Correction, 2 November 2026. Replace the cited floor with the following. Under the corrected theorem 10.1 of 2026n, the number of constraint interactions discoverable from N disclosers is bounded below by a constant times N(N-1)/2. The information those interactions carry has no proven lower bound of that order in the deposited corpus. The ordering channel of proposition OC-2 is additive to whatever the set carries, by the chain rule, and its capacity remains log N!, of order N log N. Whether "the floor dominates" the ordering channel is therefore a statement about the corrected count, not about proven information, and it holds for the count. A summary of this correction is on the papers page at https://www.susarb.com/papers; the full correction note, dated 2 November 2026, will be deposited as a version of this record.
PRIORITY FILING - INITIAL DRAFT
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). A short result connecting the temporal architecture of the Beier Tensor to the combinatorial growth claims, locating the factorial, the exponential, and the quadratic as measurements of three different objects.
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
Three growth rates circulate in the corpus and in conversation about it, and this paper fixes what each one measures so that none is claimed for the others' object. The proven floor on interaction information among N disclosing agents is quadratic. The number of subsets of a disclosure corpus is exponential. The number of distinguishable histories of a strike-timed corpus is factorial, because tamper-evident strike-time converts a set of disclosures into a sequence, and sequences of N distinguishable events are permutations. The paper's first result states the factorial cardinality of the history space as a consequence of the temporal architecture rather than as an aspiration. The second result bounds the information the realized ordering can carry at log of N factorial, which is of order N log N, an ordering channel that adds to the quadratic interaction floor without changing its growth order. The third result shows the scalar price system carries this channel at zero, because many judgment orderings are consistent with a single price path, while the strike-timed record carries it at capacity by construction. The name Beier Factorial is then given its precise referent: it names the cardinality of the history space the substrate makes navigable, the territory on which the quadratic-and-above results are proven, and the distinction between naming a space and proving a growth rate is what keeps the claim inside the corpus's evidentiary standard.
Keywords: ordering, permutation space, strike-time, interaction information, combinatorial growth, temporal architecture, OCOT
Discussions of the coordination substrate's growth reach for three rates, and the reach is usually imprecise about what is growing. Pairwise interaction structure among N agents grows quadratically, and the corpus's verified floor on interaction information is of that order. The subsets of a corpus, the collection of which disclosures a reader might combine, number two to the N, which is exponential. And the orderings of a corpus, the distinguishable sequences in which N disclosures can occur, number N factorial. The rates are not competing estimates of one quantity. They are exact counts of three different objects, and the architecture determines which objects exist to be counted.
An untimestamped corpus supports the first two counts and not the third. Its disclosures form a set; the set has pairs and it has subsets, but it has no distinguishable orderings, because nothing in the record says which came first. The Beier Tensor's temporal architecture changes the available objects. Strike-time is fixed at creation, tamper-evident, and never moves, so a corpus of Tensors is not a set but a sequence, and a sequence has an order that is part of the record rather than a reconstruction from outside it. The factorial enters here, and it enters as counting rather than as conjecture. The sections that follow state the count, bound the information it can carry, and locate the channel the scalar system cannot see.
Proposition OC-1 (Factorial cardinality of the history space). Let a corpus consist of N Tensors with distinct strike-times. The space of distinguishable disclosure histories, the possible orders in which the N disclosures could have occurred, has cardinality N factorial. An untimestamped corpus of the same N disclosures has a history space of cardinality one, since no ordering is distinguishable from any other.
Proof sketch. A history is an assignment of the N disclosures to N ordered positions, which is a permutation of N distinguishable elements, and the permutations of N distinguishable elements number N factorial. Distinctness of strike-times guarantees that every permutation is distinguishable from every other, since two histories differ exactly when some pair of disclosures occurs in opposite order and the strike-times record which order obtained. Ties have measure zero under continuous time and are resolved by any fixed tie-breaking rule without affecting the order of growth. The untimestamped case collapses the space because the record contains no information by which one ordering could be told from another; the corpus presents as a set, and a set has one history up to distinguishability. The temporal architecture is therefore what brings the factorial object into existence: the space is not large because the claims are ambitious but because permutations of distinguishable events are being counted, and the tamper-evidence of strike-time (established in the temporal-architecture results, where a single mutable timestamp was shown unable to hold honest scoring and author delay at once) is what makes the count refer to a property of the record rather than a property of a reconstruction. End sketch.
The result is deliberately modest in method and consequential in referent. Nothing is proven about value, welfare, or information growth; what is established is which object the substrate navigates. A routing engine over a strike-timed corpus is choosing among sequences, not among sets, and the space of sequences is the factorial one.
Proposition OC-2 (Capacity of the ordering channel). The information carried by the realized ordering of a corpus, beyond the information in the unordered set of its disclosures, is at most the logarithm of N factorial, which is of order N log N. The ordering channel is additive to, and does not alter the growth order of, the quadratic floor on interaction information.
Proof sketch. Write the record as the pair consisting of the unordered set of disclosures and the ordering imposed on that set. By the chain rule, the entropy of the pair is the entropy of the set plus the conditional entropy of the ordering given the set. The ordering is a random variable over at most N factorial outcomes, so its entropy is bounded by the logarithm of N factorial, and Stirling's approximation puts that logarithm at order N log N. The bound is a capacity: it is attained when orderings are uniform and shrinks as the ordering becomes predictable from the set, which is the empirically interesting case, since predictable sequence is structure and structure is what the coordination measure prices. Additivity to the interaction floor follows from the decomposition itself: the quadratic bound on interaction information is a statement about the set's content, the ordering term is a statement about the sequence, and the two occupy different terms of one identity. The ordering channel therefore enlarges what the record carries without inflating the proven growth order of interaction information, which remains quadratic from below. End sketch.
The bound disciplines the enthusiasm the factorial invites. A space of N factorial histories does not deliver N factorial value; observing which history occurred delivers at most the logarithm of the count, and the logarithm of a factorial is quasilinear. What the substrate gains from its temporal architecture is a real channel of order N log N stacked on a floor of order N squared, and the honest arithmetic is that the floor dominates. The factorial's role is different from a growth rate, and Section 5 states it.
Proposition OC-3 (Erasure and recovery of judgment order). The price path timestamps transactions, not judgments, and the causal order of judgment events is not recoverable from the price path, because many judgment orderings are consistent with a single realized path. The strike-timed record recovers the judgment order exactly. The ordering channel of Proposition OC-2 is therefore carried at zero by the scalar system and at capacity by the Protocol.
Proof sketch. A judgment event is the formation of an assessment; a transaction is its optional, delayed, and size-confounded expression. Judgments that do not transact leave no trace on the path at all, and judgments that do transact enter it after latencies that vary by agent, mandate, and liquidity, so the order of prints confounds the order of assessments with the order of executions. The mapping from judgment orderings to price paths is consequently many to one: distinct sequences of who concluded what first can produce the same sequence of trades, and where the mapping is many to one, the inverse is not a function and the ordering is not recoverable. The strike-timed record inverts this by construction. Each Tensor's strike-time is set at the assessment and is tamper-evident, so the record is an injective register of judgment order, and reading the order off the record is exact rather than inferential. The channel that Section 3 bounded is thus not merely under-used by the scalar system; it is structurally absent from it, one more instance of the general pattern in which the compressed signal is silent on a dimension the architecture makes carryable. End sketch.
Sequence is not decoration on the corpus. Which disclosure moved which reassessment is the causal skeleton of a cascade, and cascades are where the framework's discontinuities live. A system that cannot represent order cannot represent propagation, and propagation is what the phase-transition results are about.
The name can now be given its exact referent. The Beier Factorial names the cardinality of the history space that the temporal architecture makes navigable: N factorial distinguishable sequences over a strike-timed corpus of N disclosures, a count established by Proposition OC-1 as a property of the record. It does not name a proven growth rate of information or value. The proven floor on interaction information is quadratic, the ordering channel adds a term of order N log N, and any claim above those orders is a conjecture about the territory rather than a theorem about the yield. The distinction preserves both halves of what the corpus has been trying to hold at once. The space really is factorial, which is why the name is not hyperbole, and the results really are quadratic and quasilinear, which is why the name is not a proof. A map may lawfully be larger than the region so far surveyed, provided the legend says which is which, and this paper is the legend.
The forward question the distinction opens is where between the floor and the space the realized yield sits as participation broadens. Predictable orderings compress the channel and unpredictable ones fill it, richer interaction structure raises the floor's constant, and the empirical trajectory of both is measurable from the substrate itself once density permits. The factorial marks how much room the measurements have.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
July 2026