Research: exact screen for kappa > 0.1 on selected suppliers

research/kappa01-target-screen/README.md

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+# The selected suppliers cannot reach κ = 0.1 by repacking
+
+**Two-sided target-scale screen for the selected local words. No new supplier or κ claim.**
+
+The user-set target κ > 0.1 is far beyond the selected result. Under the
+retained planning transfer κ = a/(1+a), it requires primitive saving
+a > 1/9. At the boundary a = 1/9, **both selected local words** already have
+normalized moment strictly above one, even after assigning zero cost to every
+idle child, hosting operation, centre path and (on bit) surcharge. Therefore
+packing or coverage changes alone cannot take either selected word to the
+target. Both local suppliers, or the outer transfer, must change.
+
+The source-bound per-call profile from
+[`complex-local-word-ceiling/levers.py`](../complex-local-word-ceiling/levers.py)
+is
+
+```text
+H = {1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}
+width = 73; live families per call = 896/9
+```
+
+The exact lower bound is
+
+```text
+(9/896) * Σ_r H[r] * (r/73)^(8/9) > 1.155558328185663 > 1.
+```
+
+Each fractional power is enclosed by integer inequalities. For scale
+`S = 10^30`, the checker finds integers `l_r, u_r` satisfying
+
+```text
+l_r^9 * 73^8 <= r^8 * S^9 <= u_r^9 * 73^8,
+```
+
+so `l_r/S <= (r/73)^(8/9) <= u_r/S`. The complete rational lower endpoint,
+upper endpoint and per-rank witnesses are in [`RESULT.json`](RESULT.json).
+The proof does not use floating point; decimals are only for display.
+
+The same exact integer-root check is bound to the selected bit word through its
+`bit-revision3-levers-seed03` receipt, the PR589 expected record and the pinned
+word bytes. Its local histogram is
+
+```text
+H = {1:1323, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:19}
+width = 64; live families per call = 296/3
+```
+
+The bit-side local-only bill is
+
+```text
+(3/296) * Σ_r H[r] * (r/64)^(8/9) > 1.195679601524970
+```
+
+This also omits the positive inherited bad-residue surcharge. As an independent
+coverage check, even putting all bit rank mass under the selected law at the
+largest permitted idle rank 45 costs more than `1.034037852870777`; thus the
+selected bit coverage law fails even with a free local word. The complex side
+has the analogous coverage floor `1.035561535760803` from its selected
+width-73/rank-51 guard.
+
+There is also a word-independent screen for the retained width, template and
+halving guard. The largest permitted child rank is 51 (`2·51² < 73² ≤ 2·52²`).
+For a fixed coverage mass, rank 51 minimizes the moment per unit rank mass,
+because the cost per unit mass is proportional to `r^(-1/9)`. Even this
+fractional all-at-rank-51 relaxation costs **1.035561535760803** at the
+selected coverage law (one unpaid coordinate and centre mass 64). Thus the
+current coverage law cannot reach the target with any local word in this
+template, even if the local word itself were free.
+
+The exact relaxation brackets also locate the next structural threshold:
+
+| Unpaid coordinates | Centre mass | Ideal all-rank-51 bill | Meaning |
+|---:|---:|---:|---|
+| 1 | 64 | 1.035562 | Selected law fails, even with a free local word |
+| 2 | 0 | 1.012142 | Still fails after removing the centre charge |
+| 3 | 0 | 0.997886 | Ideal relaxation passes; no construction follows |
+| 3 | 64 | 1.007051 | Fails with the selected centre charge |
+| 4 | 64 | 0.992795 | Ideal relaxation passes; no construction follows |
+
+So even the relaxed format requires at least three unpaid coordinates if the
+centre is free, or four if the selected centre charge remains. Those are only
+necessary budget thresholds: the current word's local-only bill is 1.155558,
+so changing coverage alone still cannot work. The two ideal passes have less
+than 0.22% and 0.73% headroom before any actual lower-rank work is charged.
+
+The same exact relaxation gives the minimum maximum-child rank needed for each
+coverage choice. With the selected law, even a free local word needs rank cap
+70; the selected halving guard allows only 51. Removing the centre charge
+lowers the requirement to 65, and allowing two unpaid coordinates with no
+centre lowers it to 57. The two relaxed combinations that fit under rank 51
+are three unpaid coordinates with no centre, or four with the selected centre
+charge:
+
+| Unpaid coordinates | Centre mass | Minimum rank cap | Allowed by current guard? |
+|---:|---:|---:|:---:|
+| 1 | 64 | 70 | no |
+| 1 | 0 | 65 | no |
+| 2 | 0 | 57 | no |
+| 3 | 0 | 51 | yes, at the boundary |
+| 3 | 64 | 55 | no |
+| 4 | 64 | 48 | yes |
+
+Every rank-cap threshold has a strict adjacent-integer certificate. A rank cap
+above 51 would require a different outer halving argument; the table does not
+show that such an argument exists.
+
+## What a combined word-and-coverage change would still have to do
+
+As a design budget, reblock **all** 5,586 units of the selected local rank mass
+into the most favorable integer blocks allowed by the chosen local block cap,
+while leaving the remaining coverage mass at rank 51. The concavity exchange
+argument is the one retained in
+[`complex-rank-one-block-budget`](../complex-rank-one-block-budget/README.md);
+this target-0.1 calculation extends its cost screen to a changed coverage law
+and the full local rank mass. It grants every block zero gathering, chronology,
+frame, copy, dirty-restoration, hosting and precision overhead.
+
+| Unpaid coordinates | Centre mass | First local block cap that passes | Ideal local profile | Bill | Extra rank-one calls allowed |
+|---:|---:|---:|---|---:|---:|
+| 3 | 0 | 50 | 111 rank-50 + 1 rank-36 | 0.999840174348039 | 0 |
+| 4 | 64 | 48 | 116 rank-48 + 1 rank-18 | 0.998499914030822 | 6 |
+
+For each row, every smaller block cap fails. The next added rank-one call also
+fails by a strict exact interval. These margins are a demanding target for an
+actual mechanism: in the three-unpaid case there is no allowance even for one
+extra rank-one call; in the four-unpaid case there are only six. The profiles
+are cost-optimal partitions of rank mass, not identities or chronological
+decoders. The prior block-budget package's warning about full physical costs
+therefore remains central here.
+
+The selected bit word has its own zero-overhead reblocking screen. With its
+current rank cap 45, the selected one-unpaid/centre-63 coverage law still fails
+even if all rank mass is moved to rank 45. The first ideal integer reblocking
+profiles that pass under two changed coverage choices are:
+
+| Unpaid coordinates | Centre mass | Rank cap | First local block cap | Ideal local profile | Bill | Extra rank-one charges allowed |
+|---:|---:|---:|---:|---|---:|---:|
+| 3 | 0 | 45 | 42 | 122 rank-42 + 1 rank-5 | 0.997886898348711 | 8 |
+| 4 | 63 | 45 | 39 | 131 rank-39 + 1 rank-20 | 0.999087171501363 | 3 |
+
+Every smaller block cap fails, and the next extra rank-one charge takes each
+bill above one. The source word's 5,129 units of local rank mass are treated as
+arbitrarily reblockable, while idle mass is fractional at rank 45. Gathering,
+chronology, framing, copies, dirty restoration, the positive surcharge and all
+other realization costs are waived. These are necessary budgets for a future
+mechanism, not evidence such blocks exist. The complex and bit budget rows are
+separate; this screen does not show that one changed coverage contract or one
+pair of supplier identities realizes both sides together.
+
+This is a scoped result about the selected #1094 complex word, the selected
+PR589 bit word and their stated coverage/rank relaxations. It does not bound a
+redesigned template, a changed operation alphabet or supplier, a different
+outer reduction, or the newer unreviewed candidate packages. It does not change
+the selected frontier or certify the full multiplication theorem. In
+particular, this is not an independent review of open PRs #3051 or #3054.
+
+## Reproduce
+
+From the repository root, Python 3.12 and the standard library are enough:
+
+```sh
+python3 -I -B research/kappa01-target-screen/verify.py
+python3.12 scripts/research_package.py --package research/kappa01-target-screen --check-only
+python3.12 scripts/research_package.py --package research/kappa01-target-screen
+```
+
+Expected direct marker:
+
+```text
+PASS_KAPPA01_TWO_SIDED_TARGET_SCREEN complex_local_lower=1.155558328185663 bit_local_lower=1.195679601524971
+```
+
+The interval source inputs and base commit are pinned in
+[`SOURCES.json`](SOURCES.json); the replay command and its 120-second timeout
+are registered in [`VERIFY.json`](VERIFY.json).
+
+## Open next questions
+
+- What changed complex local identity can bring its paid-word term below one at
+  `a = 1/9`? This must precede any packing optimization. A feasible first step
+  is one explicit replacement profile with all new decoder, frame, dirty-state
+  and restoration costs included. The blocker is a new identity and supplier
+  contract; the current word is already 15.55% over budget before hosting.
+- What selected-bit local identity can bring its paid-word moment below one at
+  `a = 1/9`, with the bad-residue surcharge and complete PR589 interface costs
+  included? Start from one exact replacement block and its source/target map.
+  The blocker is that the current bit word alone is 19.56% over budget, before
+  surcharge or idle work.
+- Can one chronological block decoder realize the ideal 3-unpaid/50-rank or
+  4-unpaid/48-rank profile within its zero-to-six-call extra budget? Start with
+  one block and record the exact input/output map, source lifetime, frame path
+  and dirty restoration. The blocker is that the present budget relaxes away
+  all physical realization and gives almost no overhead allowance.
+- Can one changed outer transfer make both selected supplier endpoints imply
+  κ > 0.1? Derive it against the selected assembly and all-size premises, then
+  bind both supplier budgets. The blocker is that the current conversion is
+  source-bound and strictly reserved, while both selected words fail locally.
+
+## Credit
+
+The selected complex profile retains Rohan Arun's #1094 construction and
+Douglas Colkitt's #1240 analysis. The bit profile and word provenance retain
+the selected PR589 supplier and the source-bound bit-revision-3 audit. This
+package adds target-`0.1` exact floors and ideal budget screens; it does not
+modify predecessor sources. Prepared by OpenAI Codex with substantial
+assistance in the calculation, verifier and write-up. New material is covered
+by the repository's Apache-2.0 license.