# Fourier discretization and algebra construction feasibility

This edition preserves the earlier computational-model analysis and adds two bounded operational controls. All source theorems remain independently unverified.


This edition extends the [graph and constructor review](FEASIBILITY-2026-10-09-v4.md). All source results remain claims at pinned revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`; no trusted proof run or production implementation is inferred.

## A finite algorithm may activate after the useful horizon

The uniform k-server construction canonicalizes a finite rational metric and simulates its finite table constructor from scratch for `floor(log2(t+1))` elementary transitions at request t. If the constructor needs T_P transitions, activation is exactly `t*=2^T_P-1`. Earlier requests always use the first label. The constructor is guaranteed finite in the source argument; its running time was not measured here.

The displayed movement constant is `B=(t*-1)D+(2a+2)kD`, with D the metric diameter and a the constructed coefficient. The distinct-start zero-additive clause belongs to the separate policy-existence theorem. It does not eliminate startup loss from this uniform implementation. Request processing polynomial in input encoding and log request number does not guarantee polynomial activation delay or useful short-horizon loss.

The [exact movement reference](tools/kserver_reference.py) evaluates three ordinary named policies and a conventional offline DP. A three-point line starts at `(0,1)` and alternates requests 1 and 2. At 30 requests, fixed-first-label cost is 30 and the attained offline optimum is 2. At only two requests the optimum is 1; the test preserves this horizon distinction. These are finite controls, not an implementation of the source policy. They motivate a replay/startup assurance product and defer a literal backend until useful constructor and latency measurements exist.

[Uniform introduction](../source/openai-math/preprints/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026/build/source/sections/01-introduction.tex), [schedule and constants](../source/openai-math/preprints/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026/build/source/sections/04-uniform-computation.tex), [formal scope](../source/openai-math/lean/docs/110.md), [137 checks](snapshots/2026-10-08-baseline/movement-reference-validation-2026-10-09-v1.json), [saved examples](snapshots/2026-10-08-baseline/computational-model-examples-2026-10-09-v1.json).

## An exact transport sensitivity benchmark

The cube example uses `u=max(-x1,x1,a)` and `v=max(-x1,x1,a+(a/2)x2)`, for `0<a<1/2`. Both pushforward targets have outer masses `(1-a)/2` and central mass a. Only the central atom moves, by `b=a/2`. Its target distances are exactly `W2^2=ab^2` and `W1=ab`; the maps reassign source mass `b/2`, giving squared L2 map distance `b/2+ab^2`.

The [new reference](tools/transport_sharpness_reference.py) clips affine halfplanes and integrates the nine cell intersections with rational polygon areas. It compares these independent geometric results with the displayed formulas. For `a=4^-j`, the square of the constant required for a half-exponent bound is `2^(j-1)(1+a^2)`, which grows without bound. For the one-third exponent, the sixth power of the ratio is `(1+a^2)^3/16`. The source dimensions above two factor out; the implemented geometry is two-dimensional.

The manuscript also gives a usable explicit domain formula, improving the earlier dossier's unresolved-constant statement: `C_*^2=12dR^2(1+sqrt(162))^2+28L^2 P_K/volume(K)`, where P_K sums coordinate projection volumes. For `K=Y=[-1,1]^2`, using `R^2=L^2=2`, `P_K/volume(K)=1` and `sqrt(162)<=13` gives `C_*^2<=9464`; the rational choice C=98 is sufficient if the source theorem is accepted. This can be very loose and proves no pointwise, discrete-source, entropic or learned-map guarantee. The reference is a benchmark control rather than a general transport solver.

[Sharpness construction](../source/openai-math/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/build/source/sections/sharpness.tex), [domain constant](../source/openai-math/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/build/source/sections/gradient.tex), [131 checks](snapshots/2026-10-08-baseline/transport-reference-validation-2026-10-09-v1.json).

## Decision derandomization and compiler readiness

LogspaceEquality compares binary language classes using finite-control machines, endmarked input, local binary work tapes and fresh fair bits. Work space counts all traversed writable cells, including blanks. The randomized classes require a polynomial worst-case clock on every coin tape and the specified acceptance gap. Language equality does not preserve a random output law, generate cryptographic entropy or derandomize arbitrary ML.

The paper's compiler assumes supplied positive time/space parameters a,b and specializes one fixed deterministic separator library. It does not decide those semantic resource promises. The selected Comparator target proves the class-equality claim; the compiler and its numerical resource bounds are not separately selected.

The explicit bounds use `P=1000(d_code+2)^4(a+b+2)`, `rho=100(P+2)`, `S=1+(t+1)(2rho+1)(g+2)`, `K=4S`, `c=2S+1`, and `H=2^((u+100)^2)*(q_D+1)`. Four auxiliary work tapes are named. Conservatively taking only `d_code,a,b>=1`, `t>=4`, `g>=2`, `u>=1` and `q_D>=1` yields declared `K>=5,184,032,084`, `c>=2,592,016,043` and `H>=2^10202` (at least 3,072 decimal digits). These are lower envelopes for deliberately loose declared upper-bound parameters, not lower bounds on actual runtime/space or proof of impracticality for redesigned simulations. No valid program is asserted to attain minimum description length, and no compiler/library transition list was executed.

[Parameter arithmetic](snapshots/2026-10-08-baseline/logspace-compiler-parameters-2026-10-09-v1.json), [compiler source](../source/openai-math/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/build/sections/compiler.tex), [formal scope](../source/openai-math/lean/docs/103.md).

## Semantic evidence and earning boundary

[41 source-level definition observations](snapshots/2026-10-08-baseline/definition-hole-review-queue-2026-10-09-v2.json) compare KServer, Brenier and LogspaceEquality challenges and selected solution definitions. No intended-model mismatch was detected there. A lexical control also finds the displayed logspace definition/structure prefix identical after comment/whitespace removal. This does not check Lean elaboration, imported dependencies, the proof or executable software.

The [thirteen-contract registry](prototypes/curated-contracts-2026-10-09-v3.json) now includes those model distinctions, with [12 linter checks](snapshots/2026-10-08-baseline/semantic-contract-validation-2026-10-09-v1.json). Seven configurations/24 declared entries remain queued. The commercial lead stays one shared evidence/model-assurance workflow, with movement replay and transport calibration as narrower modules. Pricing, demand and recurring delivery remain unvalidated.

## Infinite convolution versus a finite circular grid

Family 371's source products and odd-power recursion use infinite Fourier convolution. The finite reference works on physical rational sparse coefficients and computes the exact polynomial before folding frequencies modulo an odd grid. For a coordinate band K, power p, N>(p+1)K is a sufficient retained-projection condition and N>2pK a sufficient full-reconstruction condition. Exact instance comparison may pass with a smaller grid. The cubic two-wave fixture distinguishes phantom central mass, correct projection and full recovery across N=3,5,7.

The source blowup theorem selects arbitrarily large odd powers on a twelve-dimensional torus, with k>8 and Gaussian alpha>k+6. It gives positive blowup probability, with no uniform lower probability or numerical lifespan supplied by the inspected statement. The cubic axis fixture checks multiplication semantics only. No time integration, profile certificate or finite-grid-to-continuum blowup proof is implemented. [Finite reference](tools/audit_fourier_aliasing.py), [110 controls](snapshots/2026-10-08-baseline/fourier-aliasing-validation-2026-10-09-v1.json).

## Elementary positivity and a terminating witness

Family 169's witness-only selected interface includes a proof for every number of color variables. The formal solution selects a successful finite table whose raw encoding space is C(2n-1,n-1)^(n^n). At n=3 that is 10^27; at n=4 it is 35^256, a 396-digit integer. Encoding size is not an actual runtime lower bound or a requirement to enumerate it. The paper's separate packet/history procedure terminates, but no polynomial practical bound was selected. No elementary unimodality or preferred geometric Hessenberg basis follows from the reviewed statements.

The conventional exact finite reference caps six vertices and two million work units. It expands proper colorings in n variables, verifies symmetry, converts to the elementary basis and reconstructs every finite coefficient. A supplied witness can fail or be incomplete. Unknown on exhaustion is distinct from a negative coefficient. [106 controls](snapshots/2026-10-08-baseline/chromatic-basis-validation-2026-10-09-v1.json), [new product hypothesis](opportunities/039-positive-basis-algebra-audit/2026-10-09-v1.md). Sage already computes related functions and basis conversions; reproducible review must add useful work beyond that existing baseline.

## Remaining representation/dynamics boundaries

Family 238's one-card uniformity does not imply joint permutation uniformity. The selected overlap/angle definitions use genuine representations, complete multiplicity-copy sums and actual signed tensor projections. Full-mixing companion claims have existential absolute constants and dyadic protocol assumptions; selected endpoints alone do not provide a finite production sweep count or cryptographic security.

Family 145's ordinary-to-multiple mixing claim is asymptotic for one invertible probability action, with no selected finite rate. Family 158 leaves six versus seven plane colors unresolved and gives no explicit finite six-required graph here. Family 297's nonseparable transfinite C*-algebra construction is not an executable sampled-matrix backend. [All 65 selected definition observations](plans/definition-semantics-review-2026-10-09-v3.md) are source-level evidence only. [24 manuscript section records](MANUSCRIPT-REVIEW-2026-10-09-v4.md) do not establish full paper coverage.

## Shared-switch permutation systems

Family 238's local-access corollary uses a fixed absolute number L of complete sweeps on n=2^d slots and Ld adaptive shared-bit queries per point. The trace-smoothing companion supplies all-size TV <=(1/2)n^-5 for 2v>=P, with one existential absolute P. The selected CoordinateTrace covers that mixing bound; actual query-code, arbitrary-sized cycle restriction and a PRG bridge remain additional obligations. Its source solution takes P=2u after u=2m and m>=max(14/a,2/b), with positive contraction coefficients a,b imported and not numerically validated here.

Exact four-card TV is 1/192 at four sweeps, 1/768 at five, and 1/3072 at six. The trace target 1/2048 fails for the first two and passes for six on this instance only. Exact eight-card one-sweep TV is 283/315 despite uniform one-card marginals. [176 finite controls](snapshots/2026-10-08-baseline/queryable-permutation-validation-2026-10-09-v1.json), [calibration](prototypes/coordinate-four-card-calibration-2026-10-09-v1.json).

A deterministic b-bit seeded permutation has at most 2^b outputs and statistical TV >=max(0,1-2^b/n!). For dyadic n, n! >=(n/2)^(n/2), so the billion-slot 2^30 case with 256 seed bits has TV >=1-2^(-15,569,256,192). The symbolic calculation does not allocate that huge integer. At 64 slots the exact lower bound already exceeds 0.999. This is a statistical support limitation, not a security assessment or evidence against practical reproducible seeded goals. [21 support controls](snapshots/2026-10-08-baseline/permutation-entropy-validation-2026-10-09-v1.json).

An immutable lazy fair-bit provider preserves the intended randomness model but its storage grows with revealed switch keys. A counter/key provider may avoid that cache while using a different full-law model. PyTorch's current sampler materializes an index list and has explicit pad/drop/rank behavior; Random123 provides existing stateless pseudorandom counters. These are substantial baselines, not demonstrated performance competitors here. [PyTorch source](https://github.com/pytorch/pytorch/blob/main/torch/utils/data/distributed.py), [Random123](https://github.com/DEShawResearch/random123).

Cycle deletion to m<=n slots preserves a permutation but may require n-m+1 full permutation calls in the worst case. Uniform source permutations induce uniform smaller ones; tiny exact controls verify that identity through four slots. An approximately uniform full law transfers at most its TV error under a deterministic pushforward, but marginals need not remain exactly uniform: the four-to-three one-sweep example has marginals 5/16,5/16,3/8 in one row. No arbitrary-size formal bridge, data-loader batching guarantee or end-to-end cost is verified. [Application and rejection conditions](opportunities/040-queryable-permutation-framework/2026-10-09-v1.md).

The older primary cell-probe context is checked in [version-one PDF](https://arxiv.org/pdf/2512.02724v1). Its experimental HTML showed a different title date, so version-specific context uses the PDF and keeps submission/title dates separate. [Retrieval notes](snapshots/2026-10-08-baseline/permutation-external-source-evidence-2026-10-09-v1.json). No external proof was independently verified.

## Provider cost and contraction audit

The new [provider reference](tools/permutation_provider_reference.py) supports canonical decimal wire and consistent seeded or local stored bits. Two local processes and restart/inverse controls pass, while incomplete, duplicate, empty and mismatched stores remain untouched. Plan isolation and actual-size read allocation were hardened. It does not establish independent fair-bit law, network durability, source compiler correctness, cryptographic security or a universal mixing threshold.

The first synthetic run exposed an 8 MB read allocation even for a tiny file. After repair, the [repeated 256-slot run](prototypes/permutation-provider-benchmark-2026-10-09-v2/measurements.json) has 43,318-byte traced Python sparse peak and 35,972 bytes of persistent cache after 16 points. Full materialization retains 293,486 file bytes for 2,048 switches, versus a 2,048-byte packed int64 representation. This is a concrete reason to reject a memory promise for the present JSON backend. The timings are single dependent phases and use a simple Python oracle; no production speed comparison is inferred. The seeded mode retains no switch rows but uses a different randomness model. Compact storage, realistic sparse tasks and end-to-end support costs remain pending.

The [Harmonic import audit](plans/coordinate-contraction-review-2026-10-09-v1.md) follows the correct TraceModel import. Level and dimension coefficients are minima of eventual bounds and finite logarithmic gaps. The gap uses a classically selected positive convolution exponent and a minimum over the permutation group; this selected proof does not expose a validated production numerical P. Do not substitute a duplicated Casimir path or the six-sweep four-card success for the universal witness. [Current dossier](opportunities/040-queryable-permutation-framework/2026-10-09-v2.md).

## Exact small-state gap versus a universal numerical witness

The [finite calibration](plans/coordinate-contraction-review-2026-10-09-v2.md) checks source-oriented butterfly/palindrome composition against an independent swap oracle. For four slots all 24 permutations occur among 256 coin strings, and the minimum probability 1/32 yields explicit minorant mass 3/4 with exponent zero/gap 5/8. The centered regular operator has a stronger exact norm 1/4, verified by symmetry, B^2=(1/4)B and an attaining zero-sum eigenvector. Conservative norm/minorant TV bounds reach the finite target at 13/37 sweeps; the direct law passes at six. These are different valid finite statements, not conflicting estimates of a universal threshold.

The source-selected minorant is not evaluated, numerical eventual thresholds/contractions remain unselected, and the regular matrix is factorial in slot count. The utility therefore caps at four slots. This strengthens a bounded quantitative assurance module; it does not justify a practical unrestricted sampler or change the unfavorable measured JSON-provider storage result. Source proof/program acceptance and actual independent randomness remain open. [50 controls](snapshots/2026-10-08-baseline/palindrome-minorant-validation-2026-10-09-v1.json), [current dossier](opportunities/040-queryable-permutation-framework/2026-10-09-v3.md).

## Public event-law and network-flow bridge

The exact event-law utility makes the fixed-margin distinction operational. A documented input's probability-order event has exact probabilities 5/143 under intended conditional independence versus 3/7 under uniform feasible tables, crossing a declared 1/20 threshold. This is a model-substitution consequence, not an existing-library defect. Bound conditioning is explicit and does not automatically represent real structural-zero mechanisms. The live SciPy import failed locally; R was not executed. [Concrete public report and limits](prototypes/public-contingency-law-review-2026-10-09-v1.md).

The source bounded-flow corollary is broader than a table sandbox: lower-bound shifts and private vertices yield a binary-size bijection on signed integer arc vectors, including loops/parallel arcs. Its sampling accuracy has polynomial 1/tau cost and does not inherit exact uncapped sampling or logarithmic-only tolerance dependence. Selected formal metadata contains table statements, with no separate flow bridge/sampler in this selection.

The exact reduction/reference passes 460 controls, and NetworkX feasibility integration adds 13. A public five-node/nine-arc planning model becomes a 14-by-14 table with total 3,422, exceeding the exact reference's dimension-six/total-128 cap. Count and reference feasibility therefore remain unknown while an existing solver provides a separately checked feasible vector. The source FPRAS/sampler is not implemented. Extra global cost constraints, multicommodity models, path-decomposition counts and operational failure probabilities require separate bridges. [Dossier 041](opportunities/041-bounded-flow-scenario-analysis/2026-10-09-v1.md).