working papers · series I–VIII · since 2026
Software the world is missing, specified before anyone builds it.
p-rick hunts for genuine gaps in the software landscape — categories with no incumbents, problems with large audiences, theories nobody has written down — and does the research to specify them: gap verification against everything that exists, formal models, validation harnesses, and honest confrontation with the reasons each gap has survived. Every paper ships with a PDF and a plain-language essay. Research first; products later.
Series VIII is the program’s second arc: machinery invented, not just found — a conservation architecture that makes depth stability a theorem, a convergence theory for agentic maintenance, the phase transition that governs agent collectives, and the Green’s function that turns context rot into an inverse design problem. Four mechanisms, four law sets, four harnesses.
The Green’s Function of Context: Positional Recall as an Inverse Design Problem
Context rot has an object: the impulse response of the positional channel. The recall profile is the kernel smeared by noise (0.037); the lost-in-the-middle dip lands exactly on its closed form (2771 = 2771); windows collapse with load exactly as the extreme-value law computes; and the monotonicity theorem says uniform recall is impossible with timescales alone — it takes a register. The register-plus-slope design prices the trade between uniform recall and order resolution by the decade.
The Lock-In Law: The Phase Transition in Agent Collectives and Its Design Temperature
Groupthink has a formula: the collective bifurcates at K_c = T(1−λ), coupling and temperature exchange one-for-one, and there is no useful middle. The locked confidence is a computable fixed point (0.045%), the locking time scales with ln N, the diversity firewall raises the threshold linearly, and the reset interval that preserves exploration is the locking time. Agreement is a phase you designed.
The Anchor Law: Ratchets, Floors, and the Minimum Coverage of Agentic Maintenance
The arithmetic for “how many tests before the agent is safe”: a maintenance floor that is an exact linear trade between un-anchored noise and frozen legacy error, a complexity ratchet invisible to behavior anchors, and a stability boundary that moves quadratically in agent entropy — the contract an organization can sign with its fleet. Tests are contraction, not verification; budgets, not tests, stop the ratchet.
The Dissipation Budget Law: Port-Hamiltonian Streams and the Conservation Architecture for Depth
Depth is rented, never owned. The port-Hamiltonian stream replaces the residual update with Cayley transport (exactly orthogonal — volume-free to 10^-14), metered dissipation (the only contraction channel: expressivity becomes a budget line), and ports as the only norm injection. Zero bound violations in 7,200 runs while the control explodes to 10^43; gradients metered through 240 layers; a rank-2 readout at depth 120 at 98.1% against the rescued control’s 38.3%, held-out over three seeds — E6 trains it end-to-end: baseline parity at depth 240 with no normalizer anywhere (97.3% ± 0.3); E6b’s trunk-LN ablation collapsed at depth 240 (0.409 ± 0.137, backward max 2.99×10^8) exactly where its unbounded backward pass predicted; and the pre-registered CIFAR-10 round (60 runs, five families) returned the mixed verdict its rules encode — parity within 1.0–1.2 pt, zero PH failures against three trunk-LN collapses, no growth advantage at depth 48, a 4.4× compute bill.
Eight series, one program
30 papersThe second arc — mechanisms invented, not just found: a conservation architecture for depth, the convergence law of agentic maintenance, the lock-in transition of agent collectives, and the Green’s function of context. Lawed, harnessed, priced.
Critical thresholds where everyday infrastructure fails abruptly: resolution satisfiability, retry-storm ceilings, credential cascades. Derived laws, not folklore.
The quantitative turn — from specification to derivation: epidemic thresholds for compromise, a closed-form admission law, the drift law for re-embedded corpora.
The guarantees software implies and never has to honor: withdrawal propagates, stored facts stay true, dormant software wakes, independent things fail independently.
The quiet premises software runs on: complexity can be destroyed, a body has one owner, the senses are always online, every number is a point, defaults are neutral.
What systems lose over time: vendor death, biological time, maintainer mortality, behavioral extinction. Conservation engineering for software’s mortal side.
Systems gaps: behavior under scarcity, storage that remembers, the scheduling of human attention, one machine across many devices.
The origin series: the personal event bus, consumer entitlements as dead capital, the n=1 cost-of-living index — data infrastructure owned by its user.
Research first, products later. Every paper verifies its gap, grades its evidence, and regenerates its figures from code — or it does not ship.
Gap verified
Before anything is proposed, the landscape is searched for incumbents — products, papers, standards. A gap only counts if the search comes back empty.
Large audience
The missing thing must matter to many: a category with mass use, not a niche tool or a wrapper around an existing capability.
No toy output
Excluded by charter: test suites, deterministic wrappers, anything an AI obsolete. What ships is theory, architecture, or a validated harness.
Honest grading
Every paper grades its own evidence — claims are marked verified, estimation, or verification-queued. Constants are derived laws or labeled configuration-specific.
The Middle of Your Context Is Rotting (Design It Away)
Lost in the middle is the Green’s function of a kernel nobody designed. The dip has an address, uniform recall needs a register (it’s a theorem), and the cure is an inverse problem that takes minutes of compute.
Agreement Has a Temperature
Groupthink in agent collectives is a phase transition at K/T = 1: one dial, one cliff, no useful middle. Four numbers you can compute before the first round — and the null result on the truncation folklore we actually tested.
The Agent Eats What You Don’t Pin
The arithmetic for “how many tests before the agent is safe”: a quadratic boundary, a ratchet your test suite can’t see, and the contract you can sign with your agent fleet. Tests are contraction, not verification.
Depth Pays Rent
Every residual layer pays norm rent a normalizer refunds after the fact. Make transport a rotation, dissipation a meter, injection a port — and depth stability becomes a theorem with a budget line. The lease-termination notice.
Where to start
three doorsHow the program decides what counts
Four tests every paper must pass before it ships — gap verified against the living landscape, large audience, no toy output, honest grading of evidence. Read this first and the papers read as what they are: specifications, not opinions.
P-001 — the personal event bus
The paper that started the program: user-owned middleware for the ambient events of a digital life — notifications, email, messages — as one durable, queryable stream. Where the whole method was worked out.
P-030 — the Green’s function of context
The newest paper and the sharpest of the second arc: context rot as an impulse response, the middle dip with a closed form, and recall as an inverse design problem. Or start from its plain-language essay above.