From 886222c1d3e2c2d878b263dc8d165ef03d5bac7f Mon Sep 17 00:00:00 2001
From: Hiroki KUMAZAKI <hiroki.kumazaki@gmail.com>
Date: Sun, 18 Oct 2026 11:05:51 +0900
Subject: [PATCH] Bind P104 outgoing ports to actual recipient histories and
 admission

---
 .../CERTIFICATE.json.gz                       |  Bin 0 -> 96984 bytes
 research/p104-external-port-closure/PROOF.md  |  110 ++
 research/p104-external-port-closure/README.md |  160 +++
 .../p104-external-port-closure/SOURCES.json   |   54 +
 .../p104-external-port-closure/SUMMARY.json   | 1198 +++++++++++++++++
 research/p104-external-port-closure/TASK.json |  143 ++
 .../p104-external-port-closure/VERIFY.json    |   21 +
 research/p104-external-port-closure/oracle.py |  175 +++
 research/p104-external-port-closure/verify.py |  395 ++++++
 9 files changed, 2256 insertions(+)
 create mode 100644 research/p104-external-port-closure/CERTIFICATE.json.gz
 create mode 100644 research/p104-external-port-closure/PROOF.md
 create mode 100644 research/p104-external-port-closure/README.md
 create mode 100644 research/p104-external-port-closure/SOURCES.json
 create mode 100644 research/p104-external-port-closure/SUMMARY.json
 create mode 100644 research/p104-external-port-closure/TASK.json
 create mode 100644 research/p104-external-port-closure/VERIFY.json
 create mode 100644 research/p104-external-port-closure/oracle.py
 create mode 100644 research/p104-external-port-closure/verify.py

diff --git a/research/p104-external-port-closure/CERTIFICATE.json.gz b/research/p104-external-port-closure/CERTIFICATE.json.gz
new file mode 100644
index 0000000000000000000000000000000000000000..e6347a3d29194c7308dab5e5fff9855c9258d2d5
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diff --git a/research/p104-external-port-closure/PROOF.md b/research/p104-external-port-closure/PROOF.md
new file mode 100644
index 0000000000..9cae0ff3bf
--- /dev/null
+++ b/research/p104-external-port-closure/PROOF.md
@@ -0,0 +1,110 @@
+# Conditional dependency, admission and endpoint proof
+
+Hiroki Kumazaki with substantial OpenAI Codex assistance. Apache-2.0.
+
+## 1. Literal closure
+
+Give every initial bank a distinct state version. Each transport/sign depends
+on its preceding bank version; ADD depends on both the preceding target and
+current source; zero has no value predecessors; erase depends on its current
+private value. The least predecessor-closed set containing an observation is
+its graph ancestor set. This proves the SSA algorithm's literal minimality.
+
+For any touched private lifetime, also require its matching erase. Iterate
+predecessor closure and this requirement to a fixed point. Each iteration only
+adds vertices in a finite graph, so it terminates. Any legal set satisfying
+these two conditions contains every iteration, hence contains the result.
+This minimality is graph-theoretic, not a lower bound against cancellation or
+alternative schedules. The independent reverse scan propagates live demands
+through precisely the same literal instructions without constructing this DAG.
+
+The 675 observations use the exact published PR3423 port-index hash. At an ADD
+observation, predecessor closure retains both the current recipient history and
+the actual source history. Running retained instructions in their original
+order therefore yields each observed ideal value, by induction. Private erase
+checks prove exact zero scalar rows, rather than silently discarding dirt.
+
+The original 146 endpoint observations require 265 inputs. Requesting the
+original endpoints of those 265 inputs adds 215 events and no new inputs.
+This is a selected-output projection. Other outgoing effects are not implied
+by preserving its selected endpoints; they are explicitly listed separately.
+
+## 2. Native premise and normalized rows
+
+Assume the relevant initial and current chart maps `A_H` are unitary
+automorphisms of `B9`, and every used native transport has its archived operator
+`T_(H->K) = A_K A_H^-1`, and that each complete native primitive is unitary,
+preserves `B9 = E Z[i]^512`, and is exact for the admitted module on the common
+grid. These are the existing chart/module implications, not consequences of
+our file hashes. The new check binds all used occurrences to their source data.
+
+Put `z_s = A_initial(s)^-1 v_s(0)`. Induction gives
+
+`v_b(t) = A_frame(b,t) sum_s r[b,s,t] z_s`.
+
+Transport changes the outer frame only. Common-frame ADD performs the literal
+rational row addition. Sign negates a row. Private zero creates the zero row;
+erase is allowed only for the zero row. These arguments quantify over arbitrary
+independent dirty initial arrays, not just the three numerical fixtures.
+The final formula keeps both frame factors; neither initial X frames nor final
+Y frames can be dropped.
+
+The exact retained matrix shows normalized restoration of 213 non-X banks and
+an identity X diagonal plus a rank-9 Y map on the 52 X endpoints. The rank is
+computed by corrected rational pivot elimination, with duplicate/dependent-row
+regressions. A separate explicit factorization expresses all 52 rows using
+nine named rows and verifies every coefficient; a 9-by-9 minor of those rows
+has determinant `-1/128`. These give the matching upper and lower rank bounds. The separate literal fixture interpreter
+checks projection/full ideal-word agreement but is not the universal proof;
+the induction and complete exact rows provide that statement.
+
+## 3. Minimal independent dyadic product entrance
+
+Restrict each independent initial bank to `2^k_s B9`, `k_s >= 0` an integer.
+The chart automorphism premise makes each normalized `z_s` independently
+arbitrary in that same module. For any one row, its output lies in `B9` for all
+such inputs if and only if `r[b,s] 2^k_s` is an integer for every source s.
+
+Sufficiency follows from integer linear closure of B9. For necessity, set every
+other input to zero and choose `z_s = 2^k_s E e_0`. Applying `E^-1` to that row
+leaves its nonintegral scalar coefficient in the zeroth coordinate. The outer
+chart is an automorphism, so it cannot repair nonmembership. All coefficients
+are real dyadic rationals; the maximum reduced-denominator exponent over every
+retained row is consequently the coordinatewise-minimal k_s.
+
+The computed maxima are two for all 104 Y inputs, one for the 38 listed N
+inputs, and zero for the other 123. Reducing any positive exponent by one is
+rejected using its actual coefficient witness (142 cases). This is a condition
+for the retained *prefix invariant*. It does not characterize correlated inputs
+or domains admitting only the final cut. N33/N34's omitted full-word half-reads
+are not licensed by this projection's weaker entrance.
+
+Every retained ADD includes its before and after rows. Under this entrance both
+are in B9, hence their difference—the actual increment—is also in B9. Thus
+rational coefficient divisions are exact on the common grid. Native prefix
+exactness uses the separate inherited module premise; scalar rows alone do not
+verify a native circuit's phase or intermediate rounding.
+
+The event-312 witness is stronger than an odd-integer half: `N100(x)=2*x_0`
+is in B9 because its only nonzero Newton coefficient is `2/(1+i)=1-i`.
+No instruction has touched N100 before its half-read. All other banks remain
+zero beforehand. Its arithmetically exact half `x_0` has coefficient
+`1/(1+i)=(1-i)/2`, outside the Gaussian integers. `4*x_0` is the positive
+2B9 control. This satisfies every original 63-bank entrance assumption and
+exhibits an independent external admission obligation.
+
+## 4. Conditional payload width
+
+Let H bound each real and imaginary physical input component across the 265
+inputs. Each 512-entry complex vector has Euclidean norm at most `32H`.
+Unitary initial chart normalization preserves this norm. Triangle inequality
+and the exact row identity give `||v_b||_2 <= 32H sum_s |r[b,s]|`.
+The retained maximum is `127/2`, first at event 3589, so `2032H` bounds each
+payload component. Unitary complete native prefixes preserve the entrance norm;
+partially written disjoint components inherit a bound from their before/after
+stage. This reproduces the retained norm estimate on the actual recipient
+slice, not a new global estimate or a physical storage theorem.
+
+H itself, initial domain preparation, physical allocation, scratch layout,
+movement, controls, exposed codecs and recursive interfaces still need a paid
+caller. The unchanged-profile `599/584` stop is unaffected.
diff --git a/research/p104-external-port-closure/README.md b/research/p104-external-port-closure/README.md
new file mode 100644
index 0000000000..4d897e52df
--- /dev/null
+++ b/research/p104-external-port-closure/README.md
@@ -0,0 +1,160 @@
+# Closing the P104 open program's external recipient histories
+
+This answers [PR3423's 675-port handoff][handoff] at
+`15d28c4de37dc47ea07cd7173c8b318fca7a4d75`, using the unchanged literal PR2586
+word retained on canonical main `6f100d4bb5083f8c35d96fd95a539811c62f245c`.
+The result is a source-bound dependency and conditional numerical interface,
+not a physical supplier or a new multiplication exponent.
+
+## Result
+
+The 63-bank open program's 675 outgoing ADDs reach 83 distinct recipient banks.
+Their actual histories require substantially more inputs:
+
+| Observations required | Events with private cleanup | Initial banks |
+| --- | ---: | ---: |
+| All 675 recipient states immediately after their port ADD | 2,907 | 263 |
+| Final endpoints of the 83 recipients and 63 original banks | 3,812 | 265 |
+| Also preserve the original endpoint histories of all 265 dependencies | 4,027 | 265 |
+
+A state-version dependency graph and a separately written reverse-liveness
+scan agree on the exact event and input sets. These are least closures in the
+literal read/write graph with all touched private lifetimes closed. They are
+not minima over algebraic cancellations, rewritten programs or different
+implementations. Merely adding the eight erases is insufficient: their forty
+cleanup predecessors must be included too.
+
+The final 4,027-event projection has 2,310 ADDs, 1,597 native transports,
+104 signs, eight private allocations and eight zero-checked erasures.
+It reaches all 265 original endpoints without introducing more inputs.
+It omits 1,371 word events and 93 initial banks. **696 omitted outgoing ADDs** to those 93 banks
+are explicitly inventoried in [SUMMARY.json](SUMMARY.json); this projection
+is not a replacement for the whole word on those omitted banks.
+
+## Additional admission and an exact-half counterexample
+
+Conditional on the inherited native chart identities and Gaussian Newton
+module invariance, the coordinatewise-minimal independent dyadic product
+entrance for **every retained atomic state** is:
+
+- All 104 initial Y banks in `4B9`.
+- The following 38 N banks in `2B9`:
+  `2,4,7,9,18,20,27,29,35,36,40,44,45,46,48,49,50,54,58,59,64,65,66,70,74,75,79,80,81,83,84,85,87,88,89,92,98,100`.
+- The other 123 dependencies in `B9`: 52 X, 32 D, 29 G, eight Q, and N33/N34.
+
+Here `B9 = E Z[i]^512`, with
+`E = [[1,0],[1,1+i]] tensor ... tensor [[1,0],[1,1+i]]` (nine factors).
+One common grid `2^-f` can scale every displayed module.
+Relative to PR3423 this adds 202 initial banks: 52 more strengthened Y inputs,
+38 half-read N inputs, two ordinary N inputs, 52 X, 27 D, seven Q and 24 G.
+It does not claim that correlated final-cut domains require these individual
+scales; PR3426 studies such cancellations separately. The whole-word product
+theorem of PR3406 is credited, not rediscovered: this result specializes the
+entrance to the actual recipient closure, where N33/N34 need only `B9`.
+
+The first added incoming boundary is event **312**, `X100 += N100/2` at the
+empty frame. Set `N100(x)=2*x_0` on the 512 binary addresses, and all other
+initial banks to zero. Every input belongs to `B9`, and all PR3423 63-bank
+admissions hold. The division is exact in Gaussian integers, but its output
+`X100(x)=x_0` has Newton coefficient `(1-i)/2` at index one, outside `Z[i]`.
+The source coefficient is `1-i`. Amplitude four is the positive control.
+This exposes an additional recipient admission obligation; it is not a defect
+in PR3423, which explicitly left recipient histories open.
+
+## Composed endpoints and widths
+
+Write `A_F = Delta_F`, with the empty-frame map equal to identity. Under the
+native covariance premise, at every retained state the exact physical value is
+
+```
+v_b(t) = A_frame(b,t) * sum_s r[b,s,t] * A_initial(s)^(-1) * v_s(0).
+```
+
+The certificate retains the complete rational rows, actual initial frames and
+actual final frames. Initial X banks are not at the empty frame. Final frames
+also vary; using PR3423's common frame at event 4516 for the later endpoints
+would be incorrect.
+
+All 213 non-X banks in the 265-bank projection return their normalized initial
+rows. Each of the 52 X endpoints has its own initial X coefficient one, only Y
+increments, and no residual N/D/Q/G dirt. The actual complete 52-by-104 Y map
+has rational rank 9: every row has a retained nine-row factorization and a
+9-by-9 minor has determinant `-1/128`. This is a different, larger operator from PR3423's
+rank-one **subset of port increments**; the remaining histories are included
+here. [CERTIFICATE.json.gz](CERTIFICATE.json.gz) provides every endpoint row
+and the before/increment/after rows for all 675 ports.
+
+If every initial real/imaginary component of the 265 physical inputs is bounded
+by `H`, unitary chart maps give `32 H * ||r||_1` as a payload bound. The maximum
+row norm is `127/2`, at event 3589 on X46, giving **2032 H**. This is the retained
+whole-word norm bound, now bound to the recipient histories; no sharper global
+width claim is made. Native prefix unitarity extends it to complete native
+stages. Physical storage, temporary layout, caller heights, ownership, routing,
+control, exposed representation conversions and recursive charges remain unpaid.
+
+The inherited unchanged-profile `599/584` stopping decision remains. This
+interface certificate does not justify another supplier-construction round.
+
+## Evidence and reproduction
+
+The checker imports no predecessor executable. It binds the unchanged WORD and
+NATIVE files and all native occurrence/frame/rank records. Native operator
+semantics are **assumed**, not proved by matching those records. PR3382's module
+theorem and PR3416's broader native-covariance certificate are credited external
+premises; PR3423's 392-occurrence check alone does not cover this larger slice.
+
+The bounded check includes eight structural/semantic rejection groups, 142
+one-bit-weaker input-scale controls, two dependent-row rank regression cases,
+an exact rank factorization/minor, three independent literal scalar fixtures
+comparing the projection with the full ideal word, and the exact-array Newton
+counterexample/positive control. The full-word comparison is scalar algebra
+under the chart premise, not a rounded native execution on unadmitted inputs.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-external-port-closure
+python3.12 -I -O -B research/p104-external-port-closure/verify.py
+```
+
+Runtime uses only the standard library. `verify.py --emit` prints the complete
+certificate; `verify.py --summary` prints its summary.
+`oracle.py --full` prints the independently reconstructed closure.
+[SUMMARY.json](SUMMARY.json) is the readable summary; [PROOF.md](PROOF.md) states
+the conditional implications. Exact base, source hashes, context heads, overlap
+coverage, environment and command outcomes are in SOURCES.json and TASK.json.
+No heavy native replay, Lean check or selected-frontier replay is claimed.
+
+## Open next questions
+
+1. Can a paid caller supply this 265-bank entrance and physical layout, including
+   the 52 additional `4B9` and 38 `2B9` admissions? Start with the explicit input
+   list and event-312 witness; stop at any unproved divisibility or ownership
+   premise. Current H and input modules are assumptions, not preparation costs.
+2. If using the projection inside the original word, how will its omitted
+   outgoing effects be supplied and paid? Start with the exact omitted-ADD list
+   in SUMMARY.json. The selected endpoints alone do not preserve those effects.
+   Any changed supplier/profile must first pass a compatible full cost screen.
+
+## Credit and scope history
+
+Hiroki Kumazaki, with substantial OpenAI Codex assistance for analysis, code,
+execution, writing and separate AI-agent review. No independent human or
+formal proof-kernel review is claimed. Apache-2.0.
+
+Credit Clear1nsight (PR3423/3382 and the explicit handoff), ywz65 (PR2586 word
+and native recipes), Arham Tahir (PR2665 binding and PR2674 atomic row/grid
+bounds), Tekalign (PR3025 literal wrapper/ledger), Ross Richardson (PR3068
+period domain), Nick Petalas (PR3406 product domain, PR3416 native covariance,
+PR3426 correlated entrance), and the original geometry/native/compiler lineage.
+Jose Manuel's unchanged-profile stop and Vasek's PR2744 audit remain credited.
+The main `p104-whole-word-internal-grid` package already proves the whole-word
+row norm and an alternative sufficient enlarged fixed grid.
+
+This refines the current P104 producer/consumer question by taking the released
+PR3423 handoff, rather than repeating the preceding PR3044 audit or reopening
+fixed-cover matching searches. Entry overlap covered 353 open PRs and 2,277
+Discussion records with complete pagination; literal numeric searches have
+false-positive matches. Focused PR3423/675-port and canonical-thread checks
+found no other published closure for this handoff. Coverage excludes unpublished
+work and exhaustive historical diffs. The canonical claim is recorded in TASK.
+
+[handoff]: https://github.com/CrocSwap/integer-mult-bounds/discussions/2750#discussioncomment-19009429
diff --git a/research/p104-external-port-closure/SOURCES.json b/research/p104-external-port-closure/SOURCES.json
new file mode 100644
index 0000000000..ff008ed3d8
--- /dev/null
+++ b/research/p104-external-port-closure/SOURCES.json
@@ -0,0 +1,54 @@
+{
+  "schema_version": 1,
+  "base_commit": "6f100d4bb5083f8c35d96fd95a539811c62f245c",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/p104-rounded-collector/inputs/NATIVE.json.gz",
+      "usage": "input",
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diff --git a/research/p104-external-port-closure/SUMMARY.json b/research/p104-external-port-closure/SUMMARY.json
new file mode 100644
index 0000000000..34385fe76f
--- /dev/null
+++ b/research/p104-external-port-closure/SUMMARY.json
@@ -0,0 +1,1198 @@
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+      4712,
+      4713,
+      4716,
+      4718,
+      4720,
+      4722,
+      4724,
+      4727,
+      4734,
+      4735,
+      4736,
+      4737,
+      4738,
+      4739,
+      4742,
+      4743,
+      4744,
+      4745,
+      4746,
+      4747,
+      4752,
+      4753,
+      4754,
+      4755,
+      4940,
+      4943,
+      4992,
+      4994,
+      4996,
+      4998,
+      5000,
+      5003,
+      5005,
+      5007,
+      5008,
+      5011,
+      5013,
+      5015,
+      5017,
+      5019,
+      5025,
+      5026,
+      5029,
+      5030,
+      5031,
+      5032,
+      5035,
+      5040,
+      5041,
+      5048,
+      5049,
+      5050,
+      5051,
+      5052,
+      5053,
+      5056,
+      5057,
+      5058,
+      5059,
+      5060,
+      5061,
+      5066,
+      5067,
+      5068,
+      5069,
+      5073,
+      5074,
+      5077,
+      5078,
+      5079,
+      5080,
+      5083
+    ],
+    "events_sha256": "f0c3adbc3eafd025739cfcb33bcb5e366cf71c3736ebfcbc7176dc0d83812679",
+    "first_two": [
+      {
+        "event": 373,
+        "operation": [
+          "add",
+          "X34",
+          "N34",
+          "1/2",
+          []
+        ]
+      },
+      {
+        "event": 374,
+        "operation": [
+          "add",
+          "X33",
+          "N33",
+          "-1/2",
+          []
+        ]
+      }
+    ],
+    "source_by_role": {
+      "G": 358,
+      "N": 96,
+      "Q": 196,
+      "Y": 46
+    },
+    "target_bank_by_role": {
+      "G": 3,
+      "N": 38,
+      "X": 52
+    },
+    "target_by_role": {
+      "G": 6,
+      "N": 138,
+      "X": 552
+    },
+    "target_count": 93,
+    "target_roles": [
+      "G2",
+      "G3",
+      "G9",
+      "N10",
+      "N12",
+      "N15",
+      "N17",
+      "N24",
+      "N26",
+      "N37",
+      "N38",
+      "N39",
+      "N41",
+      "N42",
+      "N43",
+      "N47",
+      "N5",
+      "N51",
+      "N52",
+      "N53",
+      "N55",
+      "N56",
+      "N57",
+      "N60",
+      "N61",
+      "N62",
+      "N63",
+      "N67",
+      "N68",
+      "N69",
+      "N71",
+      "N72",
+      "N73",
+      "N76",
+      "N77",
+      "N78",
+      "N82",
+      "N86",
+      "N90",
+      "N91",
+      "N96",
+      "X0",
+      "X10",
+      "X101",
+      "X102",
+      "X11",
+      "X12",
+      "X15",
+      "X16",
+      "X17",
+      "X21",
+      "X24",
+      "X25",
+      "X26",
+      "X30",
+      "X31",
+      "X33",
+      "X34",
+      "X37",
+      "X38",
+      "X39",
+      "X41",
+      "X42",
+      "X43",
+      "X47",
+      "X5",
+      "X51",
+      "X52",
+      "X53",
+      "X55",
+      "X56",
+      "X57",
+      "X6",
+      "X60",
+      "X61",
+      "X62",
+      "X63",
+      "X67",
+      "X68",
+      "X69",
+      "X71",
+      "X72",
+      "X73",
+      "X76",
+      "X77",
+      "X78",
+      "X82",
+      "X86",
+      "X90",
+      "X91",
+      "X93",
+      "X96",
+      "X97"
+    ]
+  },
+  "one_bit_lower_controls": 142,
+  "port_cut": {
+    "component_bound_factor": "2032",
+    "event_ids_sha256": "cae6eba5c0ce88e7a9affa9d7bf04b94822a4f34c63952ea8562243fc03e6e82",
+    "events": 2907,
+    "inputs": 263,
+    "max_row_event": 3589,
+    "max_row_l1": "127/2",
+    "op_counts": {
+      "add": 1754,
+      "erase": 8,
+      "sign": 52,
+      "transport": 1085,
+      "zero": 8
+    }
+  },
+  "ports": 675,
+  "recipients": 83,
+  "rejection_controls": [
+    "omit_actual_incoming_half_read",
+    "omit_private_erasure",
+    "wrong_recipient_frame",
+    "dirty_private_erase",
+    "wrong_half_read_leaves_dirty_N100",
+    "native_rank_binding",
+    "weak_N100_B9_in_actual_prefix",
+    "omit_observed_port"
+  ],
+  "scope": "Literal endpoint projection and conditional normalized algebra/module admission. Native operator semantics, physical admission, omitted-bank effects and full supplier costs remain outside this check.",
+  "status": "PASS_P104_EXTERNAL_PORT_CLOSURE",
+  "terminal_Y_minor_determinant": "-1/128",
+  "terminal_Y_rank": 9,
+  "terminal_cut": {
+    "component_bound_factor": "2032",
+    "event_ids_sha256": "fb4f9c43fa11d7eac431be79e4f137508364dc9201036889195f9230f0a941a5",
+    "events": 3812,
+    "inputs": 265,
+    "max_row_event": 3589,
+    "max_row_l1": "127/2",
+    "op_counts": {
+      "add": 2225,
+      "erase": 8,
+      "sign": 104,
+      "transport": 1467,
+      "zero": 8
+    }
+  }
+}
diff --git a/research/p104-external-port-closure/TASK.json b/research/p104-external-port-closure/TASK.json
new file mode 100644
index 0000000000..b9d5174b85
--- /dev/null
+++ b/research/p104-external-port-closure/TASK.json
@@ -0,0 +1,143 @@
+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 6,
+  "base_commit": "6f100d4bb5083f8c35d96fd95a539811c62f245c",
+  "original_card": {
+    "id": "p104-side-correction",
+    "question": "Can the proposed P104 joint endpoint flags be realized by a chronological correction program with coefficients, dirty restoration and an affordable complete bill?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/p104-native-channel/README.md",
+      "research/p104-native-channel/PROOF.md",
+      "docs/research/community-efficient1000-review.md",
+      "research/p104-joint-unit-flags/README.md"
+    ],
+    "methods": [
+      "Construct one side-correction word at an explicit chronological cut with the needed frames still available.",
+      "Derive a lower bound on the side correction and hosting costs that decides whether this channel can beat the selected budget."
+    ],
+    "acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read the revision-6 family record and #2981 static joint flags, alongside the reviewed #1991 channel. #2981 is an author-checked source archive, not an independently validated executable correction or a refutation of program-level lifetime floors.",
+      "scope": "Source-bound orientation only. Read the named package and review before relying on its local evidence; this command is not its replay or supplier admission."
+    },
+    "first_experiment": "After checking live handoffs, pin one named collection from #2981 and attach acquisition/delivery times, source dependencies and coefficient requirements. Test legal access and arbitrary dirty restoration; price any completion against the full channel/endpoint/hosting allowance. Stop with the exact missing implication or a scoped counterexample rather than expanding the static cover census.",
+    "bottleneck_link": "Completing an already realized changed channel tests a concrete construction opportunity, instead of refining the incumbent packing. A positive local result still needs global and all-size admission.",
+    "allowed_variations": [
+      "One correction factorization, one chronological cut, or one explicit hosting obstruction.",
+      "A changed correction schedule with an updated full bill and native frame contract."
+    ],
+    "scope_change_triggers": [
+      "Recompiling the already certified low-rank channel without closing a new obligation.",
+      "Treating a pending ancestor, local histogram or unpriced side correction as an admitted global supplier.",
+      "Repeating a correction or physical conversion already supplied by a pending follow-up without adding independent coverage or a different unresolved obligation.",
+      "Repeating the 208-flag static matching or calling a static nested cover a chronological correction program. Independent audit of that certificate is distinct and remains useful."
+    ],
+    "overlap_terms": [
+      "P104",
+      "I-G/2",
+      "side correction",
+      "native channel",
+      "1972",
+      "1991",
+      "2586",
+      "endpoint transport",
+      "physical dual",
+      "relay composition",
+      "2981",
+      "joint unit flags",
+      "acquisition delivery",
+      "static nesting"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #2981 reports static joint endpoint flags and a matching of 208 departure and arrival flags; archival intake is not independent verification.",
+        "evidence": "research/p104-joint-unit-flags/README.md",
+        "remaining": "Independent flag check and a chronological coefficient program with dirty restoration and a complete cost.",
+        "status": "author-checked",
+        "source_commit": "3d24b92d8ba057b57cb2d480170ec407520e6882",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Source-bound projection of PR3423\u2019s 675 outgoing ports through actual recipient and dependency endpoint histories, with a minimal independent dyadic product entrance and inherited conditional payload bound. Native semantics and physical admission remain premises.",
+  "scope_change_reason": "Take the explicit current PR3423 consumer-join handoff rather than reopen the fixed-cover correction construction or repeat the prior PR3044 replay.",
+  "bottleneck_link": "An open-program saving cannot omit external receiver preparation, history or width costs. Source-bound closure exposes the real interface required for complete pricing.",
+  "budget_minutes": 60,
+  "stop_rule": "One source-bound port-cut and terminal dependency closure; no new word, matching census, supplier admission or frontier change. Stop at the first concrete unproved boundary and record it.",
+  "outcome": "completed",
+  "evidence": "SUMMARY.json and CERTIFICATE.json.gz: separate SSA and reverse-liveness checks agree on the 2907-event/263-input port closure, 3812-event/265-input endpoint closure and 4027-event/265-input history completion. Eight rejection groups, 142 weakened-admission controls, rank-nine factorization and nonzero minor, exact-half Newton counterexample. All 675 original ports and 696 omitted outgoing ADDs are distinguished.",
+  "author": "Hiroki Kumazaki",
+  "ai_assistance": "Substantial OpenAI Codex assistance in source analysis, implementation, execution, proof writing and separate AI-agent review; no independent human or formal proof-kernel review claimed.",
+  "open_questions": [
+    {
+      "question": "Can a paid caller prepare the265-bank entrance and physical layout?",
+      "why_it_matters": "The original63-bank contract alone fails at event312, while2032H assumes supplied input heights.",
+      "first_step": "Use SUMMARY.json initial_scale_bits and event312 witness to bind the52additional4B9 and38additional2B9 admissions to one actual producer with charged conversion and ownership.",
+      "blocked_by": "No physical input producer, allocation, routing or recursive invoice is supplied by this projection."
+    },
+    {
+      "question": "How are the696omitted outgoing ADDs to93excluded banks preserved and paid when embedding this projection?",
+      "why_it_matters": "Preserving all265requested endpoints does not preserve the omitted banks\u2019 side effects.",
+      "first_step": "Start with SUMMARY.json omitted_outgoing_adds.event_indices in the pinned WORD; bind their recipient histories or a proven replacement and apply the full cost screen.",
+      "blocked_by": "No omitted-bank numerical/physical admission or improved supplier profile; unchanged-profile599/584stop remains."
+    }
+  ],
+  "coordination": {
+    "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2750#discussioncomment-19009569",
+    "claim_comment_id": "DC_kwDOU-508s4BIhAh",
+    "handoff_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2750#discussioncomment-19009429",
+    "coverage": {
+      "open_prs": 353,
+      "open_prs_fetched_at": "2026-10-18T01:45:32.767995+00:00",
+      "discussion_records": 2277,
+      "discussions_fetched_at": "2026-10-18T01:49:10.723561+00:00",
+      "complete_pagination": true
+    },
+    "nearest_work": [
+      "PR3423 supplies63-bank program/openports; this closes actual recipient histories.",
+      "PR3406 supplieswholewordproductdomain; this specializesactual265inputprojection.",
+      "PR3416 supplies nativecovariance, an inherited premise here.",
+      "PR3426 suppliescorrelatedfinalcutdomain; not duplicated.",
+      "PR2674 and p104-whole-word-internal-grid supplyretained127/2norm and enlarged-grid alternative."
+    ],
+    "limits": "No unpublished work or exhaustive historical diffs; numerical term searches include false positives.",
+    "publication_refresh": {
+      "checked_at": "2026-10-18T02:05:17.575164+00:00",
+      "main": "6f100d4bb5083f8c35d96fd95a539811c62f245c",
+      "query": "\"port\" \"P104\" in:title,body",
+      "open_pr_matches": [
+        2375,
+        3372,
+        3423
+      ],
+      "focused_discussion": 2750,
+      "comment_count": 5,
+      "decision": "PR2375 is unrelated enumeration tooling; PR3372 is a tail-coefficient cancellation; PR3423 is the claimed predecessor. No additional matching recipient-closure contribution found."
+    }
+  },
+  "environment": {
+    "python": "Python 3.12.13",
+    "platform": "Linux-7.0.0-38-generic-x86_64-with-glibc2.43",
+    "dependencies": "standard library only",
+    "native_replay": "not run; covariance/module semantics retained as explicit premises",
+    "selected_replay": "not run; selected record unchanged",
+    "checks": [
+      "Registered Python 3.12 replay",
+      "Isolated optimized Python 3.12 replay",
+      "Separate SSA/reverse-liveness closures",
+      "Three full/projection scalar fixtures",
+      "Exact-half Newton negative and positive arrays",
+      "Duplicate/dependent-row rank regressions and exact factorization/minor"
+    ]
+  },
+  "review_correction": "Prepublication AI-agent review found a stale-iterator error in the new rank helper. It was fixed before publication; corrected rank is nine, with duplicate/dependent-row regressions and an exact full factorization plus determinant -1/128. Closure and admission counts were unaffected."
+}
diff --git a/research/p104-external-port-closure/VERIFY.json b/research/p104-external-port-closure/VERIFY.json
new file mode 100644
index 0000000000..36d75f735d
--- /dev/null
+++ b/research/p104-external-port-closure/VERIFY.json
@@ -0,0 +1,21 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p104-rounded-collector/inputs/NATIVE.json.gz",
+    "research/p104-rounded-collector/inputs/WORD.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Source-bound 675-port closure in separate SSA and reverse-liveness engines; all 265 endpoint rows, exact rank-nine factorization and minor, minimal dyadic B9 product entrance, eight rejection groups, 142 weakened-admission controls and exact-half Newton witness. Native/module semantics are explicit premises; no supplier admission."
+    }
+  ]
+}
diff --git a/research/p104-external-port-closure/oracle.py b/research/p104-external-port-closure/oracle.py
new file mode 100644
index 0000000000..59cf52bb23
--- /dev/null
+++ b/research/p104-external-port-closure/oracle.py
@@ -0,0 +1,175 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Data-only reverse-liveness oracle for PR3423's external-port interface.
+
+Imports no predecessor code. This computes least literal read/write ancestor
+closures, not cancellation-minimized semantic dependencies. C erasure obligations
+are closed to a fixed point, including all restoration predecessors.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction
+import hashlib
+import json
+from pathlib import Path
+
+DEFAULT=Path(__file__).resolve().parents[2]/'research/p104-rounded-collector/inputs/WORD.json'
+WORD_SHA='cec1fb3fbd0067b500ad5119a62ba60685be718e769374d7ec8abdd2beaa7e50'
+Y={f'Y{i}' for i in [*range(22,33),*range(63,104)]}
+INTERNAL=Y|{f'{c}{i}' for c in ('D','G') for i in range(27,32)}|{'Q7'}
+
+def must(ok,label):
+    if not ok:raise ValueError(label)
+
+def digest(obj):
+    return hashlib.sha256(json.dumps(obj,sort_keys=True,separators=(',',':')).encode()).hexdigest()
+
+def hist(names):
+    return dict(sorted(Counter(n.rstrip('0123456789') for n in names).items()))
+
+def newton(values):
+    values=[(Fraction(a),Fraction(b)) for a,b in values]
+    for j in range(9):
+        for x in range(512):
+            if x & (1<<j):
+                a,b=values[x];c,d=values[x^(1<<j)];values[x]=(a-c,b-d)
+    for x,(a,b) in enumerate(values):
+        for _ in range(x.bit_count()):a,b=(a+b)/2,(b-a)/2
+        values[x]=(a,b)
+    return values
+
+def run(path):
+    raw=path.read_bytes();must(hashlib.sha256(raw).hexdigest()==WORD_SHA,'source pin')
+    word=json.loads(raw);program=word['program']
+    frames={r:tuple(f) for r,f in word['initial_frames'].items()};fh={};live={};lifetimes=[];ports=[]
+    for i,e in enumerate(program):
+        op,dst=e[:2]
+        if op=='zero':
+            must(dst not in frames,'fresh private lifetime')
+            before=None;frames[dst]=tuple(e[2]);live[dst]=i
+        elif op=='erase':
+            must(dst in live and frames[dst]==tuple(e[2]),'private erase frame')
+            before=frames.pop(dst);lifetimes.append((live.pop(dst),i,dst))
+        else:
+            must(dst in frames,'bound target');before=frames[dst]
+            if op=='transport':
+                must(before==tuple(e[2]),'transport preframe');frames[dst]=tuple(e[3])
+            elif op=='sign':must(before==tuple(e[3]),'sign frame')
+            elif op=='add':must(before==frames[e[2]]==tuple(e[4]),'ADD frame')
+            else:raise ValueError('unsupported opcode')
+        fh[i]={'event':i,'before':None if before is None else list(before),'after':None if dst not in frames else list(frames[dst])}
+        if i<=4516 and op=='add' and e[2] in INTERNAL and dst not in INTERNAL and dst!='C':
+            ports.append({'event':i,'target':dst,'source':e[2],'coefficient':str(e[3]),'frame':e[4]})
+    must(not live,'open private lifetime');must(frames=={r:tuple(f) for r,f in word['final_frames'].items()},'final frames')
+    must(len(ports)==675,'port census');recipients={p['target'] for p in ports}
+
+    def backward(final,goals,forced_erases):
+        needed=set(final);events=set()
+        for i in range(len(program)-1,-1,-1):
+            e=program[i];op,dst=e[:2]
+            if i in goals:needed.add(goals[i])
+            if i in forced_erases:needed.add(dst)
+            if dst not in needed:continue
+            if op=='erase':
+                must(i in forced_erases,'value dependency crosses a dead lifetime')
+                events.add(i)
+                continue
+            events.add(i)
+            if op=='add':needed.add(e[2])
+            if op=='zero':needed.remove(dst)
+        return sorted(needed),sorted(events)
+
+    def report(final,goals):
+        initial,rawids=backward(final,goals,set());forced=set();last=None;steps=0
+        # Force every touched private lifetime's erasure and all its value ancestors.
+        ids=rawids
+        while last!=forced:
+            last=set(forced)
+            for start,end,role in lifetimes:
+                if any(start<=i<=end and (program[i][1]==role or
+                    (program[i][0]=='add' and program[i][2]==role)) for i in ids):
+                    forced.add(end)
+            initial2,ids=backward(final,goals,forced);steps+=1
+        names=sorted(initial2);extra=sorted(set(names)-INTERNAL)
+        selected_lifetimes=[{'zero':a,'erase':b,'role':r} for a,b,r in lifetimes if b in forced]
+        def block(events):
+            return {'count':len(events),'opcodes':dict(sorted(Counter(program[i][0] for i in events).items())),
+                    'event_ids':events,'event_ids_sha256':digest(events),
+                    'frame_history_sha256':digest([fh[i] for i in events])}
+        return {'value_only':block(rawids),'lifecycle_closed':block(ids),
+                'value_initial_roles':initial,'value_initial_count':len(initial),
+                'initial_roles':names,'initial_count':len(names),'initial_by_role':hist(names),
+                'initial_roles_sha256':digest(names),'additional_initial_roles':extra,
+                'additional_initial_count':len(extra),'additional_initial_by_role':hist(extra),
+                'private_lifetimes':selected_lifetimes,'lifecycle_added_event_ids':sorted(set(ids)-set(rawids)),
+                'fixed_point_scans':steps}
+    cut=report(set(),{p['event']:p['target'] for p in ports})
+    end=report(recipients|INTERNAL,{})
+    # Every borrowed initial bank acquires its original terminal-history
+    # obligation. This is a projection onto those endpoints, not an assertion
+    # that the projected program preserves updates to excluded banks.
+    terminal=recipients|INTERNAL;history_rounds=[]
+    while True:
+        history=report(terminal,{})
+        added=set(history['initial_roles'])-terminal
+        history_rounds.append({
+            'terminal_count':len(terminal),'terminal_roles':sorted(terminal),
+            'initial_count':history['initial_count'],
+            'value_only_events':history['value_only']['count'],
+            'lifecycle_closed_events':history['lifecycle_closed']['count'],
+            'additional_terminal_roles':sorted(added),
+            'event_ids_sha256':history['lifecycle_closed']['event_ids_sha256']})
+        if not added:break
+        terminal.update(added)
+    history['terminal_roles']=sorted(terminal)
+    history['terminal_count']=len(terminal)
+    history['history_rounds']=history_rounds
+    excluded=sorted(set(word['initial_frames'])-terminal)
+    history['excluded_initial_roles']=excluded
+    history['excluded_initial_count']=len(excluded)
+    history['excluded_initial_by_role']=hist(excluded)
+    retained=set(history['lifecycle_closed']['event_ids'])
+    omitted_outgoing=[{'event':i,'operation':e} for i,e in enumerate(program)
+                      if i not in retained and e[0]=='add' and e[2] in terminal
+                      and e[1] not in terminal]
+    outgoing_targets=sorted({x['operation'][1] for x in omitted_outgoing})
+    history['omitted_outgoing_adds']={
+        'count':len(omitted_outgoing),'events':omitted_outgoing,
+        'events_sha256':digest(omitted_outgoing),
+        'source_by_role':hist([x['operation'][2] for x in omitted_outgoing]),
+        'target_by_role':hist([x['operation'][1] for x in omitted_outgoing]),
+        'target_roles':outgoing_targets,'target_count':len(outgoing_targets),
+        'target_bank_by_role':hist(outgoing_targets)}
+    earliest=[{'event':i,'operation':program[i]} for i in cut['value_only']['event_ids']
+              if program[i][0]=='add' and program[i][1] in recipients and program[i][2] not in INTERNAL]
+    # Reachable exact-half counterexample: every other initial bank is zero.
+    must(word['initial_frames']['N100']==[],'N100 entry frame')
+    contacts=[i for i,e in enumerate(program) if e[1]=='N100' or (e[0]=='add' and e[2]=='N100')]
+    must(contacts[0]==312 and program[312]==['add','X100','N100','1/2',[]],'N100 first contact')
+    witness=[]
+    for amplitude in (2,4):
+        src=[(amplitude*(x&1),0) for x in range(512)]
+        out=[(a//2,b//2) for a,b in src]
+        sc=newton(src);oc=newton(out)
+        must(all(z.denominator==1 for pair in sc for z in pair),'source Newton membership')
+        failure=[i for i,pair in enumerate(oc) if any(z.denominator!=1 for z in pair)]
+        must(bool(failure)==(amplitude==2),'counterexample/positive control')
+        witness.append({'amplitude':amplitude,'source_newton_coefficient_1':[str(z) for z in sc[1]],
+                        'output_newton_coefficient_1':[str(z) for z in oc[1]],'output_bad_indices':failure,
+                        'half_arithmetic_exact':all(a%2==b%2==0 for a,b in src)})
+    return {'status':'PASS_INDEPENDENT_REVERSE_PORT_DEPENDENCY_CLOSURE','word_sha256':WORD_SHA,
+            'word_events':len(program),'initial_word_banks':len(word['initial_frames']),
+            'internal_banks':len(INTERNAL),'outgoing_ports':len(ports),'outgoing_ports_sha256':digest(ports),
+            'recipient_banks':len(recipients),'recipient_by_role':hist(recipients),'recipients':sorted(recipients),
+            'port_cuts':cut,'final_endpoints':end,'history_complete':history,
+            'first_added_incoming_events':earliest[:12],
+            'N100_exact_half_control':witness,
+            'scope':'Least literal read/write dependency closures, including private lifetime restoration and a fixed point of original endpoint-history obligations. These are projections; omitted outgoing side effects are not preserved. Not a minimum modulo cancellation, native semantic replay, width invoice, or supplier admission.'}
+
+def main():
+    p=argparse.ArgumentParser();p.add_argument('--word',type=Path,default=DEFAULT);p.add_argument('--full',action='store_true');a=p.parse_args();d=run(a.word)
+    if not a.full:
+        for label in ('port_cuts','final_endpoints','history_complete'):
+            for part in ('value_only','lifecycle_closed'):del d[label][part]['event_ids']
+    print(json.dumps(d,sort_keys=True,indent=2))
+if __name__=='__main__':main()
diff --git a/research/p104-external-port-closure/verify.py b/research/p104-external-port-closure/verify.py
new file mode 100644
index 0000000000..67bcd79118
--- /dev/null
+++ b/research/p104-external-port-closure/verify.py
@@ -0,0 +1,395 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Literal P104 recipient join, conditional on native covariance/B9 invariance.
+
+Hiroki Kumazaki with substantial OpenAI Codex assistance. No predecessor imports.
+"""
+import argparse
+from collections import Counter
+from copy import deepcopy
+from fractions import Fraction
+import gzip
+import hashlib
+import json
+import subprocess
+import sys
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+WORD = 'research/p104-rounded-collector/inputs/WORD.json'
+NATIVE = 'research/p104-rounded-collector/inputs/NATIVE.json.gz'
+ROLES = set(['Y'+str(i) for i in (*range(22,33),*range(63,104))] +
+            ['D'+str(i) for i in range(27,32)] + ['G'+str(i) for i in range(27,32)] + ['Q7'])
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def unique(pairs):
+    d = dict(pairs)
+    need(len(d) == len(pairs), 'duplicate JSON key')
+    return d
+
+
+def read(p):
+    b = p.read_bytes()
+    return json.loads(gzip.decompress(b) if p.suffix == '.gz' else b, object_pairs_hook=unique)
+
+
+def canonical(x):
+    return json.dumps(x,sort_keys=True,separators=(',',':')).encode()
+
+
+def digest(x):
+    return hashlib.sha256(canonical(x)).hexdigest()
+
+
+def add(a,b,c):
+    out = dict(a)
+    for k,v in b.items():
+        out[k] = out.get(k,Fraction())+c*v
+        if not out[k]:
+            del out[k]
+    return out
+
+
+def row_json(r):
+    return {k:str(v) for k,v in sorted(r.items())}
+
+
+def dyadic_bits(q):
+    d = q.denominator
+    need(d & (d-1) == 0, 'non-dyadic coefficient')
+    return d.bit_length()-1
+
+
+def build(word, native):
+    """Build state-version DAG while propagating exact normalized scalar rows."""
+    frames={k:tuple(v) for k,v in word['initial_frames'].items()}
+    initial=sorted(frames)
+    node={k:-i-1 for i,k in enumerate(initial)}
+    rows={node[k]:{k:Fraction(1)} for k in initial}
+    deps={node[k]:() for k in initial}
+    owners={node[k]:k for k in initial}
+    records={}
+    lifetime={}
+    pairs={}
+    occurrences={}
+    for rec in native['transports']:
+        for i in rec['events']:
+            need(i not in occurrences,'duplicate native occurrence')
+            occurrences[i]=rec
+    need(word['chart']==native['chart'],'native chart binding')
+    for i,op in enumerate(word['program']):
+        kind,dst=op[:2]
+        if kind=='zero':
+            need(dst not in node,'private allocation aliases live bank')
+            node[dst]=i; frames[dst]=tuple(op[2]); rows[i]={};deps[i]=()
+            lifetime[dst]=i
+        else:
+            need(dst in node,'unknown target')
+            before=node[dst]
+            if kind=='transport':
+                need(frames[dst]==tuple(op[2]),'old frame mismatch')
+                rec=occurrences[i]
+                need(rec['old']==op[2] and rec['new']==op[3] and rec['recipe']['rank']==op[4], 'native occurrence binding')
+                frames[dst]=tuple(op[3]);deps[i]=(before,);rows[i]=dict(rows[before])
+            elif kind=='sign':
+                need(frames[dst]==tuple(op[3]) and Fraction(op[2])==-1,'sign frame/coefficient')
+                deps[i]=(before,);rows[i]={k:-v for k,v in rows[before].items()}
+            elif kind=='add':
+                src=op[2];need(src in node,'unknown source')
+                need(frames[dst]==frames[src]==tuple(op[4]),'unequal ADD frames')
+                c=Fraction(op[3]);need(c!=0,'zero ADD')
+                deps[i]=(before,node[src]);rows[i]=add(rows[before],rows[node[src]],c)
+                records[i]={'event':i,'target':dst,'source':src,'coefficient':str(c),'frame':op[4],
+                            'before':before,'source_version':node[src]}
+            elif kind=='erase':
+                need(dst in lifetime and frames[dst]==tuple(op[2]),'private erasure lifetime/frame')
+                need(not rows[before],'nonzero private erasure')
+                deps[i]=(before,);rows[i]={};pairs[lifetime.pop(dst)]=i
+                del node[dst];del frames[dst]
+            else:
+                raise ValueError('unknown opcode')
+            if kind!='erase':node[dst]=i
+        owners[i]=dst
+    need(not lifetime,'unclosed private lifetime')
+    need(frames=={k:tuple(v) for k,v in word['final_frames'].items()},'terminal frames')
+    need(set(occurrences)=={i for i,o in enumerate(word['program']) if o[0]=='transport'},'complete native occurrence set')
+    return {'deps':deps,'rows':rows,'owners':owners,'records':records,'pairs':pairs,'final':node,'initial':initial}
+
+
+def ancestors(dag,seeds):
+    seen=set(); stack=list(seeds)
+    while stack:
+        i=stack.pop()
+        if i in seen:continue
+        need(i in dag['deps'],'unknown state version')
+        seen.add(i);stack.extend(dag['deps'][i])
+    return seen
+
+
+def close(dag,seeds):
+    result=ancestors(dag,seeds)
+    while True:
+        erases={end for start,end in dag['pairs'].items() if start in result}
+        new=ancestors(dag,result|erases)
+        if new==result:return result
+        result=new
+
+
+def require_closed(dag,selected,seeds):
+    need(set(seeds)<=selected,'missing observation')
+    for i in selected:need(set(dag['deps'][i])<=selected,'missing actual dependency')
+    for start,end in dag['pairs'].items():
+        if start in selected:need(end in selected,'missing private cleanup')
+
+
+def evaluate_slice(word, dag, selected, values):
+    """One independent scalar fixture through the projected literal program."""
+    state={k:Fraction(values[k]) for i,k in enumerate(dag['initial']) if -i-1 in selected}
+    for i,o in enumerate(word['program']):
+        if i not in selected:continue
+        kind,dst=o[:2]
+        if kind=='zero':state[dst]=Fraction()
+        elif kind=='add':state[dst]+=Fraction(o[3])*state[o[2]]
+        elif kind=='sign':state[dst]=-state[dst]
+        elif kind=='erase':need(state.pop(dst)==0,'fixture private dirt')
+        elif kind!='transport':raise ValueError('fixture opcode')
+    return state
+
+
+def profile(word,dag,selected):
+    initials=sorted(dag['owners'][i] for i in selected if i<0)
+    bits={k:0 for k in initials}; witness={}
+    maximum=(Fraction(-1),-1)
+    for i in sorted(selected):
+        row=dag['rows'][i]
+        norm=sum(map(abs,row.values()),Fraction())
+        if norm>maximum[0]:maximum=(norm,i)
+        for k,v in row.items():
+            b=dyadic_bits(v)
+            if b>bits[k]:bits[k]=b;witness[k]={'event':i,'coefficient':str(v)}
+    return {'initial_inputs':initials,'initial_scale_bits':bits,'scale_witnesses':witness,
+            'events':sorted(i for i in selected if i>=0),
+            'op_counts':dict(sorted(Counter(word['program'][i][0] for i in selected if i>=0).items())),
+            'max_row_l1':str(maximum[0]),'max_row_event':maximum[1],
+            'component_bound_factor':str(32*maximum[0])}
+
+
+def rational_rank(matrix):
+    pivots={}
+    for row in matrix:
+        row=list(row)
+        for i in range(len(row)):
+            value=row[i]
+            if not value:continue
+            if i in pivots:row=[a-value*b for a,b in zip(row,pivots[i])]
+            else:
+                pivots[i]=[a/value for a in row];break
+    return len(pivots)
+
+
+def rank_witness(terminal_rows):
+    names=['X1','X100','X103','X13','X14','X18','X19','X20','X22']
+    columns=[1,2,14,3,7,4,5,33,22]
+    basis=[[Fraction(terminal_rows[k].get('Y'+str(j),'0')) for j in range(104)] for k in names]
+    square=[[row[j] for j in columns] for row in basis]
+    aug=[row+[Fraction(i==j) for j in range(9)] for i,row in enumerate(square)]
+    determinant=Fraction(1)
+    for col in range(9):
+        pivot=next((i for i in range(col,9) if aug[i][col]),None)
+        need(pivot is not None,'singular rank witness')
+        if pivot!=col:aug[pivot],aug[col]=aug[col],aug[pivot];determinant=-determinant
+        value=aug[col][col];determinant*=value
+        aug[col]=[v/value for v in aug[col]]
+        for i in range(9):
+            if i==col:continue
+            value=aug[i][col]
+            aug[i]=[a-value*b for a,b in zip(aug[i],aug[col])]
+    inverse=[row[9:] for row in aug]
+    need(determinant==Fraction(-1,128),'rank witness determinant')
+    coefficients={}
+    for name,row in terminal_rows.items():
+        if not name.startswith('X'):continue
+        target=[Fraction(row.get('Y'+str(j),'0')) for j in range(104)]
+        coef=[sum((target[columns[k]]*inverse[k][j] for k in range(9)),Fraction()) for j in range(9)]
+        need(all(sum((coef[k]*basis[k][j] for k in range(9)),Fraction())==target[j] for j in range(104)),
+             'rank-nine full matrix factorization')
+        coefficients[name]=[str(c) for c in coef]
+    return {'basis_rows':names,'minor_Y_columns':columns,'minor_determinant':str(determinant),
+            'coefficients':coefficients,'identity':'Every terminal X-to-Y row equals its nine coefficients times the nine named full Y rows.'}
+
+
+def check_terminal(dag,terminal):
+    for k in terminal:
+        row=dag['rows'][dag['final'][k]]
+        if k.startswith('X'):
+            need(row.get(k)==1 and all(src==k or src.startswith('Y') for src in row),
+                 'terminal dirty-input cancellation')
+        else:need(row=={k:Fraction(1)},'terminal normalized restoration')
+
+
+def audit(word,native):
+    dag=build(word,native)
+    ports=[i for i,o in enumerate(word['program']) if i<=4516 and o[0]=='add' and o[2] in ROLES and o[1] not in ROLES and o[1]!='C']
+    need(len(ports)==675,'675 actual outgoing ports')
+    need(digest(ports)=='04b3a0bc022c336c9b177eeed49036f1257ab81ea9d0459b8170075b002a24a4', 'PR3423 published port-index binding')
+    targets={word['program'][i][1] for i in ports}
+    terminal=sorted(targets|ROLES)
+    port_set=close(dag,ports)
+    end_seeds=[dag['final'][k] for k in terminal]
+    end_set=close(dag,end_seeds)
+    # Joining both sets is necessary: final cancellation must not drop port observations.
+    joined=close(dag,ports+end_seeds)
+    for s,seeds in ((port_set,ports),(end_set,end_seeds),(joined,ports+end_seeds)):
+        require_closed(dag,s,seeds)
+    p=profile(word,dag,joined)
+    all_inputs=p['initial_inputs']
+    returned=close(dag,ports+[dag['final'][k] for k in all_inputs])
+    need({dag['owners'][i] for i in returned if i<0}==set(all_inputs),
+         'terminal-history closure introduces new inputs')
+    require_closed(dag,returned,ports+[dag['final'][k] for k in all_inputs])
+    returned_profile=profile(word,dag,returned)
+    need(returned_profile['initial_scale_bits']==p['initial_scale_bits'],'returned entrance contract changed')
+    check_terminal(dag,all_inputs)
+    terminal_rows={k:row_json(dag['rows'][dag['final'][k]]) for k in terminal}
+    check_terminal(dag,terminal)
+    scales=p['initial_scale_bits']
+    for i in joined:
+        for name,c in dag['rows'][i].items():
+            need((c*2**scales[name]).denominator==1,'product entrance insufficient')
+    # Each smaller independent dyadic factor has a reachable nonintegral coefficient.
+    lower_controls=[]
+    for name,b in scales.items():
+        if not b:continue
+        witness=p['scale_witnesses'][name]
+        need((Fraction(witness['coefficient'])*2**(b-1)).denominator>1,
+             'one-bit-lower admission was not rejected')
+        lower_controls.append(name)
+    matrix=[[Fraction(terminal_rows[k].get('Y'+str(j),'0')) for j in range(104)]
+            for k in terminal if k.startswith('X')]
+    matrix_rank=rational_rank(matrix)
+    need(matrix_rank==9,'complete terminal Y-map rank')
+    need(rational_rank([[Fraction(1),Fraction(1)],[Fraction(1),Fraction(1)]])==1,
+         'duplicate-row rank control')
+    need(rational_rank([[Fraction(x) for x in row] for row in [[1,2,3],[2,4,6],[0,1,1]]])==2,
+         'dependent-row rank control')
+    port_rows=[]
+    for i in ports:
+        rec=dag['records'][i]; before=dag['rows'][rec['before']]; after=dag['rows'][i]
+        inc={k:Fraction(rec['coefficient'])*v for k,v in dag['rows'][rec['source_version']].items()}
+        port_rows.append({k:v for k,v in rec.items() if k not in ('before','source_version')}|
+                         {'before_row':row_json(before),'increment_row':row_json(inc),'after_row':row_json(after),
+                          'before_l1':str(sum(map(abs,before.values()),Fraction())),
+                          'after_l1':str(sum(map(abs,after.values()),Fraction()))})
+    # Independent integer fixture execution checks actual slice/full joins; no native claim.
+    all_nodes=set(dag['deps'])
+    for seed in (0,17,73):
+        values={k:((j*37+seed*11)%43-21)*(2**p['initial_scale_bits'].get(k,2)) for j,k in enumerate(dag['initial'])}
+        sliced=evaluate_slice(word,dag,returned,values)
+        full=evaluate_slice(word,dag,all_nodes,values)
+        for k in all_inputs:
+            expected=sum((v*values[src] for src,v in dag['rows'][dag['final'][k]].items()),Fraction())
+            need(sliced[k]==full[k]==expected,'slice/full terminal fixture')
+    result={'schema_version':1,'scope':'Conditional chart-normalized recipient closure; native covariance and B9 automorphisms inherited.',
+            'ports':port_rows,'recipients':sorted(targets),'terminal_targets':terminal,
+            'port_cut':profile(word,dag,port_set),'terminal_cut':profile(word,dag,end_set),'joined':p,
+            'history_complete':returned_profile,
+            'history_terminal_rows':{k:row_json(dag['rows'][dag['final'][k]]) for k in all_inputs},
+            'history_terminal_frames':{k:word['final_frames'][k] for k in all_inputs},
+            'terminal_rows':terminal_rows,'terminal_Y_matrix_rank':matrix_rank,'terminal_Y_rank_witness':rank_witness(terminal_rows),
+            'one_bit_lower_admission_controls':lower_controls,'terminal_frames':{k:word['final_frames'][k] for k in terminal},
+            'initial_frames':{k:word['initial_frames'][k] for k in p['initial_inputs']},
+            'private_lifetimes':[[a,b] for a,b in dag['pairs'].items() if a in joined]}
+    return result,dag,returned,ports,[dag['final'][k] for k in all_inputs]
+
+
+def controls(word,native,dag,joined,ports,end_seeds):
+    rejected=[]
+    def fail(label,fn):
+        try:fn()
+        except (ValueError,KeyError):rejected.append(label)
+        else:raise ValueError('accepted control '+label)
+    fail('omit_actual_incoming_half_read',lambda:require_closed(dag,joined-{312},ports+end_seeds))
+    start,end=next((a,b) for a,b in dag['pairs'].items() if a in joined)
+    fail('omit_private_erasure',lambda:require_closed(dag,joined-{end},ports+end_seeds))
+    wrong=deepcopy(word);wrong['program'][312][4]=[1]
+    fail('wrong_recipient_frame',lambda:build(wrong,native))
+    wrong=deepcopy(word);wrong['program'][2038][3]='1'
+    fail('dirty_private_erase',lambda:build(wrong,native))
+    wrong=deepcopy(word);wrong['program'][312][3]='1/4'
+    fail('wrong_half_read_leaves_dirty_N100',lambda:check_terminal(build(wrong,native),['X100']))
+    wrong=deepcopy(native);wrong['transports'][0]['recipe']['rank']+=1
+    fail('native_rank_binding',lambda:build(word,wrong))
+    need(word['program'][312]==['add','X100','N100','1/2',[]],'half-read witness source')
+    derived=profile(word,dag,joined)['initial_scale_bits']
+    weak=dict(derived);weak['N100']=0
+    def validate_entrance(scales):
+        for i in sorted(joined):
+            for name,c in dag['rows'][i].items():
+                need((c*2**scales[name]).denominator==1, 'admission failure at event '+str(i))
+    validate_entrance(derived)
+    fail('weak_N100_B9_in_actual_prefix',lambda:validate_entrance(weak))
+    need(Fraction(1,2)*2==1,'strengthened half-read positive control')
+    # Endpoint-correct omission is detected by retaining the observations themselves.
+    fail('omit_observed_port',lambda:require_closed(dag,joined-{ports[0]},ports+end_seeds))
+    return rejected
+
+
+def summarize(result):
+    payload=(json.dumps(result,sort_keys=True,indent=2)+'\n').encode()
+    out={'status':'PASS_P104_EXTERNAL_PORT_CLOSURE',
+         'certificate_payload_sha256':hashlib.sha256(payload).hexdigest(),
+         'ports':len(result['ports']),'recipients':len(result['recipients'])}
+    for tag in ('port_cut','terminal_cut','history_complete'):
+        p=result[tag]
+        out[tag]={'events':len(p['events']),'inputs':len(p['initial_inputs']),
+                  'op_counts':p['op_counts'],'event_ids_sha256':digest(p['events']),
+                  'max_row_l1':p['max_row_l1'],'max_row_event':p['max_row_event'],
+                  'component_bound_factor':p['component_bound_factor']}
+    out['initial_scale_bits']=result['history_complete']['initial_scale_bits']
+    out['actual_recipient_counterexample']=result['independent_closure']['N100_exact_half_control']
+    out['rejection_controls']=result['rejected_controls']
+    out['one_bit_lower_controls']=len(result['one_bit_lower_admission_controls'])
+    out['terminal_Y_rank']=result['terminal_Y_matrix_rank']
+    out['terminal_Y_minor_determinant']=result['terminal_Y_rank_witness']['minor_determinant']
+    omitted=result['omitted_outgoing_adds']
+    out['omitted_outgoing_adds']={k:v for k,v in omitted.items() if k!='events'}
+    out['omitted_outgoing_adds']['event_indices']=[a['event'] for a in omitted['events']]
+    out['omitted_outgoing_adds']['first_two']=omitted['events'][:2]
+    out['scope']='Literal endpoint projection and conditional normalized algebra/module admission. Native operator semantics, physical admission, omitted-bank effects and full supplier costs remain outside this check.'
+    return out
+
+
+def main():
+    ap=argparse.ArgumentParser(description=__doc__);ap.add_argument('--emit',action='store_true');ap.add_argument('--summary',action='store_true');args=ap.parse_args()
+    for ent in read(HERE/'SOURCES.json')['files']:
+        if ent['usage']=='input':need(hashlib.sha256((ROOT/ent['path']).read_bytes()).hexdigest()==ent['sha256'],'source pin '+ent['path'])
+    word,native=read(ROOT/WORD),read(ROOT/NATIVE)
+    result,dag,joined,ports,end_seeds=audit(word,native)
+    result['rejected_controls']=controls(word,native,dag,joined,ports,end_seeds)
+    independent=json.loads(subprocess.check_output([sys.executable,'-I','-B',str(HERE/'oracle.py'),'--full'],text=True))
+    for a,b in (('port_cut','port_cuts'),('terminal_cut','final_endpoints'),('history_complete','history_complete')):
+        need(result[a]['events']==independent[b]['lifecycle_closed']['event_ids'],'SSA/reverse-liveness event sets')
+        need(result[a]['initial_inputs']==independent[b]['initial_roles'],'SSA/reverse-liveness initial roles')
+    result['independent_closure']={k:independent[k] for k in ('outgoing_ports_sha256','N100_exact_half_control')}
+    result['independent_closure']['event_sets_and_initial_inputs_agree']=True
+    inputs=set(result['history_complete']['initial_inputs'])
+    omitted=[{'event':i,'operation':o} for i,o in enumerate(word['program'])
+             if i not in joined and o[0]=='add' and o[2] in inputs and o[1] not in inputs]
+    need(omitted==independent['history_complete']['omitted_outgoing_adds']['events'],
+         'omitted outgoing effects agree')
+    result['omitted_outgoing_adds']=independent['history_complete']['omitted_outgoing_adds']
+    if args.summary:print(json.dumps(summarize(result),sort_keys=True,indent=2))
+    elif args.emit:print(json.dumps(result,sort_keys=True,indent=2))
+    else:
+        need(result==read(HERE/'CERTIFICATE.json.gz'),'retained certificate mismatch')
+        need(summarize(result)==read(HERE/'SUMMARY.json'),'retained summary mismatch')
+        print('PASS_P104_EXTERNAL_PORT_CLOSURE',digest(result))
+
+
+if __name__=='__main__':main()