Prove a global stage-thinning ceiling for the retained paired path (no new κ)

research/controller-thinning-global-ceiling/.gitattributes

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+inputs/PR4610-ARITHMETIC.json -text

research/controller-thinning-global-ceiling/FROZEN-SOURCE.json

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+{
+  "repository": "CrocSwap/integer-mult-bounds",
+  "pr": 4610,
+  "commit": "1eb8da6cfb35512c0cc2533413e5ca42f48e13c6",
+  "path": "research/kappa-round3-warp-20261022/ARITHMETIC.json",
+  "snapshot": "inputs/PR4610-ARITHMETIC.json",
+  "git_blob": "c89e009ca4a5ad316c69f0371d885d72045ba415",
+  "sha256": "ab10f6a4a8a4f589d4b094aac663e002c43ec44774e6bd181c51915c7fa7e521",
+  "credit": "Cao Yifan (intqwq), substantial OpenAI Codex assistance; Apache-2.0; cumulative PR4470/predecessor attribution retained in original source."
+}

research/controller-thinning-global-ceiling/PROOF.md

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+# A ceiling for every thinning of the retained paired-path moment
+
+Prepared for Cinna Davis with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Scope and conclusion
+
+Freeze Cao Yifan's PR4610 round-three profile at
+`1eb8da6cfb35512c0cc2533413e5ca42f48e13c6`:
+
+```
+H = {4:28, 7:7, 8:47, 9:326}
+P = {1:2201, 2:412, 3:510, 4:127, 5:40, 8:8}.
+```
+
+There are 112 ports, fresh block width 8, and one paid rank-one query per
+port at each thin stage. Keep the predecessor's paired-path scalar word,
+all its middle idle windows, a prefix of initial idle windows and the
+complementary suffix of terminal idle windows. Allow **every** pair count
+`p>=1`, every split `a+b=p-1`, and every binary subset of the `2p+1` full
+stages. Specialize every removed stage multiplier to the identity in both
+the idle words and the advertised main targets. Keep all productive helper
+losses, paid native jobs, owned centres and thin queries.
+
+Under this retained endpoint/invoice contract, no such word has positive
+raw surplus at any saving `0.0030217759 <= C < 1`. The p16, a7/a8 word
+has positive surplus at `C=0.0030217758`. Thus the best raw crossing is
+in that interval of width `10^-10`; p16 with the existing alternating
+full-stage pattern attains it. This is an infinite-family **moment theorem**,
+not a new operator-admission theorem or a global barrier for other words.
+
+The endpoint pattern is an explicit hypothesis inherited from the path
+construction: one pair has all stage factors; the other pairs have the
+successive prefix or suffix factors below. We do not infer a new universal
+native/phase identity from finite tests. The verifier independently checks
+the literal source's exact rational projectors and the complete formula
+for every split and stage subset through p5. Those tests bind the stated
+contract to the retained examples; the reductions and calculus below are
+the infinite part of the theorem.
+
+## 1. Arbitrary-mask formula
+
+Fix `0<t=1-C<1`. Put `f(c)=(8c)^t`, `h(c)=(1+8c)^t`, `f(0)=0`,
+and `A(t)=H(t)-P(t)`. A full stage contributes `A(t)`; a thin stage
+costs 112, with no productive helper credit. The main target at the joined
+end has rank `1+8k`, twice per port, where k is the total full-stage count.
+
+On the prefix side start with `c=s_0`. For its jth window let
+`o=s_(2j-1)`, `e=s_(2j)`. Its main target contributes `h(c+o+e)`,
+its initial idle contributes `f(c+o)`, and its retained middle idle costs
+`f(e)`. Increment c by `o+e`. On the suffix side read stages in reverse,
+start with `c=s_(2p)`, and use the same formula. The remaining middle odd
+stage is a separate bit. These three bits and the p-1 pairs partition all
+2p+1 stages.
+
+Thus the complete surplus is
+
+```
+F = k A(t) + 224 [h(k) + sum_windows(h(c+o+e)-f(c+o)-f(e))]
+              - 112(2p+1-k).
+```
+
+At t=1 this simplifies to `48k-112`. In particular it is incorrect to
+keep old helper credit after thinning or omit any thin query. The source
+binding checks this rank-mass identity on every enumerated mask.
+
+## 2. Three dominance moves eliminate all noncanonical masks
+
+These moves compare raw surplus, not runtime or equal input counts. Each
+keeps k and hence the entire local helper/paid contribution unchanged.
+It produces another member of the allowed path moment family. Therefore
+a positive surplus anywhere would imply a positive canonical surplus.
+
+**Even-only to odd-only.** Replace `(o,e)=(0,1)` by `(1,0)`.
+Later cumulative counts and all targets agree. The surplus gain divided
+by 224 is `f(c)+f(1)-f(c+1)>=0`, by subadditivity of x^t. This includes
+equality when c=0.
+
+**Remove an empty pair.** For `(0,0)` remove that window and its two
+thin stages. Its contribution is `g(c)=h(c)-f(c)<=1`, including g(0)=1.
+The query saving is 224, so the surplus gain is `224(1-g(c))>=0`.
+The side length and p each decrease by one; no full stage or helper is lost.
+
+**Split a double-full pair.** Replace `(1,1)` by two successive `(1,0)`
+pairs, adding one window and two stages, with k unchanged. The extra thin
+query charge is 224. The surplus gain divided by 224 is
+
+```
+h(c+1) - f(c+2) + f(1) - 1
+  = (8^t-1) - [(8(c+2))^t-(8(c+1)+1)^t] > 0.
+```
+
+Both differences on the right integrate the strictly decreasing derivative
+`t*x^(t-1)` over an interval of length 7. The first interval starts at 1;
+the second starts at `8(c+1)+1>=9`. This proves strict positivity.
+
+Apply the first move to even-only pairs and split every double-full pair.
+Then remove all empty pairs. The process terminates with only `(1,0)`
+pairs. It preserves the three separate bits `l=s_0`, `r=s_(2p)`, and m
+at the intervening odd stage, each in {0,1}. Splitting may increase p;
+no finite pair-count cutoff is used.
+
+## 3. Eight concave sequences replace the infinite mask search
+
+Write `S=a+b`, `B=l+r+m`, `k=S+B`, and `g(j)=h(j)-f(j)`.
+For a normalized mask,
+
+```
+F(S,a,b;l,r,m) = k A(t)
+  + 224 [h(k) + sum_(j=1)^a g(l+j) + sum_(j=1)^b g(r+j)]
+  - 112(2S+3-k).
+```
+
+The positive sequence g(j) is strictly decreasing for integer j>=1.
+For fixed S and bits, the largest side sum is obtained by merging the
+two decreasing lists `g(l+1),g(l+2),...` and `g(r+1),g(r+2),...`, taking
+their S largest terms. This uses only integer argument order, not floating
+comparisons. Let their sum be G(S).
+
+For each of the eight bit triples, the resulting upper envelope has increment
+
+```
+F(S+1)-F(S) = A(t)-112
+  +224 [h(S+B+1)-h(S+B) + (G(S+1)-G(S))].
+```
+
+The h increment strictly decreases and the next merged g term never increases.
+Thus F is strictly discretely concave. A strict positive preceding increment
+and strict negative following increment certify its unique integer maximum
+over **all S>=0**. No tail search or unexplored large-p assumption remains.
+
+At `C=0.0030217759` the exact interval certificate gives:
+
+| l,r,m | Maximizing S | Upper bound for F |
+|---|---:|---:|
+| 0,0,0 | 18 | -7.78098 |
+| 1,0,0 | 17 | -5.66497 |
+| 0,1,0 | 17 | -5.66497 |
+| 1,1,0 | 16 | -3.54895 |
+| 0,0,1 | 17 | -4.23203 |
+| 1,0,1 | 16 | -2.11602 |
+| 0,1,1 | 16 | -2.11602 |
+| 1,1,1 | 15 | -0.00000722 |
+
+Displayed bounds are rounded upward; exact outward enclosures and adjacent
+increments are in RESULT.json. All eight maxima are strictly negative.
+For bits (1,1,1), S=15 means p=16, with sides 7 and 8 in either order.
+Its surplus is strictly positive at the stated lower endpoint.
+
+## 4. Exclude every larger C, rather than only one sampled exponent
+
+Let `L(t)=P(t)-H(t)`. Its derivative divided by 9^t is
+
+```
+sum_(r=2,3,4,5) n_r (r/9)^t log r
+ - 7(7/9)^t log 7 - 39(8/9)^t log 8 - 326 log 9,
+n_2=412, n_3=510, n_4=99, n_5=40.
+```
+
+For `1/4<=t<=1`, upper-bound the positive terms at t=1/4 and
+upper-bound each negative term at t=1. The exact resulting bound is below
+`-9.53`. Hence L strictly decreases on this whole interval. Each h(k)
+increases in t; so does each g(j), since `x^t log x` increases in x>=1.
+Consequently every normalized F increases in t on [1/4,1]. Negativity
+at `t=1-0.0030217759` excludes all smaller t down to 1/4.
+
+For `0<t<=1/4`, P(t)>=3298 and H(t)<=2*408, since `9^(1/4)<2`.
+Thus L(t)>=2482. Concavity gives `h(k)<=1+8kt` and `g(j)<=t`
+for integer j>=1. Therefore every normalized surplus satisfies
+
+```
+F <= -112 + k(112+1792t-2482) + 224S(t-1) < 0.
+```
+
+This also covers k=S=0. Combining the two ranges and the dominance
+reduction excludes every `0.0030217759<=C<1` in the stated family.
+
+## 5. Even free owned centres do not reach C=0.01
+
+As an explicitly unrealized optimistic relaxation, remove all eight paid
+rank-eight owned-centre jobs, keeping every other charge and target. The
+same dominance proof applies. The local derivative L' becomes still smaller,
+and the small-t lower bound becomes 2474, which still exceeds 560.
+
+Reoptimizing **all** pair counts, splits and thinning masks, rather than
+keeping p16, puts the best relaxed crossing in
+`(0.0080596238,0.0080596239)`, at p5 with both side lengths 2 and all three
+separate flags full. The eight maximizing S values at the upper endpoint,
+in the bit order used above, are `7,6,6,5,6,5,5,4`; all maxima are strictly
+negative, and their adjacent differences have the required strict signs.
+The p5 surplus is positive at the lower endpoint. Exact certificates are
+in RESULT.json. The same monotonicity and small-t argument excludes every
+larger C<1 in this relaxed family.
+
+Thus even deleting the entire owned-centre bill cannot reach C=0.01 by
+retuning this path. A C>=0.01 attack needs another changed hypothesis.
+This is a planning obstruction, not an implementation of free centres.
+
+## Implication and limits
+
+The existing p16 allocation already attains this family's best raw crossing
+with the frozen round-three inventory. More pair counts, asymmetric splits,
+partial thinning, or mixtures of even and odd full stages cannot beat it.
+The scope does not cover changed idle masks outside the prefix/suffix contract,
+changed scalar words, nonbinary replacements, altered queries, different local
+profiles, noncommuting stage actions, or a different supplier interface.
+
+No new kappa, selected-frontier change, BIT improvement, compiler admission,
+absolute-phase proof, physical implementation or multiplication theorem follows.
+The remaining productive direction is to change one of those hypotheses.
+Credit Cao Yifan and the source's full PR4470/native/controller lineage,
+including Cinna Davis, zxv, Jacob Alan Sussman and the retained predecessors.

research/controller-thinning-global-ceiling/README.md

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+# Arbitrary stage thinning cannot improve the retained p16 moment
+
+**No new kappa.** With Cao Yifan's frozen PR4610 round-three local profile,
+the retained paired-path moment has its best raw crossing in
+**(0.0030217758, 0.0030217759)**, attained by the existing p16 controller.
+The result covers every pair count, every prefix/suffix split and every
+binary choice of full versus query-only stages, under the stated endpoint
+contract. It is a written infinite-family reduction with an exact arithmetic
+certificate, rather than an optimality inference from a finite search.
+
+Read [PROOF.md](PROOF.md) for the hypotheses and proof. Three concavity
+moves reduce every stage mask to eight canonical families. Their surplus
+sequences are discretely concave, so exact adjacent differences certify
+their maxima over all pair counts. A separate derivative and small-exponent
+bound excludes every larger saving C<1. Changed words, idle topologies,
+query alphabets and local profiles remain open.
+
+An optimistic corollary removes all eight owned-centre charges and reoptimizes
+the whole family. Its best raw crossing is only
+**(0.0080596238, 0.0080596239)**, at p5. Thus even free centres cannot reach
+C=0.01 through this path. Free centres are a hypothetical relaxation;
+this does not supply a construction at that larger value.
+
+The source binding checks the literal path's rational projector identities and
+all 12,744 full/thin stage subsets, for every split through p5. This bounded
+operator check supports the invoice binding; the proof supplies the infinite
+claim. It does not prove native or absolute-phase admission for unexecuted words.
+
+Source commits, Git blobs and SHA256 pins are in [SOURCES.json](SOURCES.json)
+and [FROZEN-SOURCE.json](FROZEN-SOURCE.json). The PR4610 arithmetic is retained
+byte for byte, including its original line endings. We use its local profile
+as the theorem's fixed input, without adopting every claim in that artifact.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/controller-thinning-global-ceiling
+```
+
+The registered replay requires Python 3.12 and the standard library only.
+It verifies the committed [RESULT.json](RESULT.json), source/profile pins,
+outward rational intervals and charge controls. A second replay rejects four
+actual source/profile mutations, including an unpaid owned-centre profile
+with its hashes maliciously rebound. `--write` regenerates the
+receipt and is an authoring command, not the publication replay.
+
+No selected record, BIT construction, compiler, production verifier or
+canonical mathematical source is changed. This establishes no higher K,
+unconditional multiplication theorem or independent human acceptance.
+
+## Open next questions
+
+Publication refresh found concurrent [PR4658](https://github.com/CrocSwap/integer-mult-bounds/pull/4658):
+it certifies C<0.003022 for three pinned profiles through p64 using a max-plus
+DP. This contribution instead proves the reduction for unbounded p, sharpens
+the round-three bracket and adds the globally retuned free-centre corollary.
+PR4658 was inspected at the description level; no independent replay is claimed.
+
+The concurrent [typed quotient interface, PR4654](https://github.com/CrocSwap/integer-mult-bounds/pull/4654),
+is outside this theorem. [PR4660](https://github.com/CrocSwap/integer-mult-bounds/pull/4660)
+reports that extracting its B pairs and keeping an A-only path fails for all p
+and C. Its remaining constructive target is a controller that keeps both types
+active, or a changed centre/phase schedule. These are provisional source claims,
+not newly replayed native constructions in this package.
+
+Find a profitable change outside the retained path: a different scalar word,
+idle topology, nonbinary replacement or native query interface. First emit its
+complete endpoint contract and paid/target moment. An optimistic moment gain
+is a reason for native replay, not supplier admission.
+
+Credit Cao Yifan (PR4610), Cinna Davis and the cumulative PR4470/controller
+lineage, zxv, Jacob Alan Sussman, and all retained source authors. Prepared
+for Cinna Davis with substantial OpenAI Codex assistance; no subagents.
+Apache-2.0. See PROOF.md for precise scope and predecessor credit.