Bind one selected Toom candidate to its dirty-input interface

research/selected-toom3-typed-block/PROOF.md

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+# One literal selected block does not have the same-width Toom-only interface
+
+## Source and exact boundary
+
+Fix canonical main `20ecf2891d80b5acf18d980b2164fb44ea7faa90` and the
+selected #1094 archive pinned in `SOURCES.json`. The question is the next
+experiment explicitly left open by Gonzalo Mier's #3974, head
+`23bb4de61d629e3f103a65baa7882b34042ab10f`: bind one of its nine-climb windows
+to ordinary degree-two Toom coefficient inputs. This is one window, not a
+reclassification of all 254 reported windows.
+
+Use the first forward phase-A group under the earliest-deadline rule adapted
+from Josie Elliston's #1817. Its nine `L` registers, in order, are
+
+`546, 560, 547, 568, 548, 558, 549, 570, 464`.
+
+The first eight climbs are `(old,new,rank)=(0,4,1)`, where frame 4 is `[459]`;
+the last is `(0,5,1)`, where frame 5 is `[75]`. Frame zero is empty. The original
+zero-based event indices are `9297,9299,9301,9304,9313,9315,9317,9322,9330`.
+Their legal windows meet at skeleton gap 1190; the last event's window starts
+at 42, the others at zero. All nine destinations are distinct. Moving the
+climbs to that gap changes no event on their own registers and crosses no
+phase boundary. This is the inherited local movement model, not a new global
+hosting proof.
+
+Let `W=Q(i)^512`, the nine-bit index space of one physical payload. Write
+`Z_j` for the independent unframed coordinates immediately after the entry
+adapters, so physical register `j` is `T_start(j) Z_j`. These are **not** raw
+initial values: for example register 512 has a residual pre-wrapper. Entry
+adapters are invertible; frame coordinates do not impose a zero or clean-data
+promise on a productive helper. The selected nine helpers start at frame zero,
+have ordinary identity entry adapters, and end at full frame.
+
+The pinned `INDEPENDENT-LOCAL-REPLAY.json` explicitly declares replay on
+"every arbitrary data/helper column", with all 704 input columns and 480
+helpers; this is the selected local replacement contract. The verifier binds
+that receipt and independently checks the entire entry prefix consists of
+7,344 shears, 56 nonzero scales, 464 typed frame adapters and 240 invertible
+monomial residual pre-wrappers. There are no entry clears or copies. The first
+1,190 phase-A events before this group are also invertible scalar operations.
+Thus the full pre-cut map is invertible on all productive columns, and the
+nine payload projection is surjective. This establishes independence from
+the actual prefix and its arbitrary-dirty contract, not from an assumed
+clean-helper initialization. A globally restricted clean-input caller would
+be a different contract; it is not excluded by the dimension argument.
+
+For the additional input-form receipt, the verifier checks that this first
+phase-A prefix contains no climbs or adapters, and every scalar source and
+target has the literal common frame recorded by the trace. Thus propagating
+plain rational rows in the unframed coordinates is valid. The independent
+surjectivity argument above already suffices for the rank obstruction. The
+rational replay gives the actual incoming forms at the chosen gap:
+
+`Z_546,Z_560,Z_547,Z_568,Z_548,Z_558,Z_549,Z_570,Z_464+Z_512`.
+
+Their coefficient minor on the nine named variables is the identity. Thus
+these are nine freely variable W-valued payloads, even when all other entrance
+coordinates are fixed. None can be discarded because a helper was called
+"cold": that word describes its entrance frame, not a zero value.
+
+At this boundary the demanded map is the **linear** operator
+
+`F = diag(T_[459],T_[459],T_[459],T_[459],T_[459],T_[459],T_[459],T_[459],T_[75])`.
+
+It must overwrite all nine named destinations, preserve every other productive
+payload, and return any additional scratch in its required initial state. The
+selected event is a frame transform, not an independently named bilinear
+product of two coefficient variables.
+
+## Exact operator reconstruction
+
+For each of the two transitions the pinned serialized recipe says
+`T = L (C_[1] tensor I_256) R`, with R applied first. L and R are monomial
+maps with explicit binary permutation, translation, quadratic and Z/4 phases.
+The one-axis child is exactly
+
+`C = ((1+i) I + (1-i) X)/2`.
+
+The verifier decodes the recipe directly on every basis vector, using Gaussian
+integer numerators and a single factor 2. Independently, it evaluates the
+retained canonical-frame constructor on all 512 basis vectors for each frame.
+All 1,024 columns agree, including global phase. A separate sparse Gram
+calculation gives `T* T = I` exactly. Both operators have rank 512, and F has
+rank **4,608**. For example
+
+`T_[459] e_0 = ((1+i)e_0 - (1+i)e_459)/2`,
+
+`T_[75] e_0 = ((1+i)e_0 - (1+i)e_75)/2`.
+
+These explicit outputs catch replacing an omitted dirty input by zero.
+The implemented checks also reject altered global phase and translation.
+The retained constructor is predecessor code, not claimed newly independent;
+the direct recipe decoder and Gram certificate are separate implementations.
+
+## Declared Toom-only substitution contract
+
+The rejected contract asks for the formal exact local operator on `W^9`,
+not an encoding of a bounded finite set into wider-precision scalar words.
+The latter would require its own precision and decoding contract. The
+rejected contract has exactly these restrictions:
+
+1. Each polynomial has three coefficients in a vector space of dimension 512
+   over `Q(i)`. Both triples are produced by affine maps of the nine incoming
+   payloads. They may have arbitrary fixed coefficients and constants.
+2. The usual five Toom evaluation products, interpolation and output decoding
+   receive data **only through these six coefficient blocks**. The coefficient
+   multiplication may be any bilinear law; it need not be commutative.
+3. The nine final payloads are functions of the coefficient blocks and fixed
+   constants only. Productive inputs, other registers and scratch may not
+   supply additional data to that output decoder.
+
+Restriction 3 is material. Retaining or bypassing original payloads changes
+the contract. Five recursive calls in a general circuit do **not** imply this
+restriction. No lower bound on arbitrary five-child circuits is claimed.
+
+**Proposition.** This contract cannot realize the named selected block.
+
+**Proof.** Its affine encoding `E:W^9 -> W^6` has a nonzero linear kernel of
+dimension at least `9*512-6*512=1536`. For any nonzero kernel vector h,
+`E(x+h)=E(x)`, so every downstream output is the same at x+h and x. But the
+selected F is invertible, hence `F(x+h)-F(x)=F(h)` is nonzero. This contradicts
+exact equality on arbitrary dirty inputs. The argument permits nonlinear
+interpolation or decoding as long as it receives no extra input data. If one
+polynomial is fixed, only three coefficient blocks depend on x and the kernel
+has dimension at least **3,072**. Equivalently the possible linear output ranks
+are at most **3,072** and **1,536**, respectively, against 4,608. QED.
+
+**Sharper ordinary-decoder corollary.** If the final decoder is affine linear
+and receives only the five product blocks, the possible output differences
+lie in the image of a linear map `W^5 -> W^9`, of dimension at most **2,560**.
+Thus the usual product-only Toom interpolation/output decoder has an output
+deficit of **2,048** dimensions. An additional affine side channel entering
+that decoder must have dimension at least 2,048. This corollary has the extra
+product-only/affine-decoder restrictions; the more generous six-coefficient
+contract above permits direct coefficient reads and still fails.
+
+If both operands are homogeneous linear encodings, an additional elementary
+control applies: the Toom output is homogeneous quadratic, so its value at
+2x is four times its value at x. The nonzero selected block doubles instead.
+Affine constants remove that degree objection, but do not remove the input
+kernel obstruction above.
+
+Noncommutativity is not the obstruction. Ordinary Toom convolution works over
+an associative algebra over the rationals when its indeterminate and
+interpolation scalars are central: each ordered term is `A_i B_j`, and no
+reversal is used. What is missing here is a typed coefficient embedding of
+nine independent payloads, not a scalar identity.
+
+## Larger rational-interface obstruction
+
+There is a separate extension beyond affine encoders: allow **rational
+encoding and rational decoding** over K=Q(i). This does not assume, or imply,
+that an arbitrary set-theoretic decoder is rational.
+
+Let x=(x_1,...,x_N) be algebraically independent over K, N=4608, and let
+E=(E_1,...,E_m) be rational functions in K(x). All fixed coefficients and fixed
+side information belong to K. If independent other dirty coordinates remain
+visible as parameters, adjoin them to the base field and work at the generic
+full cut; invertibility of the full prefix keeps the nine selected payloads
+algebraically independent over that field. No special zero-valued helper
+slice is required, since such a slice could annihilate a denominator.
+Suppose D is rational over K, every denominator
+of D remains nonzero after substitution E(x), and `D(E(x))=F x` as a generic
+rational identity, where F is the exact invertible selected operator. Then
+
+`K(x)=K(F x) subset K(E_1,...,E_m) subset K(x)`.
+
+Indeed F inverse recovers each x_j from the coordinates of F x. The fields
+are equal, but the middle field has transcendence degree at most m over K,
+whereas K(x) has transcendence degree N. Therefore **m >= 4608**. Equivalently,
+the chain rule on the generic domain gives `J_D(E) J_E=F`; the left side has
+rank at most m and the right side rank 4608. This differential proof uses
+characteristic zero and the same generic-denominator condition.
+
+The two coefficient triples contain m=3072 scalar rational functions. A fixed
+operand leaves m=1536. If the decoder receives only the five product blocks,
+m=2560, so the product-only deficit is **2048 even for a nonlinear rational
+decoder**. Every input-dependent bypass, retained register or side channel
+visible to the decoder must be counted in m. Adding d rational side-channel
+coordinates thus requires d at least 1536, 3072 or 2048 respectively; these are
+information dimensions, not operation/child/native-cost lower bounds.
+
+Ordinary Toom evaluation and interpolation are rational scalar operations.
+Any fixed finite-dimensional bilinear coefficient algebra, including a
+noncommutative one with structure constants in K, has polynomial multiplication
+in its scalar entries. Variable inversion is rational where the element is a
+unit and the corresponding denominator is nonzero. Thus this larger model
+covers rational coefficient preparation, those products and rational output
+recovery, while leaving nonrational operations, wider coefficient spaces,
+finite-set/precision encodings and changed output contracts open. A denominator
+vanishing identically on the encoded image does not define an available
+rational decoder there. Pointwise injectivity over the countable field K is
+not a replacement for the formal generic identity.
+
+A small verifier fixture demonstrates a legal nonlinear rational change of
+coordinates `(x/y,y/z,z)` with exact inverse `(uvw,vw,w)`, and rejects the
+lost-scale map `(x/y,y/z)` and the identically undefined decoder denominator
+`u-v` under encoding `(x,x)`. These fixtures check boundary failure modes;
+the written function-field/chain-rule argument proves the general theorem.
+
+## What can reopen the route
+
+Adding an affine side channel of scalar dimension d changes the encoding to
+`W^9 -> W^6 x Q(i)^d`. Injectivity then requires **d >= 1536**; with a fixed
+factor it requires **d >= 3072**. These are information dimensions, **not**
+native operation, child-count, storage, or time lower bounds. Already-retained
+productive data could supply such a channel; its exact use and legal costs
+must be specified. Wider coefficient spaces, nonrational data
+encoding, a different output contract, or a new coupled recursive primitive
+are also outside this proposition. A wider space must be priced at its actual
+recursive size, rather than silently counted as the same five children.
+
+The original block already pays nine rank-one frame children and its named
+recipes. There is no completed replacement here. Extra data acquisition,
+encoding, scratch, phase/frame conversion, dyadic precision and halving,
+interpolation's non-dyadic divisions, native hosting, decoding and restoration
+all remain unknown for a changed interface. Unknowns are not zero. In
+particular the predecessor's 19 additions and 13 rational scales do not price
+those obligations or establish legal native division by three.
+
+**Expansion decision:** pause the same-width Toom-only substitution on this
+actual group. Reopen on one explicit interface carrying the missing data,
+with all nine output identities and complete restoration, then price that
+module. Do not extrapolate the 254 event count into implemented savings. This
+result changes neither the selected paid profile nor either supplier cap,
+and establishes zero realized blocks and zero kappa improvement.
+
+## Constructive comparison: homogeneous lane packing
+
+The first eight selected maps do have an exact different tensor structure:
+
+`diag(T_[459],...,T_[459]) = I_8 tensor T_[459]`.
+
+With index `(lane,x)` encoded as `512*lane+x`, the same canonical frame `[459]`
+on twelve bits realizes this identity. The verifier compares its full Pauli
+tableau with the nine-bit tableau plus three idle bits, then checks its exact
+zero-state vector to fix the global phase. Its Fourier B-block rank is **one**,
+with payload dimension **4,096** and eight times the idle extent. This is a
+constructive operator equality, not an eight-to-one paid-call conversion.
+
+The grouping therefore has a precise alternative to Toom, but the rank-one
+child must now act on eight times the payload, or a different native operator
+must be supplied. No rule identifying that enlarged input with a unit-priced
+rank-eight child follows from this identity. #1779's diagonal-product screen
+and #1836's joint-shear screen also explicitly leave physical interfaces open;
+#1836 is pinned at `0a62e3f1c0ea493ddb83f25b91c5548c87978bcc` for this comparison.
+A paid enlarged-payload native call would be the next discriminating module.
+This comparison supplies its exact operator and width, without assigning
+unproved copy, layout, precision, restoration or supplier costs to zero.

research/selected-toom3-typed-block/README.md

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+# A typed boundary test for one selected Toom-3 candidate
+
+**One actual nine-climb group is an invertible 4,608-dimensional linear map.
+It cannot factor through the 3,072-dimensional input boundary of a same-width,
+Toom-only coefficient module. The obstruction extends to rational encoders
+and rational decoders by transcendence degree. No replacement or new kappa
+is claimed.**
+
+This answers the first source-binding experiment left by Gonzalo Mier's
+[PR #3974](https://github.com/CrocSwap/integer-mult-bounds/pull/3974), head
+`23bb4de61d629e3f103a65baa7882b34042ab10f`. It pairs fast bilinear algorithms
+with linear-algebraic information bounds and typed reversible interfaces.
+
+The first forward phase-A group contains registers
+`546,560,547,568,548,558,549,570,464`. Eight climb from the empty frame to
+`[459]`, one to `[75]`. The actual incoming forms have a nine-by-nine identity
+minor, so arbitrary dirty inputs cannot be identified with only three
+coefficients per operand. We reconstruct both 512-dimensional Clifford maps
+on **all 1,024 basis columns**, including phases and halving, and independently
+check their exact Gram matrices. The selected block rank is **4,608**.
+
+| Explicit coefficient interface | Capacity in scalar dimensions | Deficit |
+|---|---:|---:|
+| One variable coefficient triple, one fixed factor | 1,536 | 3,072 |
+| Two affine variable coefficient triples, decoder may read both | 3,072 | 1,536 |
+| Five product blocks, affine or rational output decoder | 2,560 | 2,048 |
+| Actual selected nine-payload boundary | 4,608 | 0 |
+
+The rational extension uses `K(x)=K(Fx) subset K(E(x))` and transcendence
+degree, or equivalently the generic Jacobian chain rule. Decoder denominators
+must remain nonzero after substitution and all input-dependent side data
+must be counted. The original affine-encoder result permits an arbitrary
+decoder; the rational-encoder extension requires rational decoding.
+
+The obstruction assumes outputs receive data only through the coefficient
+blocks. General circuits may retain original inputs or use bypasses; five
+child calls alone do not imply this factorization. The deficits are algebraic
+information dimensions, not native cost lower bounds. Wider coefficient
+spaces and changed interfaces remain open. Noncommutative Toom convolution
+itself remains valid with central interpolation scalars.
+
+[PROOF.md](PROOF.md) gives the exact contract, source event indices, input
+forms, proof, missing obligations and expansion decision. [RESULT.json](RESULT.json)
+is regenerated by the bounded verifier. **Zero source blocks are replaced;
+selected kappa remains 0.001672660197076918.**
+
+## Reproduce
+
+```
+python3 scripts/research_package.py --package research/selected-toom3-typed-block --check-only
+python3 scripts/research_package.py --package research/selected-toom3-typed-block
+```
+
+The two registered runs (normal and optimized Python) bind the selected
+archive, reconstruct the first eligible group, derive its input minor, check
+all basis columns and exact Gram matrices, compare the result, and exercise
+six failure controls, including a legal birational-coordinate fixture and
+rejected lost-scale/undefined-denominator boundaries. They do not replay the complete selected supplier.
+The canonical-frame evaluator is retained predecessor code; the recipe
+interpreter and sparse Gram check are new separate implementations. No
+independent human or formal-kernel acceptance is claimed.
+
+## Open next questions
+
+Can one explicit bypass or wider-coefficient construction preserve the nine
+independent dirty payloads while delivering five **legal** children, with
+complete frames, precision, output and scratch restoration costs? First write
+that interface and verify all nine outputs on basis inputs, then compare the
+complete paid profile against the selected supplier. The present Toom-only
+contract is paused; another larger event-window census cannot repair it.
+The checked alternative `I_8 tensor T_[459]` packs the first eight lanes into
+a 4,096-dimensional, twelve-bit payload, but its Fourier rank remains one.
+Its larger native call and full invoice remain open. Structured obligations
+and the stop rule appear in `TASK.json`.
+
+## Sources, overlap and credit
+
+Canonical base is `20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priority revision
+6. Parent entry overlap checked 679 open PR bodies and rendered active/handoff
+Discussions on 2026-10-19; broader closed/private/Discussion coverage remains
+partial. Target #3974 was reread at its exact head before this experiment.
+This package does not claim novelty from an empty keyword search.
+
+Credit Gonzalo Mier / #3974 for the Toom screen and explicit missing binding;
+Josie Elliston / #1817 for phase-local dependency windows; Jose Manuel
+(@jmbarrancoml, as credited in #1779, submitted via 1337isnot1337) for the
+diagonal batching screen; Arham Tahir / #1605 for rank-shape budgets; Discussion
+#1987 contributors for the Karatsuba antecedent; and the selected #1094,
+Clifford compiler, PR514/PR433 and Rohan Arun / Jeremy Kothe / Hao Yan source
+lineage. Predecessor attribution is preserved rather than treating archive
+retention as scientific acceptance.
+
+Prepared for Clear1nsight with substantial OpenAI Codex assistance in source
+inspection, proof, implementation and exposition, with separate same-team mathematical
+review of the rational-interface extension. Apache-2.0. Frozen sources
+are unchanged.