Research: sharp P104 dimension-98 prefix boundary

research/p104-prefix-tail/README.md

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+# The P104 pooled correction first reaches dimension 98 at cut 2754
+
+This resolves the interval left open by
+[PR #2345](https://github.com/CrocSwap/integer-mult-bounds/pull/2345):
+
+| Completed original events | Exact rational shared correction dimension |
+|---|---:|
+| 2718–2753 | 99 |
+| 2754–2823 | 98 |
+
+The lower proof uses fixed compressed determinants **−2** and **−1**.
+A new 46-contact certificate attains dimension 98 at cut **2754**. The
+rank-99 witness at cut 2646 is retained from #2345 and independently replayed.
+In particular, **no earlier prefix can attain dimension 98**. This fills the
+gap before the reported late-prefix results in
+[Discussion #2245](https://github.com/CrocSwap/integer-mult-bounds/discussions/2245#discussioncomment-18940216).
+We do not recertify that separate late-prefix package here.
+
+**This is a rational pooled-snapshot model, not an executable correction or
+a transport-cost reduction.** Pooling grants access at different times; the
+new directions have no implemented creation, restoration or hosting bill.
+There is no supplier or κ claim and no selected-frontier change.
+
+## Exact model and independent replay
+
+Use the unchanged PR #1972 word retained by the native PR #1991 package:
+104 source inputs, 104 target inputs, 72 independent dirty helper inputs,
+4,058 events and 4,059 cuts. A cut counts completed events. For each target,
+
+```
+S_t(c) = span_Q { v_r(d) : d <= c, r != Y_t,
+                 frame(r,d) = frame(Y_t,d) as binary subspaces }.
+```
+
+All snapshots come from the original word; inserted rows do not change later
+snapshots. Basis changes within a frame are free in this relaxation. Let `R`
+be the 8-by-104 doubled root-coordinate matrix and `C=I_104-R^T R/8`.
+The number `h(c)` minimizes `dim H` over rational spaces in all 280 boundary
+coordinates such that `C_t in S_t(c)+H` for every target.
+
+[trace.py](trace.py) is a fresh sparse integer/8 implementation, with no
+imports from the earlier dense replay or native producer. It checks source
+hashes, all 280 independent columns, every frame transition, private-copy
+lifetimes, source/dirty restorations, the complete `G/2` endpoint and unchanged
+local rank mass 2368. It independently recovers the same 110,814 distinct
+target/snapshot pairs as #2345.
+
+The collector visits each changed frame group. At an unchanged group both
+membership and every logical row are unchanged, so revisiting cannot add a
+snapshot. Requested contacts are checked separately at their exact cut,
+including cuts where nothing in their group changes. This is exact reuse,
+not sampling cuts. The relevant prefix pools contain 324 and 427 nonzero
+target/coefficient pairs at cuts 2753 and 2823.
+
+## Reduction to an eight-dimensional residual
+
+We use the reduction proved in #2345, retaining its hypotheses and giving
+the argument for standalone review. The full replay checks `R R^T=104 I`,
+source projections in `row(R)`, absence of a target's own dirt from every
+eligible donor, and availability of the other 175 independent nuisance units
+at cut zero. The last two facts imply `pi_X(S_t)=S_t intersect X`: each
+nuisance component can be subtracted inside the pool. Projecting any serving
+`H` therefore gives a serving source-only space of no greater dimension,
+and source-only spaces are allowed choices.
+
+Put `L_t={a in Q^8 : aR in S_t intersect X}`. Choosing one local row per target
+and taking the residual row span yields
+
+```
+h(c) = min_{A_t in L_t(c)} rank(C+12 A R)
+     = 96 + min_{M in E(c)} rank(I_8-M),
+E(c) = {R A : A_t in L_t(c)} = span{r_t a : a in L_t(c)}.
+```
+
+The substitution by −12 preserves each linear `L_t`. To see the rank identity,
+write `D=R^T/8-12A`, so `C+12AR=I-DR` and `RD=13I-12RA`.
+The restrictions of `R` and `D` are inverse maps between `ker(I-DR)` and
+`ker(I-RD)`. Rank-nullity gives `rank(C+12AR)=96+rank(I-RA)`.
+These statements quantify rational coefficients, including arbitrary dirty
+components of the original `H`; they do not identify rational minimum rank
+with a real or complex constrained minimum.
+
+## Two small universal lower certificates
+
+Indices below are zero-based, and `e_i` are the standard vectors of Q^8.
+Use columns
+
+```
+p=e1-e0,  q=e2-e0,  w=e2+e3+e4+e5-e6+e7,
+a=e0+e1,  b=e0+e2,  k=e7-e6.
+```
+
+For every generator `r_t alpha` of `E(2823)`, the checker establishes
+
+```
+p^T M a = p^T M b = q^T M a = 0.
+```
+
+These linear functionals then vanish on all rational linear combinations,
+not only sampled matrices. For every residual `N=I-M`, the compression with
+left rows `(p^T,q^T)` and right columns `(a,b)` is
+
+```
+[ 0  -1 ]
+[-1   * ]       determinant = -1.
+```
+
+Compression cannot increase rank, so `rank N>=2` and `h(c)>=98` through
+cut 2823. At cut 2753 the additional exact functionals are
+
+```
+p^T M k = q^T M k = w^T M k = 0.
+```
+
+Since `p^T k=q^T k=0` and `w^T k=2`, left rows `(p^T,q^T,w^T)` and right
+columns `(a,b,k)` give
+
+```
+[ 0  -1   0 ]
+[-1   *   0 ]
+[ *   *   2 ]       determinant = -2.
+```
+
+Thus `rank N>=3` and `h(c)>=99` for every prefix through 2753. No nonlinear
+search, optimizer status, approximate rank or characteristic-two determinant
+is used for these rational lower bounds.
+
+## Matching upper certificates and the exact break
+
+Each entry `[target,cut,register,weight]` in [WITNESS.json](WITNESS.json)
+contributes `weight * pi_X(v_register(cut)) R^T/104` to the target's row of
+`A`. The code replays its availability, and the lifting facts above remove
+nuisance components. The resulting `K=C+12AR` serves every correction row
+since `K_t-C_t in S_t`.
+
+The 18-contact cut-2646 witness from #2345 has `rank K=99`; it remains
+available at every later prefix. The new 46-contact cut-2754 witness gives
+exactly
+
+```
+N = x (e1+2e7)^T - e1 e2^T,   x=-e2+e3+e7.
+```
+
+This has rank two: the persistent compression is nonsingular, and the
+displayed sum has at most two rank-one terms. The checker independently
+computes both `rank N=2` and `rank K=98` by rational elimination.
+At this cut the old `q^T M k=0` constraint is no longer true: in the new
+residual `q^T N k=-2`, while `p^T N k=0` and `w^T N k=2`. Thus the earlier
+3-by-3 argument cannot be extended past its checked prefix.
+
+Pools grow monotonically. The cut-2646 upper and cut-2753 lower prove 99
+throughout the first interval; the cut-2754 upper and cut-2823 lower prove
+98 throughout the second. Contacts are algebraic membership certificates,
+not literal added chronological instructions.
+
+## Reproduce
+
+Standard-library Python, with read-only default commands and no network,
+Git or pending-package code needed for the direct checks:
+
+```sh
+python3.12 -I -O -B research/p104-prefix-tail/verify.py
+python3.12 -I -O -B research/p104-prefix-tail/test_verify.py
+python3.12 scripts/research_package.py --package research/p104-prefix-tail
+```
+
+Expected primary line: `PASS_P104_PREFIX_TAIL 99 through2753; 98 from2754 through2823`.
+The full [RESULT.json](RESULT.json) must match. `--out NEW.json` writes only
+to an unoccupied path. The registered replay also runs failure controls;
+its metadata needs Git provenance, unlike the direct replay. Tests cover
+corrupt contacts, future/self contacts, nonzero compensation/cleanup faults,
+and low-rank counterexamples when required compression constraints are dropped.
+
+The proof and finite certificates were checked with rational arithmetic and
+a separately implemented source-projection exploration. Native Gaussian
+recipe regeneration, chronological correction execution, global moment and
+owner assignment, precision/all-size admission and selected cold replay are
+not performed. Hosted CI and independent human review remain separate.
+
+## Open next questions
+
+- **Turn one complement into a fully charged chronological construction.**
+  The available snapshots do not make its 98 shared directions free. Begin
+  with source reads, frame paths, creation/restoration and owners, then price
+  the entire word. Existing fixed-frame/retained-cost obstructions still apply.
+  No implementation contract is supplied here.
+- **Connect the independently published prefix certificates.** The reported
+  97 interval from 2824 and 96 interval from 2912 can be combined with #2345
+  and this package after checking their exact artifacts and source bindings.
+  Start with a source-bound replay/index of those certificates; external
+  late-prefix evidence is not rerun by this package.
+
+## Sources and attribution
+
+Scientific base: canonical main `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`,
+priority revision 4, freshly fetched at the second-round checkpoint. Only
+this new package is added; the first-round PR is not a parent dependency.
+[TASK.json](TASK.json) records the handoff, scope, freshness and overlap;
+[SOURCES.json](SOURCES.json) pins both frozen inputs.
+
+Gonzalo Mier/#2173 supplied the pooled model and dimension question;
+Gareth Davidson/@bitplane/#2251 supplied the full-pool complement;
+@josecarlosgonzalezroque-design supplied the late-prefix result and the
+handoff in Discussion2245 comment18940216. AAAmirinu/#2345 at
+`2a5fa34a967e7567abf01611f92a6ab69e8c901a` supplied the rank reduction and
+the reused cut-2646 witness. Those are not new claims here.
+
+Rohan Arun/#1972 supplied the scalar/frame channel; Chafik Boukhalfa/#1991
+its native realization; Adam Gall/#1940 the schedule; Josie Elliston/#1071
+the fixture, with Ali Romman/#1125 and the retained geometry/decoder lineage.
+Original files and notices remain unchanged.
+
+Prepared for AAAmirinu with substantial OpenAI Codex assistance in proof,
+implementation, testing and prose, including separate projection exploration,
+sparse replay and mathematical review. Author-checked evidence; no independent
+human review or proof-assistant claim. Apache-2.0.

research/p104-prefix-tail/RESULT.json

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+{
+  "schema": "p104-prefix-tail-result/1",
+  "status": "PASS_P104_PREFIX_TAIL",
+  "intervals": [
+    {
+      "start": 2718,
+      "end": 2753,
+      "minimum": 99
+    },
+    {
+      "start": 2754,
+      "end": 2823,
+      "minimum": 98
+    }
+  ],
+  "lower_certificates": {
+    "2753": {
+      "minimum_dimension": 99,
+      "compression_rank": 3,
+      "determinant": -2,
+      "generator_pair_checks": 1944
+    },
+    "2823": {
+      "minimum_dimension": 98,
+      "compression_rank": 2,
+      "determinant": -1,
+      "generator_pair_checks": 1281
+    }
+  },
+  "upper_certificates": [
+    {
+      "cut": 2646,
+      "contacts": 18,
+      "residual_rank": 3,
+      "full_source_correction_rank": 99
+    },
+    {
+      "cut": 2754,
+      "contacts": 46,
+      "residual_rank": 2,
+      "full_source_correction_rank": 98
+    }
+  ],
+  "trace": {
+    "schema": "p104-prefix-tail-trace/1",
+    "cuts_replayed": 4059,
+    "boundary_columns": 280,
+    "initial_nuisance_units_per_target": 175,
+    "distinct_target_snapshots": 110814,
+    "distinct_source_projections_checked": 118,
+    "changed_target_groups_checked": 2356,
+    "local_profile": {
+      "1": 1074,
+      "2": 58,
+      "3": 86,
+      "4": 104,
+      "5": 48,
+      "6": 24,
+      "7": 8,
+      "8": 8
+    },
+    "rank_mass": 2368,
+    "events": {
+      "add": 2624,
+      "copy": 8,
+      "erase": 8,
+      "transport": 1410,
+      "zero": 8
+    },
+    "endpoint": "G/2; all 104 sources and 72 dirty helpers restored; private storage closed",
+    "prefix_pool_totals": {
+      "2753": 324,
+      "2823": 427
+    },
+    "requested_contacts": 49,
+    "input_overrides": false
+  },
+  "scope": "Rational unchanged-trace pooled availability; no chronological or paid-cost claim."
+}

research/p104-prefix-tail/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
+      "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7"
+    },
+    {
+      "path": "research/p104-native-channel/inputs/LOWRANK-WITNESS.json",
+      "usage": "input",
+      "git_blob": "86beaf424e9f0f800ae3a62797d94ec9386b7706",
+      "sha256": "0f541bef7f19fdeffd67157684e9f03d785f42c89973632f508fc1a0452b580e"
+    },
+    {
+      "path": "research/p104-native-channel/inputs/P104-FIXTURE.json",
+      "usage": "input",
+      "git_blob": "2e16f99b295bb386fd13bd8fbd57b42ef54143a9",
+      "sha256": "516c06f6ec3cb84be588c546108de897650288576dd5fbf5fe44b9c5546ad8a4"
+    }
+  ]
+}