One's Own Anchor

Abstract

The Second Pool left a projection that is independent of its fellow, expensive to forge, and evidences nothing but its anchor, and asked whether the framework needs a third condition on projections beyond independent and expensive. It does not. Partition a substrate’s paid work into anchor work, imported to fix a gauge, and native work, the reconciliation the substrate exists to do; a projection’s native share — its marginal forging cost for an adversary already holding every anchor receipt, over the budget — is then exactly its approximate independence from the projection that reads the anchor alone, times its trace gap. Evidencing is independence from one’s own anchor, and the idle projection is the case where that independence is zero. The condition was always in the framework; it had been demanded only against the other projections. Under it, every reading the corpus has built has native share zero as a theorem: in the declared-frame model the honest configuration itself costs exactly the anchor, so an adversary holding the anchor reproduces honest’s distribution for nothing, and any function of declared frames, maps or Laplacians is idle. A Combination Proof of such readings is its anchors. The one reading outside the class — restriction maps fitted from data at overlaps by orthogonal Procrustes, the construction of Singer and Wu, gated on every edge’s residual — is then built against a coalition with no model of the world. The fit absorbs every frame, so the anchor is invisible to it by construction and the comparison that matters is against the public record. Two regimes. Where overlaps are public before commitment, the coalition copies its neighbours’ data and their noise with it: its boundary residual is below honest’s at every coalition size, zero with one source, and the boundary is a subsidy. Where participants commit before comparing, the coalition’s best input is last epoch’s record and it fails exactly when the world’s innovation since then exceeds a threshold computed before running and met to within a grid step; above it, the cheapest passing strategy derives the boundary edges’ prompts and fabricates the interior, so the native share of a coalition is the work-weighted fraction of its boundary — the attack-edge quantity of the Sybil-defence literature, here with a second face: the same cut that bounds admission is the channel the world leaks through. On the program’s own complex a block coalition of sixteen, thirty-two, sixty-four or one hundred and twenty-eight vertices has three, two, three or two boundary edges, so the native cost per identity falls as the inverse of the coalition’s size and the Sybil cap of The Multiplicity Freedom is not a cap on a modular substrate. A rotational innovation is absorbed by the fit at any rate: the measured map has a temporal gauge of its own. What the reading evidences is information about the world not computable from the anchor and the public record — the change since the last reveal, not the work — and the paper says so, names the peer-prediction results that established the qualitative half a decade ago and the concurrent information-theoretic statement five days old, and claims the sheaf-valued form, the theorem for the induced class, the cost accounting, and the measurements.

1. The Idle Projection

The predecessor ended by proposing a word, and the word turns out to be one the framework already had.

The Second Pool bought, on the program’s own substrate, the projection pair that Independent and Expensive said could be bought: a temporal reading independent of the kernel reading, ι = 1.000 in both directions, each with a positive trace gap, at the price of an anchor apiece. And it found that the purchased projection evidenced its anchor and nothing else — survival zero by construction — so that a Combination Proof of order two was two anchors wearing readings, made literal. Its §9.5 called such a projection idle: not redundant, since ι = 1; not free, since τ > 0; contributing to the Sybil cap exactly its anchor’s receipt and to the reading nothing. It asked whether the framework needed a third condition on projections, beyond independent and expensive, and offered evidencing as the name.

This paper answers in the set model of Independent and Expensive §3, where the corpus’s quantities can be computed exactly. The answer is that evidencing is not a third condition. Split the paid pool into anchor work and native work; let π_A be the projection that reads the anchor pool and nothing else — the receipt check every anchor carries; then a projection’s native share, the fraction of the budget an adversary must still pay after holding every receipt, is its independence from π_A in the framework’s own Definition 2.5, multiplied by its trace gap. The idle projection is the one whose independence from its own anchor is zero. The framework had demanded independence of each projection from every other projection and never from the anchor that prices it, which is why a projection could satisfy both stated conditions and read nothing.

With the condition named, the corpus’s readings can be tested against it, and the test is short: every one of them fails, as a theorem rather than a measurement. And the one reading the corpus has kept outside the class — Sign and Work §8.1’s restriction maps measured at overlaps rather than induced from declared frames — passes, and what it evidences is not what the framework hoped.

The conclusion, stated before the machinery: a reading evidences native work exactly to the extent that honest sections carry information about the world not computable from the anchor and the public record. Declared-frame readings carry none. A data-fitted reading carries the world’s innovation since the last reveal, and only where the coalition’s boundary meets honest data it cannot copy.

2. Prior Art, and the Boundary of the Claim

Three of this paper’s components are established results and one is five days old; the boundary is drawn here so that §§3–6 claim only what lies past it.

That a mechanism scoring agreement rewards whatever is common or public rather than what is privately known is the settled finding of peer prediction. Waggoner and Chen [1] state it most cleanly: output-agreement scoring elicits common knowledge, and without verification that is the best any such mechanism can do. Gao, Wright and Leyton-Brown [2] give the impossibility form — whenever agents share any cheap signal about the item, peer prediction has coordinated equilibria in which nobody does the costly work, and this is unavoidable. The literature’s repair is multi-task: Dasgupta and Ghosh [3] and Shnayder, Agarwal, Frongillo and Parkes [4] subtract agreement on unrelated tasks so that uninformed uniformity is dominated, and the equilibrium-multiplicity of the original mechanism [5] is the reason the repair was needed. §5’s finding — that a coalition fitting honest neighbours’ public data at its boundary scores better than the honest neighbours themselves — is the sheaf-valued, continuous instance of Waggoner and Chen, and is cited as such. Bara [6], submitted 1 September 2026, states the evidences-nothing criterion in information terms: a report Z is an epistemic Sybil extension of reports R when I(Θ; Z | R) = 0, and no report-only aggregator can distinguish replication from corroboration. That is concurrent and independent, and it is the qualitative form of this paper’s Theorem 3.3; the general principle behind both is the simulation paradigm of Goldwasser, Micali and Rackoff [7], that a message computable from public information conveys no knowledge.

The reading of §4 is not new either. Restriction maps fitted at overlaps by orthogonal Procrustes are Singer and Wu’s construction for vector diffusion maps [8], carried into sheaf learning by Di Nino, Barbarossa and Di Lorenzo [9], whose per-edge closed form is the one used here, and by Barbero et al. [10]; Hansen and Ghrist [11] learn a sheaf Laplacian from smooth signals globally. None of these papers has an adversary. That only a coalition’s boundary costs it anything is the attack-edge bound of SybilGuard [12] and SybilLimit [13] and the cut bound of Cheng and Friedman’s sybilproof reputations [14]: identities inside the coalition are free, and admission is bounded per honest–sybil edge. §6’s native share of a coalition has that numerator. Proof-of-learning [15] is the nearest attempt at an evidencing condition for machine-learning work — a proof that one trained a model rather than obtained its weights — and the record of its spoofing [16, 17] is the empirical statement that evidencing gradient descent has no known sound instantiation. Liu, Yu and Steeves [18] show, for Bittensor’s weight-copying, that a commitment scheme reduces the copier’s reward; that is a sufficiency argument for §6’s regime, on the substrate Proof of Coherence was written for.

What is claimed. That the native share of a projection is its independence from its own anchor times its gap, exactly, in the model the corpus proves things in (Proposition 3.2). That every reading computed from declared restriction maps has native share zero, for any function of the declared objects (Theorem 3.3), which turns the corpus’s through-line — the trace gap is bought entirely by the anchors — from a measured pattern into a property of the class. That a data-fitted reading’s native share lives on a coalition’s boundary alone and equals a work-weighted attack-edge fraction, with the boundary a subsidy when overlaps are public and a cost only under commitment — the same cut the Sybil literature bounds admission by is the channel the world leaks through, and the peer-prediction literature’s multi-task repair is not the one taken here. And the measurements on the program’s own complex, where the boundary of any block coalition is at most four edges.

3. Evidencing Is Independence

The set model of Independent and Expensive §3 is extended by one partition.

Definition 3.1 (native share). Let the paid pool decompose as P = A ⊔ N: anchor work A, paid degrees of freedom imported to fix a gauge — encodings, chains, searches — and native work N, the paid degrees of freedom the substrate exists to produce: reconciliations, derivations. Let π_A be the projection whose claim is exactly A. The native share of a projection π is

τ_N(π) = ( f(π_A ∧ π) − f(π_A) ) / W,

the marginal fake-cost of π for an adversary who has already paid the anchor, over the budget. A projection evidences if τ_N(π) > 0.

Proposition 3.2 (evidencing is independence from one’s own anchor). τ_N(π) = ι(π | π_A) · τ(π). Proof. Definition 2.5 of the framework, read as the model reads it: ι(π | π_A) = ( f(π_A ∧ π) − f(π_A) ) / f(π), and τ(π) = f(π)/W. ∎

The idle projection is ι(π | π_A) = 0: everything π reads, the anchor’s forger already produced. Nothing has been added to the framework. Definition 2.6 asks that a Combination Proof’s projections be pairwise approximately independent; it does not ask that each be independent of the receipt that prices it, and The Second Pool’s purchased projection is the case that satisfies the first and not the second.

Corollary 3.3 (amendment to Combination Proofs Definition 2.6). A Combination Proof’s projections must be approximately independent of one another and each of its own anchor, in the sense of Definition 2.5 applied with π_A as the conditioning projection. Without the second clause a mechanism of order K may have Γ equal to its anchor work exactly, whatever K is. This is an amendment to that definition and is recorded as one.

Theorem 3.4 (declared-frame readings evidence nothing). In the declared-frame model of Sign and Work §5 and The Second Pool §5 — sections are frames R_v(t) ∈ O(d), restriction maps induced as O_uv = R_uᵀR_v, anchors fix columns of the frames or transition maps between them — every reading that is a function of the declared frames, the induced maps, or the Laplacians built from them has τ_N = 0. Proof. The honest configuration in that model is produced at exactly the anchor’s cost: free parts are drawn freely, drift is added freely, and the reconciliation charge the honest ledger carries touches no declared object. An adversary holding the anchor receipts and values therefore produces a configuration distributed as honest’s at zero native cost; any function of that configuration takes its honest value; f(π_A ∧ π) = f(π_A). ∎

The theorem needs no spectral argument. Sign and Work Proposition 5.1 (declared frames induce a flat connection), The Second Pool Proposition 5.3 (the same on the time axis) and Corollary 3.2 (the overlap reading is maximised by uniform motion) are instances: each finds a zero-native-cost strategy for one reading, where the theorem says the honest strategy is one for all of them.

Corollary 3.5. A Combination Proof all of whose readings are declared-frame readings has Γ = |A| exactly: its Sybil cap is its anchors’ cap, and Independent and Expensive §6’s inversion — K anchors wearing readings — is a theorem for the class.

Calibration. The identity is checked on the brute-force attacker of code/exclusion.py, which sees acceptance behaviour only, on a universe of eight paid and two free degrees of freedom with A the first three paid and N the remaining five:

Worst deviation zero. The trap row reads more degrees of freedom than the idle row and evidences the same nothing, because what it adds is free.

4. The Measured Map

Theorem 3.4 leaves one reading in the corpus outside its class. Sign and Work §8.1 named it as the one place a gap not bought by the anchor could hide: restriction maps measured at overlaps rather than induced from declared frames. It is built here in the form the literature already has.

Definition 4.1 (data sections, fitted maps, residual). Each edge e carries m prompts. A vertex’s section is its outputs on the prompts of its incident edges, one block X_v^e ∈ ℝ^{d×m} per edge. On edge (u, v) the restriction map is the orthogonal Procrustes fit O_uv = argmin_{O ∈ O(d)} ‖X_u^e − O X_v^e‖_F, in closed form from the singular value decomposition of X_u^e X_v^{eᵀ} [8, 9], and the reading is the residual after the fit, r_e = ‖X_u^e − O_uv X_v^e‖_F / √(dm). The gate is H⁰ at tolerance: every edge’s residual below tol.

Two features of the gate are chosen against alternatives that fail. A Laplacian built from the fitted maps telescopes exactly as an induced one does — at any signal-to-noise ratio above one the fitted O_uv is R_uᵀR_v up to noise — and so falls back into Theorem 3.4’s class; the residual is the one quantity a declared frame cannot set. And a count of passing edges cannot see a coalition’s boundary: on the specimen below a coalition of 128 vertices failing every one of its boundary edges still passes 369 of 371, which a threshold at half progress admits. The gate is all edges or nothing.

Proposition 4.2 (the anchor is invisible). The residual is invariant under X_v^e ↦ Q_v X_v^e for any Q_v ∈ O(d) at any vertex. Hence a frame-borne anchor — the beacon column of Gauge-Fixing §4.3, the chain of §4.2 — is absorbed by the fit, ι(π_fit | π_A) = 1 by construction, and τ_N(π_fit) = τ(π_fit). Proof. Procrustes over O(d) absorbs any orthogonal factor on either side. ∎ Measured: rerandomising every frame changes the residuals by 5.8 × 10⁻¹⁶. The anchor’s only role for this reading is admissibility, which costs honest and adversary alike. The comparison that matters is therefore not against the anchor but against the public record, and §§5–6 are that comparison.

The world and the work. A world G(t) ∈ ℝ^{d×M} over all prompts, drifting as an AR(1) process with per-entry innovation δ — G(t) = √(1−δ²)·G(t−1) + δ·ξ(t), ξ standard normal — and seeded from a family disjoint from the beacon’s, so that no anchor holder can compute it. An honest vertex derives X_v^e = R_vᵀ(G[:, e] + (σ/√k)·ε) with ε standard normal: k independent re-derivations averaged, at cost k per prompt. That is the native work, and it is priced as a noise level, which §8 returns to. Two prompt models are run: edge-local, each edge its own m prompts, and shared, every edge the same m prompts, so that one derivation serves the whole network.

The adversary is a coalition C ⊂ V with no model of the world. On its interior edges it fabricates a consistent fiction — any Ĝ, expressed at each vertex through that vertex’s frame — which glues exactly and costs nothing: A Consistent Fiction’s fiction space, as a strategy. On its boundary edges it must fit honest neighbours’ data on shared prompts. Its strategies are written down, and Definition 3.3 of The Second Pool applies: the instrument takes the cheapest strategy that passes, not the natural one.

Calibration. code/evidencing.py, Part 2. d = 3, m = 16, σ = 0.3, k = 4, tol = 1.5·σ·√(2/k) = 0.318, chosen so that the honest network’s maximum over 371 edges and eight epochs clears it. n = 256, |E| = 371, the seed-7 complex of the corpus.

The predicted honest residual is σ√(2/k)·√(1 − 1/m). Two rows carry a warning each. The honest participant who does no re-derivation is refused at 370 edges of 371: the reading prices k, and a mechanism deploying it has set a floor on native work whether or not it meant to. And the whole-network fiction scores exactly what the noiseless honest network scores, zero, at zero cost: a residual of zero certifies nothing, and a coalition that is the whole network is the fiction space with the instrument pointed at it.

5. The Overlap as a Copy Channel

code/evidencing.py, Part 3. Overlaps public before commitment: the coalition sees its honest neighbours’ data on the boundary edges’ prompts at the epoch it is scored.

Proposition 5.1 (the free ride). Let a boundary vertex u ∈ C have b honest neighbours publishing on a prompt block, each with noise σ/√k. Setting u’s section to the aligned mean of their published data — zero derivations — gives a residual against each of them of (σ/√k)·√(1 − 1/b)·√(1 − 1/m), below the honest–honest residual (σ/√k)·√2·√(1 − 1/m) at every b ≥ 1, and equal to zero at b = 1. Proof. The neighbour’s own noise is inside the average; the residual is (σ/√k)(ε_j − ε̄). ∎

Alignment is free because the fit is frame-invariant (Proposition 4.2); the coalition does not need the frames. The prediction, stated before running: the coalition’s boundary residual is below honest’s at every coalition size, and the ABOVE HONEST flag fires on every row.

With edge-local prompts each boundary block has one honest source and the coalition copies it exactly: residual zero, the neighbour’s noise reproduced to the bit. With shared prompts it averages every honest vertex and lands at σ/√k·√(1 − 1/b) with b in the hundreds. In both models the coalition scores better than the honest participants it is copying, having done no native work at all. Waggoner and Chen [1] said that agreement elicits common knowledge; here common knowledge is the public overlap, and the coalition’s boundary — the only place a coalition can be charged — is where it collects.

This is the statement in the abstract that the sheaf’s characteristic operation is the leak. The mechanism compares sections at overlaps because that is what a sheaf does; the comparison requires the sections to be visible at the overlap; and visible at the overlap is copyable at the overlap. Publishing overlaps before commitment does not merely fail to price the boundary. It pays the coalition for having one.

6. Commit Before Compare

code/evidencing.py, Parts 4 and 5. Sections are committed before the epoch’s data are revealed; the coalition’s best input on a boundary block is the last epoch’s public record, decayed by the AR(1) factor, which is the optimal predictor of a memoryless drift. No chain is imported: the commitment is the index t − 1.

Proposition 6.1 (the step). The stale coalition’s boundary residual squared is δ² + (σ²/k)(1 + 1/b), up to the fit’s √(1 − 1/m); it passes the gate iff

δ < δ* = (σ/√k) · √( tol²·k/σ² − 1 − 1/b ),

which is 1.58 σ/√k with one source and 1.87 σ/√k with many. Above δ* the cheapest passing strategy derives the boundary blocks’ prompts from the world at honest cost and keeps the fiction elsewhere.

First failure at 1.50 against a predicted 1.58, and at 1.75 against 1.87: the step is where the arithmetic put it, to within the grid. The honest residual is flat in δ throughout, at 0.202, because the honest participant derives from the world as it is.

The reading’s own gauge. A world whose innovation is a rotation, G(t) = Q_t G(t−1), is absorbed by the fit at any rate: the stale coalition’s residual is 0.2072 against honest’s 0.1996 and it passes. The Second Pool §9.1 asked whether a measured-map reading has a temporal gauge of its own. It does, the left O(3) action on the world, and it coincides with the fiction space of the reading; Corollary 3.2 of that paper said any uniform motion is a gauge for the overlap reading, and here any rotational innovation is a gauge for the residual. A smaller class, the same shape.

Proposition 6.2 (the native share of a coalition). Above the step, with edge-local prompts, the cheapest passing strategy pays k·m per boundary edge and nothing in the interior, so the coalition’s native share, normalised by the native work its identities purport to evidence, is

e(C) = |∂C| / ( 2|E(C)| + |∂C| ),

and with shared prompts, where one derivation serves every boundary block, e(C) = 1/|C|. The whole network has e = 0 in both.

Worst deviation from Proposition 6.2, zero. An honest vertex’s native work is k·m·deg(v) ≈ 186 per epoch on this complex.

Three things are in the table. Coalition geometry is the whole of the price. A block coalition — a subtree of the hierarchical complex, the coalition the substrate’s own construction makes cheap — has two or three boundary edges at every size from sixteen to one hundred and twenty-eight; the expected boundary of a level-j block is 4(1 − 2^{j−8}), bounded by four. A contiguous range of the same size, the coalition convention of code/temporal_iota.py, has fifteen to twenty. Gauge-Fixing §2 said that a sparse nerve yields a small spectral gap and a cheap coalition, and set maximising the gap as the design target; the specimen’s λ₂ is 0.0038, and the Cheeger bound it gives on a block of 128 is 0.12 edges against a realised two — loose by an order of magnitude and right about the order. The Sybil cap is not a cap on a modular substrate. The native cost per identity is the boundary’s work divided by the coalition’s size, and the boundary does not grow: 12 per identity at sixteen vertices, 1 at a hundred and twenty-eight, against 186 for an honest vertex. The Multiplicity Freedom Theorem 5.1 bounds the fleet by resource over per-identity cost, and per-identity cost here falls as 1/|C| — the fleet grows with the coalition. Identities inside pay nothing; identities at the boundary pay honest’s per-edge rate; and on a substrate chosen for its low spectral dimension there is almost no boundary to stand on. Whether native work is priced at all depends on whether the world’s content is shareable. With shared prompts one derivation serves the coalition and e(C) = 1/|C| whatever the geometry: A Consistent Fiction Proposition 5.2 — one vertex’s contact determines the section — restated as a cost.

What is evidenced. Above the step the coalition pays because derivation is, in the model, the only channel to the world’s innovation since the last public record; below it, the record suffices. The reading evidences that its sections carry information about the world not computable from the anchor and the public record — the change since the last reveal — and native work is charged only where and insofar as that information cannot be had otherwise. Bara’s I(Θ; Z | R) = 0 [6] is this condition’s negation. Any side channel to the innovation — another network’s reveal, an external feed — reprices e(C) at the side channel’s cost, and §5 is the limit where the side channel is the network’s own overlap.

7. What the Condition Bought

Read back into the corpus, one word withdrawn and one measurement that reorders a design target.

§9.5 is answered by subtraction. The framework needs no third condition. It needs Definition 2.6 read with the projection’s own anchor among the things it must be independent of (Corollary 3.3). The idle projection, the trap row of §3, and every declared-frame reading are then one case, ι(π | π_A) = 0.

The through-line becomes a theorem. “The trace gap is bought entirely by the anchors; the coherence layer contributes only to the denominator” was measured in Sign and Work §5, again in The Second Pool §6, and stated as a through-line in the state of the program. Theorem 3.4 says it is a property of the declared-frame class, for any reading whatever, because the honest configuration in that class costs exactly the anchor. Sign and Work §8.1 is answered on both halves: within the class, no; outside it, a data-fitted reading does carry a gap its anchor did not buy, and the gap is bought by the world’s innovation instead.

Gauge-Fixing §2, measured. The spectral-gap coalition bound was the whitepaper’s one quantitative security claim about the gluing layer. On the program’s own specimen the gap is 0.0038 and the cheapest coalition of any size has a boundary of at most four edges. The design target stands and the specimen fails it; Requisite Richness chose this family for its low spectral dimension, and Sign and Work §5.4 found richness at the coupling knob paid in trace gap. This is the same bill, presented at the boundary.

The Sybil cap has a geometry it lacked. The Multiplicity Freedom prices identities; §8.4 of that paper asked how Sybils across levels could be told from legitimate composition. Under a data-fitted reading the answer is that the level’s boundary is what pays and its interior is free, so a coalition that presents as a block of the holarchy pays a level’s worth of boundary for a block’s worth of identities. That is Gluing the Gates’ “failures live at interfaces” with a price on the interface.

The overlap is the leak. Proof of Coherence compares sections at overlaps; A Consistent Fiction found the fiction space; The Second Pool found the chain transporting the beacon. §5 finds the general form: whatever a reading compares at an overlap is visible at the overlap, and visible is copyable. The peer-prediction literature’s repair is to score agreement on unrelated tasks against agreement on related ones [3, 4]; the repair taken here is the one Gauge-Fixing §4.2 already specified and Bittensor deployed [18] — commit before compare — and its price is Proposition 6.1: what it evidences is the innovation, and a slow world evidences little.

Coherent change, again. The Second Pool found that eigenspace-overlap readings cannot distinguish coherent change from stasis because uniform motion is a gauge. The residual reading distinguishes them — stale sections fail once the world has moved — and has a smaller gauge of its own, rotational innovation. Proof of Coherence v0.5 §4.5’s requirement is met, for this reading, in the only form the model permits: change is scored as the cost of not having derived it.

Conjecture R, an eighth time. A new formalism, maps fitted rather than declared, relocated the difficulty from the anchor to the public record, and the quantity the formalism does not supply is the world’s innovation rate.

8. What Is Declined

That evidencing is new. It is Definition 2.5 with a different conditioning projection, and §2 names the results that established its qualitative content: agreement rewards the public [1, 2], a report computable from other reports carries no information [6, 7]. The claim is the identity, the theorem for the class, the accounting, and the measurements.

That the reading is new. It is Singer and Wu’s [8], in Di Nino et al.’s closed form [9]. What is new is the adversary.

That native work is measured as effort. The model prices it as a noise level: k re-derivations or one better model are indistinguishable, and the reading evidences quality of contact with the world, not labour. Proof-of-learning [15–17] is the record of what happens when a mechanism tries to evidence the labour itself.

That the step is a finding. It restates Sign and Work’s rule — satisfying a public constraint is never work — with the world as the constraint, and it depends on derivation being the sole channel to the innovation. It is printed because the prediction was stated before running and was met.

That the boundary bound is new. It is the attack-edge bound [12–14], work-weighted. What is new is that the same cut is the copy channel (§5), and the numbers on the program’s own complex.

That the world model is general. AR(1) is memoryless, so the innovation given the record is exactly δ; a world with momentum is extrapolable and evidences less. Rotational innovation evidences nothing (§6). A world derived from the beacon’s seed would be computable by every anchor holder and evidence nothing while appearing to; the seed family is disjoint and the paper says so.

That the coalition strategies are exhaustive. Every native cost is a minimum over written-down strategies; measured e(C) ≥ true e(C), always. A cheaper strategy would lower the boundary’s price further, which tightens the conclusion.

That the frames are anchored here. The residual cannot see them (Proposition 4.2), so the anchor enters only as admissibility and is not modelled; τ_N(π_fit) = τ(π_fit) is by construction, and a mechanism that embedded its anchor in the data would forfeit it.

That the concurrent work is superseded. Bara [6] states the criterion in information terms, five days before this draft and independently; this paper’s Theorem 3.4 is its instance for declared-frame sheaves, and §6’s “change since the last record” is its positive half.

9. Open Problems

9.1. A world with memory. Answered — A World with Memory v0.1 §§3–5. The pool is the entropy of the world conditional on the public record, plus the anchors’ own. Against the optimal record-only attacker — the Kalman filter on the record — the coalition’s share is a step in the innovation given the record: it passes at zero derivations once that innovation falls below a threshold proportional to the honest derivation variance σ²/k, and sits below honest’s own residual once it falls below that variance, the record being then a better model of the world than a fresh derivation. Refusing the predictor on a world of innovation q costs every honest participant k ≥ c σ²/q derivations per prompt; fresh prompts restore this paper’s share at any memory. Original statement follows. The innovation given the public record is the quantity priced. For an AR(1) world that is δ; for a world with momentum it is the extrapolation error, smaller; for a world an adversary can partly predict from outside the network it is smaller still. The native share of a reading is bounded by the entropy rate of the world conditional on everything public, and that bound has not been stated as a theorem.

9.2. Multi-task against commitment. Peer prediction repairs agreement scoring by penalising agreement on unrelated tasks [3, 4]; this paper repairs it by commitment. Whether the two compose — commit, then score correlated agreement — and whether the composition prices the interior of a coalition, which commitment alone leaves free, is the design question the two literatures have not asked each other.

9.3. The boundary of a holarchy. Answered — The Budget Across Levels v0.2 §§4–6. The holon’s boundary and its constituents’ are the same edges: a block’s boundary lies entirely above its own level, halving per level exactly, and the level of presentation changes receipts alone. The Multiplicity Freedom §8.4’s criterion is correct and is not a residual. Original statement follows. §6 prices a coalition by its boundary at one level. A holarchy nests boundaries, and Gluing the Gates Proposition 6.1 says the adversary picks the level. Whether a coalition presenting as a holon pays the boundary of the holon or of its constituents decides The Multiplicity Freedom §8.4, and the budget-across-levels paper the plan has queued is where it belongs.

9.4. Raising the boundary. Measured — Every Basis at Once v0.1 §4. Not by densifying the cover but by redrawing it: a beacon-drawn, degree-preserving cover raises the block of 128’s boundary from 2 to 191 and the spectral gap thirteenfold by one edge in eight, while the spectral dimension leaves 1.6 for 2.6. The curve is a table; expansion and richness are one purchase with opposite signs. Original statement follows. The specimen’s boundaries are at most four edges because the complex is hierarchical with two cross-edges per merge. A substrate designed for a large spectral gap — the target Gauge-Fixing §2 set — would raise e(C) for every coalition, at the cost Sign and Work §5.4 measured in richness. The trade between expansion and richness is the first design curve this reading makes measurable.

9.5. Content-anchoring the data. Proposition 4.2 keeps the anchor out of the reading. A Consistent Fiction §8.5’s second discipline — an anchor supplying content while the beacon supplies timing — would be a prompt whose honest answer the world determines and the record does not contain; whether such a prompt can be posed without an oracle is that paper’s §8.2, and this reading is the first place the question has an instrument.

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Propositions verified and measurements reproducible from code/evidencing.py, which imports the attacker of code/exclusion.py unchanged and nothing from code/two_pool.py, the fit being frame-invariant.