Certify infinite-order growth of swapped source-code charts

research/source-code-chart-order/ARCHIVE.json

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research/source-code-chart-order/README.md

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+# Swapping source-code charts destroys periodic restoration
+
+Scoped source-algebra obstruction. No new supplier or kappa is claimed.
+
+Ali Romman's [PR1862](https://github.com/CrocSwap/integer-mult-bounds/pull/1862)
+constructs a valid order-six source code. It explicitly leaves changing-chart
+restoration open. On the actual selected 112-port query matrix, alternating its
+compensation and query port sets gives an **infinite-order** product. The product
+has a real eigenvalue strictly between 170 and 171. Repeating this pair cannot
+restore every source, even after arbitrarily many repetitions.
+
+This is a concrete, quantitative witness, not a correction to PR1862. David
+Arias's [PR1973](https://github.com/CrocSwap/integer-mult-bounds/pull/1973)
+uses these same two admissible charts to check a different statement, the
+three-chart F-B-F exchange defect. Its all-length winding theorem concerns a
+fixed code. This package instead derives the repeated source-only pair's
+characteristic polynomial and growth. It imports neither predecessor's code.
+
+## Exact instance and result
+
+The selected source has an 8 by 112 query matrix R, with
+
+```text
+K = R^T/4,   RK = 14 I_8,
+A = [0,7,20,27,36,46,57,111],
+T = [21,37,47,76,81,87,92,110].
+```
+
+Let P and S be K restricted to its A and T rows respectively. Let L_A and L_T
+be R restricted to those columns. Both eight-port minors are orthogonal with
+squared norm four. Thus
+
+```text
+RP = RS = L_A P = L_T S = L_A K = L_T K = I_8,
+L_A S = L_T P = 0.
+```
+
+PR1862's ordered code for a compensation section P and logger L is
+
+```text
+Z = K - P(RK-I),   Q = R-L,
+G(P,L) = (I-PL)(I+ZQ) = I+ZQ-PR.
+```
+
+Set G_A = G(P,L_T), G_T = G(S,L_A), and C = G_T G_A. Products act on
+column vectors, so G_A executes first. Both charts meet exactly the predecessor's
+algebraic hypotheses RP=I, LK=I and LP=0.
+
+The checked full characteristic polynomial is
+
+```text
+char_C(t) = (t-1)^88 (t^3-171t^2+27t-1)^8.
+```
+
+In particular, trace C is 1456. Each individual code has exact order six,
+whereas C has infinite order over Q and Q(i). The same-chart pair G_A G_A has
+order three and provides a positive restoration control.
+
+## Proof by three sections
+
+Put U = K-P-S. The three sections satisfy
+
+```text
+L_A U = L_T U = 0,   RU = 12 I_8.
+```
+
+The 112 by 24 matrix E = [P S U] therefore has the left inverse
+
+```text
+D = stack(L_A, L_T, (R-L_A-L_T)/12),   DE = I_24.
+```
+
+Every source splits uniquely into a vector in ker D and a vector E(u,v,z).
+The complement has dimension 88. Both codes fix it pointwise because each of
+R, L_A and L_T vanishes there. On the section coordinates, the two codes act
+by the following matrices tensored with I_8:
+
+```text
+H_A = [[-12,  -1,  -156],
+       [  1,   1,    12],
+       [  1,   0,    13]],
+
+H_T = [[  1,   1,    12],
+       [ -1, -12,  -156],
+       [  0,   1,    13]].
+```
+
+For example, G_A P = -12P+S+U, G_A S = -P+S, and
+G_A U = -156P+12S+13U. These follow directly from the ordered code formula.
+Swapping P and S gives H_T. Their product is
+
+```text
+H_T H_A = [[   1,   0,    12],
+           [-156, -11, -2016],
+           [  14,   1,   181]].
+```
+
+Its trace is 171, sum of principal two by two minors is 27, and determinant
+is one. Its characteristic polynomial is p(t)=t^3-171t^2+27t-1. The direct-sum
+decomposition proves the displayed full characteristic polynomial and
+trace C = 88+8*171 = 1456.
+
+The integer signs p(170)=-24311 and p(171)=4616 put a real eigenvalue lambda
+strictly between 170 and 171 by continuity. A finite-order matrix has only
+roots of unity as eigenvalues, so C cannot have finite order. For any induced
+real operator norm, ||C^k|| is at least lambda^k, hence greater than 170^k for
+positive k. This describes worst-case growth in the source-only port algebra.
+It is not a physical precision bill or a claim that every rational source
+vector expands at this rate.
+
+There is also a short order obstruction independent of the cubic calculation.
+For a finite-order 112-dimensional matrix in characteristic zero, each
+eigenvalue has modulus one, so the absolute value of its trace is at most 112.
+The measured exact trace 1456 already excludes finite order.
+
+## Reproduce and evidence
+
+From the repository root, Python 3.10 or later, standard library only:
+
+```sh
+python3 -I -B research/source-code-chart-order/verify.py
+python3 -I -O -B research/source-code-chart-order/verify.py
+python3 -I -B research/source-code-chart-order/test_verify.py
+python3 scripts/research_package.py --package research/source-code-chart-order --check-only
+```
+
+Expected verifier line:
+
+```text
+PASS_SOURCE_CODE_CHART_ORDER orders=6,6,infinite trace=1456 eigenvalue=170..171 columns=336 controls=4 no_new_kappa
+```
+
+Standard and optimized registered replays pass, including the source pins,
+336 column comparisons, four semantic controls and eight focused tests.
+All 45 research-workflow tests pass with no skips; bytecode compilation with
+`-Wall` also passes. Exact commands and unrun obligations are in TASK.json.
+
+The registered hook uses Python 3.12:
+
+```sh
+python3.12 scripts/research_package.py --package research/source-code-chart-order
+```
+
+The verifier authenticates the selected PR1094 archive and two exact members
+before parsing. It propagates all 112 source columns to the forward centre
+cut. It does not replay helper or private-input columns. It independently
+constructs the two 112 by 112 codes, proves DE=I, checks both section actions
+and the fixed complement, and compares the matrix calculations with a
+separate ordered-factor column interpreter in 336 cases. All checks remain
+active under `-O`.
+
+Four semantic controls detect a broken minor, overlapping compensation/logger
+supports, a changed section basis and a wrong active coefficient. The focused
+tests also reject an archive mutation and a forged receipt, and exercise CLI
+failure behavior. `RESULT.json` records the deterministic result. Default
+replay is read-only; `--write` regenerates only this package's receipt.
+
+[SOURCES.json](SOURCES.json) records Git blobs and hashes at main
+`d48fcf5eedae9d8e2296598a1868a0b1a37a83f1`.
+[ARCHIVE.json](ARCHIVE.json) additionally pins raw members and decompressed
+payloads. These payloads equal PR1862's historical PR1042 inputs. The query
+matrix digest also equals PR1973's independently extracted digest.
+The archive, selected record, frozen predecessors and `upstream/` are unchanged.
+
+## Scope and task selection
+
+Chooser seed `4a0e6caaf3f4e5d0026f7998e13a6835` proposed
+`complex-new-helper-inventory` under priority revision 3. The scope changed
+to a restoration audit of a proposed source-code interface. It does not meet
+the original card's new-inventory construction gate. The binding complex
+supplier motivates the question, but this is a scoped obstruction, not an
+exponent improvement.
+
+The card's ceiling positive control and ideal levers were run. Centre-free
+work has a favorable 2.80-times ceiling before the source coding, chart
+routing, restoration and other costs of realizing it. This audit rejects only
+periodic restoration by repetitions of the specific pair C. It supplies no
+candidate invoice and credits no omitted work as a realized saving.
+
+[The coordination claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1976)
+records the scope and overlap checks. A fully paginated refresh covered
+404 open PRs and 625 Discussion records without warnings. Coverage timestamps
+and pinned predecessor heads are in TASK.json. This is not a code-diff-wide
+search or a global novelty certificate.
+
+No physical route between these charts is constructed. Abstract admissibility
+does not establish legal native-frame reads or equivalence to a stage in the
+nine-call controller. In particular, the result does not rule out coherent
+conjugate transport, explicit inverse codes, different chart sequences,
+port-dependent repairs, changed endpoints or PR1969's joint-query interface.
+The inherited all-size, native/compiler, hosting, precision/tape and analytic
+obligations remain open. No full `make frontier1094-verify`, native recipe
+replay, controller admission or proof-assistant verification is claimed.
+
+## Open next questions
+
+Can an actual paid stage transport replace this raw swapped-chart product
+with a restoring source transition? First derive one transport U from the
+controller and check the full-source map G_T^-1 U G_A, not only its query
+compression. Then price all reads, frame moves, inverse work and endpoint
+restoration. The missing typed native transport and complete invoice block
+any supplier conclusion. The structured question is in TASK.json.
+
+## Verification design and credit
+
+Power-of-ten inspection found no recursion in the new checker. Every algebra
+loop is bounded by the pinned 112-port/8-query dimensions. The source loop is
+capped at 30,000 events, files at 8 MiB, and archive member count at 2,000.
+Functions are below 60 physical lines; failures propagate as exceptions with
+no writes on ordinary replay. Python's rational/dictionary allocation and
+unittest's bounded method dispatch are deviations from rules 3 and 9. Pinned
+source and matrix sizes bound live state. Tests own and discard their temporary
+directories, and subprocesses have 30-second timeouts. If these scripts become
+runtime components, replace them with fixed-capacity storage and direct calls.
+`-Wall`, bytecode compilation and standard/optimized replays check the
+interpreted code; no static type analysis is claimed.
+
+Credit Ali Romman for PR1862's source-code construction and its explicit
+changing-chart warning, inspected at
+`bb29ea73ad9be9a0c3d53cdcb1d22b2ee9ffe42f`. Credit David Arias for PR1973's
+fixed-code winding and universal F-B-F obstruction, inspected at
+`165f08a694b26b0f4c521ce5e1132ede0b7362a2`. Those are valid predecessor
+results, not claims introduced here.
+
+The selected local source and assembly retain Jeremy Kothe's PR1042 and
+mutable-leaver work, Rohan Arun's PR1094 and k8/slot work, Hao Yan's controller,
+Chafik Boukhalfa's frame work, icekylinx and the earlier gauge44/split-centres
+contributors. Original notices remain in the unchanged archive and source
+packages. No predecessor executable or text is vendored. Nick Petalas.
+New work is Apache-2.0 under the repository license.

research/source-code-chart-order/RESULT.json

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+{
+  "status": "PASS_SOURCE_CODE_CHART_ORDER",
+  "query_sha256": "70055070e6f0465dcb9f99773769809f032043e7a1cba4f0c0fe5e5c365f0213",
+  "ports": 112,
+  "query_rank": 8,
+  "A": [
+    0,
+    7,
+    20,
+    27,
+    36,
+    46,
+    57,
+    111
+  ],
+  "T": [
+    21,
+    37,
+    47,
+    76,
+    81,
+    87,
+    92,
+    110
+  ],
+  "individual_chart_order": 6,
+  "pair_fixed_dimension": 88,
+  "pair_active_dimension": 24,
+  "pair_action": [
+    [
+      "1",
+      "0",
+      "12"
+    ],
+    [
+      "-156",
+      "-11",
+      "-2016"
+    ],
+    [
+      "14",
+      "1",
+      "181"
+    ]
+  ],
+  "pair_characteristic_polynomial_low_to_high": [
+    -1,
+    27,
+    -171,
+    1
+  ],
+  "full_characteristic_polynomial": "(t-1)^88 * (t^3-171*t^2+27*t-1)^8",
+  "pair_trace": 1456,
+  "pair_order": "infinite over Q and Q(i)",
+  "real_eigenvalue_interval": [
+    170,
+    171
+  ],
+  "polynomial_at_endpoints": [
+    -24311,
+    4616
+  ],
+  "column_oracle_cases": 336,
+  "semantic_failure_controls": 4,
+  "same_chart_pair_order": 3,
+  "scope": "Source-only port algebra, not native chart transport or a complete supplier.",
+  "selected_frontier_changed": false
+}

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