Compress H6 compensation with exact same-frame signed captures

research/h6-mixed-pair-capture/CONTEXT.json

@@ -0,0 +1,57 @@
+{
+  "schema_version": 1,
+  "canonical_base": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "pending_context": {
+    "pr": 4027,
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/4027",
+    "head": "1bcc4f8261d2dc322c98fa9e44c4ced43cd1cd95",
+    "author": "Ross Richardson (RossRichardson)",
+    "usage": "Conditional unchanged K profile and semantic bills; not imported or executed by the primary coefficient proof.",
+    "proof_path": "research/h6-u9-asymmetric-compensation/PROOF.md",
+    "proof_sections": "4-5",
+    "K_component_primitive_invoice_forward": 243600112,
+    "K_component_primitive_invoice_dual": 243602560,
+    "full_local_histogram_before": {
+      "1": 454625280,
+      "8": 4336,
+      "9": 32,
+      "15": 227312640,
+      "16": 59760,
+      "17": 1600
+    }
+  },
+  "claim": "https://github.com/CrocSwap/integer-mult-bounds/pull/4027#issuecomment-6265404853",
+  "predecessors": [
+    {
+      "name": "Dominik Scholz",
+      "source": "PR606 H6 geometry and factor"
+    },
+    {
+      "name": "DreamingOfClouds",
+      "source": "H6 model disclosure on discussion1167"
+    },
+    {
+      "name": "H6 compensation network contributors",
+      "source": "Frozen canonical bucket schema, arbitrary-dirt contract, native frame compiler and scalar tariff"
+    },
+    {
+      "name": "Ross Richardson",
+      "source": "PR4027 exact asymmetric U9 K factor, full bucket splice and outstanding physical obligations"
+    }
+  ],
+  "license": "Apache-2.0; no predecessor source is copied or modified in this package.",
+  "publication_overlap": {
+    "utc": "2026-10-19T11:20:43Z",
+    "canonical_main": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+    "open_pr_bodies_including_drafts": 744,
+    "open_issues": 1,
+    "recent_updated_discussions": 30,
+    "parent_pr_comments": "All current comments at refresh; no competing same-scope claim or objection found.",
+    "related_distinct_prs": [
+      3649,
+      3655,
+      4027
+    ],
+    "limitation": "Bounded discovery coverage; not a proof of novelty across every closed PR or old discussion."
+  }
+}

research/h6-mixed-pair-capture/PROOF.md

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+# Same-frame signed-column factor for H6 compensation
+
+## 1. Inputs and precise claim
+
+Use the doubled-coordinate H6 ports of the frozen PR606 lineage: two pure
+E8 projective squared-norm 16 shells of 1080 ports and mixed ports
+`s+=(u,v), s-=(u,-v)` for 120 projective squared-norm 8 root lines in each block.
+There are 30,960 ports. Define `tau=t·s/4`,
+`f(z)=(z^3-4z)/48`, `K_ts=f(tau)`, and `C=I-K`.
+
+The inherited binary-label isometry satisfies
+`p(t)·p(s)=1+tau (mod2)`. Every source has line `P_s=span(p(s))`,
+every target has cap `Q_t=p(t)^perp`, in dimension 17. Compatible means
+`p(t)·p(s)=0`, hence **odd tau**. Compatible C entries are nonzero;
+the result removes proportional columns, not zero columns.
+
+We prove a new indexed factor `DG=C` by changing only mixed-source columns
+within a fixed target-label bucket. It has 123,632,640 ordinary helpers,
+instead of 227,312,640. Under the complete K-word and frame/erasure contracts
+of `h6-compensation-network/contract/THEOREM.md`, it gives the same exact
+arbitrary-dirt data map with the helper path `[1,15,1]`. This is a scoped
+factor/construction proof conditional on those inherited contracts, not a
+formal or unconditional multiplication theorem.
+
+## 2. Same actual source frame
+
+In doubled E8, `v-(-v)=2v`, so v and -v have identical lattice coset mod2.
+Equivalently every root pairing signature is unchanged. The source's
+`binary_labels` selects an invertible mod-two root Gram basis and applies
+an injective affine isometry. Thus `p(s+)=p(s-)`, with identical packed
+labels, not merely isomorphic abstract lines.
+
+The source pure fibers have 135 labels of multiplicity 8 per block; the
+mixed fibers have 14,400 labels of multiplicity 2. Root, pure and zero cosets
+are disjoint and exhaust the 256 E8/2E8 classes. The primary enumeration
+uses all 120 pairing signatures. Independently, for doubled E8 vectors q,r,
+`(q-r)/2` belongs to doubled E8 iff the coordinate differences are all 0
+or all 2 modulo 4 and their sum is 0 modulo 8. Hence a direct coset key is
+`(min(q mod4, (q+2*ones) mod4), sum(q) mod8)`.
+The independent implementation confirms the exact same pure partition.
+
+The canonical native compiler uses `ce.cgate(H,p)` as its source
+representative. Equality of p therefore gives equality of actual source
+operators, including phase. At a target bucket all targets likewise have
+the same actual cap representative. Reusing a surviving helper's existing
+zero→line→cap→FULL recipes introduces no new native conversion family.
+This argument inherits the correctness of those recipes; the present
+replay does not recompile or independently verify all native operators.
+
+## 3. Complete two-column classification
+
+For target mixed pair `(a,b),(a,-b)`, put `A=u·a/4`, `B=v·b/4`.
+Each lies in `{-1,0,1,2}` for canonical root lines. Its C block is
+
+```
+delta - [[f(A+B), f(A-B)],
+         [f(A-B), f(A+B)]],
+```
+
+where delta is the identity only for `A=B=2`, and otherwise zero.
+The sixteen attainable ordered classes are recorded exactly in the receipt.
+
+* If B=0 and A odd, the columns are equal; take sigma=+1.
+* If A=0 and B odd, f is odd, so the columns are opposite; take sigma=-1.
+* If one pairing is 2 and the other odd, the block has rank 2 with determinant
+  of absolute value 3/32. Keep both original helpers.
+* All remaining blocks are zero, including the diagonal block after I-K
+  cancellation. No compatible nonzero entry is discarded.
+
+The rank 1 entries all have magnitude 1/16. For a pure target `(w,0)`,
+the two columns are equal; for `(0,w)` they are opposite. In a compatible
+pure target bucket, Cauchy gives `|u·w/4|<=sqrt(8)`, and odd integrality
+then gives magnitude 1 pairing, hence every C entry has magnitude 1/16.
+The whole eight-port target bucket has the same pairing parity.
+
+For a nonzero rank 1 block write its columns as `[v,sigma*v]`. Replace
+`G=I_2, D=[v,sigma*v]` by `G=[1,sigma], D=v`. Exactly
+`D G=[v,sigma*v]`. This changes no other block; full-rank mixed blocks
+keep the original identity capture, and every pure-source helper is kept.
+Summing over disjoint source-label/target-label blocks proves **DG=C**
+for the complete indexed matrix.
+
+## 4. Arbitrary dirt, frames and both orientations
+
+Allocate an independent ordinary helper for each surviving factor row h.
+Let its arbitrary logical entrance be e_h; correlations with other inputs
+are allowed. Its G row uses only sources at one common line P_h, and its
+D column uses only targets at one common cap Q_h. Nonzero C compatibility
+gives `P_h subset Q_h`. Perform these complete phases:
+
+1. At ZERO, `y -= D e`.
+2. Helpers climb to their source line and capture `Gx` there. Both reads
+   of a signed pair are emitted and paid; helper values are `e+Gx`.
+3. Run the **complete unchanged** arbitrary-input K word on its disjoint
+   data and private registers. It maps y to `y+Kx`, returns logical x,
+   places sources at FULL and targets at their actual caps, and restores
+   its own auxiliary dirt under its paid contract. Its internal source
+   mutation is allowed. Do not move sources to FULL before this word.
+4. Helpers climb line→cap and add `D(e+Gx)` to target buckets.
+5. Helpers climb cap→FULL and subtract `Gx` at FULL.
+
+The helper ranks are 1,15,1, and both operands in every addition use
+the identical actual frame. Data paths and K children are unchanged.
+The forward output is
+
+`y-De+Kx+D(e+Gx)=y+(K+DG)x=y+x`,
+
+and each helper returns to e. This proves every independent dirt column
+cancels. Its **physical** endpoint is FULL applied to e, not its physical
+entrance e. Any wrapper restoring literal physical entrance, copied-total
+erasure or recursive reserve remains a separately paid inherited obligation.
+
+Dualize every scalar addition in the same order, replacing
+`a+=c*b` by `b-=c*a`, keeping the frame history. Initially the prefix
+gives `h=e+D^T y`, capture subtracts `G^T(e+D^T y)` from x, K subtracts
+`K^T y`, scatter removes `D^T y` from h, and cleanup adds `G^T e` to x.
+Thus `x -> x-(K+DG)^T y=x-y`, y is unchanged, and helpers return to e.
+The complete K interface, including arbitrary sources, targets, dirt and
+private erasure, is essential in both orientations.
+
+`word.py` checks all source/target/helper basis columns in one seed 83 fixture
+using exact sparse rationals and source-line/cap chronology. K is an
+explicit exact macro contract there, not an implemented native K tape.
+The full phasewise proof above is separate from that finite replay.
+
+## 5. Exhaustive reduced census and complete scalar invoice
+
+For every root line, among 120 root lines exactly 63 are orthogonal,
+56 have odd pairing, and one is the same line. Every root has 72 compatible
+pure target-label buckets. Each pure source has 64 compatible root lines,
+64 pure-label buckets, 448 pure-port pairings of magnitude 1 and 64 of
+magnitude 3. The two enumerations check the complete tables, including every
+one of the 1,166,400 pure/pure entries. Products of these checked degrees,
+with section 3's complete class split, give full-network counts.
+
+For each mixed source label:
+
+| Target caps | Number | Old helpers/cap | New helpers/cap |
+| --- | ---: | ---: | ---: |
+| Mixed rank 1 | 2*63*56=7056 | 2 | 1 |
+| Mixed rank 2 | 2*56=112 | 2 | 2 |
+| Pure rank 1 | 2*72=144 | 2 | 1 |
+
+Old helpers per mixed label are 14,624; new are 7,424. Thus removed helpers
+are `M=14400*7200=103680000`. Pure-source helpers remain
+`270*8*(120*64+64)=16727040`. Total helper counts are
+`227312640 -> 123632640`.
+
+Each removed mixed-target helper removes 2 D entries, and each removed pure
+target helper removes 8. Therefore removed D entries are
+`14400*(7056*2+144*8)=219801600`, all of magnitude 1/16.
+`D_nnz:480337920 -> 260536320`; the 3,363,840 entries of magnitude 5/16
+are retained. Crucially **G_nnz stays 227312640**: two reads feed one helper.
+There is no halving of the G term or a free source-access operation.
+
+The pinned signed Horner tariff is 4 for ±1, 8 for ±1/16, 11 for ±5/16.
+Each D entry occurs in prefix and scatter; each G entry in capture and
+cleanup. Hence the complete component primitive allowance is
+
+`16*D_nnz + 6*D_5 + 8*G_nnz`.
+
+It falls from 9,524,090,880 to 6,007,265,280, saving 3,516,825,600.
+This includes every changed capture, cleanup, prefix and scatter; helper
+frame children are counted separately. Native coordinate passes, actual
+address evaluation, tape movement and physical dispatch are not part of
+this semantic scalar tariff. These are prescribed upper invoices, not
+minimum gate counts or measured speedups.
+
+## 6. Conditional K and full-host deltas
+
+PR4027 at the exact head in CONTEXT.json supplies the unchanged asymmetric
+K histogram and scalar bills. Conditionally reusing those inputs, the new
+full local histogram is
+
+`{1:247265280,8:4336,9:32,15:123632640,16:59760,17:1600}`.
+
+Rank mass falls by `17M=1762560000` from 3,865,333,216 to 2,102,773,216.
+The conditional combined scalar invoices are 6,250,865,392 forward and
+6,250,867,840 dual. No ordinary or private K child is otherwise changed.
+
+The certificate also applies the unchanged fractional full-row convention
+from the base pricing source: for L stages, `m=16L+1`, `q=floor(m/17)`
+and `pad=m mod17`. Removing M helpers removes formal stock `ML/q`, and
+mass/rank terms `(2ML,1)`, `(15ML,15)` and `(ML*pad/q,pad)` if pad>0.
+Their mass sums to `mML/q`, so the formal coverage deficit is unchanged.
+Stages 3,5,7 and all pad work are checked against the independently extracted
+source function; repeated ranks are combined for comparison. This is a
+conditional accounting identity, not a physical stock/reserve certificate.
+
+## 7. Range, coverage and open implementation obligations
+
+At the **forward logical capture cut**, `|e|<=E`, `|x_+|,|x_-|<=X`
+give `|e+x_+ +sigma*x_-|<=E+2X`, previously E+X. Captures introduce no
+new fractional denominator. One extra magnitude bit suffices relative to
+that single old cut bound; this says nothing by itself about full-network
+approximate-child precision, dual accumulation, or physical normalization.
+Those must be propagated through every inherited child and adapter.
+
+The exhaustive domain is the fixed H6 geometry, all 16 mixed pairing classes
+and its complete counts; seed 83 only selects a bounded all-column fixture.
+There is no probabilistic search, parameter sweep or optimality claim.
+Pure-source compression, native recipe regeneration, a complete materialized
+tape, copied-total erasure, physical restoration/placement, actual reserve,
+source-access complexity and the joint BIT/complex analytic admission remain
+open. No new κ follows from a smaller unnormalized rank or scalar invoice.
+
+Credit: Dominik Scholz, DreamingOfClouds, the H6 compensation/compiler
+contributors and Ross Richardson's PR4027. Prepared by Tekalign with
+substantial OpenAI Codex assistance; Apache-2.0. Frozen sources are unchanged.

research/h6-mixed-pair-capture/README.md

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+# Signed mixed-pair capture in the H6 compensation factor
+
+The two sources `(u,v)` and `(u,-v)` share one actual H6 source frame.
+Where their `C=I-K` columns are equal or opposite, one ordinary helper can
+capture their sum or difference. This gives an exact `DG=C` factor with
+**103,680,000 fewer helpers**, retaining both paid source reads. Full-rank
+mixed buckets and all pure-source helpers remain unchanged.
+
+| Complete compensation invoice | Before | After |
+| --- | ---: | ---: |
+| Ordinary helpers | 227,312,640 | 123,632,640 |
+| Nonzero scatter entries | 480,337,920 | 260,536,320 |
+| Nonzero capture entries | 227,312,640 | 227,312,640 |
+| Component primitive allowance | 9,524,090,880 | 6,007,265,280 |
+
+These are exact semantic counts and prescribed scalar upper allowances,
+not measured wall-clock/tape costs. The helper reduction is 45.61%; the
+compensation scalar allowance decreases 36.93%. Forward logical capture
+range grows from `E+X` to `E+2X`. No complete physical precision, allocator,
+restoration, all-size supplier or new multiplication exponent is claimed.
+
+The [proof](PROOF.md) specifies the complete factor, the rank-two exception,
+same-representative frame legality, arbitrary-dirt forward/dual chronology,
+counts and inherited limits. [CERTIFICATE.json](CERTIFICATE.json) records all
+16 attainable mixed pairing classes, full reduced-geometry tables by digest,
+complete invoices and an all-input-column seed 83 fixture. The entire
+30,960-square matrix and multi-billion-event network are not materialized.
+
+## Independent evidence
+
+`verify.py` enumerates doubled E8 directly as an integer lattice and groups
+pure fibers by pairing signatures. `independent.py` uses a separate pattern
+enumeration and direct lattice cosets, never importing the primary geometry,
+factor, invoice or lifecycle code. It extracts the coefficient, Horner tariff
+and helper padding convention from authenticated frozen source ASTs.
+The two implementations agree on all 120 roots, 1080 pure points, all 256
+cosets, 14,400 root/root, 129,600 root/pure and 1,166,400 pure/pure entries.
+This is independent exact finite evidence, not a formal verification of the
+full multiplication theorem or of the inherited native compiler.
+
+The complete-fiber fixture has 24 data ports and checks every 120 original
+or 110 compressed source/target/helper entrance columns, forward and dual.
+Its helpers decrease 72→62, scatter entries 216→160, and scalar allowance
+4056→3160; capture entries stay 72. The 25 failure controls include matched
+wrong capture/cleanup signs, full-rank compression, missing prefix/K/cleanup,
+premature source-FULL frames, wrong coefficients, free source reads, typed
+receipts and corrupted/removed source pins before execution.
+
+## Reproduce
+
+From the repository root with Python 3.12; no third-party dependencies:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/h6-mixed-pair-capture
+```
+
+Expected markers: `PASS_H6_MIXED_PAIR_CAPTURE`,
+`PASS_INDEPENDENT_H6_PAIR_CAPTURE`, `PASS_H6_CAPTURE_CONTROLS` and
+`PASS_REGISTERED_RESEARCH_PACKAGE`. The command is read-only. Standalone scripts also
+run from an exported package with its eight declared inputs, without Git.
+Receipt generation requires removing the old receipt explicitly and running
+`verify.py --write`; the generator refuses any existing destination.
+
+Exact canonical base: `20ecf2891d80b5acf18d980b2164fb44ea7faa90`.
+[SOURCES.json](SOURCES.json) pins every live input by Git blob and SHA256;
+the verifier seals the inventory itself. The selected record and task
+catalogue are historical provenance. [TASK.json](TASK.json) records OS seed 83,
+domain, stopping rule and the refined independent-verification card.
+
+## Conditional asymmetric K comparison
+
+Ross Richardson's [PR4027](https://github.com/CrocSwap/integer-mult-bounds/pull/4027)
+at `1bcc4f8261d2dc322c98fa9e44c4ced43cd1cd95` supplies the unchanged
+asymmetric K histogram and scalar bills in [CONTEXT.json](CONTEXT.json).
+Conditionally adding those bills gives 6,250,865,392 forward and 6,250,867,840
+dual component primitives, before real/imaginary doubling and native adapters.
+The complete local rank mass becomes 2,102,773,216
+from 3,865,333,216. The certificate also gives formal removed row-stock,
+rank mass and padding terms for stages 3,5,7 under the unchanged parent
+row convention. Role counts and capture range change too, so these local
+differences do not imply a joint moment or κ improvement. No pending K
+implementation is imported or independently admitted by this package.
+
+## Open next questions
+
+Can one compress an eight-column pure-source fiber with an affordable capture
+matrix? Those 16,727,040 helpers are unchanged; first factor one fiber exactly
+and price dense capture, scatter and range before expanding the search.
+
+Can the reduced factor pass full approximate-child and physical admission?
+First propagate error through the whole K splice, then bind source access,
+restoration, bank placement and pads to actual reserve and the joint BIT
+moment. The inherited copied-total/erasure and tape contracts remain blockers.
+The structured questions and first steps are in [TASK.json](TASK.json).
+
+## Credit and license
+
+Dominik Scholz's [PR606](https://github.com/CrocSwap/integer-mult-bounds/pull/606)
+supplies H6 geometry/factor; DreamingOfClouds supplies the H6 proposal.
+The frozen [compensation network](../h6-compensation-network/README.md),
+including PR1830's lineage, supplies the bucket, compiler and dirt contract.
+Ross Richardson's PR4027 supplies the asymmetric K work and explicitly
+identifies the remaining helper burden. Their native/compiler and upstream
+attributions remain with the original sources, which are unchanged.
+
+Prepared by Tekalign (`tekuuu`), with substantial OpenAI Codex assistance in
+the mathematics, implementations, proof, controls and packaging. Machine
+checks are not human peer review. New work is Apache-2.0.