Research: census of staged entries on PR3734's other helpers; none found that pays (no new κ)

research/staged-entry-census-3734/FINDING.md

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+# Method and findings
+
+All statements concern PR3734's BORROW-WORD-43 words (`base` and `extra`), read through PR3734's
+`audit_core.py` (frame bill, hygiene) and `audit_verify.py` (arbitrary-dirty semantics).
+
+## 1. The staged entry, exactly
+
+A helper `h` has entrance frame `E` (a subspace of F_2^9) and terminal `FULL`; its hosted width is
+`9 − dim E`. Every gate touching `h` must have a frame containing `h`'s current frame. Raising the
+entrance to `P = E + span(u)` requires every contact of `h` before its frame contains `P` (the
+*blockers*) to disappear. An entering helper may not be read by the zero-frame precondition and
+compensation wrappers, nor be a correction source (`audit_core.hygiene`).
+
+Let `M_c` be the scalar map of the program prefix before event `c` (corrections, wrappers, A, cut,
+B with warps), with `h` read but not written before `c`. Delete every contact of `h` before `c` (they
+must all be reads) and insert releases `r += a_r h` at `c`. For any `w` supported on ordinary
+non-restored helpers other than `h`, put `v = e_h + w`. If `a = M_c v − e_h` and no warp or
+correction sees a nonzero value of `v`'s trajectory, then the new prefix equals the old prefix times
+the input shear `S = I + w e_h*`. Because the completed target map annihilates helper inputs, the
+target map is unchanged, and cleanup is regenerated as the literal inverse. PR3605 and PR3734 are
+the case `a = w` (`M_c v = v`). A release `r += a_r h` may then slide later, as long as no event
+reads, scales or warps `r` or writes `h` in between.
+
+Frames are the only remaining condition. A release gate needs a frame `R ⊇ P + frame(r)` inside the
+next frames of both `r` and `h`.
+
+## 2. Candidates (`code/census_blockers.py`, `code/candidates.py`)
+
+- 213 width-9 ordinary helpers: all are read by the compensation wrapper (207) or are correction
+  sources (6).
+- For 38 width-9 helpers, every blocker is a compensation read before the producer.
+- For 54 width-8 helpers, every blocker is a read before the producer: 46 target reads in B,
+  24 helper reads in A, and mixed cases.
+
+Each helper admits up to 15 raises `P` with this property, 416 (helper, raise) pairs in total.
+
+## 3. Exact identities exist (`code/scalar_census.py`, float screen)
+
+Strict identities (`M_c v = v`) exist on part of the cut window for 25 width-8 candidates and for all
+38 width-9 candidates. The width-9 identities are dense combinations: no two compensation columns are
+proportional, and 207 columns span the 112 target coordinates. Every identity used later is
+re-checked exactly with fractions (`construct.exact_fixed`).
+
+## 4. Fixed other frames: no construction (`code/pass5.py`)
+
+For each (helper, raise, cut), a register `r` is *compatible* when some slot of its commuting range
+hosts `R = P + frame(r)` inside both next frames. The linear program asks for `w` with `a_r = 0`
+on every incompatible register (warp and correction functionals included), minimising `Σ|a_r|`.
+
+Results: 24 sampled cuts per helper plus the producer cut, 9,862 cases per provider, none with a
+solution (`evidence/FIXED-FRAMES-*.json.gz`).
+
+Validation: on PR3734's parent word the same pass finds 544 at 16 of 72 sampled cuts, with releases
+`672 += 544/2` and `676 −= 544/2` before old B693 and PR3734's exact bill. On PR3133's word it finds
+644 and 660 with PR3605's donors and coefficients (`evidence/CONTROL-*.json.gz`). Its later placements
+give a cheaper local bill than PR3605's, as PR3642's retimings already do.
+
+## 5. Re-solved frames: exact feasibility test
+
+`repair.repair(minimal_B=True)` gives every B and restore gate exactly the join of its registers'
+current frames, warps following (`U` = current lender frame, `E` = image). By induction over the
+events, every assignment that keeps A, the entrances and the terminals contains it pointwise. The
+terminal constraints (target port hyperplanes, restored-helper terminals) are upper bounds, so if
+this assignment fails, every assignment fails.
+
+`minimal_all.minimal_all` extends this to A outside the prelude. The last A gate of a target is
+forced to the target's cut level, and that of a centre to its centre frame, as `frame_bill` requires.
+On PR3734's word both assignments pass, with the same rank mass and a costlier histogram: the
+control is that they are not too strict.
+
+- **B windows** (40 width-8 helpers, `code/pass7.py`): 6 cuts, release vectors = pure retiming of
+  the early reads plus a greedy sparse strict solution, up to 4 placements per recipient. Out of
+  2,160 constructions, only 472 is feasible (4 cuts, same releases); all 1,992 pure retimings are
+  infeasible (`evidence/RESOLVED-B-WINDOWS-base.json.gz`).
+- **A windows** (38 width-9 and 14 width-8 helpers, `code/pass8.py`): 5 cuts, sparse strict solutions
+  with a dense fallback. 414 rows, none feasible. On a sample of the rejections, the failure is a
+  target leaving its port hyperplane at the end (33) or exceeding its cut level (27)
+  (`evidence/RESOLVED-A-WINDOWS-base.json.gz`).
+
+## 6. Helper 472
+
+- **Construction.** Blocker B31 `130 += 472/2` at frame `[51]`. Producer B538 `472 += 283` at frame
+  `span(508,81,51)`. Identity `v = e472 − e624 − e628/2` holds exactly at the producer cut. Releases
+  `624 −= 472` and `628 −= 472/2` go just before B538, with entrance `span(429,51)` (width 7).
+- **Audit.** Frames are repaired downstream (`repair.repair`) and cleanup is regenerated. Both
+  providers pass frame bill, hygiene and semantics (`words/W472-*.json.gz`).
+  - Bill: `{1:2455,2:716,3:222,4:104,5:20,8:8}` against PR3734's `{1:2440,2:725,3:224,4:102,5:20,8:8}`.
+  - Widths: `{7:54,8:95}` against `{7:53,8:96}`.
+- **Cascade.** Nine registers change their climbs: 472, 624, 625, 626, 628 and the partners 426,
+  433, 435 and 437. Each partner goes from `{1:7,2:1}` to `{1:9}`.
+  - The cause: `429 ∈ P` enters 624 and 628 before their B999 gates at `span(496,68,34,17)`, and
+    `span(508,81,51) ∩ span(496,68,34,17) = span(51)`.
+  - At the producer cut, among 313 active donors, `{624, 628}` is the only donor set of size three
+    or less (`code/donors472.py`).
+- **Pure retiming.** Moving the read `130 += 472/2` to just before B538 is scalar-correct but
+  frame-infeasible: target 130 passes through `span(290,51)`, outside 472's producer frame.
+- **Pricing.** `code/ceiling.py` adapts PR3815's exact dual (same ten-pattern basis, found optimal for
+  both inventories by `code/basis472.py`):
+  - every owner fill of the 472 inventory and bill has root `< 0.0017305031` (tight to 1e-10);
+  - the parent inventory's bracket is `[…740, …741]`, as in PR3734's PROOF.md and PR3815.
+  - At PR3862's exponent, the 8→7 price is `0.987736781…`, while 472's local saving is at most
+    `0.986194…`.
+- **Recovery attempts.**
+  - The B-frame optimiser (`code/frameopt.py`) and a parallel session's A+B optimiser leave the bill
+    unchanged.
+  - A retiming search over 4,200 slots (parallel session, local-only) finds 0 decreases.
+  - Strength of the B optimiser: from minimal frames it recovers PR3734's own bill to within two
+    rank-1 climbs. An exact frame solve of the 472 word was not run.
+
+## 7. Value scale (ideal fixed-inventory LP, `code/ideal.py`)
+
+| change | Δ ideal ceiling |
+|---|---|
+| best possible 8→7 or 9→8 (one climb shortened by one) | +4.4e-7 to +4.8e-7 |
+| one rank-2 climb split into 1+1 | −1.2e-7 |
+| PR3734's 544 signature `{1:+3,2:+1,3:−2}` with 8→7 | +1.6e-7 (realised by PR3734's packing: 2.95e-8) |
+| 472 as built | −7.5e-8 |

research/staged-entry-census-3734/README.md

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+# Staged entries on the other helpers of PR3734: none found that pays
+
+PR3605 staged helpers 644 and 660 and PR3734 staged helper 544, each lowering one hosted width
+from 8 to 7. This package asks whether the same mechanism lowers any other helper of PR3734's word,
+from 9 to 8 or from 8 to 7, and whether it pays. **No new κ.**
+
+## Result
+
+On PR3734's word (both providers), no profitable staged width reduction was found:
+
+- **The identity is not the obstacle.** An exact prefix identity exists for 25 width-8 helpers and
+  for all 38 width-9 helpers whose blocking contacts are reads of the helper before its first producer.
+- **With the other frames fixed, no construction exists.** The general form allows releases into any
+  frame-compatible register, targets included. 9,862 (helper, raise, cut) cases were checked per
+  provider, with no compatible identity. The same code, run on the parent words, rediscovers the three
+  published staged entries: 544 on PR3682's word, and 644 and 660 on PR3133's word, with the same
+  donors and coefficients.
+- **With frames re-solved, only helper 472 becomes width 7.** Feasibility is checked by an exact
+  minimal-assignment test. 472 is staged by `624 −= 472` and `628 −= 472/2` before its producer B538,
+  with entrance `span(429,51)`. The word passes PR3734's frame bill, hygiene and full arbitrary-dirty
+  semantics on both providers.
+  - The frame cascade costs `Δbill {1:+15, 2:−9, 3:−2, 4:+2}` per call.
+  - An exact dual certificate (2,551 patterns) shows that every owner fill of this inventory and bill,
+    for any clock and any replica count, has its root below `0.0017305031`. The parent inventory's
+    ceiling is `0.001730578437266246818740` (attained by PR3802 and PR3815). With these frames, 472's
+    staged entry therefore lowers the reachable κ by at least `7.5e-8`.
+  - In PR3862's width-unit terms, 472's signature saves at most `0.98619` of local cost per unit
+    against an 8→7 price of `0.98774`, a net of `−0.00154` per unit. PR3734's signature
+    `{1:+3, 2:+1, 3:−2}` nets `+0.00326`.
+- **No width-9 helper can be staged, and no other width-8 helper either.** A width-9 helper must also
+  leave the zero-frame compensation. For every placement tried, the exact test fails: a target leaves
+  its port hyperplane, or exceeds its cut level.
+
+**Why.** For 544, 644 and 660, the donors' next frames already contained the raised entrance. For
+every other candidate, the producer's frame and the recipients' next frames meet only in the old
+entrance. The new direction must therefore propagate, and it reaches either a target that excludes
+it or a partner whose rank-2 climb it splits. For 472 it enters 624 and 628 before B999 and splits
+the rank-2 climbs of 426, 433, 435 and 437. Among 313 active donors, {624, 628} is the only donor set
+of size three or less, so the cascade is forced. A local frame optimiser on B, an A+B optimiser and
+a retiming search (4,200 slots, run by a parallel session; local-only) recover nothing.
+
+[FINDING.md](FINDING.md) gives the exact staged-entry condition, the census passes, the exactness
+argument for the feasibility test, and the 472 analysis.
+
+## Reproduction
+
+```sh
+python3.12 -B scripts/research_package.py --package research/staged-entry-census-3734
+```
+
+`verify.py` uses the standard library only and takes about 15 s. It rebuilds and audits the 472
+word on both providers, with three controls (flipped release, releases after the producer, pure
+retiming that is scalar-correct but frame-infeasible). It rebuilds 544, 644 and 660 on their
+parent words, checks the exact certificate (tight to 1e-10, with the parent bracket and a perturbed
+dual as controls), and authenticates and re-summarises the recorded census.
+
+The census itself uses numpy 2.5.4 and scipy 1.18.1. It is recorded in `evidence/` with its commands
+(`evidence/CENSUS.json`). `python3.12 -B research/staged-entry-census-3734/replay_subset.py` reruns
+a subset in about 2.5 minutes and must reproduce the recorded rows: the three positive controls,
+fixed frames for 457, 472 and 640, re-solved B windows for 472, and A windows for 461. It is not
+registered, because float decisions may differ across BLAS builds. The full census took about 20 min
+per provider for fixed frames, about 1 h on five cores for B windows and 12 min for A windows.
+
+## Scope and limits
+
+- The census samples cuts: 24 per helper plus the producer cut with fixed frames, and 5 or 6 per
+  helper with re-solved frames, with at most 4 placements per release recipient. Infeasibility is
+  proved per placement, not over all placements.
+- Release vectors tried: greedy sparse strict solutions, an L1-weighted solution, a dense fallback and
+  pure retimings. The exhaustive enumeration of donor sets of size three or less was run for 472 only.
+- Fixed throughout: the other helpers' entrances and terminals, the prelude and the native warp maps.
+  Joint staged entries are not searched, and the clock is not changed.
+- Re-solved costs come from local optimisers, not an exact frame solver. The 472 certificate holds for
+  the constructed frames. Neither optimiser (B frames; A and B frames) improves them. But the B
+  optimiser, started from minimal frames, only recovers PR3734's own bill to within two rank-1 climbs,
+  about 1e-7 in κ, which exceeds the 7.5e-8 gap. A cheaper frame assignment of the 472 scalar word is
+  therefore not excluded.
+- Against PR3945's 4→5 widening, with 472's cascade assumed unchanged on that word, the ideal ceiling
+  also drops by `7.5e-8`. This is a float estimate from a local profile inferred from PR3945's
+  published histogram; PR3945's words were not used.
+- All of PR3734's conditional hypotheses are inherited. No selected-frontier change, no
+  unconditional theorem and no formal proof is claimed.
+
+## Open next questions
+
+The three structured questions are in [TASK.json](TASK.json):
+
+- **Joint staged entries.** Do two or more helpers staged together pay where single ones do not, as
+  for 644 and 660 in PR3605?
+- **Absorbing the cascade.** Can the partners' terminals or entrances take the 472 cascade, or does
+  472 pay on PR3945's word?
+- **Unsampled cuts.** The census samples cuts and placements; running every cut is a matter of compute.
+
+## Sources and credit
+
+Stacked on PR3734's head `fa041037a3374eaa08feeca0b26a5d6893194147`; only this directory is added. The
+following are imported from PR3734's package, not copied, and their attribution is kept:
+
+- `audit_core.py` and `audit_verify.py` (Sagar Thakrar, PR3687/3693): frame bill and semantics;
+- `rational_price.py` (dataisfire);
+- the words, and the parent words in `REPLAY-INPUTS.tar.gz`.
+
+`code/ceiling.py` adapts PR3815's exact dual certificate (ChaloupeMusic) to a parameterised inventory.
+The staged-entry mechanism is due to Ross Richardson (PR3605) and dataisfire (PR3734). The width-unit
+prices are due to Sagar Thakrar (PR3862) and ToikY (PR3890).
+
+Prepared by Erwan Secousses (ChaloupeMusic) with substantial assistance from Claude (Anthropic,
+Claude Code). There is no independent human or formal review. Apache-2.0.

research/staged-entry-census-3734/SOURCES.json

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