1 has a formal statement.
Proposition T-1 (Decoupling necessity). No single-timestamp record can simultaneously preserve strict propriety of the scoring rule and permit a positive publication delay chosen by the author. A record with strike-time and transmit-time scored on the strike-time and published on the transmit-time satisfies both.
Proof sketch. Let a record carry one timestamp tau and let the cognition occur at tau_cog. Strict propriety requires that the scored claim be measurable with respect to the information set held at the instant the claim is made, so that the author maximizes expected score by reporting the belief held at that instant rather than by waiting (Savage, 1971; Gneiting and Raftery, 2007). If tau = tau_cog, then the publication instant equals the cognition instant and the delay is zero, which denies the author any positive delay. If tau > tau_cog, then the scoring set is the information held at tau, which includes everything realized over the interval from tau_cog to tau, so the author can condition the reported claim on partially resolved uncertainty, and the rule is no longer strictly proper. One timestamp therefore forces delay to zero or sacrifices propriety. The pair separates the two roles: scoring on tau_s fixes the scoring set to the cognition instant, and publishing on tau_t permits any tau_t >= tau_s, hence any nonnegative delay. End sketch.
The pair only works if the strike-time cannot be moved. If an author could reset tau_s after filing, the second branch of the proof returns through the back door: the author files at the cognition instant, watches the outcome develop, and relocates tau_s forward for claims that aged badly, recovering the ability to condition on resolved uncertainty.
Proposition T-2 (Tamper-evidence). Scoring on the strike-time is incentive-compatible if and only if the strike-time is tamper-evident.
Proof sketch. Sufficiency follows from Proposition T-1: with tau_s fixed and verifiable, the scoring set is pinned to the cognition instant and the rule is strictly proper, so honest reporting maximizes expected score. Necessity is the relocation argument. Suppose tau_s can be altered without detection. An author who has struck a set of claims observes the outcomes as they resolve and, for any claim whose realized score is poor, moves tau_s forward to an instant at which the adverse information was already available, converting a forecast that failed into a record that merely reports what was known. The expected score under this strategy strictly exceeds the expected score under honest dating, so honest dating is not a best response, and incentive compatibility fails. Tamper-evidence removes the strategy by making any alteration of tau_s detectable, which restores the fixed scoring set. End sketch.
The immutability requirement that the architecture places on records is not a preference for permanence. It is the condition under which the strike-time can carry an honest score. A record that could be silently re-dated would score the author on a claim the author did not make.
The transmit-time is chosen, and authors do not all choose alike. The benefit of delay rises with the competitive sensitivity of the underlying position: an author whose assessment concerns a large undisclosed holding gains from releasing late, while an author assessing a public question with no position at stake gains nothing from waiting. Against the benefit stands a cost. Opacity carries a reputational price, because a reader who must wait for a record learns to discount an author who routinely makes them wait. The lag the author selects balances the two.
Proposition T-3 (Lag separates types). Let an author's type be the sensitivity m of the underlying position, with the benefit of delay b(delta, m) increasing in m and the reputational cost k(delta) increasing in delta. If b satisfies a single-crossing condition in (delta, m), then the author's optimal lag delta*(m) is nondecreasing in m, and the equilibrium is separating or monotone pooling. A reader who observes the realized lag updates on the author's type, so the mutual information between type and lag, conditional on the record's content, is strictly positive whenever delta*(m) is nonconstant.
Proof sketch. The author chooses delta to maximize b(delta, m) - k(delta). Single-crossing in (delta, m) means the marginal benefit of delay rises with type, so the first-order condition b_delta(delta, m) = k'(delta) has a solution delta*(m) that is nondecreasing in m by the standard monotone-selection argument for signaling games (Spence, 1973; Mailath, 1987). Where delta*(m) is strictly increasing, the lag is a sufficient statistic for type along that range, and the reader's posterior over m shifts with the observed delta, which is the definition of positive conditional mutual information. Where delta*(m) is locally flat, a set of types pools at a common lag and the lag is informative across pools but not within them. End sketch.
The result is that the lag is a channel. It is worth being careful about its sign. A long lag is not a confession. It can mark a legitimate competitive reason to wait, and it can mark an author holding a record back past the point a candid actor would have released it. Which reading applies depends on the joint distribution of sensitivity and lag in the relevant domain, and the architecture does not assign the lag a fixed valence. What it asserts is narrower and defensible: the lag is informative, a reader is entitled to condition on it, and an author who releases at the cognition instant has sent a different signal from an author who held the same assessment for a year. The silence result of the corpus made the analogous point for the binary choice to disclose at all (Beier, 2026aj). The lag refines that choice into a continuum and shows the continuum is read.
The cost of opacity in Proposition T-3 was written as a fixed function k(delta). It is not fixed. A long lag is conspicuous in a population that transmits quickly and unremarkable in a population that transmits slowly, so the reputational price of delay depends on what everyone else is doing.
Proposition T-4 (Endogenous compression and its rest points). Let the reputational cost of a lag be k(delta; F), where F is the population's lag distribution, and let the marginal cost k_delta(delta; F) increase as F concentrates mass near zero. The map from the population distribution F to the representative author's best-response lag is monotone decreasing. The best-response dynamic therefore admits a transparent rest point, at which most authors transmit at short lag and short lag is cheap to sustain, and an opaque rest point, at which most authors transmit at long lag and long lag is cheap to sustain. The dynamic is bistable, and movement from the opaque rest point to the transparent rest point is a coordination transition rather than a response to any mandate.
Proof sketch. When F concentrates near zero, k_delta rises, so the first-order condition of Proposition T-3 is satisfied at a smaller delta, and the best-response lag falls; this is the monotone-decreasing map from F to best response. A transparent configuration in which most mass sits near zero makes opacity expensive and short lag a best response, which reproduces the configuration, so it is a fixed point. An opaque configuration in which most mass sits at long lag makes opacity cheap and long lag a best response, which also reproduces itself, so it is a second fixed point. Between them the best-response map crosses the identity, giving an unstable interior fixed point that separates the two basins. Iterating the best response from a point in the transparent basin converges to the transparent rest point; from a point in the opaque basin it converges to the opaque rest point. Movement across the separating threshold requires a displacement of the population, which a coordinated entry of short-lag authors supplies. End sketch.
The bistability is the same shape that appears wherever the corpus treats the transition from a low-observation regime to a high-observation one (Beier, 2026q; Beier, 2026e, Theorem 5.2). An economy in which everyone hides is stable because hiding is cheap when hiding is universal, and an economy in which everyone discloses is stable for the mirror reason. The architecture's adoption mechanism is built to move a population from the first basin to the second by introducing a bloc of authors contractually required to transmit at short lag. The contractual clause in an investment management agreement does not legislate transparency for the whole population; it places enough short-lag mass to carry the population across the separating threshold, after which the best-response dynamic does the rest (Beier, 2026, The Future Transparent). Once across, a long lag becomes a costly outlier, and the pressure that compresses it is the population's, not a regulator's.
The structure answers an objection the architecture meets whenever it reaches practitioners who hold sensitive positions. The objection is that a public-disclosure requirement and a competitive need for secrecy cannot both be satisfied, so any real disclosure regime must either exempt the sensitive cases, which guts it, or override the secrecy, which no fund will accept.
Proposition T-5 (Bounded delay preserves the obligation). A disposition that delays publication by a bounded, disclosed interval delta <= delta_max preserves the disclosure obligation and the honesty of the score: the record is guaranteed to enter the commons by tau_s + delta_max, and the claim is scored against the outcome on the true strike-time regardless of when it is published. A disposition that delays by an unbounded or undisclosed interval collapses to non-disclosure.
Proof sketch. Under a bounded, disclosed delay, two guarantees hold. The record becomes public no later than tau_s + delta_max, so the disclosure obligation binds with a known deadline rather than being deferred indefinitely. The score is computed on tau_s by Proposition T-2, so the delay does not let the author improve the score by waiting. The obligation and the honesty are therefore both intact, while the author retains the interval from tau_s to tau_t to manage the exposure, which is the orderly-rebalancing window the sensitive case requires. Under an unbounded delay, no deadline guarantees publication, so the record may never appear, which is non-disclosure by another name. Under an undisclosed delay, the reader cannot observe delta and so cannot condition on it, which removes the signal of Proposition T-3 and lets the author claim a disclosure that the reader cannot place in time. The bound and the disclosure of the bound are jointly what keep delay from degenerating into concealment. End sketch.
The lever is delta_max. Set it small and the obligation bites hard, at the cost of forcing sensitive positions into the open before they can be managed. Set it large and the rebalancing is comfortable, at the cost of letting the signal age before it reaches the market. The contractual clause sets delta_max for the holdings it covers, and the choice is visible in the record, so a reader sees not only the assessment but the window the author reserved before releasing it (Beier, 2026, The Future Transparent, Ch on delayed disclosure). The fund discloses what it judged, when it judged it, scored on that instant, and releases it within a stated bound. The objection dissolves not because the secrecy is denied but because the strike-time carries the accountability that the secrecy would otherwise have destroyed.
Three questions follow directly from the model and belong to companion work. The first is empirical: the lag distribution in a live commons is measurable, and its shape over the adoption window is a test of the compression dynamic in Section 5, with the prediction that the distribution contracts toward zero after the contractual bloc enters and not before. The second concerns the interaction of the lag with the disposition types of the Transmit facet, since a private record and a delayed-public record carry different lags and different terminal states, and the joint signal of disposition and lag is richer than either alone (Beier, 2026, "The Six-Facet Beier Tensor," BT6-1). The third is the welfare accounting of the delay: a bounded delay protects orderly rebalancing but withholds a signal the rest of the market could have used, and the net effect on coordination depends on how the withheld interval compares with the resolution time of the underlying constraint, which the threshold results of the corpus are positioned to evaluate.
The two-timestamp record is the smallest structure that lets a judgment be both honestly scored and strategically released. The next paper takes the disposition that the transmit-time executes and asks what the choice among private, shared, public, and delayed is worth, and why one of those options cannot be removed.
The sixth facet of a Beier Tensor is the disposition, which fixes who is permitted to read the record: the author alone, a named group, the public, or the public after a stated delay. The companion paper on the corrected tensor established that the disposition is an independent information channel but deferred two results to here. The first is that the disposition is the map from a record to its readership set, and that an author's choice of disposition reveals the author's belief about which actors the underlying issue affects. The second concerns the scope test that the architecture asks every author to apply: choose the narrowest disposition that does not impair other actors' ability to coordinate around the same underlying issue. This paper proves that the scope test is equivalent to a coordination-completeness condition, that an author who narrows scope below that condition can be corrected by any third party who holds the information, and that this correction operates through permissionless redundancy rather than through any central authority that checks dispositions. The structural requirement that makes the redundancy defense work is a public option available to every author, and the paper proves that removing the public option makes the commons enclosable: a coalition controlling the boundaries of group-scoped records can exclude the parties an issue affects, and no third party can broaden the scope to reach them. The mandatory public option is therefore the anti-enclosure commitment, and it cannot be removed without surrendering the property that lets a permissionless commons govern its own scope. The boundary of the redundancy defense is identified: it fails when no motivated third party holds the information, which marks the cases where external mechanisms remain necessary.
Keywords: Transmit facet, disposition, scope, commons, enclosure, permissionless redundancy, coordination, OCOT
A manufacturer discovers a quality problem with a supplier's components. The problem is real, the assessment is sound, and the manufacturer files a Tensor recording it. The right disposition for that record is narrow. The parties who need the information to act are the manufacturer, the supplier, and perhaps a few downstream integrators, and all of them sit inside a bounded group. Publishing the assessment to the world would expose a commercial relationship without helping anyone coordinate who could not already be reached inside the supply chain.
A manufacturer discovers a contaminant in baby formula. The assessment is again sound, and again a Tensor records it. The right disposition for this record is the opposite. The parties who need the information reach far past the manufacturer's walls: the parents feeding the formula, the pediatricians treating the infants, the regulators issuing recalls, the hospitals preparing for cases. A disposition that kept this record inside the firm would withhold it from almost everyone whose coordination depends on it.
The two records differ in the assessment they carry, but the choice that distinguishes them is not in the assessment. It is in the disposition. The same architecture, the same facets, the same scoring, and the difference that matters is who is permitted to read the record. The disposition is where that difference lives, and the question this paper answers is how an author should set it, what the setting reveals, and what structural guarantee keeps an author from setting it to conceal rather than to coordinate.
Let the disposition of a Tensor be an element D of the set
{ private, group(G), public, delayed-public(delta) },
where group(G) names a finite set G of permitted readers and delayed-public(delta) makes the record public after the lag delta studied in the companion paper on the two timestamps (Beier, 2026, "Strike-Time and Transmit-Time"). Each disposition determines a readership set R(D): the author alone under private, the set G under group(G), the whole population under public, and the whole population after delta under delayed-public.
Fix an underlying issue, by which is meant the constraint, hazard, or condition the record assesses. Define the affected set A(issue) as the set of actors whose ability to coordinate around the issue depends on access to the information. The parents and pediatricians belong to A for the contaminant; the supply-chain integrators belong to A for the supplier defect; the world at large belongs to A in neither case beyond those who are actually affected. The affected set is a property of the issue and not of the author, and the author may estimate it correctly or incorrectly.
A disposition is a choice of readership set, and the scope test the architecture asks the author to apply compares that readership set against the affected set. The next sections make the comparison precise and then ask what happens when an author gets it wrong on purpose.
The companion paper on the corrected tensor proved that the disposition is linearly independent of the five assessment facets as an information channel (Beier, 2026, "The Six-Facet Beier Tensor," BT6-1). What that channel carries can now be stated.
Proposition D-1 (Readership selection and revealed belief). The disposition is the map D maps to R(D) from records to readership sets. An author who chooses D conditional on the affected structure of the issue reveals, through the choice, an estimate of A(issue). A reader who observes the disposition updates on the author's belief about who the issue affects, independently of what the reader learns from the assessment itself.
Proof sketch. The map from disposition to readership set is given by the definitions of Section 2 and is fixed by the disposition alone. An author choosing a disposition to satisfy the scope test of Section 4 selects R(D) to cover the actors the author believes the issue affects, so the chosen disposition is a function of the author's estimate of A. Observing the disposition therefore licenses an inference about that estimate: a record marked public signals an author who believes the affected set is broad, and a record marked group signals an author who believes it is contained. The inference is separate from the content because two authors with identical assessments can hold different beliefs about A and choose different dispositions, which is the independence established in BT6-1 read in the direction of what it reveals. End sketch.
The consequence worth drawing out is that a disposition is a contestable claim. When an author marks a record group-scoped, the author is asserting that the affected set fits inside the group. Another actor who believes the issue reaches further is in disagreement with that assertion, and Section 5 shows what the architecture lets that actor do about it.
The architecture's guidance to authors is a single criterion: choose the narrowest disposition that does not impair other actors' ability to coordinate around the same underlying issue (Beier, 2026, Good Judgment, Ch on the Transmit facet). The criterion has an exact equivalent.
Definition (Scope-completeness). A disposition D is scope-complete for an issue when its readership set contains the affected set, R(D) contains A(issue).
Proposition D-2 (Equivalence). A disposition satisfies the scope test if and only if it is scope-complete. Narrowing a disposition impairs coordination around the issue exactly when the narrowing drops an actor from the affected set.
Proof. Consider narrowing a disposition from D to D' with R(D') a subset of R(D). The actors who lose access are those in R(D) but not R(D'). Coordination around the issue is impaired precisely when some actor who needs the information to coordinate loses access, that is, when the dropped set intersects A(issue). The narrowing preserves coordination if and only if no member of A is dropped, which holds if and only if A is contained in R(D'), the definition of scope-completeness for D'. The narrowest scope-complete disposition is therefore the narrowest disposition that does not impair coordination, which is what the scope test selects. End proof.
The two opening records are the two regimes of the proposition. For the contaminant, A reaches past any group an author could name, so the only scope-complete dispositions are public and delayed-public, and the scope test forces publication. For the supplier defect, A sits inside the supply chain, so a group disposition naming the supply-chain actors is scope-complete, and the scope test permits the narrow scope that protects the commercial relationship. The criterion does not push everything toward maximal disclosure. It pushes toward the narrowest disclosure that still reaches everyone the issue touches.
An author has a private reason to violate the scope test in one direction. Narrowing scope below completeness conceals a matter the author would rather keep contained, and the cost of concealment falls on the actors dropped from the readership. A manufacturer who marks a contaminant group-scoped has under-scoped, and the parents and regulators are the dropped set. The architecture has no central authority that inspects dispositions and overrules the manufacturer. The correction comes from a different direction.
Proposition D-3 (Redundancy defeats under-scoping). Suppose an author selects a disposition that is not scope-complete. If at least one other actor holds the same information and the actor's benefit from publishing at a scope-complete disposition exceeds the actor's cost of doing so, then in equilibrium the information is published at a scope-complete disposition by that actor, with no central enforcement required.
Proof sketch. The under-scoping author cannot prevent another actor from publishing, because the publication infrastructure is permissionless: any actor with an ORCID can strike a Tensor and transmit it under any disposition the actor chooses (Beier, 2026, Good Judgment, Ch on publication infrastructure). A third actor who holds the same information faces a benefit from publishing at scope-complete disposition, which includes the scored reputational gain from a correct disclosure that the strike-time machinery records to the actor's track record, and may include direct stakes in the issue, against a cost that includes effort and any retaliation risk. If for at least one such actor the benefit exceeds the cost, that actor publishes at a scope-complete disposition, and the dropped members of the affected set receive the information. The correction is performed by the actor with the strongest motive to perform it, and the under-scoping author's narrow disposition does not bind the third actor. End sketch.
The mechanism is the one Ostrom (1990) identified in self-governing commons, transposed to information: the parties to a shared resource police its boundaries themselves when the institutional structure lets them, rather than relying on an external monitor. The regulator who suspects a withheld hazard, the journalist who receives a tip, the laboratory that runs an independent test, the insider who knows what the firm marked private, each can publish a Tensor at the scope they judge correct, and each is scored on the result. The under-scoping author is corrected not by being caught but by being routed around.
The proposition states its own boundary. The correction requires a third actor who holds the information and for whom benefit exceeds cost. When the under-scoping author is the sole holder of the information, no third actor can publish it, and the redundancy defense is empty. When every actor who holds the information faces a cost of disclosure that exceeds the benefit, which severe retaliation or coordinated suppression can produce, the defense again fails. These are the cases where permissionless redundancy is insufficient and external mechanisms remain necessary, and they are the same boundary the endogenous-governance analysis reaches from the side of the Scaling facet: the architecture's internal corrections work where outcomes are observable and identities are not suppressed, and they thin out where information is held by one party under threat (Beier, 2026, "The Endogenous Governor"). Naming the boundary is part of stating the result honestly.
The redundancy defense of Section 5 rests on a structural assumption that has not yet been made explicit: a third actor can publish at a scope that reaches the dropped members of the affected set. If the broadest disposition available to any author were a bounded group, that assumption could fail, and with it the whole correction.
Proposition D-4 (Anti-enclosure). If the public disposition is removed from the available set, so that every record must be private or group-scoped with a bounded group, then the commons is enclosable: a coalition that controls membership in the maximal available group can exclude actors from the affected set, and no author can reach the excluded actors by broadening scope. Restoring a permissionless public option available to every author removes the enclosure.
Proof sketch. Suppose no public disposition exists and the largest readership any author can select is some bounded group G_max. Any actor outside G_max receives no record under any disposition, because no disposition reaches beyond G_max. A coalition that controls who belongs to G_max can therefore systematically exclude chosen actors from all records, including actors in the affected set of an issue the coalition prefers to contain. The redundancy defense of Proposition D-3 cannot correct the exclusion, because a third actor who wishes to reach the excluded parties has no disposition that reaches them either: the third actor is bounded by the same G_max. The information is thus enclosable, held within a controlled boundary that no author can cross. Now restore a public disposition available to every author. Any actor, including a third actor seeking to reach the excluded parties, can select public and reach the entire affected set regardless of any coalition's control over group membership. The enclosure is removed, because the boundary a coalition might control is no longer the outermost boundary of the commons. End sketch.
The public option is therefore not one disposition among four that an implementation might trim for convenience. It is the disposition whose availability makes the other three safe, because it is the option a third actor invokes when a coalition tries to contain an issue inside a group. An implementation that offered only private and group dispositions would look more controllable and would in fact be capturable: whoever controlled the largest group would control what the commons could see. The requirement that every implementation offer a permissionless public disposition, and that the public option cannot be removed, is the commitment that keeps the commons from being enclosed by the parties best positioned to enclose it (Benkler, 2006).
A bounded group disposition retains a legitimate use that the anti-enclosure result does not threaten. Competing firms can share constraint assessments inside a named group without publishing to the world, in the way that the advertising-technology industry built clean rooms where rivals combine data under controlled terms. The group disposition is the clean-room construct, and it is useful for exactly the coordination the supplier-defect record needed. What the mandatory public option guarantees is that the clean room cannot become a wall: an actor outside the group who is affected by what the group is coordinating around can always publish at public scope and be heard. The group disposition serves contained coordination, and the public option prevents containment from hardening into enclosure.
The disposition completes the channels of the corrected tensor and leaves two measurable questions to later work. The distribution of dispositions in a live commons is observable, and the share of records at each disposition, tracked against the affected structure of their issues, is a test of whether authors apply the scope test or systematically under-scope, with the prediction that under-scoping concentrates in domains where a single party holds the information and faces retaliation. The interaction of the disposition with the substrate-type marker is the next structural question, because a record generated by a model and a record generated by a person may warrant different default dispositions, and the marker that distinguishes them is the subject of the following paper. The disposition also closes the metadata channels that bear directly on who reads a record, which leaves the channels that bear on how a record was produced: the engagement that went into it and the substrate that produced it. The next paper takes the substrate-type marker and asks what it is worth in a commons that mixes human and machine judgment.
A facet of a Beier Tensor may be filled or may carry a typed null, and the architecture distinguishes four kinds of absence: a facet that does not apply, a facet the author cannot supply, a facet whose content is withheld under the record's disposition, and a facet carried by the author's published standing positions rather than stated in place. The published treatment of non-disclosure in the corpus is binary, reading a missing assessment as a single present-or-absent signal. The facets are not binary, and this paper develops the information in the kind of blank. The alphabet of absence is a strictly richer channel than binary silence, because the four types are distinguishable and imply different things about the underlying state and the author, so a reader who knows the kind of blank updates on something a reader who knows only that the facet is empty cannot. The types invite a strategic substitution, in which an author dresses ignorance as irrelevance or as secrecy by choosing a less costly null type than the truth warrants, and the paper shows where that substitution is disciplined by scoring and where it is merely ignored by readers who learn the author's null types carry no information. The default-view null is shown to be a compression device that lets a record's effective content exceed its stated content by pointing to the author's separately scored standing positions, with accountability extending to those positions. The withheld null is shown to be the facet-level expression of a restricted disposition, with the strike-time binding the withheld content as it binds any facet, so that later revelation is a publication event and not a re-dating.
Keywords: typed null, absence, silence, implicature, default view, withheld content, scoring, OCOT
Two analysts each leave the Scaling facet of a Tensor empty. The first is assessing a single building's exposure to a local zoning change, a question on which systemic scale does not bear, and the empty facet means the question does not apply. The second is assessing a datacenter's water draw, a question on which systemic scale bears heavily, and the empty facet means the analyst could not work out what happens when the whole pipeline of facilities reaches operating scale. The two facets are equally blank, and they mean opposite things. The first blank is a statement that there is nothing to say. The second is a statement that there is a great deal to say and the author cannot say it.
A reader who sees only that the facet is empty cannot tell these apart, and the difference matters. The first analyst has produced a complete assessment of a question where scale is irrelevant. The second has produced an assessment with a hole in it on exactly the dimension that determines the systemic risk. Treating the two blanks as the same signal discards the information that distinguishes a finished record from an unfinished one. The architecture does not treat them as the same. A facet that is empty carries a type that says which kind of empty it is, and this paper develops what those types carry, how they are gamed, and how two of them connect to machinery the metadata papers already built.
A facet coordinate takes a filled value or one of four typed nulls (Beier, 2026, "The Six-Facet Beier Tensor," Section 2). The four are as follows. A not-applicable null records that the facet does not bear on the assessment, as systemic scale does not bear on a single-asset question. An unknown null records that the facet bears on the assessment but the author cannot supply it. A withheld null records that the author has an assessment for the facet but is restricting it under the record's disposition. A default-view null records that the facet is carried by the author's published standing positions rather than stated in place, pointing the reader to those positions for the content.
The four are distinct elements of the facet's coordinate alphabet, distinct from each other and from any filled value. A typed null is not a zero and not a default filled value; it is a labeled absence, and the label is the carrier of the information this paper develops. The pragmatic principle that the choice not to state something itself communicates, which Grice (1975) formalized for conversation, is what makes the alphabet a channel: a reader reads not only the filled facets but the kind of absence in the empty ones, and an author who marks a facet not-applicable has said something different from an author who marks it unknown, even though both have left it empty.
The published treatment of non-disclosure reads a missing assessment as a binary signal, present or absent, and shows that even the binary signal is informative in a market where silence is observed (Beier, 2026aj). The facets carry more than the binary.
Proposition N-1 (Typed absence is a richer channel). The four-type null alphabet carries strictly more information than a binary present-or-absent signal. The types are distinguishable and imply different things about the underlying state and the author, so the conditional mutual information between the underlying state together with the author's situation and the null type, given only that the facet is absent, is strictly positive.
Proof sketch. Construct two records identical except for the null type on a facet, one marked not-applicable and one marked unknown. A reader reads the first as a statement that the facet does not bear on the assessment and the second as a statement that it bears but the author could not supply it, which are different readings carrying different implications for how complete the assessment is and for what the author does and does not know. A binary present-or-absent signal collapses the two records to the same value, because both have the facet absent. The typed alphabet separates them. The separation is the strict positivity of the conditional mutual information: knowing the type, beyond knowing the facet is empty, changes the reader's posterior over the state and the author. The argument extends to the other type pairs, each of which a binary signal collapses and the typed alphabet distinguishes. End sketch.
The four types partition the single event the binary signal reports into four events a reader can act on differently. An unknown null on a load-bearing facet flags an assessment with a hole and invites another author to fill it, which is the gap the composition algebra detects (Beier, 2026, "The Composition Algebra," Proposition C-2). A not-applicable null on the same facet reports a complete assessment and invites no such filling. A withheld null reports that an assessment exists and is being restricted, which is a different invitation again. The binary signal cannot distinguish these invitations, and the typed alphabet is what lets the commons respond to each.
The types differ in how costly they are to the author. An unknown null admits a limitation. A not-applicable null admits none, asserting instead that there was nothing to know. A withheld null admits none either, attributing the blank to strategy rather than to ignorance. The cost difference creates an incentive to substitute.
Proposition N-2 (Strategic substitution and its discipline). An author has an incentive to replace a costly null type with a cheaper one, marking a facet not-applicable or withheld when the truth is unknown. Where the null type is checkable against outcomes, against the evident applicability of the facet, or against other records, a dishonest type is a detectable miscalibration that degrades the author's track record. Where the null type is not checkable, it carries no information, and a reader who conditions on the author's history of null-typing discounts it.
Proof sketch. Marking unknown costs the author an admission of limitation, while marking not-applicable or withheld avoids that admission, so an author who wishes to conceal a limitation has a reason to mark the cheaper type. The substitution is disciplined in the cases where it can be caught. A not-applicable null is checkable when the facet is evidently material to the assessment, since a facet that plainly bears on the question cannot honestly be marked not-applicable, and a reader or a scoring process can flag the contradiction. A withheld null is checkable when the content, once revealed under the disposition's terms, proves not to have existed at strike-time, since a withheld null asserts that an assessment was held and restricted, and a later revelation that finds no such assessment exposes the type as false. In these cases the false type is a miscalibration that the track record records, lowering the author's standing as any failed claim does. Where none of these checks applies, the null type cannot be verified and so carries no information a reader should weight, and a reader who has observed that an author's null types do not predict the eventual state discounts that author's null types, in the same way a reader discounts an author whose claimed engagement does not predict accuracy (Beier, 2026, "The Engagement Level," Proposition E-4). End sketch.
The discipline is uneven, and the proposition does not pretend otherwise. A not-applicable null on a clearly material facet is catchable, and an author who uses it to hide ignorance risks the catch. An unknown dressed as not-applicable on a facet whose applicability is truly ambiguous may never be caught, and there the type is simply noise that informed readers stop weighting. The withheld null sits between, disciplined by the eventual revelation that the disposition's bounded delay guarantees for delayed-public records and never guaranteed for permanently private ones. The architecture does not make every null type honest. It makes the consequential ones checkable and lets the rest decay to uninformative.
One of the four types does something the others do not. A default-view null does not report an absence so much as redirect to content held elsewhere.
Proposition N-3 (Default-view compression). A default-view null points to the author's published standing positions rather than stating the facet in place. A record's effective information set is therefore the union of its stated facets and the standing positions its default-view nulls reference, so the record carries more than it states, and the author's accountability extends to the referenced standing positions as to the stated facets.
Proof sketch. A default-view null is a pointer to a persistent, separately published body of the author's standing positions on background questions, a body that is itself scored as the author's positions are tested against outcomes (Beier, 2026, Good Judgment, Ch 11, on the data layer's default-view state). A reader who encounters the null follows the pointer and reads the referenced positions, so the content the reader obtains from the record is the stated facets plus the referenced standing positions, which is the union claimed. Because the author is accountable for the standing positions through their own scoring, the accountability the record inherits on a default-view facet is the accountability of those positions, so the author cannot escape responsibility for a facet by pointing to a standing position the author then disowns. The compression is that the author need not restate the standing positions in every record, which keeps records short while preserving their full effective content through the pointer. End sketch.
The default-view null is what lets an author build a record without rebuilding a worldview each time. A macroeconomist whose standing position on a discount-rate question is published once can reference it from a hundred records rather than restating it in each, and a reader who wants the author's full reasoning follows the reference. The standing position is scored on its own, so the reference carries scored content and not a free assertion. The data layer of the Protocol already carried a default-view state for exactly this purpose, and the default-view null is that state expressed at the level of a single facet (Beier, 2026, Good Judgment, Ch 11).
The fourth type connects to the machinery the disposition and timestamp papers built. A withheld null restricts a facet's content, which is what a disposition does for a whole record.
Proposition N-4 (Withheld nulls and the strike-time). A withheld null records that the author held an assessment for the facet at strike-time and is restricting its display under the record's disposition. The strike-time binds the withheld content as it binds any facet, so a later revelation of the withheld content is a publication event at a transmit-time and not a re-dating of the assessment.
Proof sketch. A withheld null asserts that the facet was filled at strike-time and is dispositionally restricted from display, which is the facet-level version of a record marked private or delayed-public (Beier, 2026, "The Transmit Facet," Proposition D-1). The strike-time binding established for any facet applies to the withheld content, because the content existed at strike-time by the assertion, and the tamper-evidence of the strike-time fixes when it was held (Beier, 2026, "Strike-Time and Transmit-Time," Proposition T-2). When the disposition's terms permit, the withheld content is displayed, and that display occurs at a transmit-time later than the strike-time, exactly as a delayed-public record's content becomes visible at its transmit-time. The revelation does not change the strike-time, so the withheld content is scored on the instant it was held and not on the instant it was shown. A withheld null is therefore a delayed or restricted disposition applied to one facet rather than to the whole record, and it inherits the honesty guarantees of the disposition and timestamp machinery. End sketch.
The withheld null lets an author commit an assessment without displaying it, in the way a fund commits a position assessment it cannot reveal in real time. The facet is filled and dated at strike-time, the content is restricted under the disposition, and when the restriction lifts the content appears, scored on the strike-time it was held. The honesty that the two-timestamp structure secured for whole records is secured for individual facets through the withheld null, so an author can leave a facet dark without leaving it unaccountable.
The typed null closes the facet-level structure of the corrected tensor. The facets carry filled assessments or labeled absences, the metadata carries who may read a record and how it was made, and the lineage carries how records build on one another. The four kinds of blank are not gaps in the record but content of their own: a not-applicable null reports completeness, an unknown null reports a hole that invites filling, a default-view null redirects to scored standing content, and a withheld null restricts a facet under the same terms a disposition restricts a record.
With the typed null in place, the structural foundations of the six-facet object are complete: the corrected tensor, the two timestamps, the disposition, the substrate, the engagement, the composition algebra, and the alphabet of absence. The next sequence turns from the structure of a record to the structure the records coordinate across, and asks why the framework hosts natively in some political environments and cannot be hosted in others. The first paper of that sequence takes the claim that regime type determines participation capacity and gives it the formal grounding the North Star asserts but has not yet proved.
A Beier Tensor records the substrate that produced its reasoning: a person, a model, or a combination of the two. The marker looks like bookkeeping. It is load-bearing. A commons that mixes human and machine judgment without distinguishing their origin admits a failure that neither produces alone. This paper develops the economics of the substrate marker through four results. The marker is a distinct information channel, because human reasoning and model reasoning fail in different ways and a reader who knows the substrate updates differently on the same assessment. The marker is necessary, not merely useful, because current commercial language models cannot reliably monitor their own calibration, so a model-origin claim cannot vouch for itself the way a human track record vouches for a person who bears the consequences of being wrong, and only the marker keeps a model-origin claim from borrowing a calibration presumption it has not earned. The marker defends against a lemons dynamic: model-generated assessments that are structurally correct and analytically empty can be produced at near-zero cost, and absent a truthful, costly-to-forge marker they drive down the average quality of the commons until high-effort producers withdraw. The hybrid case is not a midpoint between the two pure substrates but an inheritor of both their failure modes, which means the marker should record the division of labor across facets, and that record is naturally derived from the composition lineage. The marker is what lets the scoring machinery apply the prior each substrate warrants, and it is the commons-level counterpart to the pre-kernel architecture that the alignment papers argue a wise machine requires.
Keywords: substrate type, provenance, self-monitoring, calibration, adverse selection, human-AI exchange, hollow disclosure, OCOT
Suppose a commons of structured judgment is working. People file Tensors about constraint conditions, the assessments are scored against outcomes, and track records accumulate to the analysts who earn them. Now suppose a producer points a language model at the same commons and generates assessments at scale. The model-generated records are well formed. Each carries an Input facet that cites plausible sources, a Method facet that describes a sound-looking procedure, an Output facet that states a definite claim, and a Scaling facet that gestures at systemic effects. The records pass every structural check. They are also, in a fraction of cases the reader cannot identify in advance, empty: the citations do not support the claim, the method was never run, and the systemic assessment is a paraphrase of priors rather than an analysis of the specific constraint.
A reader facing this flood has a problem the architecture has not yet solved. The structural facets do not distinguish a careful human assessment from a fluent model imitation of one, because the imitation was optimized to match the structure. The track record would eventually distinguish them, once outcomes resolve and scores accrue, but the flood arrives faster than outcomes resolve, and in the interval the commons is diluted. The reader needs to know something the assessment facets do not carry: what produced the reasoning. The substrate marker carries it, and this paper is about why that marker has to be there and what it is worth.
Let the substrate type of a Tensor be an element s of the set
Sigma = { human, model, hybrid },
recording whether the reasoning in the record originated from a person, from a model, or from a combination. The marker is itself a claim by the author and is subject to the same honesty pressures as any facet: it can be reported truthfully, reported falsely, or left as a typed null when the origin is mixed in a way the coarse categories do not capture. The architecture treats the marker as first-class metadata, on the same footing as the timestamps and the disposition, and not as an optional annotation (Beier, 2026, "The Six-Facet Beier Tensor," Section 2).
Two background results from the corpus frame what follows. The first is that human and machine judgment trade at an exchange rate rather than being interchangeable or strictly ranked, so a commons that uses both is pricing two different inputs and needs to tell them apart to price either (Beier, 2026aa). The second is that a wise machine requires structure below the level of its trained objective, a pre-kernel layer the alignment papers argue cannot be supplied by the objective itself (Beier, 2026g; Beier, 2026ad). The substrate marker is the commons-level expression of both: it makes the exchange rate observable by labeling which input a record came from, and it supplies at the level of the shared record the provenance distinction that a model cannot supply about itself from inside its own objective.
The first question is whether the substrate is recoverable from the assessment, in which case the marker would add nothing. It is not.
Proposition S-1 (Substrate carries reliability information). Conditional on the assessment facets, the substrate type carries information about the reliability and the failure modes of the reasoning. Human reasoning and model reasoning have different error structures, so a reader who knows the substrate forms a different posterior over the assessment's reliability than a reader who does not. The conditional mutual information between reliability and substrate, given the facets, is strictly positive.
Proof sketch. The reliability of an assessment is a function of how the reasoning that produced it tends to fail. Human reasoning fails through fatigue, motivated inference, and the known catalogue of cognitive biases; model reasoning fails through fabrication of unsupported detail, deference to the framing of the prompt, and gaps where the constraint lies outside the training distribution. These failure structures are different, so the probability that a given assessment is unreliable depends on which substrate produced it, even holding the assessment's content fixed. A reader who observes the substrate conditions on the relevant failure structure and updates accordingly, which a reader without the marker cannot do. The difference in posteriors is the strict positivity of the conditional mutual information. End sketch.
The two substrates are not ranked by the proposition. A model assessment is not uniformly less reliable than a human one, and in domains where the failure modes of human reasoning dominate, the model assessment may be more reliable. What the proposition establishes is that the substrates fail differently, so knowing which one is in front of the reader changes what the reader should watch for. The marker is the channel that delivers that knowledge.
A human track record works because the score attaches to a persistent identity that carries the consequences of being wrong, and because a person can, within limits, report and improve their own calibration. A model-origin claim cannot rely on the second mechanism.
Proposition S-2 (The self-monitoring gap makes the marker necessary). Current commercial language models cannot reliably monitor their own calibration. A model-origin claim therefore cannot certify its own reliability from inside the producing system, and its reliability can be established only externally, through scoring against outcomes. The substrate marker is necessary for the scoring machinery to apply the correct prior, because absent the marker a model-origin claim is indistinguishable from a human-origin claim and inherits the calibration presumption that attaches to human track records, a presumption the model has not earned through a self-monitoring capacity it does not possess.
Proof sketch. Take as established the claim, argued in the alignment papers and not softened here, that a current commercial language model cannot be trusted to accurately report its own confidence or to detect its own errors (Beier, 2026g; Beier, 2026ad). A track record functions as a calibration signal under two conditions: the scored entity bears the consequence of error, and the entity's interim self-reports carry some information about its reliability between scored outcomes. For a human, both hold to a degree. For a current model, the first can be made to hold only by binding the output to a responsible human identity, and the second does not hold, because the self-report is unreliable by the premise. The scoring machinery must therefore treat a model-origin claim's interim self-reports as uninformative and rely entirely on outcome scoring, while it may treat a human-origin claim's self-reports as partially informative. These are different priors. Applying them requires distinguishing the two origins, which is exactly what the marker does. Without the marker, the machinery applies one prior to both, which is miscalibrated for at least one of them: either it over-trusts model self-reports or it under-trusts human ones. The marker is what makes the partition observable, and the partition is what the correct prior requires. End sketch.
The result is the formal reason the self-monitoring claim cannot be quietly relaxed. If a model could monitor itself, a model-origin claim could vouch for its own calibration and the marker would carry less weight. Because it cannot, the only valid basis for a model-origin claim's reliability is external scoring, and the marker is the thing that tells the machinery to demand that external basis rather than extending a presumption the model has not earned. The marker is not a courtesy to readers who prefer human authors. It is the load-bearing distinction without which the calibration accounting is wrong.
The flood of Section 1 has a name in economics. When buyers cannot distinguish high-quality from low-quality goods, the price they will pay reflects the average, high-quality sellers cannot earn a premium, and the market unravels toward the low end (Akerlof, 1970). The same structure threatens a commons that cannot distinguish substrate.
Proposition S-3 (Truthful marking breaks the lemons dynamic). If the substrate marker is absent or freely forgeable, the commons admits a lemons dynamic: a producer can generate model-origin assessments that are structurally correct and analytically empty at near-zero marginal cost and present them as indistinguishable from high-effort human judgment, the readers' valuation of an unmarked assessment falls toward the value of the low-cost output, and high-effort producers withdraw. A truthful, costly-to-forge marker combined with outcome scoring breaks the dynamic, because readers can condition on substrate and the track record penalizes empty assessments regardless of substrate.
Proof sketch. The empty assessments are cheap to produce and structurally indistinguishable from careful ones at the moment of reading, so without a marker a reader cannot tell them apart and prices the average. As the share of cheap empty assessments rises, the average value falls, the return to high-effort human production falls below its cost, and high-effort production contracts, which is the unraveling. A truthful marker interrupts the unraveling at the first step by letting the reader separate the substrates and price each on its own distribution rather than on the pooled average. Outcome scoring interrupts it at a second step by lowering the track record of any producer, human or model, whose assessments fail against outcomes, so the empty assessments degrade their producer's standing once outcomes resolve, removing the cost advantage of producing them at scale. For the marker to do this work it must be costly to forge, since a freely forgeable marker would let the empty-assessment producer relabel as human and reenter the pool. The detection channels that make the marker costly to forge are the ones the adversarial analysis already assembled, and the marker adds a provenance channel to that set, which has no shared blind spot with the others (Beier, 2026an; Beier, 2026y). End sketch.
The hollow-disclosure attack that the adversarial paper identified is the lemons dynamic run by an adversary rather than emerging from honest cost differences (Beier, 2026an). The defense is the same in both cases. The marker plus scoring lets the commons separate cheap imitation from costly judgment and lets outcomes punish the imitation once they resolve. What the marker cannot do alone is catch a producer who lies about substrate, and that is why it joins the existing detection channels rather than replacing them.
The third category is the one that will dominate in practice, because most real assessments will soon be produced by a person working with a model. The temptation is to read hybrid as a blend that splits the difference between the pure substrates. The failure structure says otherwise.
Proposition S-4 (Hybrid inheritance). A hybrid Tensor inherits the failure modes of both substrates rather than their average. A hybrid record is reliable on a facet only to the extent that the substrate responsible for that facet is reliable on it, so the marker should record the division of labor across facets, and that division is derivable from the composition lineage when the hybrid record is a composition of a human Tensor and a model Tensor.
Proof sketch. Consider a hybrid record in which the model produced the Input and Method facets and the person produced the Output and Scaling facets. The reliability of the Input facet depends on the model's failure modes on sourcing, not on the person's, because the person did not produce it. The reliability of the Scaling facet depends on the person's failure modes on systemic reasoning, not on the model's. A facet's reliability is therefore governed by whichever substrate produced it, and the record as a whole carries both substrates' failure modes, one per facet, rather than a blended failure mode that is the average of the two. Recording which substrate produced which facet is what lets the scoring machinery attribute a failed facet to the responsible substrate and update the field-specific track record accordingly, using the diagonal weight that the corrected tensor already carries (Beier, 2026, "The Six-Facet Beier Tensor," Definition 3). When the hybrid is assembled by composing a human-origin Tensor and a model-origin Tensor, the division of labor is already present in the composition history, and the facet-level substrate marking can be read off the lineage rather than re-entered by hand (Beier, 2026, "The Six-Facet Beier Tensor," Proposition BT6-3). End sketch.
The hybrid case is where the substrate marker meets the composition algebra, and the meeting is productive: a careful hybrid workflow leaves a lineage that records which substrate did what, and the field-specific track record can then credit the human's systemic judgment and the model's sourcing speed separately, or assign blame to whichever facet failed. A hybrid record that hides the division of labor forfeits this, and is scored as a single substrate whose failure modes are the union of both, which is the conservative treatment the marker defaults to when the lineage is absent.
The substrate marker raises questions that the measurement of a live commons will answer. The share of records at each substrate, tracked over time, is a measure of how fast machine production enters the commons, and the relative track-record trajectories of the three substrates are a test of where each is reliable, with the facet-level division in hybrid records giving the finest-grained version of that test. The interaction of the marker with the disposition is a design question the previous paper opened: a model-origin record may warrant a different default disposition than a human-origin one, because the affected-party reasoning the scope test requires is exactly the kind of judgment a model is least equipped to make about itself. The deepest open question is the one the alignment papers carry, which is whether a future system could earn the calibration presumption that current models cannot, and what the marker would have to record if the self-monitoring gap narrowed. The marker is built to make that change legible if it comes.
The provenance of a record and the depth of work that went into it are the two metadata channels that describe how a record was made rather than who may read it. The substrate marker handles the first. The next paper takes the second, the engagement level, and asks why the depth of cognitive work belongs in the record and how it should bear on the score.
A Beier Tensor records the depth of cognitive work that produced it, from a single-glance assessment to a multi-month study. The engagement level is metadata, and it is easy to mistake for a restatement of confidence or of track record. It is neither. This paper proves that the engagement level is orthogonal to both, that it is most informative exactly where the track record is least informative, that it must be kept out of the scoring rule even though it belongs in the record, and that its honesty is policed indirectly rather than by verification. The orthogonality result is that effort, stated certainty, and historical calibration vary independently in the cross-section, so a reader who knows the engagement level updates on something the confidence field and the track record weight do not carry, namely how far the assessment has already survived scrutiny and how likely it is to be revised under more. The complementarity result is that for claims which have not yet resolved, the track record speaks to the author's general calibration but not to the specific assessment's robustness, and the engagement level supplies the missing claim-specific signal, which makes it most valuable in long-horizon and novel domains where the track record is quiet. The mechanism-design result is that the engagement level must not enter the proper scoring rule, because doing so would reward a reported input rather than the calibration of a claim against an outcome, would break strict propriety, and would be gameable; the engagement level is instead a reader-side conditioning variable, on the same footing as the disclosure lag. The honesty result is that an author who inflates engagement gains no score and sees the reader benefit erode, because the track record reveals whether that author's claimed engagement predicts accuracy, and a reader rationally discounts engagement reports from an author whose engagement does not.
Keywords: engagement level, effort, proper scoring, calibration, robustness, reader inference, metadata, OCOT
An expert glances at a chart and says, with high certainty, that a battery cost curve will cross a threshold within two years. A novice spends three months on the same question and concludes, with low certainty, that it might. A second expert runs a six-month study and arrives, with high certainty, at the opposite call from the first expert's glance.
Three quantities are moving across these cases, and they are independent. The certainty each assessor states is one quantity. The historical calibration each assessor brings, the track record that says how often their stated certainties have matched outcomes, is a second. The amount of work each assessment took is a third. The first expert's glance is high certainty, high track record, and low effort. The novice's study is low certainty, low track record, and high effort. The second expert's study is high certainty, high track record, and high effort, pointing the other way. No two of these three quantities determine the third.
The architecture carries all three. Certainty lives in the confidence the facets state. Calibration lives in the track record weight that the metric uses. The amount of work lives in the engagement level, a metadata coordinate that records depth from a glance to a multi-month study (Beier, 2026, "The Six-Facet Beier Tensor," Section 2). A reader who has only certainty and calibration is missing the third quantity, and this paper is about what that quantity carries, where it carries the most, why it must stay out of the score, and how it stays honest.
Let the engagement level of a Tensor be an element e of an ordered set E, graded from minimal to maximal depth of cognitive work. The grading is ordinal: a multi-month study ranks above a day's analysis, which ranks above a glance, without the intervals between grades carrying cardinal meaning. The engagement level is metadata in the sense of the corrected tensor, framework-invariant and read identically by every analytical framework, unlike the facets, which different readers project differently (Beier, 2026, "The Six-Facet Beier Tensor," Definition 2).
Engagement can be recorded at the level of the whole record or at the level of individual facets, because an author may engage deeply on one facet and lightly on another. A quantitative analyst may spend a month on the Method facet and a minute on the Scaling facet, and the field-specific track record the corrected tensor already carries makes facet-level engagement the natural granularity (Beier, 2026, "The Six-Facet Beier Tensor," Definition 3). The results below are stated for a scalar engagement level for clarity, and each extends to the facet-level grading by applying it per facet.
The first claim to establish is that the engagement level is a channel of its own and not a relabeling of something the record already carries.
Proposition E-1 (Orthogonality). The engagement level carries information not contained in the confidence field or the track record weight. Effort, stated certainty, and historical calibration vary independently in the cross-section, so for a suitable measure of an assessment's robustness, the conditional mutual information between robustness and engagement, given confidence and track record, is strictly positive.
Proof sketch. Take robustness to mean the degree to which the assessment would survive further scrutiny without revision. Construct records that hold confidence and track record fixed while varying engagement: the same expert, with the same track record, issues a glance and a deep study on related questions, both at the same stated confidence. The glance and the study differ in how much checking the conclusion has already withstood, so they differ in expected revision under further scrutiny, which is the robustness measure. That difference is associated with engagement and not with confidence or track record, since those were held fixed by construction. A reader who observes engagement therefore updates on robustness in a way a reader who observes only confidence and track record cannot, which is the strict positivity claimed. End sketch.
The content the engagement level adds is robustness to scrutiny, and robustness is distinct from each of the other two quantities. Confidence is the author's stated probability, which a careless author can set high after little work. Track record is the author's calibration averaged over past resolved claims, which says nothing specific about how much evidence this particular claim rests on. Engagement is how much this claim has already been worked, which bears on how likely it is to move when worked further. The three answer different questions, and a reader who wants to know whether an assessment is likely to hold needs the answer engagement gives.
Orthogonality establishes that engagement carries something. The next result locates where that something is worth the most.
Proposition E-2 (Complementarity). For a claim that has not yet resolved, the track record speaks to the author's general calibration but not to the specific assessment's robustness, and the engagement level supplies the claim-specific signal the track record omits. The engagement level is therefore most informative in long-horizon and novel domains, where the track record on the specific kind of claim is short or absent and resolution lies far in the future.
Proof sketch. The track record weight is a property of the author, aggregated over the author's past resolved claims. For a claim that has not resolved, the weight gives the reader a prior on reliability conditional on the author, but it does not condition on the depth of this particular assessment, because depth is not what the weight aggregates. The engagement level conditions on exactly that depth. The two are complementary inputs to the reader's posterior on the unresolved claim, one carrying the author's general standing and the other carrying this claim's specific robustness. In a long-horizon domain, resolution is distant, so the reader must lean on pre-resolution signals for an extended period, and in a novel domain the author's track record on the specific kind of claim is short, so the author-level prior is weak. In both, the claim-specific signal that engagement provides carries proportionally more of the reader's inference, which is the located value. End sketch.
The result matters because the domains the architecture most wants to reach are precisely the long-horizon and novel ones: climate trajectories, aquifer depletion, institutional erosion, and the systemic effects that the Scaling facet exists to surface. These are the domains where outcomes resolve slowly and the track record cannot yet distinguish the calibrated from the uncalibrated within the window of decision relevance (Beier, 2026v). The engagement level is the signal that does some of that work in the interim, telling a reader that a far-horizon assessment rests on a month of analysis rather than a moment of intuition, in a setting where the track record will not say so for years.
The temptation, once engagement is established as informative, is to reward it: to give a correct high-effort call a higher score than a correct glance. The temptation has to be refused, and refusing it is the central mechanism-design result of the paper.
Proposition E-3 (Effort-blind scoring). The engagement level must not enter the proper scoring rule. Letting it enter would make the score depend on a reported input rather than on the calibration of a probabilistic claim against an outcome, would break strict propriety, and would be gameable. The engagement level is instead a reader-side conditioning variable, on the same footing as the disclosure lag, and the reader's use of it is not part of the incentive mechanism the scoring rule governs.
Proof sketch. A strictly proper scoring rule rewards a forecaster for reporting the belief the forecaster holds, by making the expected score maximal at truthful reporting and at nothing else (Gneiting and Raftery, 2007). The rule achieves this by depending only on the probabilistic claim and the realized outcome. Suppose the engagement level entered the score, for instance through a bonus to high-engagement correct calls. Three failures follow. The score would now depend on a reported input, engagement, in addition to the claim and the outcome, so it would reward marshaling effort rather than reporting calibrated belief, which is a different objective. The rule would lose strict propriety on the augmented report space, because an author could raise the expected score by adjusting the reported engagement rather than by reporting a better-calibrated probability, so truthful probability reporting would no longer be the unique maximizer. And the bonus would be gameable, because engagement is in general not verifiable, so an author could claim high engagement at no cost and collect the bonus. The conclusion is that the score stays a function of the probabilistic claim and the outcome alone, which keeps it strictly proper and effort-blind. The engagement level remains available to the reader, who may condition a decision on it exactly as the reader conditions on the disclosure lag, because the reader's inference is a private decision-theoretic act and not part of the scored mechanism (Beier, 2026, "Strike-Time and Transmit-Time," Proposition T-3). End sketch.
The separation the proposition draws is the architecture's recurring one. The score is sacred and depends only on outcomes, which is what keeps it honest and ungameable. The metadata is reader-facing and informs inference without touching the incentive. Engagement joins the lag and the substrate marker on the reader-facing side of that line. A reader rationally weights a far-horizon assessment by its engagement, and the author who produced it earns no points for the effort and loses none for its absence, earning points only when the claim proves out. Effort changes how a record is read. It does not change how a record is scored.
Because engagement is self-reported and generally unverifiable, the obvious worry is that authors will inflate it, claiming deep work they did not do in order to have their unresolved claims weighted more heavily. The worry is real and the architecture disarms it without verification.
Proposition E-4 (Honesty through the engagement-calibration statistic). An author who inflates the engagement level gains no score, by Proposition E-3, and sees the reader benefit erode over time. Define an author's engagement-calibration statistic as the association, across the author's resolved claims, between claimed engagement and realized accuracy. A reader who weights an author's engagement reports by this statistic discounts the engagement reports of an author whose claimed engagement does not predict accuracy, so inflated engagement provides no lasting reader benefit.
Proof sketch. By Proposition E-3 the score is effort-blind, so a false engagement claim cannot improve the author's score; the only benefit of inflation is to influence readers' weighting of the author's unresolved claims. That benefit depends on readers treating the author's engagement reports as informative. As the author's claims resolve, a reader can compute whether the author's high-engagement claims have in fact resolved more accurately than the author's low-engagement claims. For an honest author whose deep work does improve accuracy, this association is positive, and the engagement reports are informative and worth weighting. For an author who inflates, the claimed engagement does not track accuracy, the association is near zero, and a reader who conditions on the engagement-calibration statistic finds the author's engagement reports uninformative and discounts them. The inflation thus yields no score and a reader benefit that decays to nothing as the statistic accumulates, so it is not a profitable strategy in the long run. End sketch.
The discipline operates through the same track record that scores the claims themselves, applied now to the joint distribution of claimed engagement and outcome rather than to the claim alone. An author cannot make engagement mean something by asserting it. Engagement comes to mean something for an author only when that author's deep work has repeatedly produced more accurate calls than that author's shallow work, and a reader who has watched the record knows the difference. The signal is earned in the same currency as everything else in the system, which is the record of being right.
The engagement level completes the channels that describe how a record was produced, and it leaves two threads for later work. The facet-level grading interacts with the composition lineage, because a composed record inherits the engagement of the records it draws on, and a synthesis that cites a month-long study should carry that depth into its own Input facet rather than presenting as a fresh glance; the accounting of inherited engagement is a question for the composition algebra. The empirical thread is the engagement-calibration statistic of Section 6, which is measurable in a live commons and whose distribution across authors is a test of whether deep work pays in accuracy, with the prediction that the association between engagement and accuracy is strongest in the long-horizon domains where Proposition E-2 places engagement's value.
The metadata channels that describe who may read a record and how it was made are now in place: the disposition, the two timestamps, the substrate, and the engagement level. What remains is the structure that lets records build on records, the composition history and the lineage it indexes. The corrected tensor proved that lineage is closed and acyclic. The next paper develops the algebra on that lineage in full, and shows how provenance, emergence detection, and the inheritance of engagement and substrate all rest on it.
A Beier Tensor can build on earlier Tensors, recording the records it draws on in its composition history. The corrected tensor proved that the resulting lineage is closed under composition and acyclic. This paper develops the algebra on that lineage and shows that several capabilities the architecture promises rest on it. Citation is not a single relation but a labeled one: a record builds on, endorses, refutes, or merely references each parent, and the labeled lineage is an acyclic graph whose ancestry relation is a partial order. On that order the three emergence signatures the architecture detects become decidable predicates: convergence is independent lineages whose conclusions agree, divergence is lineages in refutation or a common source whose descendants conflict, and a gap is a region of the constraint space that no record's conclusion reaches. The predicates are well-posed once a domain ontology supplies the constraint coordinates and a similarity on conclusions, and their decidability is the formal basis for the convergence and divergence alerts and the gap mapping that the knowledge-coproduction engine performs. The paper then proves the result that keeps the convergence signal honest: agreement between two records is confirmatory only when their lineages are independent, because a shared ancestor induces correlation that makes inherited agreement masquerade as independent confirmation, so the confirmation value of agreement degrades with the overlap of the two records' ancestor sets and is full only when those sets are disjoint. Finally the paper supplies the inheritance rules the metadata papers left open: a composed record's substrate and engagement on a facet are governed by the parent whose work that facet draws on, with a depth floor that forbids presenting inherited deep work as a fresh glance, and a composition cannot disclose content that a parent's disposition restricted, which forbids laundering a private parent into a public record.
Keywords: composition, lineage, directed acyclic graph, emergence detection, independence, common cause, inheritance, OCOT
Two analysts publish assessments of the same aquifer and reach the same conclusion. The agreement looks like confirmation, and a reader is inclined to trust a conclusion two independent experts have reached. Then a third fact emerges: the second analyst read the first analyst's assessment before writing, and built on it. The agreement is no longer two independent observations converging. It is one observation and its echo. The second analyst may have added nothing the first did not supply, and the apparent confirmation is the same claim counted twice.
A commons of structured judgment has to tell the difference between convergence and echo, and it cannot do so from the conclusions alone, because the conclusions are identical in both cases. The difference is in the lineage: whether the two records share an ancestor. The architecture records that lineage in the composition history of each Tensor, and the corrected tensor proved that the lineage forms a closed, acyclic structure (Beier, 2026, "The Six-Facet Beier Tensor," Proposition BT6-3). This paper develops the algebra on that structure and shows that the echo problem, the detection of convergence and divergence and gaps, and the inheritance of provenance and depth across composed records all reduce to operations on the lineage. The closure theorem said the structure exists. What follows is what the structure does.
A citation carries an intent. A record that builds on a parent extends it; a record that endorses a parent agrees with it; a record that refutes a parent disagrees with it; a record that references a parent only points to it for context. These are different relations, and collapsing them loses the information that distinguishes agreement from disagreement in the lineage.
Definition (Labeled lineage). The lineage is the directed graph G with the Tensors as vertices and a labeled edge from each record to each parent it cites, the label drawn from { builds-on, endorses, refutes, references }. An edge points from the citing record to the cited parent.
Proposition C-1 (Acyclicity and the ancestry order). The labeled lineage is acyclic, and the ancestry relation, in which one record is an ancestor of another when a directed path runs from the second to the first, is a strict partial order on the Tensors.
Proof sketch. Acyclicity is inherited from the corrected tensor: every citation edge points to a parent whose strike-time is earlier than or equal to the citing record's, with ties broken by the transmit-time that fixes publication order, so a directed cycle would require a record to precede itself in that order, which is impossible (Beier, 2026, "The Six-Facet Beier Tensor," Proposition BT6-3). The edge labels do not enter this argument, so labeling preserves acyclicity. The ancestry relation is the transitive closure of the edge relation. It is irreflexive because a record cannot reach itself in an acyclic graph, transitive because the closure of a relation is transitive by construction, and antisymmetric because two records each an ancestor of the other would close a cycle. A relation that is irreflexive, transitive, and antisymmetric is a strict partial order. End sketch.
The order is what gives the lineage its usable structure. Ancestry is well-defined, so the question of whether two records share an ancestor has a definite answer, and the labels let that answer distinguish a shared endorsement from a shared refutation. The rest of the paper operates on this ordered, labeled structure.
The knowledge-coproduction engine the architecture specifies is built to surface three things from a growing commons: where independent work is converging, where it is in conflict, and where the constraint space has no coverage at all (Beier, 2026, cosci.app specification). Each of the three is a predicate on the labeled lineage.
Proposition C-2 (Decidability of the emergence signatures). Given a domain ontology that supplies a set of constraint coordinates and a similarity measure on Output facets, the following are decidable predicates on the labeled lineage. Convergence holds of a set of records when their ancestor sets are pairwise disjoint and their Output facets agree under the similarity measure. Divergence holds of a set of records when they are connected by refutes edges, or when they descend from a common ancestor and their Output facets disagree. A gap holds of a region of constraint coordinates when no record's Output facet covers that region.
Proof sketch. Each predicate is a finite computation on a finite graph. Pairwise ancestor-set disjointness is a reachability computation, which terminates on the acyclic graph of Proposition C-1. Output agreement and disagreement are evaluations of the supplied similarity measure on the relevant Output facets, which terminate once the measure is given. Coverage of a region is a check of whether any record's Output coordinates fall in that region, which terminates over the finite record set. Each predicate therefore reduces to terminating computations and is decidable. The dependence on the ontology is essential and is stated as a hypothesis: the constraint coordinates and the Output similarity are not produced by the lineage itself but supplied by the domain ontology that the institutional program builds for each constraint environment, and the predicates are well-posed exactly to the extent that the ontology is. End sketch.
The three predicates are the formal content of the engine's three functions. A convergence alert fires when Proposition C-2's convergence predicate becomes true of a newly completed set of independent records. A divergence alert fires when the divergence predicate becomes true. A gap map is the set of constraint regions on which the gap predicate holds. The engine's claim to detect emergence is the claim that these predicates can be evaluated as the commons grows, and Proposition C-2 establishes that they can, with the caveat that the constraint coordinates and the conclusion similarity come from the domain ontology and inherit whatever coarseness the ontology carries.
The convergence predicate of Proposition C-2 was stated with a condition that has not yet been justified: the ancestor sets must be pairwise disjoint. The echo of Section 1 is what the condition guards against.
Proposition C-3 (Independence and the value of agreement). Agreement between two records is confirmatory only when their lineages are independent. A shared ancestor induces correlation between the two records' assessments, so their agreement may be inherited rather than independently arrived at, and the confirmation value of the agreement degrades with the overlap of their ancestor sets, reaching its full value only when those sets are disjoint.
Proof sketch. Two records that share an ancestor have both drawn, directly or through intermediate records, on a common source. The common source is a common cause of the two assessments, and a common cause induces statistical dependence between them even when neither record influenced the other (Pearl, 1988). Conditional on the shared ancestor the two assessments may be independent, but unconditionally their agreement reflects the shared input as well as any independent observation, so the agreement overstates the independent evidence by the contribution of the shared ancestor. When the ancestor sets are disjoint, no common cause links the two assessments, their agreement reflects independent observation, and the confirmation is full. As the ancestor sets overlap more, more of the agreement is attributable to common input and less to independent observation, so the confirmation value falls, reaching zero in the limit where the two records share their entire lineage and the second adds nothing the first did not supply. End sketch.
The result is the lineage-level statement of a concern the corpus carries throughout: agreement among observers is informative only when the observers are independent, which is also why the combinatorial information that many disclosers produce requires their independence and collapses when they are echoing one another (Beier, 2026ag). The convergence predicate of Proposition C-2 is built to respect this by requiring disjoint ancestry before it counts a set of records as converging. A commons that ignored the requirement would mistake a chain of mutual citation for a chorus of independent confirmation, and would report consensus where it had only an echo. The lineage makes the distinction decidable, and Proposition C-3 is the reason the distinction has to be drawn.
A composed record draws on its parents, and the metadata papers left open what it inherits from them. The substrate paper established that a hybrid record's reliability is governed facet by facet by whichever substrate produced each facet (Beier, 2026, "The Substrate Marker," Proposition S-4). The engagement paper established that depth is recorded per facet and that inherited depth should not be presented as fresh work (Beier, 2026, "The Engagement Level," Section 7). The composition algebra supplies the rules.
Proposition C-4 (Substrate and engagement inheritance). For a composed record, the substrate of a facet is the substrate of the parent whose work that facet draws on, and is hybrid when the facet combines work from parents of different substrates. The engagement of a facet is at least the maximum engagement among the parents whose work that facet draws on. A composed record cannot record a lower engagement on a facet than the deepest parent that facet builds on.
Proof sketch. A facet of a composed record that draws on a parent contains the work embodied in that parent's corresponding facet, so the reliability of the composed facet depends on the substrate that produced the drawn-on work, which is the parent's substrate, or on the several substrates when several parents contribute. The substrate marking of the composed facet must therefore record the contributing parent's substrate, hybrid when more than one contributes, which is the inheritance rule for substrate and is the facet-level accounting of S-4 applied along the lineage. For engagement, the work embodied in a parent's facet is present in the composed facet, so the composed facet rests on at least that much work, and recording a lower engagement would understate the depth the facet actually carries. The engagement floor is therefore the maximum engagement among the contributing parents, which forbids presenting a facet built on a month-long study as a fresh glance. End sketch.
The rules close a laundering route. Without the substrate floor, a producer could compose from a model-origin parent and mark the result human, erasing the provenance the substrate marker exists to preserve. Without the engagement floor, a producer could build on deep work and present it as light, claiming the robustness of the parent's depth while disclaiming the inheritance. The floors make the composed record carry forward the provenance and the depth of what it was built from, so that the metadata of a synthesis reflects the metadata of its sources rather than resetting at each step.
Proposition C-5 (Disposition under composition). A composed record cannot disclose content that a parent's disposition restricted. A record composed from a parent marked private or group-scoped may be published at a broader disposition only to the extent that it does not reveal the restricted content of that parent, and revealing the parent's restricted content requires the consent that the parent's disposition encodes.
Proof sketch. The disposition of a parent fixes the readership permitted to see that parent's content (Beier, 2026, "The Transmit Facet," Proposition D-1). A composed record that drew the parent's restricted content into a Output or Input facet and then published it at public disposition would deliver the restricted content to readers the parent's disposition excluded, defeating the parent's disposition. The composition's disposition is therefore bounded on any content inherited from a restricted parent by that parent's disposition, and broadening it on that content requires the consent the parent's disposition represents. A composed record may still be published broadly on the content it adds itself and on content inherited from parents whose dispositions permit it, so composition does not force a record down to its most restricted parent on all content, only on the content that parent restricted. End sketch.
The disposition rule prevents composition from becoming a leak. A private assessment that an author shared with a named group cannot be repackaged into a public synthesis that exposes its substance, because the public synthesis would reach exactly the readers the private disposition excluded. Composition carries the disposition of its sources forward on the content it draws from them, in the same way it carries their substrate and their depth, so that the act of building on a record respects the terms under which the record was made available.
The composition algebra closes the structural account of the corrected tensor. The facets carry the assessment, the metadata carries who may read it and how it was made, and the lineage carries how records build on one another, with the inheritance rules ensuring that a composition reflects its sources rather than erasing them. The capabilities that depend on the lineage now have their foundation: emergence detection is the evaluation of decidable predicates on the labeled order, the honesty of the convergence signal is the independence requirement of Proposition C-3, and the integrity of provenance and depth across syntheses is the inheritance of Propositions C-4 and C-5.
One facet-level structure remains unformalized across the metadata sequence, and it is not metadata but a property of the facets themselves: a facet may be filled or may carry a typed absence, and the four kinds of absence carry different information. The next paper takes the typed null and shows that the alphabet of absence is itself a channel, sharpening the binary treatment of silence the corpus began with into the graded structure the facets actually use.
Akerlof, G. A. (1970). "The Market for Lemons: Quality Uncertainty and the Market Mechanism." Quarterly Journal of Economics, 84(3), 488 through 500.
Amari, S. (1985). Differential-Geometrical Methods in Statistics. Berlin: Springer.
Beier, G. C. (2026). "Strike-Time and Transmit-Time: The Information Economics of the Two-Timestamp Record." Working Paper.
Beier, G. C. (2026). "The Composition Algebra: Lineage, Independence, and the Detection of Emergence." Working Paper.
Beier, G. C. (2026). "The Endogenous Governor: How Disclosure's Scaling Field and Track Record Capital Produce Self-Regulating Combinatorial Discovery." Working Paper.
Beier, G. C. (2026). "The Engagement Level: Depth of Work as a Reader's Signal, Not a Scorer's Input." Working Paper.
Beier, G. C. (2026). "The Six-Facet Beier Tensor: A Formal Definition of the Extended Price with Disposition, Temporal, and Composition Structure." Working Paper.
Beier, G. C. (2026). "The Substrate Marker: Provenance, Self-Monitoring, and the Mixed Human-Machine Commons." Working Paper.
Beier, G. C. (2026). "The Transmit Facet: Disposition, Scope, and Why the Public Option Cannot Be Removed." Working Paper.
Beier, G. C. (2026). Good Judgment: Building the Foundation. Susarb LLC.
Beier, G. C. (2026). The Future Transparent: Pension Fund Edition. Susarb LLC.
Beier, G. C. (2026). cosci.app specification, version 2.0. Susarb LLC.
Beier, G. C. (2026aa). "The Economics of Cognitive Trade: Three Propositions on the Human-AI Exchange Rate." Working Paper. Zenodo.
Beier, G. C. (2026ad). "The Judgment Kernel: Three Propositions on the Cognitive Architecture for Wise Machines." Working Paper. Zenodo.
Beier, G. C. (2026ag). "Factorial Growth Through Coordination: Why the Interaction of N Disclosers Produces Combinatorial Information, Not Linear." Working Paper. Zenodo.
Beier, G. C. (2026aj). "The Information Content of Non-Disclosure: Silence as Signal in Structured Judgment Markets." Working Paper. Zenodo.
Beier, G. C. (2026al). "How the Improvement Mechanism Scales Across Populations: The Economics of Collective Calibration Improvement." Working Paper. Zenodo.
Beier, G. C. (2026an). "Adversarial Disclosure Economics: Attacks on the Judgment Data Stream and the Cross-Disciplinary Defense That Unification Provides." Working Paper. Zenodo.
Beier, G. C. (2026b). "The Disclosure Incentive: How Track Record Capital Resolves the Grossman-Stiglitz Paradox and Reveals the Competition-Cooperation Synthesis Across Four Disciplines." Working Paper. Zenodo.
Beier, G. C. (2026e). "Supple: How Beierspace Extends Eigenspace." Working Paper. Zenodo.
Beier, G. C. (2026g). "The Ngakpa Principle: Why AI Alignment Requires a Pre-Kernel Architecture of Wisdom and Compassion." Working Paper. Zenodo.
Beier, G. C. (2026h). "Excess Distortion in the Price System: A Rate-Distortion Analysis of Scalar Compression in Multi-Dimensional Constraint Environments." Working Paper. Zenodo.
Beier, G. C. (2026q). "The Scalar Lobby: Political Economy of the Transition from Scalar to Multi-Dimensional Observation." Working Paper. Zenodo.
Beier, G. C. (2026r). "The Incentive Compatibility of the Disclosure Incentive: A Mechanism Design Foundation for the Judgment Market." Working Paper. Zenodo.
Beier, G. C. (2026u). "Compositional Improvement Beyond the Gaussian Case." Working Paper. Zenodo.
Beier, G. C. (2026v). "A Tight Bound on the Calibration Threshold for Honest Disclosure: Order Statistics Refinement of the Strategic Selection Analysis." Working Paper. Zenodo.
Beier, G. C. (2026y). "Adversarial Detection Superadditivity and Geometric Distinguishability." Working Paper. Zenodo.
Beier, G. C. (2026z). "The Beier Tensor: A Formal Geometric Definition of the Deep Price." Working Paper. Zenodo.
Benkler, Y. (2006). The Wealth of Networks: How Social Production Transforms Markets and Freedom. New Haven: Yale University Press.
Blackwell, D. (1953). "Equivalent Comparisons of Experiments." Annals of Mathematical Statistics, 24(2), 265 through 272.
Ericsson, K. A., Krampe, R. T., and Tesch-Romer, C. (1993). "The Role of Deliberate Practice in the Acquisition of Expert Performance." Psychological Review, 100(3), 363 through 406.
Gneiting, T., and Raftery, A. E. (2007). "Strictly Proper Scoring Rules, Prediction, and Estimation." Journal of the American Statistical Association, 102(477), 359 through 378.
Grice, H. P. (1975). "Logic and Conversation." In P. Cole and J. Morgan (eds.), Syntax and Semantics, Volume 3: Speech Acts, 41 through 58. New York: Academic Press.
Mailath, G. J. (1987). "Incentive Compatibility in Signaling Games with a Continuum of Types." Econometrica, 55(6), 1349 through 1365.
Ostrom, E. (1990). Governing the Commons: The Evolution of Institutions for Collective Action. Cambridge: Cambridge University Press.
Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Francisco: Morgan Kaufmann.
Savage, L. J. (1971). "Elicitation of Personal Probabilities and Expectations." Journal of the American Statistical Association, 66(336), 783 through 801.
Shannon, C. E. (1948). "A Mathematical Theory of Communication." Bell System Technical Journal, 27, 379 through 423 and 623 through 656.
Spence, M. (1973). "Job Market Signaling." Quarterly Journal of Economics, 87(3), 355 through 374.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
July 2026