Mister-Meeseeks · Oct 14, 2026 · Oct 14, 2026
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+# A chronological P104 schedule attains the known 32-chain bound
+
+**New constructive attainment and a scoped tail invoice, not a new lower
+bound, restored supplier, or kappa.**
+
+Ali Romman's [PR1125](https://github.com/CrocSwap/integer-mult-bounds/pull/1125)
+already proves that monotone P104 fixed-term delivery needs at least
+**32 paths**, including the eight seeds. Its proof explicitly does not
+construct a schedule attaining that bound (PROOF lines 204–206).
+This package supplies that missing **globally executable 32-chain
+attainment for the exact fixture from Josie Elliston's
+[PR1071](https://github.com/CrocSwap/integer-mult-bounds/pull/1071)**.
+
+Allowing independent target read orders reduces that fixture's witnessed
+delivery bank from PR1071's **55 under a global order** to **32**, or
+**24 additional arrays instead of 47**. `SCHEDULE.json` executes all
+**416 reads on 104 targets**, and its conditional supplied-copy tail
+profile improves componentwise. The existing 32-path antichain bound is
+rechecked as a credited special case, not claimed anew. The supplied-copy
+boundary does **not** form or restore copies: those costs remain unknown,
+not free work.
+
+## Exact object and retained restrictions
+
+The fixture is `section4_explicit_P104` of the hash-pinned
+`research/biframe-decoder-ladder/RESULTS.json`. The eight pairs `(a_k,b_k)`
+are doubled-coordinate E8 roots. On the 104 live root lines,
+
+```
+D[t,k] = <r_t,a_k>,   T[s,k] = 2 <r_s,b_k> / <a_k,b_k>,
+D T^transpose = 2 G.
+```
+
+Each centre gathers in an eight-dimensional binary label hyperplane and has
+a rank-seven drop to its one-dimensional survivor `span(u_a_k)`. Its 52
+readers are orthogonal to that survivor. All eight survivor labels are
+independent over F2. These are predecessor facts, independently rechecked
+here for this fixture, including all 10,816 decoder entries and all eight
+pairing ranks. The copied-information charge **56 is not a new result**.
+
+Let `K_t` be the nonzero positions in row `D[t]`. A target chooses any
+permutation of `K_t`; immediately after its j-th read its frame is **exactly**
+the span of those j survivor labels. There are no additional spectator
+frame directions. A delivery array carries one fixed centre value throughout
+its lifetime, follows nested read frames, and has no subsequent drop, reload,
+or mixing with another centre. Equal-frame reads may share the same array.
+
+Compared with PR1071 this changes the global-order restriction. Compared
+with PR1125 it adds an explicit chronological attaining witness and tail
+invoice, not a stronger obstruction. PR1125's lower bound covers all 7168
+mixed decoders and permits additional frame directions and paid downward
+moves; our witness concerns one pinned decoder with exact prefix frames.
+It does not optimize other decoders, dynamic centre mixing, additional
+drops, or an admitted physical P104 compiler. Its target frames stay inside
+their final hyperplanes because each read survivor is orthogonal to that
+target's label.
+
+## Proof of the sharp minimum
+
+### Credited lower bound: 32 incomparable final reads
+
+The following is the minimal-prefix special case of PR1125's terminal-read
+theorem (PROOF sections 1 and 4). The independent replay rederives its finite
+premises on this fixture; the bound and antichain idea belong to PR1125.
+The exact target support distribution is
+
+| Number of centres read | 2 | 3 | 4 | 5 | 6 |
+|---|---:|---:|---:|---:|---:|
+| Targets | 12 | 24 | **32** | 24 | 12 |
+
+The 104 support masks are distinct. In particular, each of the 32 four-read
+targets has a final read at its four-dimensional support span. Because all
+eight survivor labels are independent, those 32 subspaces are distinct.
+Distinct subspaces of equal dimension are incomparable. Any monotone
+fixed-centre delivery chain can visit at most one of them, regardless of
+which centre supplies the final read. Therefore **at least 32 arrays/chains
+are necessary** for every allowed collection of target permutations.
+
+This lower bound does not assume a common prefix tree, a bound on program
+length, or the particular search algorithm used for the witness.
+
+### Upper bound: exact schedule and chronological feasibility
+
+`SCHEDULE.json` contains 116 parent edges on the eight-bit subset lattice.
+An entry `[S,k]` means that the chosen path into `S` adds bit `k` to
+`S without k`. Following parents to zero gives each target's read order.
+Every parent edge is used by at least one target. Each centre's used read
+frames are explicitly partitioned into nested chains:
+
+```
+centre                 0  1  2  3  4  5  6  7
+number of chains       5  3  4  4  4  3  4  5     total 32
+```
+
+The replay checks every frame, chain assignment, omitted/duplicate read,
+and the full scalar coefficient matrix. It then executes events in increasing
+`(frame dimension, frame mask, centre, target)` order. Each target dependency
+strictly increases dimension. Each strict copy-chain dependency also increases
+dimension; all reads at one fixed frame are consecutive within that chain.
+Thus the event order satisfies both sets of dependencies and cannot contain
+a chronological cycle. This is more than a disconnected collection of
+minimum path covers. The receipt also exhibits two targets with incompatible
+centre precedences, confirming that the changed hypothesis is actually used.
+
+The schedule plus the antichain lower bound proves **exactly 32**, in the
+precise minimal-prefix, fixed-centre, no-redrop geometry above.
+
+## Complete invoice at the declared supplied-copy boundary
+
+This section deliberately does not call its subtotal a complete supplier
+profile. Its boundary is:
+
+* one exact centre value is already available on each of the 32 arrays at
+  its singleton survivor frame;
+* each target enters at frame zero with arbitrary target value;
+* every array is completed to the full rank-nine frame after its last read;
+* every target is completed to its rank-eight final hyperplane after its
+  last read.
+
+All positive frame increments at this boundary are counted. Rank-r transports
+are denoted `H_r`.
+
+| Contribution | New per-target schedule | Replayed best-global-order witness |
+|---|---|---|
+| Delivery arrays, singleton entry to rank-nine exit | `85H1 + 17H2 + 19H3 + 20H4` | `101H1 + 27H2 + 24H3 + 28H4 + 11H5 + 4H6 + 2H7 + H8` |
+| Delivery-array rank mass | **256** | **440** |
+| Targets, zero entry to final hyperplane | `416H1 + 12H2 + 24H3 + 32H4 + 24H5 + 12H6` | same |
+| Target rank mass | **832** | **832** |
+| Scalar centre reads | **416**, coefficients `D[t,k]` | same |
+
+The new delivery subtotal is smaller **componentwise**: its saved histogram
+is `{1:16, 2:10, 3:5, 4:8, 5:11, 6:4, 7:2, 8:1}`. Therefore its tail cost is
+strictly smaller for every positive rank-cost function, not just at a sampled
+exponent. The comparator is one explicit, independently reconstructed
+55-chain cover of the predecessor's best global order. We do **not** claim
+that its tail histogram minimizes the moment among all global-order covers.
+The predecessor's exhaustive 40,320-order minimum is credited, not rerun;
+its 55-chain witness and its single-cut 19-chains-per-centre control are
+reproduced here.
+
+`RESULT.json` additionally gives certified rational intervals for the
+normalized child subtotal `sum n_r (r/73)^(1-a)` at `a=0`,
+`a=3351/2000000`, and `a=1/99`, using the pinned interval arithmetic engine
+from `complex-local-word-ceiling`. There is **no role denominator W** in
+this subtotal. The incumbent coverage-law ceiling is not applied to this
+unfinished, changed interface.
+
+### What is not free or implemented
+
+The eight original gather-to-survivor drops still have rank charge 56. The
+24 additional supplied centre values require a legal formation mechanism.
+An ordinary addition into an arbitrary dirty helper produces `centre+dirty`,
+not the boundary value required above. The verifier injects one independent
+dirty symbol into **every** supplied copy in turn; all 32 contaminations
+have a nonzero target response. Thus arbitrary dirty-copy formation or
+cleanup cannot be silently omitted.
+
+A completed construction would have to charge, and prove compatibility of,
+
+```
+C_complete = C_gather + C_original_drops + C_form_24_exact_copies
+           + C_delivery_tails + C_target_tails + C_416_scalar_reads
+           + C_restore + C_side_corrections + C_hosting_and_precision.
+```
+
+Only the two tail profiles, the exact scalar read map, and the original
+rank-seven drop geometry are established here. Source formation, dirty
+lifetime restoration, phase/frame compatibility with the side corrections,
+whole-call hosting, precision/tape work, and the all-size recursive interface
+remain unresolved. The original 5%-above-selected complete-profile gate of
+the chooser card is **not passed**. This is an explicitly declared changed-scope
+result on a central missing delivery lemma, not a supplier improvement or
+an updated frontier.
+
+PR1125's separate budget warning is unchanged: two extra ranks per call
+already exceed its T37/T16 P104 knob budgets **if the remaining paid moment
+retains or exceeds that knob profile**. Its theorem does not establish that
+cost premise for every new word. Our 32-carrier witness does not evade the
+at-most-26-path screen or discharge the remaining-cost premise; it only
+attains the previously unconstructed monotone path threshold.
+
+## Reproduce and evidence
+
+From the repository root, CPython 3.12, standard library only:
+
+```sh
+uv run --python 3.12 --no-project python -I -O -B research/p104-target-order-delivery/verify.py
+uv run --python 3.12 --no-project python scripts/research_package.py --package research/p104-target-order-delivery
+```
+
+Expected verifier marker:
+
+```
+PASS_P104_TARGET_ORDER_DELIVERY chains=32 extra_arrays=24 reads=416 lower_bound=32 controls=8 no_new_kappa
+```
+
+Default replay does not modify files; `--write` regenerates only this
+package's `RESULT.json`. Assertions are not used for acceptance. Negative
+controls reject an omitted read, duplicate read, omitted chain, wrong
+survivor frame, reversed chain, broken target path, and decoder coefficient
+mutation. The eighth control verifies nonzero dirty sensitivity for all
+copies. Positive controls rederive the exact decoder, 55-chain best-order
+witness, and eight single-cut chain numbers equal to 19.
+
+Actual focused checks ran on **CPython 3.12.12**: exact receipt generation,
+isolated optimized replay, and the registered package replay passed with
+the markers above and `PASS_REGISTERED_RESEARCH_PACKAGE`. The planning
+control printed `PASS_COMPLEX_LOCAL_WORD_CEILING`. No full selected-supplier
+cold replay, native compiler, hosted CI or proof assistant was run.
+
+The witness was discovered with OR-Tools CP-SAT 9.14.6206, one worker and
+search tie-break seed 3. An initial 180-second unrestricted optimization
+found 70 chains, not an improvement. Adding the proved rank-four lower
+bound and asking for its exact attainment produced the 32-chain certificate.
+No solver optimality report or floating-point calculation is used for
+acceptance; neither OR-Tools nor its dependencies are needed for replay.
+The universal statement is the written antichain proof plus this exact
+finite attainment, not a claim inferred from an unsuccessful search.
+
+## Provenance, novelty and credit
+
+* Exact baseline: `0b1cccf86e2f5132ed286a26ef0a681a495740ea`.
+* Independent chooser seed: `4e1451a884a6979cb599a332ccc7ad40`, track
+  `complex-mechanisms`; original card `complex-rank-shape-decoder`.
+  The full original draw is in `SELECTION.json`; `TASK.json` declares the
+  changed scope before synthesis.
+* Precise claim:
+  [round-two discussion 1929](https://github.com/CrocSwap/integer-mult-bounds/discussions/1929#discussioncomment-18925825).
+* Ali Romman / **PR1125**, head
+  `9f7c0679d64bab954b7161d8586dc626a59b92ee`: the 32-path terminal-read
+  lower bound, uniform cube-support classification across 7168 decoders,
+  and the paid-downward-move and retained-cost budget qualifications.
+  Its README and proof are pinned inputs. It explicitly does not construct
+  a schedule at 32 paths; that constructive gap is the new contribution.
+  Its substantial OpenAI Codex attribution is retained.
+* Josie Elliston / **PR1071**, head
+  `ce9531beba1ce115fdcdc44925ded995a29b8bac`: the pinned P104 fixture,
+  survivor geometry, single-cut and global-order delivery bounds, and the
+  explicit open per-target question. Its research credited substantial
+  Claude Opus 5.5 assistance; that attribution is retained.
+* themilkman216 / **PR968**, head
+  `6a99e7240b961ac5e115ae6aa0301ec7c4a45d77`: frame-pair theorem and
+  exact eight-term decoder floor. DreamingOfClouds / **PR717**, live head
+  `4a97676294f9fa413b65986e9984fddeda71e4cc`: the decoder model, P104
+  witnesses, and prior pricing questions. DreamingOfClouds / PR446 and
+  PR595 supply the inherited copied-information model. GamingPuzzled /
+  Joel Pulikkan / PR620 supplies the coordinate decoder lineage.
+* Douglas Colkitt / **PR1240** supplies the pinned exact interval calculator
+  and the selected-word planning control; no ceiling outside its hypotheses
+  is inferred here. The selected PR1094/PR589 constructions are unchanged.
+* Shared cached overlap coverage contained 579 open PRs and 547 discussion
+  records with complete pagination at 2026-10-14 14:10/14:11 UTC. The initial
+  narrow searches missed merged PR1125. A final live `P104` search over PR
+  titles and bodies found it; its exact head, README and proof were read
+  before publication, and the scope/claim was corrected to constructive
+  attainment. Live heads of PR1071, PR968 and PR717 were also rechecked.
+  This is a metadata overlap screen, not an exhaustive code-diff novelty
+  certificate.
+  The final targeted offline search used the parent's refreshed cache:
+  584 open PRs at 14:27:47 UTC and 565 discussion records at 14:29:09 UTC,
+  both with complete pagination; it recovered PR1125's local package.
+
+Prepared for Adam Gall with substantial OpenAI Codex (gpt-6-astra)
+assistance in research, proof, implementation and writing. Apache-2.0.
+No independent human proof review or formal proof-assistant verification
+is claimed. No frozen predecessor, selected record, or shared index changes.
+
+## Open next questions
+
+1. **Can the 24 extra exact copies be formed and restored cheaply on arbitrary
+   dirty arrays?** This determines whether the chain reduction survives a real
+   invoice. Start by compiling one of the rank-four-ending chains, including
+   its dirty-state correction and full cleanup. The blocker is an admitted
+   P104 copy/restoration interface, not the path-cover combinatorics.
+2. **Can side corrections coexist with the layer-by-layer target order?**
+   The schedule proves only the centre delivery. The first experiment is to
+   splice its actual 416 reads into a fully specified P104 correction word
+   and check each physical frame and chronological dependency. No such
+   complete correction word is supplied here.
+3. **What is the minimum complete tail moment at 32 chains?** The chain count
+   is sharp, but our profile is only an attaining witness. Optimize chain
+   assignments with exact rank costs while retaining the same support paths;
+   any result still needs the physical contracts above before supplier use.