[codyw912]

Sharpen default chronological width to d+1 and transfer cuts to arbitrary heights

Results: one contribution, two scoped replay packages

This PR combines the default-height width sharpening with the arbitrary-height
cut transfer and conditional width envelope. It preserves two independently
replayable package boundaries, not a shared replacement engine or two competing
theorem submissions:

• research/chronological-width-bound: for a fixed default-height mixed rule,
  log B_N = O((log N)^(d+1)), where d counts primes dividing target/paid
  ranks. Bounded visits to each source-count plateau remove one logarithmic
  factor from the d+2 bound already proved in Prove quasipolynomial bound for chronological batch widths (no new κ) #4586.
• research/adaptive-chronological-batching: extends Reduce direct-flow batch widths while preserving chronological calls #4507's least chronological
  batching policy to Remove tensor padding (199.32× smaller uniform-rank exponent) #4483's arbitrary-height forests, including components
  beneath empty parents. Separate paid positions and repeated paths retain
  separate cuts: occurrences multiply cost, not local batch size. The least
  multiplier remains lcm_(e,q) B_e/gcd(B_e,q), and normalized leaf costs are
  unchanged. Its conditional fixed-parameter theorem gives
  log B_N = O(N^z (log N)^(d'+1)), with
  z=max(0,1-g log_M(kmin)) and d' counting primes in target/paid ranks or M.
  It is quasipolynomial if kmin^g>=M, and subexponential otherwise.

Inherited results versus the contribution here

Predecessor	Inherited result	Delta here
#4436 / #4483, zxv, building on Jacob Alan Sussman	Default/arbitrary-height aggregate flow and resource identities	Do not claim a new flow or convergence theorem
#4507, Mihir Srivastava	Least per-cut LCM policy, default cut correctness, finite width improvement	Default all-rank sharpening and arbitrary-height cut-level transfer
#4586, aadb7bc59390683450b1969e4632f546929dc569	Default log B_N=O((1+log N)^(d+2))	d+1 through bounded per-plateau visits; not the first quasipolynomial classification
#4588, 2f6578907b7f8718a13921552cc3069d3908e936	Independent finite default checks	Complementary coverage, not this all-rank sharpening
#4591, 84c0fb3a39bcb1e4b8c48892bfdd8a6e7ae58a50	Arbitrary-height aggregate identities	Actual separate chronological cut divisibility, minimality and conditional width bound

#4519's finite recount, #4521's expanded counts, #4532's gate/critical-root
proof and #4533's witness retune remain distinct. The default raw-p7 finite
width improvement is reproduced predecessor evidence, not claimed as new.

Specialization and nonemptiness qualifications

Setting M=kmin,g=1 in the adaptive symbolic theorem gives the default height,
d'=d,z=0 and logarithmic degree d+1 on their common hypothesis domain.
This is not a claim that the named adaptive implementation exercised that
specialization: its ('adaptive', M, a, b) route requires positive a,b, so
g=a+b>=2. The width witness code accepts g directly and includes a g=1
toy case; that is a separate bounded witness, not an all-rank default replay.

Adaptive WIDTH.md assumes paid rank one to ensure every forest is nonempty.
The default proof needs no paid rank one: its root unit exists because
kmin^G<=C. Thus the adaptive statement as written does not subsume the whole
default hypothesis domain. The default plateau-product divisor and adaptive
level-LCM divisor are different certificates; no universal finite numerical
dominance is claimed. Wholly empty flows are not productive suppliers, and
all 405 adaptive oracle fixtures have paid rank one; wholly empty-flow
rejection was not exercised.

Executable evidence

• Default receipt: five fixtures, 102,215 chronological batch constraints,
  source/cut plateau signatures, source and contraction bounds, chain-depth
  and per-plateau-visit checks, eleven diagnostic prime valuations, exact
  all-prime integer divisibility, and four boundary controls. The retained
  Reduce direct-flow batch widths while preserving chronological calls #4507 compiler produces the widths; this is not an independently written
  default cut compiler. Its literal cut oracle is inherited, not rerun here.
• Adaptive receipt: independent literal ordered-word/paid-path oracle on 405
  nonempty flows and 19,089 classes, authenticated source-Flow and closed-form
  comparisons, 2,304 brute-force minimality cases, six compilations with
  186,476 chronological constraints, and 25 semantic controls.
• Adaptive width receipt: four additional toy compilations, 13,377 constraints,
  all-prime integer divisibility, three pinned real-profile boundary
  applications, and two controls. Real-profile checks do not materialize
  all-rank compilers. The asymptotic results are symbolic proofs, not finite
  extrapolations.
• Raw p7 at N=1209 with explicit M=2 height: least width 5,450 bits versus
  74,481 nested bits. This is a finite changed-geometry witness, not Remove tensor padding (199.32× smaller uniform-rank exponent) #4483's
  M=256 witness. Pinned Remove tensor padding (199.32× smaller uniform-rank exponent) #4483 applications give p7
  z=1-log_256(25) and native4453/p16 z=3/5, only for those exact raw
  profiles and pinned bases. No strictification, retuned Sharper #4483 witnesses: p16 2^1587000000, native4453 2^713000000, p7 2^2253000000, same κ (compiler improvement, no new κ) #4533 geometry or
  physical caller admission is inferred.
• Existing OpenAI/Anthropic cross-family reviews and replays are retained as
  AI evidence. Mathematical challenges supplied the plateau-visit and integer
  level-run arguments. No independent human or formal-kernel acceptance claimed.

Fresh consolidation replay: both registered package commands below passed on
Python 3.12 in an author-local environment. Adaptive normal/optimized cut
checks and normal/isolated-optimized width checks passed; default normal and
isolated-optimized checks passed. Receipts were compared without regeneration.
The two hooks completed serially in 74.07 seconds. This is agent-run validation,
not hosted CI, independent human acceptance or a proof-kernel check.

Reproduce both scoped packages

From the repository root, serially with Python 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/chronological-width-bound
python3.12 -B scripts/research_package.py --package research/adaptive-chronological-batching

Nix and direnv are not repository requirements. Workstation environment files
are retained locally and excluded through Git's private info/exclude, not
tracked in this PR; .gitignore remains unchanged from upstream.

Each registered hook retains its existing checks, including optimized/isolated
runs as declared in VERIFY.json. No receipt regeneration is needed. Scientific
code, receipts, frozen inputs and source pins are preserved; only package
positioning/task metadata changes accompany the copied six-file default package.

Scope, provenance and attribution

Historical canonical base f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc, priority
revision 7 and the original independent-moment cards are retained. The old
23/10000 selected-record context is historical, not a live-selection claim.
The consolidation comparison uses fetched main
fbc9ea0249746acca4f504d4fda2b84e0e36cf0d: selected conditional κ is
.00302015 from #4470; #4483 and #4507 are not selected. No selected-frontier
file changes. The default source package is preserved from
099267b65810ebe095076e211b74beeed5575b25.

The 2026-10-22 overlap snapshot covers 408 open PRs at 13:08:38 UTC, 3,852
closed PRs at 13:09:39 UTC and 3,577 Discussion records at 13:09:11 UTC,
with complete reported pagination. It is non-atomic and not exhaustive novelty
proof; the actual closest proofs/code supply the scoped comparison above.
Earlier searches and runs remain historical provenance. Canonical main and
the #4586/#4588/#4591 heads were rechecked unchanged before publication.
Package-local LOCAL-ONLY notes record the pre-publication state; this PR
provides the public artifact without claiming scientific acceptance.

No global batching optimality, tight width lower bound, new leaf exponent,
κ, convergence theorem, practical threshold or physical/routing/precision/
stock/tape admission. Native semantics, whole-job contracts, analytic transfer
and recovery remain inherited premises.

Credit Jacob Alan Sussman, zxv, Mihir Srivastava, Ryanson Jonathan, Cinna Davis
and the retained lineage, with complementary #4586/#4588/#4591 work credited
above. Prepared for codyw912 with substantial OpenAI and Anthropic assistance.
Retained scientific sources keep their own notices; new material is Apache-2.0.

Current publication revision

Branch research/20261022-adaptive-batch-width, commit
385a4436a47ea8e80cc1503948fe7a36e1b1c5af. This follow-up removes the
agent-added .envrc, devenv.nix, devenv.yaml, devenv.lock and related
.gitignore changes from the PR contents. The local files remain intact and
privately ignored. Both package READMEs describe plain Python reproduction.
The PR now adds only the two scoped research packages; scientific code,
receipts and source pins are unchanged.

An independent Anthropic review approved the environment-boundary correction.
Both registered hooks passed again through the retained local environment in
74.79 seconds, including normal/optimized commands. Committed-byte and
source-history checks passed at this head: 19 adaptive files and 12 default
files. Earlier export/integrity checks belong to commit 420b7ba9249e.
No human/proof-kernel acceptance or hosted CI success is claimed.

Hosted status

The Registered research packages run
at current head 385a4436a47e reports action_required. No hosted success is
claimed; upstream approval is required before the fork workflow can run.