# v3.7 Top-Five Revision Plan This plan implements `referee_reports/v37_referee_codex.md` while preserving `KEY_MESSAGES.md`. Work proceeds in dependency order; later results may not be repaired by adding caveats to a broken upstream object. ## Dependency map ```text primitive task technology | v task requirements + success kernels ----> scalar representation | v aggregate surplus W and boundaries B_H, B_F, C | v product-market profit and runner-up structure | v certification/entry equilibrium ----> direction-specific imitation lags | | +----------------+-----------------+ v dynamic investment equilibrium | | v v rotation strategic herding \ / v v welfare and market structure | v policy and calibration ``` ## Stage 0 — Baseline and invariants - [x] Freeze canonical v3.7 hash. - [x] Record protected messages in `KEY_MESSAGES.md`. - [x] Complete a fresh report on the actual v3.7 source. - [x] Add counterexamples found by the report to the regression suite before revising the affected theorems, so the old statements fail visibly. **Gate:** the regression suite must distinguish the old false statements from their repaired replacements; merely retaining “95 pass” is not evidence. ## Stage 1 — Exact task environment 1. [x] Keep primitive task type `tau=(n,phi,ell,epsilon)` and define requirements `h(tau), f(tau)`. 2. [x] Write exact task success as a task-specific function `p(q,tau)`; do not discard the forgiveness-dependent slope. 3. [x] Define `W(q)=integral a(tau)p(q,tau)dmu(tau)` over primitive task types. 4. [x] Re-derive `B_H`, `B_F`, and `C` with task-specific kernels. State the minimal supermodularity condition for `C>=0`. 5. [x] Restate the representation theorem: - chain iff scalar; - common `(ell,phi,epsilon)` is a sufficient scalar benchmark; - heterogeneous forgiveness generates incomparability under an explicit support condition, not as an unqualified equivalence. 6. [x] Separate “threshold at tolerance epsilon” from “location of the Gumbel limit.” **Gate:** analytic counterexamples to the old reduction and old homogeneous-forgiveness statement must pass as tests of the new scope. Every downstream formula must compile with the primitive task measure. **Status (2026-07-14): passed.** Canonical source compiles cleanly at 39 pages; the expanded suite reports 102 passes and 0 failures, including exact heterogeneous-kernel derivative, cross-partial, Bertrand-gap, and path-integral checks. The website copy is retained as the frozen v3.7 comparison rather than silently overwritten. ## Stage 2 — Certification and entry equilibrium Build the smallest economic game that derives the lag: 1. [x] A seller may serve a fragile task only after meeting a certificate. 2. [x] Incumbent and entrant choose paid verification effort; public trials arrive independently; certified deployment generates proprietary evidence. 3. [x] The statistical lower bound constrains the certificate technology. 4. [x] Allow a parameter for cross-task evidence transfer/generalization. 5. [x] Derive equilibrium certification time or cost, the incumbent/entrant asymmetry, and the condition under which `tau_F>tau_H`. 6. [x] Derive how public verification changes entry delay and private innovation incentives. **Gate:** `tau_F` must be an equilibrium object. The theorem must distinguish minimax sample complexity, realized evidence flow, and strategic waiting/testing. **Status (2026-07-15): passed.** The reduced-form data-moat claim is replaced by a strictly convex verification-capacity and entry game. The entrant lag is now the equilibrium certificate time, with an exact primitive condition for exceeding the plan-access lag; incomplete transfer, proprietary deployment evidence, public verification, and nonentry have separate roles. Unsupported lag and wedge calibrations were removed. The expanded suite reports 108 passes and 0 failures, and the 40-page source builds with resolved references and no overfull boxes. The frozen website copy remains unchanged. ## Stage 3 — Dynamic rotation 1. [x] Choose a tractable task density and cost structure that preserve the two boundaries. 2. [x] Establish existence of an optimal investment path. 3. [x] Derive the switching surface in the relevant state `(q, remaining research)`; lag windows are capitalized into annuity values in the committed-cohort benchmark. 4. [x] Prove conditions for a horizon-first path and a finite switch to reliability. 5. [x] Separate crossing-position and crossing-time comparative statics. 6. [x] State robustness to task creation as an explicit growth-rate condition. **Gate:** rotation must follow from the optimizing policy, not from assuming a pure horizon push. A numerical example may illustrate but cannot substitute for the theorem. **Status (2026-07-15): passed.** An explicit task density yields decreasing horizon value, increasing reliability value, and positive cross-boundary complementarity. A fixed-capacity laboratory chooses its endpoint and full direction sequence. Positive complementarity and the longer reliability annuity make every reliability-before- horizon inversion strictly suboptimal; strict concavity then gives a unique switch surface and a closed-form interior horizon-to-reliability rotation. The ordering proof extends to any smooth task surplus with positive cross statistic. The theorem separates research position from calendar time, gives an exact task-creation growth condition, and has a completion-date-discounted extension with an explicit ordering condition. The suite reports 118 passes and 0 failures, including exhaustive comparison of all `2^12` direction paths; the 43-page PDF builds without warnings or bad boxes. The website remains the frozen v3.7 baseline. ## Stage 4 — Strategic racing 1. [x] Replace the one-shot direction choice with a finite-state repeated or Markov game. 2. [x] Track unmatched increments or explicitly justify a one-increment state. 3. [x] Allow direction-specific breakthrough scale and cost. 4. [x] Derive the region in which both laboratories choose reliability and the region in which they differentiate. 5. [x] Remove the infinite-intensity extrapolation unless the continuous limit scales step size so total progress and surplus remain bounded. **Gate:** the herding headline must be an equilibrium of the stated dynamic game. Strict and weak inequalities must have correct equilibrium multiplicity. **Status (2026-07-15): passed.** Direction choice is now a finite embedded-state semi-Markov renewal game. A one-increment capacity constraint explicitly rules out an innovation queue; after imitation or a finite rival match, laboratories return to the tied state and choose again. Closed Bellman values produce two best-response differences that classify unique moat herding, unique horizon herding, differentiation, and coordination. Primitive moat-herding thresholds retain direction-specific scale, flow cost, discovery rate, matching rate, and imitation lag. Equality creates the correct multiplicity, and the unsupported infinite-intensity limit is removed. The companion derivation and 107-check suite verify the renewal transforms, fixed points, all four nonempty regions, thresholds, equality cases, and comparative statics. The 43-page PDF builds with resolved references and no warnings or bad boxes; theorem and proof pages were inspected visually. The frozen website source remains unchanged. ## Stage 5 — Welfare and market structure 1. [x] Start from exact private and social marginal-value integrals. 2. [x] Derive a bound or local corollary for the annuity-share wedge; carry all remainder terms consistently. 3. [x] Introduce a modeled concentration statistic: task share, HHI, entry, or margins. 4. [x] Derive separately the effects of liability and verification on: - frontier reliability; - follower reliability/diffusion; - investment; - concentration or margins. 5. [x] State the innovation-diffusion tradeoff; do not assert that faster diffusion always raises future frontier safety. **Gate:** “opposite market structures” must refer to a derived statistic, not a renamed capability gap. **Status (2026-07-15): passed.** Gross-social value is now the exact discounted stream of future boundary values, and private value subtracts the stream transferred after follower absorption. Their direction ratio is exact; a four-error bound states when the local imitation-annuity ratio signs the distortion, and the old ratio is retained only as a flat-boundary corollary. A local two-seller allocation model adds heterogeneous serving-cost advantages, deriving leader task share and two-seller HHI from the reliability gap. Liability raises investment, both systems' reliability, the gap, and leader share. Public verification lowers investment, frontier reliability, the gap, and leader share; follower reliability rises only under an explicit finite-horizon diffusion condition, while induced entry reinforces the market-share decline. The companion derivation and 115-check suite verify the exact accounting, bounds, local nesting, policy derivatives, HHI signs, follower cutoff, and entry margin. The 46-page PDF builds with resolved references and no warnings or bad boxes; the theorem, policy, notation, and proof pages were inspected visually. The frozen website source remains unchanged. ## Stage 6 — Evidence and novelty 1. [x] Refresh METR and Epoch data from dated raw snapshots. 2. [x] Re-estimate 50%/80% ratios with uncertainty and the corrected METR method, or demote the ratio from a formal fact. 3. [x] Construct a reproducible aggregate lag panel. Estimate `tau_F` only if a measure maps to the model's certification margin; otherwise remove the 14-month number. 4. [x] Add a source for frontier spending and separate model-training cost from hyperscaler capital expenditure. 5. [x] Engage the closest certification, data-enabled learning, AI task heterogeneity, and 2026 directed-technology papers identified in the report. **Gate:** every calibration input must have a reproducible source and an explicit map to the model object. “Illustrative” does not license an unsupported number. **Status (2026-07-15): passed.** `evidence/STAGE6_EVIDENCE_LEDGER.md` and the source manifest pin the current METR and Epoch measurements to dated pages, Git commits, and file hashes. The METR 50%/80% ratio remains an analytic diagnostic rather than a reported empirical estimate because the compact public summary does not supply joint bootstrap draws and the current frontier reaches the suite's unreliable region. The two-row aggregate catch-up panel records the 2025 and 2026 Epoch ECI methods and uncertainty or alternative rule, but maps neither row to the model's plan-access or certification lag; no `T_E`, wedge, switch, or herding cutoff is calibrated. The opening puzzle now uses sourced fixed-performance inference prices and frontier training compute/cost, while hyperscaler cash capex is explicitly labeled broader infrastructure context. The novelty section now states exact contrasts to the closest certification, data-learning, AI task-heterogeneity, O-ring, openness, and 2026 steering papers. All 54 citation keys resolve, the suite remains at 115 passes and 0 failures, and the 46-page PDF builds without warnings or bad boxes; the revised evidence and literature pages were inspected visually. The frozen website source remains unchanged. ## Stage 7 — Presentation and clean review 1. [x] Rebuild the abstract and introduction around one equilibrium mechanism and one main theorem chain. 2. [x] Demote identities and benchmark algebra from theorem status. 3. [x] Keep the main text compact; place technical robustness in the appendix. 4. [x] Run proof audit, counterexample search, symbolic/numerical suite, clean LaTeX build, citation audit, and page-by-page render inspection. 5. [x] Conduct a fresh referee read from the PDF without consulting the revision memo until after the verdict. **Submission gate:** no fatal technical objection; all abstract claims appear in a formal result with matching scope; current-data citations verified; clean build; fresh top-five assessment at least R&R rather than reject. **Status (2026-07-15): passed at R&R.** The abstract and introduction now present one mechanism chain and explicitly identify the conditional, modular equilibrium architecture rather than implying a global MPE. A complete label audit left standard price identities as lemmas, benchmark extensions as propositions, and the policy result as a corollary; only the derived structural and dynamic results retain theorem status. The clean proof pass tightened the representation theorem's compactness and kernel scope, added the no-purchase option to Bertrand competition, supplied an adaptive two-policy selection lower bound, stated the simulator constant on its actual domain, and aligned every annuity claim with the post-absorption and rising-lag terms. The welfare comparison now states its common-research-cost condition, and literal one-shot copying is identified as a normalization rather than a load-bearing assumption. The EdgeBench analogy was removed after primary-source review because it measures aggregate environment learning rather than the paper's execution kernel. `referee_reports/v38_referee_codex_clean.md` records the fresh major-revision R&R verdict; `v38_response_codex.md` records the response and post-revision gate. The suite reports 117 passes and 0 failures. All 53 cited keys resolve to 53 used bibliography entries. The 45-page PDF builds without warnings, undefined references, duplicate labels, or bad boxes, and every rendered page was inspected. The canonical source and PDF SHA-256 hashes are `f3377a15fa2cdf9887255c8fe79bc9c6b6a822e25d2e97df70e3d8ee6dbcdbbb` and `b4a34bb00fcb3dbb29a947e2ec07729ac472add4519cdd8ebb6bbb452791fde1`; the frozen website source remains unchanged. ## Stage 8 — Generality of the rotation centerpiece 1. [x] Let the committed-cohort annuities vary with the full frontier state and derive the exact cross-lag ordering condition. 2. [x] Derive the unique switch under own-state schedules and show exact nesting of the constant-lag theorem. 3. [x] Endogenize total research-program size and characterize its marginal and joint uniqueness conditions. 4. [x] Combine state-dependent annuity schedules with completion-date discounting. 5. [x] Add proofs, derivation notes, machine checks, scope guardrails, and an adversarial top-five referee round. **Gate:** the horizon-first rotation must not be an artifact of constant lags, fixed program size, or treating completion dates as simultaneous. Any claim about varying lags must remain inside the committed-cohort benchmark unless the exact moving-lag term is also carried. **Status (2026-07-15): passed at R&R.** Proposition 3 now permits each direction's committed-cohort annuity to depend on both frontier coordinates. Its exact program curl includes the annuity-gap complementarity force and both cross-lag derivatives; positive curl orders every project inversion. Own-state schedules nest transparently, and the polynomial benchmark yields a unique switch under an explicit curvature condition. The laboratory also chooses total program size, with a terminal-project scale margin and a sufficient Hessian condition for joint uniqueness. A fourth part combines full-state schedules with completion-date discounting. The prose maps the cross-lag terms to the certification comparative statics and explicitly excludes the exact moving-lag correction from this proposition. The suite reports 127 passes and 0 failures. All 90 labels are unique, all 70 referenced keys resolve, and all 53 cited bibliography keys match the 53 bibliography entries. The final 50-page PDF builds without warnings or bad boxes and passes PDF structural validation. The new theorem, proof, notation, and adjacent pages were inspected visually. The canonical source and PDF SHA-256 hashes are 0d346b840707824f34e31d0675f299e13933a062f57d17ee90cb5a77f5ded62c and 155840dd75e42d5e8747ce2cfe776a1086ad15c1ab1e7e6b32a61d6860370edf. The frozen website source remains unchanged at f527d1b6a026790f5d59134e2f9ba76497f25fcc5c1e06c7429b045eefdb0ca1. The Stage 8 referee report finds no fatal technical objection but retains an R&R verdict because modular equilibrium scope and the unmeasured certification lag remain top-five editorial risks. The persistent publication goal remains active. ## Stage 9 — Aligning the price–spending headline with endogenous scale 1. [x] Audit the opening claim against the endogenous program-size model and the protected-message charter. 2. [x] Derive the total-scale comparative static for a widening directional imitation asymmetry. 3. [x] State the exact condition under which reliability-annuity growth dominates lost horizon rent, and map program scale to resource spending under common unit costs. 4. [x] Add the corollary, proof, falsification restriction, derivation note, machine checks, and front-matter scope corrections. 5. [x] Conduct a top-five editorial desk screen of the revised title, abstract, introduction, formal spine, and scale proof. **Gate:** the opening coexistence of falling fixed-performance prices and rising frontier inputs must follow from a stated model condition. Faster horizon imitation alone must not be described as increasing total research when the endogenous-scale comparative static says otherwise. Public training-cost series must not be relabeled as the model's program control. **Status (2026-07-15): passed at send-to-referees.** Corollary 1 shows that faster horizon imitation alone reduces total research-program size, while a longer reliability annuity expands it. With both annuities moving, equation (31) is the necessary and sufficient condition for scale to rise. Under common unit costs and a positive marginal resource-cost coefficient, program resource spending inherits the scale sign. The abstract, introduction, motivating facts, result table, falsification section, and conclusion now present the opening claim as a conditional coexistence result and distinguish observed training-run cost from the model's scale control. The suite reports 132 passes and 0 failures. All 93 labels are unique, all 73 referenced keys resolve, and all 53 cited bibliography keys match the 53 bibliography entries. The 51-page PDF builds without warnings or bad boxes and passes PDF structural validation. The revised abstract, introduction, corollary, conclusion/notation page, and proof were inspected visually. The canonical source and PDF SHA-256 hashes are d14a94620845a47552422c2b32c35356bfd0cb8055d1f1679d0771b04173ec0e and c95b7cff62a375ad456e953556f615de25cbd5071c5de8ac151f3c825739aae8. The frozen website source remains unchanged at f527d1b6a026790f5d59134e2f9ba76497f25fcc5c1e06c7429b045eefdb0ca1. The Stage 9 editorial screen recommends sending the paper to external referees, but this is not an actual journal decision. The persistent publication goal remains active. ## Stage 10 — Journal positioning and submission engineering 1. [x] Verify current official submission requirements for AER, Econometrica, JPE, QJE, and ReStud. 2. [x] Rank journal fit without confusing formatting convenience with contribution fit. 3. [x] Prepare cover-letter, disclosure, AI-use, data/code, abstract, and author-decision templates without inventing author facts. 4. [x] Create an integrity-checked manuscript, source, evidence, and verification bundle. 5. [x] Record the remaining author-only decisions and preserve journal exclusivity. **Gate:** no external submission until the human author approves the target, authorship and correspondence data, conflicts and funding, exclusivity, fees, the exact AI-use disclosure, and the entire final manuscript. Journal requirements must be rechecked on the actual submission date. **Status (2026-07-15): preparation complete at the author-decision gate; not submitted.** The recommended first target is JPE, with ReStud as the theory-oriented alternative. The provisional fit sequence is JPE, ReStud, QJE, AER, and Econometrica. The current 51-page paper is closest to JPE's surfaced instructions. AER would require a substantive main-paper compression and its 100-word abstract limit; QJE would require journal-specific first-page metadata and table/figure placement. Compliant abstract drafts contain 131 words for Econometrica and 94 words for AER. `submission/` now contains the exact canonical PDF and source snapshot, administrative templates, author-only decision checklist, official-requirements audit, and a reproducibility package. The copied verification code reports 132 passes and 0 failures; the copied TeX compiles independently; both PDFs pass structural validation; and the copied manuscript PDF and source match the canonical SHA-256 hashes. No journal account was accessed and no paper was sent externally. The persistent publication goal remains active until actual acceptance. ## Stage 11 — Public website synchronization 1. [x] Replace the frozen v3.7 public manuscript, source, and checker with exact copies of the audited canonical artifacts. 2. [x] Remove obsolete page counts, check counts, calibrations, and simulated acceptance probabilities from the public landing page. 3. [x] Publish the current architecture, scope limits, evidence record, derivation audits, revision plan, and internal review history. 4. [x] Keep unapproved identity, conflict, funding, AI-disclosure, and portal-submission material outside the public site. 5. [x] Verify desktop and mobile rendering, local links, artifact hashes, production HTTP responses, and the live deployment. **Gate:** the website must present the audited manuscript rather than the frozen comparison version; every visible numerical or editorial-status claim must match the current record; public transparency must not be confused with journal acceptance or publish private author metadata. **Status (2026-07-15): live and verified.** The current site is `https://task-frontier-race.pages.dev/`. It serves the exact canonical 51-page PDF and source, the 132-pass verification suite, current architecture and scope, evidence and derivation records, and the latest internal desk screen. The landing page now states “send to external referees” as an internal verdict and explicitly says it is not a journal decision. Desktop and mobile renders were inspected, all local links resolve, the production page and sampled artifacts return HTTP 200, and the public PDF/source hashes match the canonical files. The manuscript itself was not changed in this stage, its protected messages remain intact, and the publication goal remains active.