WP-38 · Positional Dominance under Non-Contestability

Standard results in option-exercise games hold that competition erodes, and in the limit dissipates, the option value of waiting. This paper identifies the single condition under which that mechanism operates — and shows that where it fails, volatility does the opposite of what the literature predicts. The condition is a single parameter — the preemption hazard λ — and the paper commits to a closed-form critical value for it, λ̄ = ((1−δ)/δ)·γ/(1−γ), together with five further signed predictions and the observation that would refute each. The empirical claim is an interaction, not a level: every theory of rents predicts that prices rise when supply breaks; this one predicts that the slope of positional return against volatility reverses when λ does.

MODEL proved here DATA observed CALIB calibrated, not derived OPEN unresolved

Correction notice

The paper of record corrected here is Grossi (2026a) v1, DOI 10.5281/zenodo.21013066, of which The 1/3 Invariant · Positional Dominance is the HTML rendering. That correction is now carried by v2, DOI 10.5281/zenodo.21753025, which restores the dropped linear and constant terms of the value gap and withdraws the ψ apportionment; the present paper builds on v2 and does not itself retract anything. For the record: v1's Proposition 1 That proposition justified on the ground that "V earns positive rents at ." With the tabled payoffs this is false — multiplies the benefit term for both players, so and the threshold collapses. §7 supplies the correct source and §7.2 retires the invariance claim. The method used to find this is WP-30 applied to the author's own work.

This model predicts. It does not describe.

Six commitments · each one signed, each one refutable

The claim being made is not that bottlenecks are valuable when supply breaks — every theory of rents says that, and confirming it discriminates between nothing. The claim is that a specific parameter governs a specific sign, and that the sign flips when the parameter does. Below is what the model is committed to before any data is seen. Each row states what would have to be observed for the model to be wrong.

	Prediction	Falsified by
P1	A point value for the critical hazard. λ̄ = ((1−δ)/δ)·γ/(1−γ). At δ = 0.985, γ = 0.55 this is 0.0186 per period, about 20% annualised. No free parameter is fitted to produce it: β, τ, c_K and κ all cancel.	Positions with measured hazard above λ̄ that still show positional dominance at high σ, or positions below λ̄ that do not.
P2	The kink, not the level. ∂²W_J/∂σ∂λ < 0 — the slope of positional return against volatility jumps discontinuously when contestability collapses, and the jump is timed to the collapse rather than to the price move.	A level effect with no change in slope; or a slope change not timed to λ.
P3	Four signs, not four magnitudes. ∂σ/∂c_K > 0, ∂σ/∂K < 0, ∂σ/∂τ > 0, ∂σ/∂γ < 0. Carry and friction raise the threshold; capacity and private information lower it.	Any one sign reversing in estimation. These are not fitted; they follow from the implicit function theorem at the crossing.
P4	Patience cuts the other way. ∂λ̄/∂δ < 0. The long-horizon holder tolerates less preemption risk, not more — at δ = 0.999, λ̄ = 0.0012.	Long-horizon holders sustaining the strategy at hazards short-horizon holders cannot.
P5	A mirror with the opposite sign. The five episodes are all λ collapsing. Events where λ jumps on an unchanged position — custody freezes, withdrawal suspensions, exchange halts, seizure — must move ∂W_J/∂σ the other way, and holders must shift toward participation.	λ-jumps producing the same sign as λ-collapses. This is the strong test: five events sharing one sign fit many theories; the same parameter reversing the sign fits one.
P6	An existence condition that can fail. No threshold exists at any volatility unless γ > 0. Position must inform, not merely cost. A secure but uninformative position earns nothing extra however turbulent the environment gets.	A non-contestable, uninformative bottleneck that shows the pattern anyway.

What is deliberately not claimed. σ* is not predicted to be 1/3, or to be any constant. §4.1 shows it moves nearly two to one under ±20% parameter variation, and Corollary 1 makes it an explicit function of carry, capacity, friction and information. It is a property of a particular bottleneck. The dimensionless object in this model is λ̄, and even there no distinguished value is asserted — only the formula, and the sign of its derivative.

§ 1 · Environment

Primitives

Discrete time , discount factor . Volatility follows a finite Markov chain with matrix . Each period the network realises a spread , symmetric, atomless, unbounded support, . Three nodes ; node is a capacitated hub owned by ; ownership is a state variable .

Player	Choice	Information	Costs
J — positional	flow	private signal, strength	per unit moved; fixed carry
V — velocity	speed	public only, share	convex; capture capped at

§ 2 · The pivotal definition

Contestability

Definition 1 · Contestability MODEL

Let be V's action set and the ownership transition. The hub is contestable if with ; non-contestable if for all .

Definition 2 · Preemption hazard

. Non-contestability .

Under non-contestability no expenditure by V — of any magnitude, at any speed — transfers the hub. V may compete for flow rents; V cannot compete for position.

That rights over constrained network capacity confer market power, and that the constraint is what makes them valuable, is Joskow & Tirole (2000). What Definition 1 adds is the transfer map: their transmission rights are tradable, so λ > 0; the case treated here is the one where no expenditure by the rival moves the right at all.

§ 3 · Lemmas

Convex against affine

Lemma 1 · J is convex in σ

Π_J(σ) = K·E[(β σ|ξ| − τ)⁺] − c_K is convex and strictly increasing in σ, with Π_J″ > 0.

J moves only when the spread covers transport: q* = K·sgn(X)·1{β|X| > τ}. For fixed ξ, σ ↦ (βσ|ξ| − τ)⁺ is a call struck at τ — convex in σ; convexity survives expectation. F atomless with unbounded support gives the exercise region positive measure for every σ > 0, hence strictness. □

Remark. This is the Dixit–Pindyck (1994) channel in bare form: J holds a real option on the spread, and option value is convex in scale. Two forces compound — the spread widens and J's exercise region expands — which is the economic content of the convexity.

Closed form (Gaussian). For ξ ~ N(0,1), writing a ≡ τ/(βσ) and g(a) ≡ E[(|ξ| − a)⁺] = 2φ(a) − 2a·Q(a) with Q = 1 − Φ, Π_J(σ) = Kβσ·g(τ/(βσ)) − c_K. Since g(0) = 2φ(0) = √(2/π) = κ and g′(a) = −2Q(a), one has g(a) = κ − a + O(a²), so Π_J(σ) = Kβκσ − Kτ + O(1/σ). VERIFIED

Correction, 2026-08-01. An earlier draft of this lemma wrote J's flow payoff as (|X| − τ)|q| − c_K, without β, while V's retained it. That asymmetry is fatal: J's asymptotic slope becomes Kκ = 1.196 against V's β(1−γ)κ = 3.051, so ΔV → −∞ and no threshold exists at any volatility. β belongs on both sides, as in Grossi (2026a) v2 Lemma 3.4 (Π_J ∼ βκσ − τK). Restoring it also removes the scale inconsistency v2 recorded as a discarded observation: the strike ratio at σ = 0.33 is τ/(βσ) = 0.72, so the option is live.

Lemma 2 · V saturates

For

the FOC gives

. The cap binds at

; thereafter

, cost pinned at

, benefit linear in

. □

Why the cap is not a convenience

is the physical latency floor — light speed and colocation — beyond which speed expenditure purchases nothing. It is why V's advantage saturates while J's does not. This is the same bound that WP-28 audits from the timing side.

Lemma 3 · The intercept

: at no flow is profitable but carry is incurred regardless.

Figure 1 — the mechanism. (a) J's payoff is a call on the spread struck at transport cost, hence convex; V's is affine once the latency cap binds. (b) Convex minus affine is convex, negative at the origin by Lemma 3, divergent above — exactly one root. Uniqueness and the sign pattern are parameter-free. CALIB The plotted location

reflects a

chosen to match the previously reported interval

; it illustrates the mechanism, it is not a derived constant. See §7.2 and WP-29.

§ 4 · Corrected threshold

Proposition 1′

Proposition 1′ · Existence and uniqueness MODEL

Let A_J, A_V be the asymptotic slopes of Π_J, Π_V in σ. Under non-contestability, ΔV(0) = −c_K/(1−δ) < 0 and

a threshold σ* exists ⟺ A_J > A_V

With J's benefit coefficient β at k = K/K̄ = 1 and V's β(1−γ), this reduces to γ > 0 — the condition of Grossi (2026a) v2, Lemma 3.4. Above the saturation point σ̄, Π_J is strictly convex (Lemma 1) and Π_V is affine (Lemma 2), so ΔV is strictly convex there; being negative somewhere and divergent, it has exactly one root on [σ̄, ∞). □

Condition (C′) — no root below saturation. At σ̄ the two branches of Π_V meet and Π_V(σ̄) = ½ exactly, since β(1−γ)κσ̄ = 1 by construction. Hence no root lies below σ̄ iff Kβσ̄·g(τ/(βσ̄)) − c_K ≤ ½, i.e. c_K ≥ 0.6577 under the calibration (the gross option value at σ̄ is 1.1577). VERIFIED

Correction, 2026-08-01. The divergence step previously read that E[(σ|ξ| − τ)⁺] "grows superlinearly in σ while Π_V grows linearly." It does not. A call's value is convex but asymptotically affine: E[(σ|ξ| − τ)⁺] = κσ − τ + o(1). Divergence therefore cannot rest on superlinearity; it rests on the slope gap A_J − A_V = βκγ > 0, which is strictly positive precisely when γ > 0. Strict convexity plus a negative intercept is not sufficient on its own — a strictly convex function may stay negative forever.

Uniqueness below σ̄ when (C′) fails is OPEN. Numerically the root is unique for every c_K tested, including cases where it lies below σ̄; that is simulation, not proof, and the tag stays.

Corollary 1 · Signed comparative statics

The threshold rises with the cost of holding position and with transport friction; falls with capacity and with the informational advantage of position. Each sign is checkable — which the quadratic ansatz did not deliver. Estimation route in WP-31.

§ 5 · Main result

Non-contestability reverses preemption

Theorem 1 · Non-contestability reverses preemption MODEL

Let the hub be contestable, generating preemption hazard λ > 0. Write ρ(λ) = (1−δ)/(1−δ+δλ) ∈ (0,1], so that W_J = Π_J/(1−δ+δλ) and (1−δ)·ΔW_λ = ρ(λ)Π_J − Π_V. Then:

(i) ∂²W_J/∂σ∂λ < 0 — preemption risk erodes the sensitivity of J's advantage to volatility.
(ii) there is a critical hazard λ̄ above which no finite σ* exists and V weakly dominates at every volatility.
(iii) at λ = 0 the full option value accrues to J, and σ* is as in Proposition 1′.

Under hazard λ, W_J = Π_J/(1−δ(1−λ)); the convexity of Lemma 1 enters multiplied by [1−δ(1−λ)]⁻¹, strictly decreasing in λ, giving (i). W_J is scaled down uniformly while Π_V is untouched, giving (ii). Setting λ = 0 recovers (1−δ)⁻¹, giving (iii). □

Theorem 1(ii) in closed form. A threshold survives the hazard iff ρ(λ)A_J > A_V. Solving for λ:

λ̄ = ((1−δ)/δ)·(A_J/A_V − 1)

and in the normalisation of v2, where A_J/A_V = 1/(1−γ),

λ̄ = ((1−δ)/δ)·γ/(1−γ)

Under the calibration (δ = 0.985, γ = 0.55) this is λ̄ = 0.0186 per period, about 20% annualised. Independent numerical confirmation, by bisecting on sup_σ[ρ(λ)Π_J − Π_V] = 0 over an 8,000-point σ grid, agrees with the closed form to 0.056%. VERIFIED

λ̄ contains no β, no τ, no c_K, no κ. Price impact, transport cost, carrying cost and shock dispersion all cancel: λ̄ depends only on the discount factor and the information share. σ* is denominated in the units of a particular bottleneck and moves 2:1 under ±20% parameter variation; λ̄ is a pure number in (0,1), comparable across a pipeline, an exchange and a job. If a transferable constant exists in this family, it is here and not at σ*. Whether λ̄ takes any distinguished value is OPEN and is not asserted.

Patience reduces tolerance for hazard. ∂λ̄/∂δ = −(1/δ²)(A_J/A_V − 1) < 0. At δ = 0.90, λ̄ = 0.136; at 0.95, 0.064; at 0.985, 0.019; at 0.999, 0.0012. As δ → 1, J's value is divided by 1−δ+δλ → λ while V's is divided by 1−δ → 0, so the ratio collapses for any λ > 0. The long-horizon holder needs more security than the short-horizon one, not less. VERIFIED

Figure 2 — Theorem 1. As the preemption hazard rises the gap flattens and

(dots) migrates right, until above

the curve no longer crosses zero. Grenadier's dissipation result is the

regime; this paper's is

. Not a conflict — one is the boundary case of the other.

5.1 Relation to the preemption literature

Grenadier (1996, 2002) and Huisman (2001) establish that competition drives exercise toward the zero-NPV threshold. Theorem 1 does not contradict this; it identifies the condition under which it operates. Dissipation transmits through , and requires that the rival's action transfer the asset. Against a non-contestable bottleneck the transition map is constant and the mechanism has nothing to act on. Aquilina, Budish & O'Neill (2022) measure that race directly — races resolve in 5–10 millionths of a second and account for roughly a fifth of volume — which is the saturation of Lemma 2 observed rather than assumed. This also nests Budish, Cramton & Shim (2015): their rent-dissipating arms race is the case where the prize accrues to position and V's expenditure is pure carry.

§ 6 · Extension

Markov volatility

With following , . Since has non-negative entries, convexity is preserved state by state. Persistence raises relative to in the low state — J's option retains value in anticipation of the high state while V's saturating advantage does not:

This is the correct home for the "Markov persistence" component previously absorbed into the correction factor .

Scope, added 2026-08-01. OPEN Preservation of convexity by (I − δP)⁻¹ settles J's side only. Grossi (2026a) v2 records that in the genuine two-state chain V's value inherits curvature from the continuation term and is no longer affine above saturation, so the convex-minus-affine step of Proposition 1′ does not transfer and uniqueness there is open. This section should be read as the comparative-static case — σ held fixed, chain absorbing — not as a proof for the ergodic chain.

§ 7 · Corrections

What changed, and why

7.1 The intercept

Lemma 3 supplies the correct source of : the capitalised fixed cost of holding position, — pipeline O&M, vault storage and insurance, custody. The quadratic form survives as the second-order approximation to a strictly convex function with negative intercept; only its justification changes. The quadratic must keep its linear term: ΔV ≈ aσ² + mσ − b, not aσ² − b. Dropping m is the defect v2 corrects as its first item; the truncated form's root √(b/a) is a strict upper bound, not an estimate.

7.2 σ* is not invariant OPEN CLOSED

Corollary 1 makes an explicit function of and . It is a property of a particular bottleneck under particular carry, not a constant of nature. The interval should be reported as a midstream calibration, with Corollary 1's signs as the testable content. The cross-domain 1/3 conjecture is withdrawn as evidence and retained only as an open question — the concern is exactly the one formalised in WP-29: four rows in which one is proved, one is a convention ( admits any value, is chosen), and two are measurements, are not four instances of an invariant.

§ 8 · Empirical strategy

The identifying claim

The test is on the interaction, not the level

Every model predicts that prices rise when supply breaks. Scarcity rent, inventory models, Kyle-type informed trading (Kyle 1985), plain market power — all generate a positive level effect. Confirming one tests nothing.

This model predicts the slope changes. The sensitivity of the position holder's return to volatility jumps discontinuously when contestability collapses — not because the prize grew, but because the option premium stopped being dissipated by preemption risk.

A level effect is consistent with a dozen theories. A discontinuity in , timed to the collapse of and absent beforehand, is consistent with this one.

Figure 3 — what is being estimated. Under contestability positional returns rise gently with volatility. When

collapses the relationship kinks. The estimand is the kink, not the height.

Prediction: ; and from Theorem 1(ii), where is large.

8.1 Event set DATA

#	Event	Date	λ→0 via	Magnitude	Position =
1	COMEX–London gold	24 Mar 2020	Flights grounded 14 Mar; Swiss refineries closed	Spread 1–2	Metal in deliverable form
2	WTI May contract	20 Apr 2020	Cushing tankage committed; no delivery destination; see Fernandez-Perez, Fuertes & Miffre (2023)	Settled −40.32	Empty tank space
3	Winter Storm Uri	10–17 Feb 2021	Wellhead and pipeline freeze-offs	Henry Hub 23.86; Waha peak 3→>$1,200/Mcf	Firm transport, storage
4	Panama Canal drought	Nov 2023 – 2026	Draft restrictions cut daily transits	Slot auction 4m; premia $385–425k (2026)	Auctioned transit slot
5	LME nickel	8 Mar 2022	Deliverable stock cornered	Exchange cancelled trades	— (boundary case)

Figure 4 — magnitudes. Positional rent as a multiple of pre-event baseline, log scale. WTI shown separately because it is a sign inversion rather than a multiple — which is the reason for including it.

8.2 Why this set identifies

Exogeneity. Every trigger is weather, drought, pandemic logistics or freeze — outside both action sets and outside the price process.

Cross-asset. Metals, crude, gas, shipping capacity. No asset-specific story generates all five.

The sign test. Events 1 and 2 are mirror images. In gold, position means holding the good; at Cushing in April 2020 it meant holding the space. The model is indifferent — both are non-contestable node capacity. That they have opposite price signs while sharing one mechanism is a strong joint test. Any "shortage raises price" theory gets event 1 and dies on event 2.

Event 4 prices position directly. The canal auctions the bottleneck itself, so one observes a clearing price for across a period where varies with rainfall. That supports a panel, not an event study. Metering and disclosure parallels in WP-36.

Event 5 bounds the theorem. When positional rent became extreme the exchange voided the trades. is not indefinitely sustainable: at sufficient extraction the institution resets ownership by fiat. The strategy's ceiling is political, not economic.

8.3 What would falsify

: holders earn more but no more sensitively — the mechanism is scarcity rent, not option value.
Returns rising before : anticipation, so was priced and the asset contestable after all.
Event 2 loading opposite to event 1: the space/good symmetry fails and non-contestability is not the operative variable.

§ 9 · Generalization

Otium — the theorem outside markets

Strip the commodity content and three objects remain: becomes standing, becomes replaceability, becomes turbulence, becomes carry, becomes participation effort. Lemma 2 required only that effort face saturating returns; Lemma 1 only that the position hold an option on turbulence. Both hold for a wide class of social and professional positions.

Proposition A · FOMO and JOMO are not attitudes

FOMO is the correct response to ; JOMO is the correct response to . Neither is a virtue. Each is a strategy, and each is wrong in the other's regime.

Corollary A.1. Advice to disengage is sound for agents with and harmful for agents with large. Since is inversely related to the security of one's position, counsel to step back is systematically better advice for the secure than for the precarious — while being dispensed, as a rule, by the former to the latter.

This is the JOMO pack's missing derivation

The JOMO Bot states the thesis in four words in its own docstring — "The bot runs. You live." That is Proposition A with the regime left implicit. A strategy that abstains during low-information hours is exploiting , and Theorem 1 says its edge must be increasing in . That is a testable signature the pack did not previously have.

Concretely: agents/kelly_jomo_agent.py sizes with a capped Kelly fraction. Corollary 1 implies the cap should not be constant — the fraction should scale with distance above , and should go to zero when the venue is contestable. Abstention is not a stance; it is optimal only while holds, and WP-28's audit is the test of whether it does. Theorem 1 supplies why the abstention should pay; Corollary 1 says when it should not.

The classical vocabulary had the partition already. Negotium is literally nec-otium, not-leisure — business defined negatively, as the absence of the primary state. Seneca's De Otio treats leisure not as rest from work but as the condition under which judgement is possible: the regime in which is affordable. That is Lemma 1 stated as ethics — the option to act, retained, is worth more than the act. Developed in Ch. Ocio. Veblen (1899) saw the observable side without the mechanism: conspicuous leisure signals status precisely because idleness would ruin anyone contestable.

Positions where can actually be acquired: property and title; equity rather than employment; credentials; owned audience rather than algorithmic feed position; and citizenship — no rival's effort removes it, carry is near zero, it is heritable under stated conditions, and its value rises with turbulence. Which is the formal reason to acquire it before turbulence rather than during.

§ 10 · Related chapters

Where this sits

WP-28Auditing the Circadian Trader ClaimAudits the timing edge empirically. Theorem 1 supplies the derivation it was missing; Corollary 1 says when the edge should vanish. WP-29The Numerology SweepThe standard this paper applies to itself in §7.2 when withdrawing the 1/3 invariance claim. WP-30How to Audit a Mathematical ClaimThe method that found the

error in the superseded Proposition 1. WP-31The Calibration PipelineWhere Corollary 1's signed comparative statics get estimated rather than asserted. WP-32The Forced Urgency GapManufactured urgency is λ-inflation: raising perceived replaceability rather than the return to participating. Proposition A is its game-theoretic statement. WP-36No Gauge on the TankMetering and disclosure. Event 2 is what happens when the tank has no gauge and the position is the space itself. JOMO packJOMO Bot — The Circadian Trader"The bot runs. You live." Proposition A is that sentence with its regime condition restored. JOMO packkelly_jomo_agent.pyCapped-Kelly implementation. Corollary 1 implies the cap should vary with distance above σ*, not sit fixed. Book 3 · Ch 3cThe Circadian TraderThe strategy this paper derives a threshold for. Book 3 · OcioCh. OcioOtium and negotium; §9 is its formal counterpart. Book 3 · Ch 4Market Volatility ManifoldsThe volatility geometry σ inherits. Source packThe 1/3 InvariantSuperseded in part — see the correction notice above and §7.

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