Research: active p2 feedback and exact output coupling (no new κ)

research/coded-active-feedback/PROOF.md

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+# Active coefficient recovery and the remaining coupling
+
+Cinna Davis, with substantial OpenAI Codex assistance. Apache-2.0.
+
+This is a conditional component construction on the inherited physical ABI.
+It supplies neither four autonomous Gaussian jobs nor a profitable caller.
+
+## Source and conventions
+
+Use Dugongue's PR4948 backward involutive producer, Dillon's signed PR4933
+normalizers and the mixed p2 chronology retained in the passive saved-query
+package. There are four independent 112-owner main banks and two disjoint
+408-owner helper inventories. Original helpers are never scratch. Rightmost
+operators act first. On each main pair the inherited FULL is
+
+    B_s = diag(D_s,D_s) [[I,-Q C_s],[0,I]].
+
+Here Q is the central Gaussian, D1 and D3 are the two fresh eight-axis
+Gaussians, C1 and C3 are the actual normalized F-transpose codes, and
+
+    C_s^2=I, [C_s,D_s Q]=0, [C_s,D_t]=0 (s != t).
+
+Q commutes with D1,D3, which commute with each other. C_s need not commute
+with Q or its own D_s. Q^4=I, and P=Q^2 is the physical monomial Pauli.
+The source's all-width phase, chart and helper-autonomy results are premises;
+this package does not independently replay their native words.
+
+The original thin inputs are X at query0, C1 X at query2, and C3 X at
+query4. Their paid response is Q^-1 times that input; query0's response is
+subsequently multiplied by C1 before addition. Retain the two FULLs, paid
+IDLE, all bridges, signed swaps, scalar boundaries and all three queries.
+
+## The changed inputs
+
+Save h, the first FULL's Y output, before the subsequent IDLE and bridge.
+At query2 replace its argument C1 X by C1 X+h, obtaining response r2.
+At query4 replace its argument C3 X by C3 X+C1 Q^2 r2.
+Both additions use owned query copies before the destination is updated.
+No extra Gaussian query is introduced.
+
+Let x0,y0,x1,y1 denote the four original independent input banks, before
+the scalar right boundary. After query4, before the last scalar operations,
+the second pair's Y bank is exactly
+
+    D1 Q^-1 x0
+    + Q^-1 C1 Q (I-Q) D1 y0
+    + [(Q^-1 C1 Q C1-Q^-1)D1 + Q^-1 C3 D1 D3] x1.
+
+Its y1 coefficient is zero. `verify.py` checks this complete four-block
+row against chronological multiplication. No selected-column substitution
+is used in that equality.
+
+For a short independent derivation of the x0 coefficient, restrict the
+input to x0 only. After FULL1, X vanishes and h=Q^-1 C1 D1 x0.
+Query2's response is Q^-1 h. Bridge-minus sends this response to the
+second pair's Y input. FULL3 produces
+
+    X=-D3 Q C3 Q^-1 h,  Y=D3 Q^-1 h.
+
+The old part of query4's response is -Q^-1 D3 h, cancelling this Y.
+Its new part is Q^-1 C1 Q h=D1 Q^-1 x0. Thus the previously absent
+rank-nine coefficient has appeared through active use of two paid responses.
+The finite passive trace's missing-grade invariant does not apply to this
+changed trace. This observation is not independent-job credit.
+
+## The output is coupled at every positive width
+
+The y0 coefficient is an invertible left factor times (I-Q) times an
+invertible right factor. It is nonzero whenever Q != I. At every positive
+digit-column width, Q has an eigenvector of eigenvalue i (one column has
+odd central frequency and the other columns have even central frequency).
+Consequently an arbitrary y0 input changes this output. The old autonomous
+job has not been recovered.
+
+There is also an exact size for the contamination. At column count f, in
+the common 17-axis ambient address space, define
+
+    b0(f) = sum_{0<=k<=f, k=0 mod 4} binomial(f,k).
+
+Q has eigenvalue i^k on a central Walsh frequency of Hamming weight k.
+Hence, including the sixteen spectator axes and all 112 owners,
+
+    rank(Q^-1 C1 Q (I-Q)D1)
+        = 112 * 2^(16f) * (2^f-b0(f)).
+
+The fraction of the full owner-bank dimension is 1-b0(f)/2^f, tending
+to 3/4: the root-of-unity filter gives
+`b0(f)=(2^f+(1+i)^f+(1-i)^f)/4` for f>=1, and both extra terms have
+magnitude 2^(f/2). For example the rank fraction is 1/2,3/4,7/8 at
+f=1,2,3. This is ordinary
+complex linear rank, not a lower bound on recursive calls or free circuit
+cost. It does not price a correction or exclude a changed caller.
+The checker independently compares binomial sums, a four-residue recurrence
+through f=64, and central frequency enumeration through f=12.
+
+## Inventory and ordinary work
+
+The attempted complete caller still invokes exactly two source FULLs,
+one IDLE on two 112-owner banks, and three 112-query banks. Its recursive
+paid histogram is the inherited
+
+    {1:4754,2:824,3:1020,4:254,5:80,8:224}.
+
+All 816 original helper jobs remain autonomous once; no owned copy earns
+a productive target. The old proposed main rewards cannot be retained.
+
+For the explicit component, one extra owned 112-field bank can first hold h,
+then be reset and reused for r2. One extra eight-field mode workspace
+implements C1. The additional field-level upper ledger is:
+
+| Added operation | Count |
+| --- | ---: |
+| Two feedback additions | 224 scalar ADDs |
+| One inherited F-transpose recoding | 896 ADDs, 224 monomial passes, 16 mode CLEARs |
+| Q^2 on the saved response | 112 monomial passes |
+| Captures of h and r2 | 224 COPYs |
+| Initial, intermediate and final reset of the saved bank | 336 owned resets |
+| Extra owned capacity | 112 saved fields plus 8 modes |
+| Gaussian children | 0 |
+
+Query buffers already required by the three thins stay charged. Read h
+before changing its source; save r2 before releasing its query buffer.
+The native helpers and original inputs are never cleared. These are
+field-level counts under inherited owned-reset and monomial implementation
+contracts. A fixed-tape implementation, precise rounding budget, completion,
+compiler and outer bill are not supplied.
+
+## Bounded synthesis and its limits
+
+The search uses the semidirect-product algebra of the inherited gauge engine,
+adding C1^2=C3^2=I and Q^4=I. Its fresh D exponents are formal integers.
+In particular D^4=I and other additional physical source relations are not
+imposed. This is a sufficient algebra for a positive identity; a failure in
+it is not a faithful physical impossibility proof.
+
+For each feedback, allow it to be absent, or use a reduced C1/C3 word of
+length at most seven with coefficient in {-2,-1,-1/2,1/2,1,2}. There are
+91 choices per edge and 8,281 policies. The second word is followed on the
+right by Q^2. Keep both original thin-input coefficients equal to one.
+The passive left decoder may use the thirty words W and W Q^2, for reduced
+C1/C3 words W of length at most seven, and arbitrary rational scalar sums
+across all 39 recorded bank observations. This is a restricted whole-bank
+decoder grammar; arbitrary port-dependent controls are not included.
+
+After query4 only scalar operations occur. Thus the span of all observations
+equals that of the thirty preceding observations and response4, with the
+same thirty left controls. This justifies caching the pre-query basis.
+All 8,281 policies attain only the positive first-pair job in this formal
+span. `SEARCH.json.gz` retains every policy, exact rank and attainment mask;
+the full registered replay recomputes all of them.
+
+The smaller 49-policy pilot uses code-word length one, coefficients +/-1
+and decoder length three with the same Pauli placements. For all 196 target
+questions it checks either a constructive recipe or a rational separating
+functional directly against every declared generator. These checks do not
+trust the elimination pivots. Deleted recipes and false dual pairings reject.
+
+The finite span result excludes the specified recipes only in the declared
+formal algebra. Different feedback sources, earlier active changes, input
+weights, additional Pauli placements, port-dependent controls, native prefix
+observations and further physical relations remain outside it.

research/coded-active-feedback/README.md

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+# Active query feedback restores a missing coefficient, with high-rank coupling
+
+Two changed inputs to already-paid queries recover the first original input's
+missing rank-nine Gaussian coefficient in the PR4948 physical p2 caller.
+The complete output remains coupled to two other arbitrary input banks.
+**No autonomous new job, profitable complete caller or higher kappa is supplied.**
+
+Save the first FULL's Y output h. Add h to the second query's old input;
+then add C1 Q^2 times that query response to the final query's old input.
+All FULLs, the IDLE, three paid queries and 816 independent helper jobs stay
+charged. The new response path gives exactly D1 Q^-1 on the first original
+input. It escapes the unchanged-input premise of the passive grade result.
+
+The same output has second-input coefficient
+`Q^-1 C1 Q (I-Q) D1`, which is nonzero at every positive width. At digit
+width f its ordinary complex rank on the 112-owner, 17-axis ambient bank is
+`112 * 2^(16f) * (2^f - sum_{k=0 mod 4} binomial(f,k))`.
+The fraction of the bank dimension tends to 3/4. This measures coupling;
+it is not a recursive-call lower bound. The [proof](PROOF.md) gives the
+complete four-input row, all-width argument, exact recipe and ordinary bill.
+
+The exact bounded synthesis tests 8,281 feedback policies in a declared
+universal operator algebra, retaining only one of the four required jobs
+in every case. This rejects that finite formal synthesis domain. It is
+**not** a physical impossibility result: additional source relations, other
+feedback inputs and larger decoder grammars are untested.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/coded-active-feedback
+python3.12 -I -O -W error -B research/coded-active-feedback/verify.py
+```
+
+The registered normal command replays every full-search policy and rank
+(about three minutes on the author's machine). The optimized command checks
+the complete component row, rank counts through width 64, positive/negative
+controls and 196 direct span certificates on a smaller 49-policy pilot.
+Both authenticate four inherited source inputs. All guards remain active
+under optimization; no scientific module is loaded from a bytecode cache.
+Default verification writes only private temporary outputs. `--write` is
+the explicit RESULT.json authoring option.
+
+`SEARCH.json.gz` is a compressed mathematical receipt. Replay compares its
+parsed JSON with the regenerated receipt, including every case. Compression
+container bytes are not a cross-platform mathematical equality condition.
+[RESULT.json](RESULT.json) records the canonical payload digest.
+
+## Scope, sources and credit
+
+Base: canonical main `ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c`, priority
+revision 7, selected conditional kappa 0.00302015. [SOURCES.json](SOURCES.json)
+pins the gauge engine, exact original chronology and mixed derivation, and
+the involutive physical ABI proof. Their all-width native implementation
+and helper-autonomy claims remain inherited premises. No new phase audit,
+native-word emission, precision/tape implementation, full multiplier replay,
+independent human review or formal proof-kernel acceptance is claimed.
+
+The [canonical investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/4930#discussioncomment-19145683)
+and its [scope refinement](https://github.com/CrocSwap/integer-mult-bounds/discussions/4930#discussioncomment-19145848)
+precede the published package. [TASK.json](TASK.json) records the exact
+bounded scope, overlap coverage and remaining questions.
+
+Credit Dugongue/mdumitrean PR4948/4935/4929; Dillon PR4933; the passive
+saved-query authors and Chris R/Clear1nsight's packaging; Cao Yifan PR4610;
+Cinna Davis PR4694/4698/4470; Rohan Arun PR1931 and Ali Romman PR1862 for
+the involutive repair lineage; zxv, Jacob Alan Sussman and all retained
+native, Clifford, compiler and precision contributors. The inherited
+gauge engine is used unchanged except for its explicitly supplied word
+reducer. The new active recipe, complete row, coupling rank and bounded
+synthesis are by Cinna Davis with substantial OpenAI Codex assistance.
+No subagents. Apache-2.0.
+
+The submission refresh also found [PR5200](https://github.com/CrocSwap/integer-mult-bounds/pull/5200),
+head `40bb94aebbf591a99e2f48533f16efe8914fb1e2`. Its two-FULL ceiling has
+dirty identity workspace and excludes additional paid operations and owned
+COPY/CLEAR resources. This package retains three separately paid query banks
+and owned captures; its output is coupled. It is neither a counterexample
+to that ceiling nor a replay of its proof. Standalone active repair bounds
+and the passive unchanged-input result also remain outside the new recipe's
+declared source-sharing premise.
+
+## Open next questions
+
+- Can a changed query input or boundary remove the explicit y0 coupling
+  while preserving every original main and helper job? Start with the
+  complete row in PROOF.md and declare the extra work before expanding.
+  Another selected-column identity does not supply the missing caller.
+- Can a physical source relation or port-dependent control absent from
+  the formal algebra yield a complete identity? First give one explicit
+  relation and an exact all-input check. The finite search is not a
+  reason to repeat the same unaltered grammar.
+- If a complete caller survives, price its scalar work, owned storage,
+  rounding, completion, compiler, BIT and outer transfer before assigning
+  a new kappa. The preserved paid histogram alone admits no new targets.