Research: transposed JP8 work saves 32,810 backward primitives (no new kappa)

research/jp8-transposed-coefficient-work/CLAIM.md

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+# Local sprint claim: transposed private work
+
+- Track: complex supplier; fresh OS-random seed **25** (`secrets.randbelow(100)`).
+- Exact base: `c290b7b919790300e14004a75d8d24b55ca3c5b5`.
+- Current accepted result: #741/#589, conditional kappa
+  `1661202643272239/1000000000000000000`; read its certificate, construction
+  and maintainer review. This experiment deliberately uses historical #626.
+- Exact object: the inverse-transpose companion of the fixed-source JP8
+  cleanup blocks constructed in #811, including varying magnitudes and
+  positive numerator-changing source reads.
+- Method: transpose each literal private-work circuit in reverse order,
+  reverse the sign of updates into the original source, and check all dirty
+  native/private inputs, complete prefix envelopes and all costs. Compare
+  with #751's equal-coefficient reverse-Horner result. No larger search.
+- Deliverable: standalone pinned companion construction, finite exact
+  certificate, independent block check, negative controls and cost note.
+- Expected finish: 06:45 UTC on 2026-10-13; handoff before 06:54 UTC.
+
+Three candidates: (0) CK numerator-changing forward reuse; (1) JP8
+inverse-transpose continuation; (2) transport coefficient reuse into #741.
+Seed `25 % 3 = 1` selects the second. Candidate 0 repeats the recent forward
+mechanism; candidate 2 requires a wider interface audit. This claim concerns
+the backward dual, not another forward coefficient search.
+
+Checked open metadata through #811, #751 at
+`7a1a3a319da359ec8b6e83230a7fcf7733fac8e5`, #739, #793 and the review ledger.
+#751 handles 999 equal-magnitude fan-in runs; #739/#811 report no backward
+same-source join. This extends the backward circuit class to the transposes
+of their varying-magnitude continuations. #793 changes compensation order.
+There is no pinned sprint issue; the issue list only contains #650 about
+Discussions. This file is the local coordination fallback. #811 will be
+pinned at its completed historical-assembly update, with full attribution.
+
+Contributor: Ryan Haines (@Ryan-Haines), with substantial OpenAI Codex
+assistance. Apache-2.0. Result and handoff will be recorded once after checks.
+
+## Result and handoff
+
+Completed: [PR #830](https://github.com/CrocSwap/integer-mult-bounds/pull/830).
+All 1,091 backward companions pass; complete backward count is 224,470,
+saving 32,810 real primitives and 13,122 beyond #751's equal-coefficient
+result. Paired counts are 224,422/224,470. Exact complete prefixes remain
+64/101 bits, while 377 isolated prefixes grow (worst ratio 40/9), explicitly
+recorded as a limit on blanket transposition arguments. No new kappa.
+
+Literal replay covers 725 arbitrary columns in both orientations; the
+independent integer checker covers 6,776 block input columns and the full
+backward prefix recurrence. Windows/Linux, 10+4+25 mutation controls, five
+optimized/relocated/recompressed packaging tests, Ruff, Git-byte source pins
+and current selected-record integrity pass. README.md gives the commands.
+
+Final overlap refresh: 638 open PRs through #825 at 06:33 UTC; no new
+transposed-coefficient/fan-in claim was present. #816 is a CK target forest;
+#821 audits a global prefix estimate, a different object. #811 and #751
+remain at the pinned heads. No pinned coordination issue exists. Main
+advanced to `7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261`; all six consumed
+input files are unchanged, and the selected #741 frontier remains separate.
+
+Initial research commit: `0a5e2034c0d537c05a2cc3bb45f5d35ae0e3a668`.
+This is the single final claim update. The local workspace also retains
+`jp8-transposed-s25.patch` and `jp8-transposed-HANDOFF.txt`, with the final
+commit, patch hash and reproduction command, as a submission-independent
+handoff. No issue was opened merely to reserve this scope.

research/jp8-transposed-coefficient-work/EXPECTED.json

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+{
+  "schema": "jp8-transposed-private-work-replay/1",
+  "base_commit": "c290b7b919790300e14004a75d8d24b55ca3c5b5",
+  "negative_controls": [
+    "initial_clear",
+    "final_clear",
+    "missed_inverse_sign",
+    "omitted_target_write",
+    "forward_instead_of_reverse_order",
+    "inverted_scale",
+    "endpoint_neutral_excess_growth",
+    "frame_crossing",
+    "wrong_macro_dual",
+    "forward_source_write"
+  ],
+  "orientations": {
+    "forward": {
+      "columns": 725,
+      "real_primitives": 224422,
+      "prefix_l1": "4536094588619580433782972610869617/281474976710656",
+      "prefix_bits": 64,
+      "non_scalar_sha256": "1fcf711555d365968ff8b64172090960cf98559865bd858c521c2adc7b12f792",
+      "paid_profile": {
+        "1": 2512,
+        "2": 400,
+        "3": 448,
+        "4": 160,
+        "5": 70,
+        "6": 8,
+        "8": 8
+      },
+      "original_real_primitives": 257232,
+      "saved_real_primitives": 32810
+    },
+    "backward": {
+      "columns": 725,
+      "real_primitives": 224470,
+      "prefix_l1": "16753412757380464429924108070201913461/8388608",
+      "prefix_bits": 101,
+      "non_scalar_sha256": "299442a1d68206eb7f49f0c9789cc8fa410a0615cff9c1900c54b064e901aa21",
+      "paid_profile": {
+        "1": 2512,
+        "2": 400,
+        "3": 448,
+        "4": 160,
+        "5": 70,
+        "6": 8,
+        "8": 8
+      },
+      "original_real_primitives": 257280,
+      "saved_real_primitives": 32810
+    }
+  },
+  "certificate_sha256": "8b4a9f4502acc772ebb18e1fd84fba9cdaf479b9b2759a5cd921515bc1f3a171",
+  "blocks": 1091,
+  "block_input_columns": 6776,
+  "unintegrated_budget": {
+    "adapter_allowances": 437040,
+    "compiler_charge": 25934336,
+    "original_local_bound": 11334342462720,
+    "paired_candidate_local_bound": 11334342429910,
+    "backward_only_local_bound": 11334342462672,
+    "new_kappa_claimed": false
+  },
+  "comparators": {
+    "equal_magnitude": {
+      "blocks": 999,
+      "backward_saved": 19688,
+      "backward_primitives": 237592
+    },
+    "power_only": {
+      "blocks": 1015,
+      "backward_saved": 30026,
+      "backward_primitives": 227254
+    }
+  },
+  "prefix_scope": {
+    "isolated_growth_blocks": 377,
+    "worst_start": 24875,
+    "worst_ratio": "40/9",
+    "old_isolated_prefix": "9/2",
+    "new_isolated_prefix": "20"
+  },
+  "prefix_nonmonotonicity_witness": {
+    "sources": 4,
+    "coefficient": "-1/4",
+    "old_prefix": "2",
+    "transposed_prefix": "4"
+  }
+}

research/jp8-transposed-coefficient-work/INDEPENDENT.json

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+{
+  "method": "independent integer block matrices and dyadic full-prefix recurrence",
+  "certificate_sha256": "8b4a9f4502acc772ebb18e1fd84fba9cdaf479b9b2759a5cd921515bc1f3a171",
+  "blocks": 1091,
+  "block_input_columns": 6776,
+  "real_saved": 32810,
+  "isolated_prefix_growth_blocks": 377,
+  "complete_backward": {
+    "real_primitives": 224470,
+    "prefix_l1": "16753412757380464429924108070201913461/8388608"
+  },
+  "negative_controls": [
+    "initial_clear",
+    "final_clear",
+    "inverse_sign",
+    "neutral_growth"
+  ]
+}

research/jp8-transposed-coefficient-work/NATIVE.json

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+{
+  "status": "PASS_INDEPENDENT_LOCAL_SCALAR_FRAME_AND_WRAPPER_REPLAY",
+  "v": 112,
+  "helpers": 492,
+  "local": {
+    "forward": {
+      "all_exact_columns": 716,
+      "prefix_l1_upper": "4536094588619580433782972610869617/281474976710656",
+      "prefix_bits_upper": 64,
+      "charged_payload_bits": 192,
+      "private_work_erased": true
+    },
+    "backward": {
+      "all_exact_columns": 716,
+      "prefix_l1_upper": "16753412757380464429924108070201913461/8388608",
+      "prefix_bits_upper": 101,
+      "charged_payload_bits": 192,
+      "private_work_erased": true
+    }
+  },
+  "paid_profiles": {
+    "forward": {
+      "1": 2512,
+      "4": 160,
+      "8": 8,
+      "3": 448,
+      "2": 400,
+      "5": 70,
+      "6": 8
+    },
+    "backward": {
+      "1": 2512,
+      "4": 160,
+      "8": 8,
+      "3": 448,
+      "2": 400,
+      "5": 70,
+      "6": 8
+    }
+  },
+  "entering_helpers": 204,
+  "cold_hyperplane_ended_helpers": 22,
+  "negative_controls_rejected": [
+    "wrong_scale",
+    "missing_center",
+    "wrong_common_frame",
+    "wrong_child_rank",
+    "missing_pre_wrapper",
+    "missing_post_wrapper",
+    "wrapper_with_wrong_frame",
+    "wrapper_in_wrong_phase",
+    "wrapper_on_a_data_register",
+    "wrapper_on_a_target_register",
+    "wrapper_on_a_centre_register",
+    "hpe_pre_wrapper_resolved_to_full_recipe",
+    "hpe_post_wrapper_without_terminal_frame",
+    "hpe_post_wrapper_wrong_terminal_frame",
+    "hpe_tail_gate_at_full_frame",
+    "hpe_tail_climb_dropped",
+    "hpe_helper_exit_adapter_to_full",
+    "hpe_tail_gate_moved_into_full_cleanup",
+    "cold_hpe_pre_wrapper_replaced_by_identity_entry_adapter",
+    "cold_hpe_pre_wrapper_dropped",
+    "cold_hpe_post_wrapper_replaced_by_exit_adapter",
+    "cold_hpe_pre_wrapper_resolved_to_cold_full_convention",
+    "cold_hpe_wrapper_with_nonzero_entrance",
+    "cold_hpe_post_wrapper_wrong_terminal_frame",
+    "cold_hpe_premix_gate_before_pre_wrapper"
+  ],
+  "scope": "Literal forward and inverse-transpose replay on every arbitrary data/helper column, physical frame incidence, ordered paid wrappers, cleared private work and prefix coefficients. Global hosting and outer exponent are separate."
+}

research/jp8-transposed-coefficient-work/PROOF.md

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+# Transposing a cleared work stream
+
+## Endpoint identity
+
+Take a fixed-source block `y_j += q_j x`, with `x` distinct from every
+target. Targets may repeat. Its live-register matrix is `F = I + B`, where
+
+```text
+B = sum_j q_j e_(y_j) e_x^T.
+```
+
+Since no `y_j` is `x`, `B^2=0`; hence `F^(-T)=I-B^T`.
+The backward block is therefore `x -= sum_j q_j y_j`. Its source registers
+are unchanged, so the target updates commute within this block. No order
+change across block boundaries is used.
+
+The forward private-work program begins and ends with `clear(w)`. The
+pinned constructor proves its live map is `F` and its final work is zero
+for arbitrary incoming work. On the enlarged live-plus-work vector its
+matrix is consequently `P = diag(F,0)`.
+
+If the original primitive execution is `E_1,...,E_m`, then
+`P = E_m ... E_1`. Transposing gives `P^T = E_1^T ... E_m^T`, implemented
+by executing the transposed primitives in **reverse order**. Clear is a
+self-transpose coordinate projection. Private doubling and halving are
+self-transpose diagonal maps. A unit transvection transposes by swapping
+source and destination, retaining its sign.
+
+Let `D` change only the live coordinate `x` to `-x`. Then
+`D P^T D = diag(I-B^T,0)`, since `D B^T D = -B^T`.
+The original program never writes `x`, so its transpose never reads `x`
+as the source of an off-diagonal update. The conjugation is implemented
+without additional negation gates: simply reverse the sign of every
+transposed update **into `x`**. Other transposed update signs stay unchanged.
+
+Thus the concrete compiler is:
+
+1. Reverse the complete primitive list, including both clears.
+2. Keep private clears/doublings/halvings as written.
+3. Swap source and target of each signed unit update.
+4. Reverse that update's sign exactly when its new target is `x`.
+
+This realizes the inverse-transpose live block while killing arbitrary
+incoming work and returning it to zero. Each old primitive contributes one
+new primitive, so the count is identical, including all source reads and
+both clears. This algebraic argument is local to the stated register model;
+it does not establish the inherited all-size multiplication theorem.
+
+## Interfaces and complete costs
+
+All affected original events are consecutive scalar additions in the
+historical #626 JP8 `full_frame_cleanup`, at frame 1. The candidate generator
+checks each backward macro against the exact signed scalar dual of its
+forward counterpart. No source is written inside a forward block; no source
+is written inside the backward fan-in except its common target. No child,
+adapter, wrapper, frame change or private-center event lies inside a block.
+All endpoints used by the block have native terminal frame 1.
+
+The shared private stream is cleared before the next external event. Source
+and target registers agree at block boundaries for arbitrary dirty inputs.
+Non-scalar event streams, helper counts and recursive profiles stay exactly
+as pinned. The new schedule needs no extra register or restoration pass.
+
+The construction maps all 1,091 forward blocks, not a selected successful
+subset. Their total saving is 32,810 real-component primitives in each
+orientation. The backward equal-magnitude and power-only comparators save
+19,688 and 30,026 respectively. Including every unchanged scalar primitive
+gives 224,422 forward and 224,470 backward. The original adapter/compiler
+allowances are retained in the complete unintegrated budget in EXPECTED.json.
+No stronger kappa is inferred from this finite constant saving.
+
+## Prefix growth is a separate obligation
+
+Transposition preserves the endpoint and count; it does not in general
+preserve the positive prefix envelope. For a simple counterexample, let
+four independent source registers each have envelope 1 and the target have
+envelope 1. The original backward block performs four additions with
+coefficient `-1/4`. Separate preparation has envelope at most 1 and the
+target ends at envelope 2, so the maximum is 2. The transposed shared circuit
+first accumulates four sources in `w`, reaching envelope 4, then halves twice,
+subtracts into the target and clears `w`. Its endpoint is the same but its
+maximum is 4. The verifier executes this witness explicitly.
+
+The concrete corpus contains 377 such isolated growth cases. At event span
+`[24875,24886)`, common target 336 receives eleven contributions with
+magnitudes `1/4,1/4,1/2,1/8,1/4,1/8,1/4,1/2,1/2,1/4,1/2`.
+The original isolated maximum is `9/2`; the companion's maximum is `20`.
+The certificate records the literal signed circuit and both exact values.
+This is the largest isolated ratio in the corpus, `40/9`.
+
+For the complete program, propagate positive envelopes independently of
+signed matrix cancellations: additions and subtractions both add source
+envelopes; scaling uses the absolute multiplier; clear sets zero. The
+Fraction-based literal replay and the independent integer/dyadic recurrence
+agree on the exact backward value
+
+```text
+16753412757380464429924108070201913461 / 8388608.
+```
+
+It is precisely the original 101-bit bound, below `2^192`. The paired
+forward construction retains its original 64-bit bound. The finite full
+replay, rather than an invalid blanket prefix-invariance rule, discharges
+this obligation for these pinned programs.
+
+## Independent finite checks
+
+The second checker consumes literal backward circuits and original macros,
+without importing the transpose compiler or the forward constructor. Each
+block is executed on independent integer columns at scale `2^d`, where `d`
+is at least its total number of halving instructions and every reference
+coefficient denominator exponent. All intermediate columns are therefore
+integral at that scale; each division is checked for exactness. This checks
+6,776 block input columns, including dirty private work.
+
+Its complete-program prefix checker represents positive dyadics as reduced
+integer/exponent pairs and expands the original macros separately. It checks
+literal costs and the exact full backward prefix, with no Fraction arithmetic.
+Mutation controls include missing clears, inverse-sign errors and circuits
+with the same endpoint but excessive intermediate growth. These are scoped
+finite checks, not a formalization of the general multiplication theorem.