Research: exact early-prefix P104 shared correction dimensions

research/p104-prefix-minrank/README.md

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+# Exact early-prefix dimensions for the P104 pooled correction
+
+The rational shared correction dimension for the unchanged P104 trace is:
+
+| Completed original events `c` | Exact minimum `h(c)` |
+|---|---:|
+| 0–2525 | 104 |
+| 2526–2555 | 103 |
+| 2556–2571 | 102 |
+| 2572–2617 | 101 |
+| 2618–2645 | 100 |
+| 2646–2717 | 99 |
+
+This answers an early-prefix part of the open question in
+[the prefix-boundary handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/2245#discussioncomment-18940216).
+It also gives an exact eight-dimensional minimum-rank formulation for any
+prefix satisfying the checked lifting hypotheses. The earlier
+[PR #2251](https://github.com/CrocSwap/integer-mult-bounds/pull/2251) establishes
+the full-pool minimum 96; the handoff reports 97 on 2824–2911 and 96 from 2912.
+Those late-prefix results are credited, not independently recertified here.
+
+**No chronological correction circuit or paid transport saving follows.**
+The model pools snapshots from different times and permits arbitrary new
+rational shared directions for free. It does not construct, transport, host
+or restore those directions. No new supplier, exponent or selected result is
+proposed. Cuts 2718–2823 are deliberately left unresolved in this package.
+
+## Model and immutable inputs
+
+Use Rohan Arun's 4,058-event PR #1972 channel, as frozen by Chafik Boukhalfa's
+native PR #1991. A cut counts completed events, so cut zero is the initial
+state. There are 104 source inputs `X`, 104 target inputs `Y` and 72 arbitrary
+dirty helper inputs. For each target `t`, set
+
+```
+S_t(c) = span_Q { v_r(d) : 0 <= d <= c, r != Y_t,
+                 frame(r,d) = frame(Y_t,d) as binary subspaces }.
+```
+
+Here `v_r(d)` is the logical row in the **unchanged original trace**. No added
+row is fed back into that trace. Basis changes inside a binary frame are free
+in this pooled model. Let `R` be the 8-by-104 doubled root-coordinate matrix,
+`C = I_104 - R^T R/8`, and define `h(c)` as the least dimension of a rational
+space `H` in all 280 boundary coordinates with `C_t in S_t(c)+H` for every `t`.
+All coefficient fields in the result are **Q**. We make no assertion equating
+constrained rational minimum rank with real or complex minimum rank.
+
+[SOURCES.json](SOURCES.json) pins the two actual JSON inputs by Git blob and
+SHA-256 at main `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`. The input witness
+hash is `0f541bef7f19fdeffd67157684e9f03d785f42c89973632f508fc1a0452b580e`.
+No pending predecessor program is imported; frozen packages are unchanged.
+
+## Exact reduction, including arbitrary dirty components
+
+The checker reconstructs the roots and verifies `R R^T = 104 I_8`. Hence
+`C` is invertible, with inverse `I_104-R^T R/96`, and `R C = -12 R`.
+It checks the following facts against **every** original cut:
+
+1. Every eligible snapshot's source projection lies in `row(R)`.
+2. At cut zero all 175 non-source coordinate units other than `Y_t` are
+   available to target `t` in the zero frame.
+3. No eligible snapshot contains its own target input `Y_t`.
+
+Write `pi_X` for projection to the source coordinates. Facts 2–3 let us subtract
+the entire nuisance part of any eligible row using available initial units.
+Therefore `pi_X(S_t(c)) = S_t(c) intersect X`. This lifting equality is needed:
+projection containment alone does not justify replacing a shared space with
+dirty components by a source-only space. The control suite gives two small
+counterexamples when either lifting premise is dropped.
+
+Define `L_t(c) = { a in Q^8 : a R in S_t(c) intersect X }`. Any serving `H`
+gives rows `C_t-s_t in H`; projecting them yields a serving source-only space
+of dimension at most `dim H`. Conversely a source-only space is an allowed
+full-space choice. Thus the two minima agree, and choosing one local row per
+target gives exactly
+
+```
+h(c) = min_{B_t in L_t(c)} rank(C - B R).
+```
+
+The residual row span attains each candidate rank. A minimum exists because
+the possible ranks form a nonempty finite set of integers. Since each `L_t`
+is a linear space, substitute `B=-12 A`. With `D=R^T/8-12 A`, the residual
+is `I_104-D R`. The maps `R` and `D` restrict to mutually inverse maps between
+`ker(I_104-D R)` and `ker(I_8-R D)`: if `x=D R x`, then `R x=R D R x`, and
+conversely `z=R D z` gives `D z=D R D z`. These are kernel isomorphisms, not
+claims that either rectangular map is globally injective. Since
+`R D=13 I_8-12 R A`, rank-nullity gives
+
+```
+h(c) = 96 + min_{A_t in L_t(c)} rank(I_8 - R A).                 (1)
+```
+
+Let `E(c)=span_Q { r_t a : a in L_t(c) }` be the space of 8-by-8 matrices,
+where `r_t` is column `t` of `R` and `a` is a row. The matrices `R A` are
+exactly `E(c)`; row choices for distinct targets are independent.
+
+## A linear test that certifies the early staircase
+
+Let `U(c)` be the union span of all coefficient rows in all `L_t(c)`, of
+dimension `r`, and also use `U` for any full-row-rank basis matrix. Every
+`M in E(c)` has row space contained in `U`. Therefore `M z=0` on `ker U`,
+so `I-M` fixes that `(8-r)`-dimensional space pointwise:
+
+```
+rank(I-M) >= 8-r.
+```
+
+Equality holds **if and only if** `U M=U`. For necessity, equality forces
+`image(I-M)=ker U`, so `U(I-M)=0`. For sufficiency, this identity puts the
+image inside `ker U`, while the fixed-point property supplies the reverse
+inclusion. Thus feasibility of `U M=U`, `M in E(c)`, is an ordinary rational
+linear-system question. Infeasibility strengthens `h(c)>=104-r` by one.
+
+The verifier reconstructs every distinct eligible source projection through
+cut 2717, forms `U` and `E`, and checks exact coefficient/augmented ranks at
+the following right endpoints:
+
+| Cut | `rank U` | `dim E` | `U M=U` feasible | Certified lower `h` |
+|---:|---:|---:|:---:|---:|
+| 2525 | 0 | 0 | yes | 104 |
+| 2555 | 2 | 8 | no | 103 |
+| 2571 | 2 | 13 | yes | 102 |
+| 2617 | 4 | 27 | no | 101 |
+| 2645 | 5 | 33 | no | 100 |
+| 2717 | 6 | 45 | no | 99 |
+
+[WITNESS.json](WITNESS.json) supplies respectively 1, 4, 6, 14 and 18 weighted
+contacts at the five nontrivial **left** endpoints. A contact
+`[target, cut, register, weight]` contributes its source coefficient
+`weight * pi_X(v_register(cut)) R^T/104` to the corresponding row of `A`.
+Each contact's availability and every dirty-input lifting premise are replayed.
+The checker computes the exact rank both of `I_8-R A` and of the actual
+104-by-104 matrix `K=C+12 A R`. Their ranks agree with (1); `row(K)` is a
+serving space because `K_t-C_t` lies in `S_t(c)`.
+
+The source-only 104-dimensional space is the trivial first upper witness.
+Pools grow with the prefix, so `h(c)` is nonincreasing. Matching a left-end
+upper bound to a right-end lower bound certifies every cut in each interval.
+This is not a floating-point or finite-field search inference.
+
+## Reproduce and inspect
+
+Standard-library Python, read-only by default; the registered runner uses 3.12:
+
+```sh
+python3.12 -I -O -B research/p104-prefix-minrank/verify.py
+python3.12 -I -O -B research/p104-prefix-minrank/test_verify.py
+python3.12 scripts/research_package.py --package research/p104-prefix-minrank
+```
+
+Expected primary output: `PASS_P104_EARLY_PREFIX_MINRANK intervals=6 cuts=4059 columns=280`.
+[RESULT.json](RESULT.json) is compared in full, including the six bounds,
+110,814 distinct target/snapshot pairs and unchanged local profile. The dense
+replay checks 280 independent columns on a checked denominator-eight integer
+lattice, all frame transitions, owned private lifetimes, source/dirty helper
+restoration and the original `G/2` endpoint. This is an algebraic trace replay;
+it does not rerun the predecessor's native Clifford recipes.
+
+`build.py --out NEW.json` optionally regenerates the contact certificate using
+an independently written eight-coordinate rational projection search. It
+imports no mathematical routines from the acceptance verifier. Its output must
+equal `WITNESS.json`; it alone is not a check of arbitrary dirty inputs.
+All output options refuse to replace an existing file.
+
+Controls cover incorrect contact coefficients, unavailable/self or future
+contacts, wrong paid rank, missing initial compensation and final cleanup,
+the rectangular rank identity, and failures of both lifting hypotheses.
+Checks use explicit exceptions/unittest assertions and run with `-O`.
+
+## Open next questions
+
+- **Resolve cuts 2718–2823.** Apply (1) to the exact small operator spaces.
+  The one-step projector obstruction may leave a gap; seek rank-one/two
+  residual certificates or stronger dual obstructions. This would complete
+  the interval before the predecessor's `h=97` result at 2824. The blocker
+  is constrained rational minimum rank, not another root/Gram calculation.
+- **Realize an affordable chronological correction.** Choose a serving space
+  and provide actual creation, frame paths, source reads, dirty restoration,
+  owner scheduling and a complete recursive invoice. A dimension such as 99
+  or 96 cannot be substituted for a paid transport count. Fixed-path
+  obstructions and precision/all-size obligations remain applicable.
+
+## Credit and scope
+
+Gonzalo Mier / #2173 supplied the pooled-snapshot model and lower96/upper97
+question. Gareth Davidson (@bitplane) / #2251 supplied the rational complement
+construction and full-pool minimum96. The prefix-boundary handoff by @josecarlosgonzalezroque-design at
+Discussion #2245 comment 18940216 supplied the prefix formulation, earliest96
+cut and 97 interval, and explicitly left earlier prefixes open. This package
+extends that algebraic problem; it does not claim their model, 96 lower bound,
+or late-prefix result as new.
+
+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 their retained geometry/decoder
+lineage. Their original files and notices are unchanged.
+
+AAAmirinu with substantial OpenAI Codex assistance in mathematics, code,
+checking and writing, including separate AI-assisted exploration and proof
+review. Evidence is author-checked, not independent human review or formal
+proof-assistant verification. Apache-2.0. No native regeneration, global
+hosting/moment, precision/all-size proof or selected cold replay is claimed.

research/p104-prefix-minrank/RESULT.json

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+{
+  "schema": "p104-prefix-minrank-result/1",
+  "status": "PASS_P104_EARLY_PREFIX_MINRANK",
+  "cuts_replayed": 4059,
+  "boundary_columns": 280,
+  "distinct_target_snapshots": 110814,
+  "intervals": [
+    {
+      "start": 0,
+      "end": 2525,
+      "minimum": 104,
+      "union_rank_at_end": 0,
+      "operator_rank_at_end": 0,
+      "projector_feasible_at_end": true
+    },
+    {
+      "start": 2526,
+      "end": 2555,
+      "minimum": 103,
+      "union_rank_at_end": 2,
+      "operator_rank_at_end": 8,
+      "projector_feasible_at_end": false
+    },
+    {
+      "start": 2556,
+      "end": 2571,
+      "minimum": 102,
+      "union_rank_at_end": 2,
+      "operator_rank_at_end": 13,
+      "projector_feasible_at_end": true
+    },
+    {
+      "start": 2572,
+      "end": 2617,
+      "minimum": 101,
+      "union_rank_at_end": 4,
+      "operator_rank_at_end": 27,
+      "projector_feasible_at_end": false
+    },
+    {
+      "start": 2618,
+      "end": 2645,
+      "minimum": 100,
+      "union_rank_at_end": 5,
+      "operator_rank_at_end": 33,
+      "projector_feasible_at_end": false
+    },
+    {
+      "start": 2646,
+      "end": 2717,
+      "minimum": 99,
+      "union_rank_at_end": 6,
+      "operator_rank_at_end": 45,
+      "projector_feasible_at_end": false
+    }
+  ],
+  "local_profile": {
+    "1": 1074,
+    "2": 58,
+    "3": 86,
+    "4": 104,
+    "5": 48,
+    "6": 24,
+    "7": 8,
+    "8": 8
+  },
+  "scope": "Rational unchanged-trace pooled snapshots only; no chronological correction or paid cost result."
+}

research/p104-prefix-minrank/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"
+    }
+  ]
+}