Prove prime supply for subexponentially many axes (no κ)

research/subexponential-prime-dimension/PROOF.md

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+# Prime supply with subexponential dimension
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+## Statement and inherited reduction
+
+Let ell and d be integers satisfying
+
+```text
+ell >= 2^18,       2 <= d <= 2^floor(sqrt(ell)/4),       D=d-1.
+```
+
+For either t=2^ell or t=2^(ell-1), the half-open interval
+
+```text
+I(t,d) = (t*(1-1/(3d)), t*(1-1/(4d))]
+```
+
+contains at least D primes. Thus the same two-scale caller can select D
+distinct odd prime lengths in original axis order. Retain the separate
+premise target_halo+2w <= 2^(ell-3) to obtain its period guard. The band
+also retains the source-ratio guards and T/S<3/2 from ToikY's
+[PR4029](https://github.com/CrocSwap/integer-mult-bounds/pull/4029).
+
+The new result is the displayed parameter domain. Source guards, ordering,
+product accounting and the prime-selection algorithm retain their
+predecessors' credit. No producer implementation or multiplication theorem
+is proved here.
+
+All logarithms below are natural. Put y=log(t). We use the following
+interval estimate from PR4029's PROOF.md, section "Imported analytic input
+and effective cutoff", as a cited premise:
+
+```text
+pi(b)-pi(a) >= t/(12*d*y) - 20*t*sqrt(y)*exp(-sqrt(y)/2).       (1)
+```
+
+Its derivation requires only d>=2 and t>=4, before imposing the old
+polynomial dimension restriction. Its analytic input is Fiori, Kadiri and
+Swidinsky, [Corollary 22, arXiv:2206.12557v2](https://arxiv.org/pdf/2206.12557v2):
+
+```text
+|pi(x)-Li(x)| <= 9.2211*x*sqrt(log(x))*exp(-.8476*sqrt(log(x))), x>=2.
+```
+
+Neither predecessor theorem is reverified by the finite checker. To make
+the right side of (1) strictly larger than D, it suffices to prove
+
+```text
+480*d*y^(3/2)*exp(-sqrt(y)/2) < 1,                             (2)
+t > 24*d*D*y.                                                (3)
+```
+
+Indeed, (2) makes the error strictly smaller than t/(24*d*y), leaving
+strictly more than t/(24*d*y), which exceeds D by (3).
+
+## A dimension bound independent of a polynomial degree
+
+The elementary estimates 1/2<log(2)<7/10 give
+
+```text
+ell/4 <= y <= ell,       log(d) <= (sqrt(ell)/4)*log(2)
+                                     < 7*sqrt(ell)/40.
+```
+
+For the first lower bound, ell>=2 implies
+(ell-1)*log(2) > (ell-1)/2 >= ell/4. Set z=sqrt(ell), so z>=512.
+The left side of (2) is at most
+
+```text
+480*z^3*exp(-3*z/40).
+```
+
+Let f(z)=log(480)+3*log(z)-3*z/40. At z=512,
+
+```text
+f(512) < 7 + 3*9*(7/10) - 3*512/40 = -25/2 < 0.
+f'(z) = 3/z - 3/40 <= f'(512) = -177/2560 < 0.
+```
+
+Here log(480)<7. These inequalities establish (2) on the entire domain,
+without an enumeration over ell or d.
+
+For (3), D<d and y<=z^2 give
+
+```text
+log(24*d*D*y) < 5 + 7*z/20 + 2*log(z),
+log(t) >= z^2/4.
+```
+
+Use log(24)<5 and define g(z)=z^2/4-5-7*z/20-2*log(z). At z=512,
+
+```text
+g(512) > 512^2/4 - 5 - 7*512/20 - 2*9*(7/10)
+       = 326696/5 > 0.
+g'(512) = 512/2 - 7/20 - 2/512 = 327227/1280 > 0.
+g''(z) = 1/2 + 2/z^2 > 0.
+```
+
+Therefore g stays positive for all z>=512, proving (3) and the theorem.
+
+The logarithm estimates use classical elementary inequalities. Integrating
+1/t>1/2 on [1,2] proves log(2)>1/2. Positive exponential partial sums
+of degrees three, six and three at 7/10, 7 and 5 exceed 2, 480 and 24,
+respectively. The verifier checks these strict rational comparisons.
+
+## Strict enlargement of the original dimension region
+
+For every ell>=2^32, the new cap contains d<=ell^334 from PR4029.
+It suffices to prove
+
+```text
+334*log(ell) < (sqrt(ell)/4-1)*log(2),                        (4)
+```
+
+because floor(u)>u-1. With z=sqrt(ell)>=65536, the right side minus
+the left side of (4) is strictly greater than
+
+```text
+h(z) = z/8 - 7/10 - 668*log(z).
+```
+
+At z=65536, h(z)>8192-7/10-668*16*(7/10)=7097/10>0.
+Its derivative 1/8-668/z is positive there and increases afterward.
+This proves (4) for all ell>=2^32.
+
+The enlargement is strict. At ell=2^32, d=2^12000 exceeds
+ell^334=2^10688 and is below the new cap 2^16384. Only exponent
+comparisons are needed to certify this witness. The old improved
+polynomial cutoff in PR5149 is a different domain comparison; no least
+cutoff or optimal dimension constant is claimed.
+
+## Evidence boundary
+
+The standalone verifier computes the displayed rational endpoints and
+derivative signs. It also checks the first error endpoint by the separate
+integer inequality
+
+```text
+8^38 > 480*512^3*3^38,
+```
+
+using e>8/3 and 3*512/40>38. A separate quota calculation compares
+the exact exponents ell-1=262143 and 5+256+18=279, since
+24*d*D*y < 2^5*2^256*2^18 at the first endpoint. These corroborate
+the endpoint arithmetic; the derivatives above prove the unbounded range.
+
+The proof does not execute a giant prime search, materialize D lengths,
+prove a practical cutoff, or implement an encoded caller. The imported
+prime-counting theorem is mathematical input, not a finite certificate.
+Native operations, physical costs, complete numerical recovery, compiler
+closure, precision/tape and selected-supplier admission remain separate.
+Independent human and formal proof-kernel review are unclaimed.

research/subexponential-prime-dimension/README.md

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+# Subexponentially many axes in the fine prime band
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped mathematical proof with exact endpoint certificates.
+The many-axis prime band from [ToikY's PR4029](https://github.com/CrocSwap/integer-mult-bounds/pull/4029)
+contains at least D=d-1 primes at either retained target scale whenever
+
+```text
+ell >= 2^18,       2 <= d <= 2^floor(sqrt(ell)/4).
+```
+
+This removes the polynomial dimension restriction in that analytic supply
+argument. For every ell>=2^32 the new cap contains its entire old region
+d<=ell^334. At ell=2^32 it also admits d=2^12000, while the old cap is
+2^10688. The separate halo premise is retained. No new kappa is claimed.
+
+[PROOF.md](PROOF.md) gives the interval reduction, new endpoint and
+monotonicity argument, and the strict enlargement proof. The predecessor's
+interval estimate and the Fiori–Kadiri–Swidinsky prime-counting theorem are
+cited inputs. The new result is the dimension region, rather than an
+audit of either predecessor or a prime-producer implementation.
+
+## Reproduce
+
+From repository root, with CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error research/subexponential-prime-dimension/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B -W error research/subexponential-prime-dimension/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/subexponential-prime-dimension
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/subexponential-prime-dimension --check-source-history
+```
+
+Another CPython 3.12 installation may replace that interpreter path. The
+direct verifier prints a generated JSON record binding all six package
+files, followed by `PASS_SUBEXPONENTIAL_PRIME_DIMENSION`. It writes nothing
+and needs no Git, network or predecessor runtime input. The package and
+source-history checks separately validate the recorded historical pins.
+
+The checker uses exact fractions and integer powers. It checks the four
+strict exponential partial sums, the stated endpoint and derivative
+fractions, two alternative integer endpoint calculations, the polynomial
+containment certificate and the strict enlarged-domain witness. It also
+corroborates 24 dyadic endpoints and rejects nine invalid or unsupported
+certificate parameters. Rejection of a smaller cutoff certificate does
+not mean prime supply fails at that cutoff. The universal statement rests
+on the written proof and imported analytic theorem.
+
+## Scope and credit
+
+The `uniform-transfer-audit` card changes to an analytic source-supply side
+result connected to `analytic_recovery`. It discharges none of the five
+selected assumptions in full. The selected PR6030 baseline returned
+`PASS_RESEARCH_BASELINE_BINDINGS_ONLY`; that is metadata orientation,
+rather than a supplier replay. The working base is
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8.
+
+Credit ToikY / PR4029, head `726c9a3a1dc58d2b1223dd681eb712255be407c1`,
+for the exact band, caller binding, interval reduction and prior supply
+domain. Credit Fiori, Kadiri and Swidinsky for their imported analytic
+theorem; Jose Manuel / PR3935 for the tensor interface; Tekalign / PR4024
+for the CRT source premise; unpaidmoon / PR4142 for the separate guard;
+Cinna Davis, Chafik Boukhalfa, Josie Elliston, the source08 contributors,
+Harvey–van der Hoeven and Agarwal–Cooley for the retained selector and
+source-construction lineage. Nick Petalas's PR5149 and PR5152 concern the
+older polynomial cutoff and product envelope. Their results are credited
+context, without being replayed here. Classical exponential estimates and
+monotonicity arguments retain their mathematical credit; no global
+literature-priority claim is made.
+
+Recent PR6444 improves a setup algorithm at the old abundance domain.
+This result supplies a distinct analytic region and does not implement
+its open encoded interface. No giant primes are computed. Encoded source
+production, all-size physical tensor integration, numerical recovery,
+stock, precision/tape, native and compiler obligations remain open or
+inherited. No selected cold replay, Lean or Docker is run. Independent
+human and formal proof-kernel review are unclaimed.
+
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19372253)
+records the same target. Live PRs and recent handoffs, current main and
+merged object/formula/general-region searches, recent account work and
+local reservations were checked. Arbitrary historical diffs and unpublished
+work remain outside coverage. [TASK.json](TASK.json) records overlap
+freshness and the explicit scope change. [SOURCES.json](SOURCES.json)
+pins historical provenance; no predecessor code is imported or modified.
+
+## Open next questions
+
+1. Can the original endpoint errors admit a larger coefficient of sqrt(ell)
+   in log(d), with an explicit cutoff? This would improve the analytic
+   capacity. Start with both unweakened endpoint errors. A uniform error
+   comparison and quota argument are still needed; no optimum is claimed.
+2. Can a changed complete transform use the expanded dimension region to
+   lower its dominant paid bill? Start by deriving that caller's precision,
+   halo, source-loading and tensor cost laws. The present source-supply
+   theorem alone neither supplies that caller nor changes kappa.
+
+## Code inspection
+
+The verifier has no recursion, subprocesses, writes or dynamic code
+generation. Every function is under 60 lines. All loops have fixed caps:
+ten series terms, four logarithm certificates, 24 endpoint probes, nine
+rejection controls, or six package files of at most 256 KiB. Explicit
+exceptions remain active under `-O`. Scalar certificate parameters are
+validated before arithmetic. CPython's integer and container allocation
+is a managed-runtime deviation from fixed-storage execution; root exponents
+are capped at 32, and the largest alternative power is 8^38.
+These caps belong to finite corroboration, not to the theorem. Warning-strict
+normal and optimized checks provide the local diagnostics. No static
+analyzer or literal fixed-storage execution is claimed.

research/subexponential-prime-dimension/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
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+      "usage": "provenance",
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+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
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+      "usage": "provenance",
+      "git_blob": "f3f736734a0d209a5fb383494b2253cc8f7702e4",
+      "sha256": "68ba22bcf761370c9370f770ed65de0de699feccb89c9011e470e00542ae72e9"
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+    {
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+    },
+    {
+      "path": "research/many-axis-prime-cutoff/README.md",
+      "usage": "provenance",
+      "git_blob": "7c79fab9a66e8d63602bbf0f65d3589e9f71e032",
+      "sha256": "786d4a5fbb1da3ba988f86d445c28e33000363e7e3cf962a536b61e4fdcc8e11"
+    },
+    {
+      "path": "research/many-axis-prime-product-loss/README.md",
+      "usage": "provenance",
+      "git_blob": "e93107c8c3f7b2f0a07560d8547cd9759e1c5525",
+      "sha256": "e939dd08789a9a2a429ef8c7ca9883589fd5d4ca5bdff3da06178352c89eeb2a"
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+  ],
+  "external_mathematical_input": {
+    "url": "https://arxiv.org/pdf/2206.12557v2",
+    "authors": [
+      "Andrew Fiori",
+      "Habiba Kadiri",
+      "Joshua Swidinsky"
+    ],
+    "result": "Corollary 22",
+    "role": "Imported theorem; not executed, vendored or reverified."
+  }
+}